u w? 2%. ~l d 2 45.1.} 0 1.5 Rik—32' . L a.“ t 1 .2 5;. »:. “.44; a. a. , .2 "r H‘ 1:. 3 .4 film W, m. . mm. M d‘ .=...m.$_.....;.= (.3 a 11.3. u.«.... :3 . § r66 '5 9 Ph van:- 1 of; “for the Deg TE ,0 Dissertation A m 'lCHlGAN SI mes LDR . .. i ,. N. 0. WA T JENN 'ROEER 4 7... 9 1 n I . ‘3 ’t - qu ‘. w. Mn. LIBRARE» ' .r l - - Imengan $117.2; L LIIVCI‘Sf :y This is to certify that the thesis entitled INVISCID COMPRESSIBLE FLUID FLOW IN ' A CURVED MEMBRANE . SHELL presented by Robert Jennings Waldron, Jr. has been accepted towards fulfillment of the requirements for Mathematics r L_ degree in ’ I: g! 0‘! Date- épt'ember 20, 1974 V .V ~3 ' 1-1 - ' . fi 111%. 6-7639 ,. ‘ 3:.‘LSC1D. C03: A C CR V. Robe Atkin, membra 21:2:15 of radius " 15:5 fluid which is l Eafia'de: Stokes equa Iiithe c'ynamic forn‘ ::::-:-i' the membran I“: 6 small, (5: aF 33526!“- form Xliseflfl) 3 “=15: and r form p< 3% a is the angle a E .1211 real Valued wave The Object of ti 5.36ch“ to the it e Qt 121:0 order term 5 ; grim"eaxlecl that s ...*.‘. correctiOn T 1 he correcuor SPedal . , ca$55 of u MMfluid, t} '2?ch V . PTODIQm Of a 4 1‘33?- 3‘61 OCItY is COP, 37? u canes with C01)“ Al', ABSTRACT INVISCID, COMPRESSIBLE FLUID FLOW IN A CURVED MEMBRANE SHELL BY Robert J. Waldron, Jr. A thin, membrane, cyclindrical shell of radius "a" is coiled into a torus of radius "R". The shell is filled with a compressible, inviscid fluid which is flowing axisymmetrically as it enters the coil. The Navier Stokes equations are used to describe the fluid motion while the dynamic form of Sanders' shell equations describes the motion of the membrane shell. All variables are exPanded in a series about 5 small, 5 = aR-l. Wave solutions are sought giving variables the general form x”), 9,130: {XO(¢», r)+ 5Xl(zp,r)+. . . )eXp {i(s 66-1- 30} where '1’ and r form polar coordinates in any cross section of the tube, 9 is the angle around the torus, 3 is the frequency and + 523 +58 +... 8:80 is the real valued wave number. 1 2. The object of the analysis is to find the first non—zero correction to the frequency equation so = f( 3) where we interpret the zero order terms as representing the straight tube flow. The work revealed that 81 = O forcing the determination of the second order correction s2 = g(so,fl). The correction, s2, has been determined in closed form for the special cases of wave motion of a fluid in a rigid tube, the free flowing jet of fluid, the vibrations of an empty tube, and the inter- action problem of a fluid flowing in an elastic shell. The corrected phase velocity is compared to the straight tube phase velocity for all four cases with comments made on any significant differences. zagecial case is 9 war: followed by a C $62121 findings. F‘ m... :*|I Robert J. Waldron, Jr. A special case is examined in which the first correction, 31’ is not zero followed by a comparison of the present work with exPerimental findings. Finally comments are made on further areas of study. m'scm comp: A C UR \' E Robert J . Mic 1n Partial f1 DOC INVISCID COMPRESSIBLE FLUID FLOW IN A CURVED MEMBRANE SHELL By Robert Jennings Waldron, Jr. A DISSER TA TION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1974 'va To my parents for their encouragement and to my wife for her patience and support. ii I €115 like -I .v-h“fl O5 gouu“\' uh- ‘ 1 w oka ~03. v I‘Ch‘kafhs I: 0 . .1--V:~j “‘1' 'f‘i’t #ch 5-0“ 7":‘:& I «~-.1:s for :nallv, 1 6 :"A ‘”“:Of this 2.: 7'2?“ k ' “H“! tAl ACKNOWLEDGMENTS I would like to give my thanks to Mary Reynolds and Kathy Trebilcott who bore the burden of manuscript preparation. Special thanks must go to Jeff Gorney for the plots and graphs. My appreciation must also be extended to the Department of Mathematics for their support during this process. Finally, I must indicate my appreciation to Professor Charles Martin who guided and encouraged me through the com- pletion of this degree. His efforts were never ending. Again, many thanks. iii "45‘. :5 Figures induction 1.". Bialcgical Cons: 1.1 Wave Propagatl 1.3 sttory of the P; )4 Outline of Pres: ,Tlaiematical Formu 1.1 Introduction 1.2 Field Equations 3-3 Field Equations l-l Geometry of tb 3'3 Fluid Shell Prr: 3-6 Traveling War. {nature COII‘CCIlL 3". Introduction 3'3 Shell Displacer 3'3 Corrections to 3'4 AlliSymmetric 3'5 First Order D: M Second Order l . unratiOns of the F o" 1 Introduction 1..) " Zero Order P: a 2.1 - , 4.4 .. Frequency qu. "J . I ,_ mud Order "sens m Interactior Jo SCUSsiOn to requency E. I. II. III. IV. TABLE OF CONTENTS List of Figures Introduction 1- 1 Biological Considerations 1. 2 Wave Pr0pagation Analysis 1- 3 History of the Problem 1.4 Outline of Present Work Mathematical Formulation of the Problem 2. 1 Introduction , Z. 2 Field Equations for the Thin Shell 2. 3 Field Equations for the Fluid 2. 4 Geometry of the System 2. 5 Fluid Shell Problem - Complete Equations 2. 6 Traveling Wave Solutions Curvature Correction for the Empty Shell 3. 1 Introduction 3. 2 Shell Displacement Equations of Motion 3. 3 Corrections for Arbitrary n 3. 4 Axisymmetric Frequency Equation 3. 5 First Order Displacem ents 3. 6 Second Order Correction - Axisymmetric Case Vibrations of the Fluid 4. 1 Introduction 4.2 Zero Order Pressure and Veloctiy Relations 4. 3 Discussion for Arbitrary n 4. 4 Frequency Equation for Inviscid Axisymmetric Flows 4- 5 Determination for First Order Functions 4. 6 Second Order Correction V. General Interaction Problem 5-1 Introduction 5.2 Discussion for Arbitrary n 5. 3 Frequency Equation for Inviscid, Axisymmetric Flow iv vi 24 25 29 35 38 45 6O 63 67 76 82 96 112 113 120 135 145 160 165 166 175 TA Iézera‘. Interaction 5.4 First Order Pr 3.5 Second Order C Currents and Cons 2.1 introduction :1 Ccmments on 1 til Comparison \L 1". Final Conside V. VI. TABLE OF CONTENTS General Interaction Problem 5. 4 First Order Pressure, Velocities and Displacements 5. 5 Second Order Correction Comments and Considerations 6- 1 Introduction 6. 2 Comments on the Size of m 6. 3 Comparison with Experimental Work 6. 4 Final Considerations BibliOgraphy 201 219 230 231 237 248 250 . 5533.181“ 1 , Ergty Tube Frequt' »:,_anc'cg for the E -. iron Tube Displa : Centrected Displace Carrected Displace ' l3. . ‘ I «Jetted Dlsplaci 1 arrested Disnlac = .crrected Dis place . r. . _ .. arrected Displam I". s v.rrected Phase \ ., P. - turrected Phase \ " Pl ‘ - vnetted Wave N r. _ “will Tube Corr " “mill Tube Corre wII'ECted phase 1 . Pg- 'SWS Sir I Q h I it .. Zero Order Velo: !,. Zero Order Velo 5 ‘ Ol'l’ected V810ci - e. .rrected V9100 orrected VEIOC mummbwwv— 15 . l6. 17. 18. 19- 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. LIST OF FIGURES Geometry Empty Tube Frequency Evaluation CI) and cg for the Empty Tube Empty Tube Displacement Ratio: Zero Order Corrected Displacement Ratio: = -600 ,V=0 Corrected Displacement Ratio: d): 300 ,V=0 Corrected Displacement Ratio: 4): 300 ,v=.25 Corrected Displacement Ratio: 21): - 60O ,V=.25 Corrected Displacement Ratio: 1b: 30° ,v:. 5 Corrected Displacement Ratio: ti): - 600 ,U:. 5 Corrected Phase Veloctiy V20 Corrected Phase Velocity Vz. 15 Corrected Wave Number V2.15 Empty Tube Corrected Phase Velocity V=3-»,/8 Empty Tube Corrected Wave Number Vz3-i/8 Corrected Phase Velocity V2.5 Corrected Wave Number so vs B/c cp/c andc /c vs R/C Zero Order Velocity Ratio for the Rigid Tube Case Zero Order Velocity Ratio for the Stress Free Case Corrected Velocity Ratio: Stress Free to: 30°, 20:2.20 Corrected Velocity Ratio: Stress Free 4): 30°, 20:5.52 Corrected Velocity Ratio: Stress Free {Dz-60°, 20:2.20 Corrected Velocity Ratio: Stress Free rim-60°, Corrected Velocity Ratio: Rigid Tube lb: 30°, Corrected Velocity Ratio: Rigid Tube it: 30°, Corrected Velocity Ratio: Rigid Tube JIM-60°, Corrected Velocity Ratio: Rigid Tube ' tin-60°, Corrected Phase Velocity vi =5.52 :3.83 =7.02 =3.83 I‘0 1 1 1 l=7.02 36 78 80 81 86 88 90 92 94 95 98 101 103 105 106 108 109 137 138 142 144 149 151 152 154 155 156 158 159 163 5 far interaction 1 .. Phase Velocity L. Pztsszble interactic : s: for Acoustic M: can :for Interaction E ‘ Tobe Modes anse Velocity for 45. velocity Ratios: A ‘1 i'eizcity Ratios: A i- velocity Ratios: ‘~ ieioctiy Ratios Witt Ratios ; ‘J: . ' 'ElOClllj Ratios 3‘ msPlacement Ra», A 5- ’l'elocity Ratios: A : "I : "l Replacement Rat . Displacement Ral '- Replacement Ra Corrected Phase Corrected Phase Corrected Phase Corrected Phase Corrected Phas« hase VEIOcity-z .orrected Ph a: Corrected phas 31. 32. 33. 34. 35. 36. 37. 38. 39- 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. S3. 54. 55. 56. LIST OF FIGURES 80 for Interaction Problem Phase Velocity Possible Interaction of Frequency Components 80 for Acoustic Modes so for Interaction Problem Tube Modes Phase Velocity for 'Ilibe Modes Velocity Ratios: Acoustic Mode #1, m=100 Velocity Ratios: Acoustic Mode #2, m=100 Velocity Ratios: Acoustic Mode #1, m=. 01 ,V=0 Velocity Ratios: Acoustic Mode #1, m=. 01 ,V:. 5 Veloctiy Ratios: Tube Mode m=. 01 Velocity Ratios: Tube Mode m=100, V=0 Velocity Ratios: Tube Mode m=100 , V2.05 Displacement Ratio: First Acoustic Modes Displacement Ratio: Second Acoustic Modes Displacement Ratio: Tube Mode m=.01 Displacement Ratio: Tube Mode m=100 Corrected Phase Velocity: Acoustic Mode m=. 01 Corrected Phase Velocity: Acoustic Mode m=100 Corrected Phase Velocity: Tube Modes 11:. 5 Corrected Phase Velocity: Tube Modes V:. 5 Corrected Phase Velocity: Tube Modes V=0 Phase Velocity: Flexural Waves Corrected Phase Velocity: Axial Waves Corrected Phase Velocity: Pressure Waves vii 180 181 189 192 193 197 198 204 205 206 207 210 212 213 215 216 217 218 222 224 226 228 229 239 244 247 in recent ‘19 .0‘1" “ .ne use Of alas originating :les include the tizrithedic prothc- ins and the desi; “in; various parar 2:21:92» -nts signal 2.3g in basic bio This thesis tithes its origi fetal situations. is. In section Tx‘s'atich are Pe eat-ill concern 0t Excel analysis 1 Section 1 rite present pl" ion of analyses will, in secti< 398 :n " l the remit CHAPTER I INTRODUCTION In recent years there has been much work completed relating the use of applied and engineering mathematics to problems originating out of biological or medical situations. Examples include the use of stress analysis in the study of orth0pedic protheses, the development of artificial heart valves and the design of electronic equipment for deter- mining various parameters in the blood. All of these develoPments signal the arrival of mathematics and engin- eering in basic biomedical problems. This thesis is concerned with a theoretical prdblem which has its origin or motivation in a variety of bio- logical situations, namely fluid flow in curved. elastic tubes. In section 1.1 we will discuss the basic biological facts which are pertinent to our prdblem. In section 1.2 we will concern ourselves with a description of the mathe- matical analysis to be employed in the thesis. Section 1.3 contains an outline of the development of the present prdblem, tracing the improvement and vari- ation of analyses employed in studying our situation. Finally, in section 1.4 we will preview the analysis to be used in the remaining chapters of the thesis. 1 Cur concern 1 :12:- ccr-non to a v 21; were propagati friars example i: re: include the :[particularly ll sell as the movem: is of the inner t Seedless to 391165 toward vari :2: “O r ' we Physical s “3% to delineate -. :5 5‘18 ten. Onl- ten of wave prO estlc tube, hOpin . aRCe 0f Curve "‘1 f0! this ane Cit-er . ' partlcularj e‘.. We find in l;lb Biolggical Considerations Our concern in this thesis is with a fundamental problem common to a variety of biological fluid flows, namely wave propagation in fluid-filled, elastic tubes. An obvious example is the circulation of blood. Other examples include the peristaltic motion of the urinary tract (particularly in regard to reflux in small children) as well as the movement of fluid in the semi—circular canals of the inner ear. Needless to say, there is a wealth of effort being directed toward various aspects of these larger problems. Since the physical situations are so complex, investigators are forced to consider one or two pertinent situations hoping to delineate their contribution to the total behavior of the system. Our concern will be with the particular prdblem of wave pr0pagation in a curved, fluid-filled elastic tube. h0ping to gain some understanding as to the importance of curvature in wave pr0pagation. The moti- vation for this analysis springs from the circulatory system. particularly that portion containing the aortic arch. We find in the mammalian circulation that blood is collected and returned via the veins to the right side of the heart. From here it is pumped into the lungs to have carbon dioxide removed and oxygen added. From the lungs riled is collecte: :sjezted periodica 251:3: vessel to r 25:2:rves updard : Nature is rat?- aziting midday thr’ :2. called the ac ifixlcm {:03 C 2' t0 4 cm. It 1 as: pronounced- l'ost effort transnational e “-‘E-een based on iris apparent that tier- of curvatu:| Etaig‘nt tube model .; ~e "ll . rved flOws . The physiC Eiriilen are km" tie-tailed in PaI lesity of blood ' to 1.29/0m3 ' Early the same 1 Zilecit' wt les is Oh the blood is collected in the left chambers of the heart and ejected periodically into the rest of the ciruclation. The first vessel to receive the ejected blood is the aorta which curves upward to the left and then downward again. The curvature is rather pronounced and the aorta exhibits branching midway through the curve. In this region of the aorta, called the aortic arch, the diameter of the vessel is from 1 cm. to 3 cm. and the radius of curvature is 3 cm. to 4 cm. It is apparent that the curvature here is rather pronounced. Most efforts at modeling blood flow have followed fibre conventional engineering approaches, that is, the models have been based on a variety of straight tube conditions. It is apparent that some attention should be given to the problem of curvature, if only to determine to what extent straight tube models effectively forecast the behavior of the curved flows. The physical parameters essential to modeling such a problem are known in physiological circles, but they will be detailed in part for the sake of completeness. The density of blood vessel walls is a variable ranging from 1.0 to 1.29/cm3. Since the density of blood itself is nearly the same it is reasonable to conclude that the ratio of desities is one. Elastic defo: its so Poisson' s "gatewall is cor 5511c :cdulus nay stall is stretch:- :;':::.:ernal press far; of the dog t‘: we «LO to 20.4 X the blood '1 furious cells in it'sicle blood vol ’frfloe in Sic-all « ‘Jlible, while in '111 not be alters at. conSider ' th‘ "Elite; the flow N There is :26. "Sean. Elastic deformations take place at nearly constant volume so Poisson's ratio can be taken as .5. In addition the tube wall is composed of various fibers so that the elastic modulus may vary, depending on the degree to which the wall is stretched. Certain fibers are not stretched until high internal pressure is applied 50 that in the femoral artery of the dog the value of Young's modulus may vary from 6 1.2x10 to 20.4 x104 dynes/cmz. The blood is also a composite, being a suspension of various cells in plasma. The cells form 45$Q to 5034 of the whole blood volume and contribute to the viscosity. For flow in small diameter vessels, the viscosity is variable, while in vessels which are larger, the viscosity will not be altered significantly. For the prdblems we will consider, the viscosity will be treated as a constant, making the flow Newtonian. There is still much discussion concerning the sizes of these various parameters. The difficulties accompanying experimental determination of these parameters are manifold since excising a sample of tissure from the living subject alters its condition. On the other hand, examination within the subject restricts the experimenter to only those vessels and tissues which are readily accessible. As can be seen there is great need for exper- imental techniques which can better cope with the prdblems inherent to testing living subjects. Further information :‘ss details of th- .:::ety of sources refer l‘ which 6 Iii vessels is es; on the details of the circulation can be obtained from a variety of sources. In particular, a review article by Frasher [ 1] which deals with the physiology of the larger blood vessels is especially helpful. Since the int :sgatmn of waves - to consider so: wares propagating ::::si:‘er the pIOpE retire freely. Tl 2:: of motion with. 1:31: the vibrati .5.-.1 P. -....w waves in S tits direction. lagers-ted form, t are 2 is the di: «a related to t‘: Risency of vibrat Elan} is call a... . ‘A. ~ on! ‘ “'a U. The SUbstit 3‘)“ 1"‘0o h n k ‘ flu) ""‘On. From th : \ |.\‘ length and ph 11:2: I System is V If the prof ligation. e 9 it: w a . contlnuOus 1.2 wave Propagation Analyses Since the intent of this work is to study the propagation of waves in a curved, fluid filled tube, we pause to consider some of the basic approaches to the study of waves propagating through a medium. The first step is to consider the prOpagation of waves in a medium.which is vibrating freely. This means that we find the medium in a state of motion without benefit of any external force to maintain the vibration. We assume the movement of the inbration waves in some direction and assume harmonic motion in this direction. Hence we might write all our variables in separated form, the wave portion denoted by exp[i[kz-wt]} where Z is the direction of prOpagation, k is a wave number related to the inverse of the wave length, at is the frequency of vibration and the ratio wyRe(k) [RE denoting Real part] is called the phase velocity associated with the‘wave. The substitution of the wave form given above into the equations of motion for the system results in an equation k = f(uD which we call the frequency or dispersion equation. From this relation we can tell the wave number, wave length and phase velocity once we know at what frequency, at the system is vibrating. If the prdblem is such that the direction of propagation, e.g. z, is not limited, we find that k = f(w) gives a continuous representation for k in terms of w- :2: ievelop that ‘ a t: small wall Iii ate a particula t" :1- ie characteri .‘. :21: conditions, :‘is a special ca 35(1). we! PYOb: macerned with 'iration is finit :csing boundary .sais to a situati 4' : exists at Iaproblem like litits apart an I”? L) . ' ‘15 = Z 1" LA 11:1 r It may deve10p that under certain conditions, e.g. long ‘waves or small wall inertia, the relation k = f(w) re- duces to a particular form.which we call a mode of vibration. This mode characterizes the vibration we find under those certain conditions, i.e. long waves or small wall inertia, and is a special case of the larger situation given by k = f(w). Other problems associated with free vibrations are concerned with situations in which the direction of \dbration is finite, say along a string of length L. Imposing boundary conditions at the ends of the string leads to a situation where the relationship between k and u) exists at an integral number of points. A solution to a prdblem like this, e.g. = c Zy’tt with ends Y'zz L units apart and fixed, is given by . y(x,t) = 23[A cos(nwcL-1t) + B sin(nwcL-1t)] sin nWL-lz. n21 n n FOr each integer n the term inside the sum is a solution and superposition allows us to say the sum is a solution given the prOper convergence. If now we are to consider a system in which there is a function forcing the vibration to continue, then we would expand the forcing term in a Fourier series like that above and solve a series of prdblems in which the solution also has such an expansion. Tne problem u mated with the f ::.iei in one dire :aience of the wet ‘ V severe to be a. 1.1m the direct u... I15 rom the 355 case in sect;- 33-3‘1' mlation 1 iv +L. . “ “9 assmnptmr led to its rad; e of an inf in "1 in turn Gives The problem under consideration in this work is that associated with the free vibrations of a medium which is unbounded in one direction. Here we find a continuous dependence of the wave number on the frequency. If the problem were to be altered later so as to have a finite domain in the direction of propagation we find only minor variations from the original problem. Love [.2] discusses such a case in section 199 of his text. He finds the frequency relation for the longitudinal vibrations of an infinite circular cylinder. For the finite case the new fkequency relation is satisfied by the original equation imder the assumption that the length of the tube is long compared to its radius. Hence it is possible to relate the Ltdblem of an infinite medium to that of the finite case, which in turn gives rise to solutions of forced vibrations of the same system. r. -i We begin th’ 215515 centers on tratzons of strai straight to}; 12:9 approach thC‘ :1 generally their”. is approach the azlztcns are song? sued in this ap: £112: procedure a: 4:21: theories to {'flalak [ 4]. There are ~ 1 I ‘I Mlle ar set of eq t“. :3. Vol ‘ . 0 ' city 13 a 1513?; “Y “‘31)? be 8( o .D ‘ “3‘ ‘ “es" as meas ist-tl '04 tiIHes 5:14: ‘eh'tonian at 1.3 History of the Problem we begin the discussion by stating that the bulk of the analysis centers on various prOblems associated with vibrations of straight tubes. There are two basic approaches to the straight tube problem as outlined by Rudinger [3 ]. In one approach the problem is treated as being non-linear, but generally there is only one dimension studied. In the other approach the equations are linearized and wave solutions are sought. In general all three dimensions are studied in this approach. Our analysis will follow the latter procedure and so we leave the discussion of non- linear theories to the article by Rudinger as well as one 'by Skalak [ 4]. There are good physiological reasons for assuming a linear set of equations to describe the wave motion of a fluid in a tube. For blood flow in the aorta the average flow velocity is about 25 cm/Sec while the pulse wave velocity may be 800 cm/sec. Next, the distension of the arteries, as measured by Poiseuille, is not large, being about 1.04 times normal values. Third, the blood, though non-Newtonian at the low shear rates of the capillaries can be treated as Newtonian in the faster motion associated 'with larger vessels. Finally, the tube wall, though non- linear, can be approximated by a linear, visco-elastic material if need be. So, we may c :aperfect assump :12. The advanta 25:35 closed form :‘szlzticns which ; We begin on: 1:25 Yong [ 5] w" Etetatical deters.1 Lifltid filled ti 1:16.“. 33 10 So, we may conclude that a linear model although not a perfect assumption, does not grossly distort the system. The advantages are numerous as now we are able to find closed form solutions as well as utilize superposition of solutions which is essential to vibration analyses. we begin our discussion of prior work with Thomas Young [ 5] who made one of the first attempts at a mathematical determination of the phase velocity associated with fluid filled tubes. Young's phase velocity can be written as _ 1/2 CpO — (hE/2apo) ‘where h and E are the thickness and Young's modulus for the tube respectively, a is the tube radius and p0 is the fluid density. Later, other investigators found this velocity after assuming the fluid to be inviscid, incompressible and the waves to have long wave length. Korteweg [(5] considered a compressible fluid and studied the velocity of sound in a fluid filled tube. He found his phase velocity to be c = iigifefl/Z p1 hE I( The new parameter, K, is the bulk modulus of the fluid and 1/2 is the velocity of sound in the fluid. the ratio (K/po) Hence Korteweg found that the phase velocity was less than the velocity of sound. For an incompressible fluid, ?:.‘ becomes ”if i," 3339 ' 5 node . Later: Lafi‘l: the linearized miss. one be“ 3: 133‘ ea 2 is as be :hisson's ”ti mifferent from‘ :3: only tube par From this it: others imprOl "lStosity, taperil 5571 to better k ‘4 lyedence. E ml for some i In the 1‘ 3:: .erreport, w 9:} . «was exten ‘ 81 {3'1 . °1°glcal w~ ' C 5: ' - -r191d tube 11 (K/po) becomes infinite and Cpl is then equivalent to c , Young's mode. P0 Later, Lamb ['7] studied the modes of free vibration for the linearized fluid in a membrane shell. He found two modes, one being C , Young's mode, and the other p0 given by _ 2 1/2 cp2 — (E/P(1-v )) where E is as before and p and v are the tube density and Poisson's ratio for the tube respectively. This mode was different from the earlier modes in that it was associated with only tube parameters. From this point the analyses became quite numerous with others improving Lamb's work by the addition of viscosity, tapering, branching, etc. Useful analogies were drawn to better known electrical concepts such as resistance and impedence. Electrical transmission line theory became a tool for some investigators. In the 1950's, several reports, collected in one larger report, were written by J. R. Womersley [ 8 ]. His work was extensive and one of the most complete treatments of biological wave propagation. He examined an infinitely long rigid tube which had a region of steady pressure. He solved the linearized Navier Stokes equations and the continuity equation under the conditions of fluid viscosity, incompressibility and axial symmetry. Assuming tr sztesstre and ti start. he conga 15225 a phase v fixaatity of f1 0 (I, I mersley illeo with a vis serptimis, Wome 7-“:~ the phase v 3: were introd 2. if 21bration bu. .7472: b Y Cp \‘h " s x h“: 13:93 in free 1" add: tired other . “e elastic to Q9 and corr .5 I la » “Ed ~ 0 I? “Oils 0f t “the . as. * stressed 12 Assuming traveling wave solutions WOmersley found the pressure and two fluid velocities up to an arbitrary constant. He computed average flow velocity, quantity of flow and a phase velocity. His values for average flow and quantity of flow compared quite well with experimental results. WOmersley then proceeded to study the elastic tube filled with a viscous fluid. Under several simplifying assumptions, WOmersley arrived at a frequency equation which gave the phase velocity directly. When the assumptions of Lamb were introduced, his value reduced to Lamb's mode given by C Lamb's mode is independent of the frequency p2' of vibration but with viscous terms added, Wbmersley found the phase Velocity of Lamb was altered significantly by chances in frequency. In addition to the work mentioned, Wbmersley con- sidered other effects: a thin boundary layer: added mass to the elastic tube; corrections for the non-linearity of the tube; and corrections for the approximate boundary conditions he used. There were many aspects to Wbmersley's report and so his work prompted many new investigations, dealing with portions of the prdblems he discussed. Numerous refinements ‘were made to his models with allowance being made for the pre-stressed condition of the tube and for its visco-elastic nature. One of the T 1;“; filled tube W1 r a; considered a C‘ :‘rzpic, elastic ii. ”the linear: aims of mtior. zsjecti'zely and t3 :ieszribe various The main e: Zine discussion < ”tiled elastic tub. 313% shall 0' their results, *‘ “a; "‘ 'm ‘5 reason We be: results her m that the tli' “59 VEIOCitY wa «be c132, Lam}; ““0 equal to Zt ”which did in “her than one 3r» ts“ . ' .~..sslb1e flu has depended 0 £11 ‘ aC0u . Stlc S 13 One of the more complete analyses of the infinite fluid filled tube was advanced by Rubinow and Keller [9 ]. They considered a complex model involving a homogeneous, isotropic, elastic tube filled with a viscous, compressible fluid. The linearized elasticity equations and linearized equations of motion were used for the tube and fluid respectively and they allowed for a complex impedance matrix to describe various boundary conditions at the outer wall. The main effort of Rubinow and Keller was extended to the discussion of axisymmetric vibrations of a thin walled elastic tube containing an inviscid fluid. The results we shall Obtain in latter chapters duplicate many of their results, under slightly different conditions. For this reason we will discuss a few of the Rubinow and Keller results here. For Poisson's ratio equal to zero Rubinow and Keller found that the tube had a mode in which the non-dimensional phase velocity was one. In dimensional terms it was found to be C Lamb's mode. They also found for Poisson's p2' ratio equal to zero that there was another mode for the tube which did not exist for non—dimensional frequencies greater than one. This mode for small frequencies was merely the mode found by Korteweg, Cpl' Since they assumed a compressible fluid Rubinow and Keller foundrother modes which depended on the acoustic speed of the fluid. For small acoustic speeds these modes were similar to those of 1: tube. However ties were distinc ‘s'arger values c :: the modes re: 1'16! pre‘ lens cm a". as well as d We have ; been obtains all find later j size: is YEferr. ration“ above 2 straight tube A featu: EES'EPtiOn of a {Link “ml are not S if." 1‘ . ‘1 late Seml‘Ci 3‘41: Er are par r- . J‘k which has 14 the tube. However, for large acoustic speeds the acoustic modes were distinct from the tube modes of Lamb and Korteweg. As larger values of the acoustic speed were considered only the tube modes remained. Their paper considered various other problems connected with viscoelasticity of the tube ‘wall as well as different boundary conditions. we have presented a few of the many results which have been Obtained in the study of linear theories. we will find later in the study of curved flows that the results of straight tube flows play an integral part. The reader is referred to the articles by Rudinger and Skalak (mentioned above) for further discussion of linear theories in straight tubes. A feature common to most of these models is the assumption of a straight, cylindrical tube. However, it is apparent that there are important regions of fluid flows which are not straight. The situations of the aortic arch and the semi-circular canals of the inner ear mentioned earlier are particular examples. So we now discuss the ‘work which has been done on curved tube flows. One of the first analyses directed to the prOblem of curvature was put forth by Dean in two papers, [10], [11]. The Object of Dean's work, as well as those who followed his lead, was to examine the secondary flow in the cross section of the curved tube, determining what effects were introduced by the curvature. This meant that the streamr lines and particle paths were examined in the cross section. Dean began 7.22225. incompres swirled the no "-‘g in a par szcla‘s nurber, airs of the ten Ereter D sma iiSrivastava [l 1 '10 be of moder I a. . .H '. 3:633? Wtion , It Was t ‘ l"Steady mot :1. tube, In :-. A: ' l...- was of the am Mn oi a cu; I 5:1 a pplied p 1123:» . 15 Dean began by studying the steady motion of a viscous, incompressible fluid flowing in a rigid torus. He examined the non-linear equations of fluid motion by expanding in a parameter D = 2(Re)2a/R where Re is the Reynolds number, a is the tube radius and R is the radius of the torus. Dean analyzed his prOblem for the parameter D small. Subsequent investigations by McConalogue and Srivastava [12] and Barua [13] considered the size of D to be of moderate and large values respectively. A feature common to all of these works was the assumption of steady motion. It was then the interest of others to investigate the unsteady motion in the cross section of a curved rigid tube. In the first work, by Lynne [14], we find an analysis of the unsteady, secondary motion in a cross section of a curved tube. The fluid is viscous, and subject to an applied pressure gradient. The fluid is assumed incompressible and the variables are all expanded in a series about the ratio (a/R) small, a and R taken to be the same as in Dean's analysis. Lynne utilized two basic parameters 1/2 E = W(a/R) /a6 and RS = wza/VRB where w is a typical velocity along the pipe, B is the frequency of applied pressure and v is the kinematic viscosity of the fluid. The parameter E is always small and RS is a type of Reynolds number. Lynne found that E... 2 , [51 2‘ IR is S? 5 may layer 1“ 1:5 is inviscid. yrme four: 2;;tetic expansi' :: homer bound firstlayer in add 2::zcrrtal diamete .fjc d .. .etermined t :gzsie to that ; Iron is from th In an ana? ‘43 Studied puls; ‘33 “v- 9“ 0f the w :3va -.ation at N lie stream ft \ '-: \§ 16 ‘when 262/RS is small, the viscous effects are confined to a boundary layer near the wall while the remainder of the fluid is inviscid. Lynne found in his analysis, which involves matched asymptotic expansions between the boundary layer and core, that another boundary layer is formed at the edge of the first layer in addition to a free layer existing on the horizontal diameter of the pipe. For unsteady motion Lynne determined that the motion of the cross section is opposite to that predicted by steady motion, i.e. unsteady motion is from the outside of the curve to the inside. In an analysis similar to Lynne's, Zalosh and Nelson [15] also studied pulsating flow in the curved tube. This study was part of the work done by Zalosh in his 1970 Ph.D. dissertation at Northeastern University. This work solved for the stream functions associated with the secondary flow inaacross section of the tube, the fluid being under the in- fluence of a sinusoidally varying pressure gradient. As ‘with Lynne's work, the tube was assumed to be rigid. All variables are expanded in a power series for the ratio (a/R). Then equations are found for the first term in the stream function expansion and for the first two terms in the expansion of the longitudinal velocity. Unfortunately, there seems to be a lack of consistency in their expansion of the differential operators, leaving one to suspect the accuracy of their equations for higher order terms in the expansion. the first as: parts. or 37:27.3] of the fie longitudi: is terns toge fixertial eqt flzzgitudinal ref on the a fiirtinate tha rseizure-2:1 Si N. n. 1 esa'its the In all 1578110 inf fie"’all or. #159 an El; :E‘C‘us wo rk: (13' tberef In 19 azurved. f] edisrieuh falls and 15: Fatal salil 5:: 0n the us ‘43-. .. Ulr- 17 The first term of the stream function expansion had two parts, one steady, the other varying at twice the frequency of the applied pressure gradient. The first term of the longitudinal velocity is found and these two first order terms together become inhomogeneous terms for the differential equation for the second term in the expansion of longitudinal velocity. As stated earlier some doubt is placed on the accuracy of this expansion. It is also unfortunate that the second term in the stream function was not computed since the zero order system taken by itself represents the straight tube, not the curved tube. In all of the analysis for the curved tube presented to date no information had been gathered as to the influence of the wall on the motion, i.e. no work had been done to include an elastic wall in the model. In addition, these previous works concentrated on the streamlines and particle paths, therefore excluding wave propagation analyses. In 1971 Daras [16] investigated wave propagation in a curved, fluid filled elastic tube. Acknowledging the difficulty associated with wave prOpagation in toroidal shells and flow in rigid tubes, Daras pr0posed to consider several salient features of the problem, hoping to shed some light on the essential aspects. The first task Daras undertook was an expansion of the wave number k as _ 2 so. = k—ko+6k1+6k2+ .6 a/R- Daras conc 15C k for a ' 0 a. is the wave m tiles and solvii 1:5 and a membr. : Ls interpreted :fnaras was abl ””c/«tu' no». “OTIS for t For F I C'-. vi He tl ismtric ax: tibiaiHEd son. bikini [17] Daras t} m fine-d cur 23V" . .mt tube 11 is intrOdUCEd “on equalti "’"Qlated with " msidered t 351:: ' . in depth a 18 Daras concluded that the effort involved in deter- mining kO for a general prOblem was too great, since k0 is the wave number associated with the straight tube prOblem and solving for k0 involved equations for a fluid and a membrane. The next term in the expansion of k is interpreted as being a correction for the curvature and Daras was able to conclude that k1 = 0 so that any corrections for the straight tube wave number kO are of order 52. He then found kO for some special cases (axisymmetric axial and pressure waves plus flexural waves) and Obtained some agreement with the experimental work of Van Buskirk [17]. Daras then reviewed work previously completed on fluid filled curved tubes. He examined an axisymmetric straight tube in which curvature effects are introduced through a modified stress law, and effects of secondary flow are introduced as an equivalent viscous stress in the axial momentum equation. Finally, he considered basic prOblems associated with entrance effects in a curved tube. In all, he considered the highlights of several problems leaving more in depth analysis for others. figtllne 0f Pl" The presen‘ is: problem whic‘ sterzination Of ‘5 :-zero correctiO red elastic tub :euized Navier :ifor the tube is rations of Sande aessary for non- In Chapte: Esplations of 1 its: non-dimensi X”: 3,1’, t) = { 19 1.4 Outline of Present work The present analysis proposes to investigate the first prOblem which Daras only touched upon, namely a determination of the zero order wave number and the first non-zero correction for wave propagation in a fluid—filled, curved elastic tube, coiled in a torus. we consider linearized Navier StOkes equations for a compressible fluid and for the tube we utilize the linear, membrane shell equations of Sanders [18] augmented with acceleration terms necessary for non-steady prOblems. In Chapter 2 we begin by establishing the geometry and equations of motion for the conditions stated above. After non-dimensionalization we expand a typical variable as X(W.e,r,t) = {XO(¢,r) + oxl(w,r) + --- lexP{i[seo'l-6t]} 'where w and r are polar coordinates in a cross section of the tube, 9 is a coordinate down the tube, t is the time variable, 6 is a non-dimensional frequency, 6 = aR-l, a is the tube radius, R is the radius of the torus and s = s + 551 + 6252 + --° is the non-dimensional wave 0 number. we note that if exp{i[se - Bt]} is used instead of the exponential mentioned above we Obtain for our zero order prOblem a flow which is essentially two dimensional and which exists only in a plane cross section of the tube. This would be the expansion used, for example, by Lynne in studying secondary flow patterns. Inserting :Eferential Oper- A first and s as apron term appropriate 1 the inhomogei :5;‘.er two cont :ziitions for t In chapl its tube is vib: tsolv e the eq' .eexistenCe c . examine thle ri- deteminat 1' ”9- find the next 20 Inserting the expansion in 6 for all variables and differential Operators results in a series of prOblems labeled zero, first and second order accordingly as 60, 61 or 62 is a common term to the equation. The zero order equations are apprOpriate to straight tube flow and their solutions are the inhomogeneous terms of higher order equations. Chapter two contains all the equations and boundary conditions for the zero, first and second order prOblems. In chapter three we consider the situation in which the tube is vibrating but without the fluid inside it. ‘we solve the equations of motion apprOpriate to such a system and then determine a frequency equation written as so = f(B). This equation is the condition necessary for the existence of non-trivial displacements in the shell. ‘we examine this frequency equation and then proceed to the determination of the first correction to so, namely 81' we find, as did Daras, that 31 a 0 and so we deter- mine the next correction, 32. A requisite for the value of $2 is the value of all first order displacements. These we determine, as ‘well as their effect on the displacements found in the zero order. we are then able to make some conclusions as to the effectiveness of the straight tube model in predicting the motion of a curved tube. Chapter four contains an analysis of the system in which the fluid is of prime importance. This situation are be rigid a: iii to be flowint afisc‘ a frequean prcpriate to the fajita that the f ieinterface. F ei-rand that the file fluid medit Analysis ' :1 the acoustic tenor of the W :traases the vit it the wave nu? .: ricompressib 1e 21 reduces to two cases. In the first case we consider the tube to be rigid and in the second case we consider the fluid to be flowing without the tube. In each situation 'we find a frequency relation by satisfying boundary conditions appropriate to the particular case. For the rigid tube we require that the fluid velocities vanish at the fluid- tube interface. For the fluid flowing without the container we demand that the fluid stresses vanish at the outer edge of the fluid medium. Analysis of the two frequency equations indicates that the acoustic speed of the fluid is essential to the behavior of the waves. we find that as the acoustic speed increases the vibrations propagate at only high frequencies with the wave number being purely imaginary when the fluid is incompressible. The remainder of chapter four is concerned with determining that 31 E O, with the solution to the first order velocities and with the second order correction 32. After determining each of these quantities we pause to examine their effect on lower order terms, commenting on the comparison between straight tube vibration and vibration which has been corrected for curvature. In chapter five we move to a more difficult prOblem, namely the interaction prOblem in which the fluid and tube vibrate together. Utilizing the zero order fluid terms which we have found we solve for the shell displacements 3555 of the f riarv conditio 2': the velociti a: if horoqeneor it to a frequc Analysis 513359 equatio .‘-:. rigid tube iT-Etion is a t1 * Wilding th. stat the other ‘I‘ a, EStitrait-2a The De) velociti‘ Correcti. gist Order t 3“ . V‘s ‘ s '1 \ Omake S “‘ations . 22 in terms of the fluid stresses acting of the shell. The boundary conditions require that we match the fluid velocities with the velocities of the shell. The result is a set of homogeneous equations and for a non-trivial solution leads to a frequency equation, 30 = f(5). Analysis of this equation reveals that it is composed of those equations we investigated earlier in the empty tube, rigid tube and stress free cases. The new frequency equation is a transcendental equation and so we analyze it by expanding the wave number s in several power series 00 about the other frequency equations which we have previously investigated. The next step is determining 51, the first order fluid velocities, and shell displacements, and the second order correction, 52’ We examine the effects of these higher order terms on their zero order counterparts and are able to make some comments on straight tube and curved tube vibrations. The first portion of chapter 6 is devoted to a special case which was not considered in the previous chap- ters. we have found a non-dimensional parameter, m, which appears in our equations and all our expansions about 6 small have been based on the assumption that m is of order one compared to 6. we consider the very real poss- ibility that m is 0(6) and then proceed to analyze the resulting equations. We find that, contrary to the earlier 1m.“ :ation, the firs- z... Using this 1 rs i.e. fluid ‘ uting out diff e ‘ Mb; .t.: J ens 0 We come lucl fittinental works Iigation in cur filirities with 3 future conside 23 situation, the first order wave number, 51' is not identically zero. Using this fact we find the remaining first order terms, i.e. fluid velocities and shell displacements, pointing out differences and similarities with earlier situations. We conclude chapter six with some notes on recent experimental works which have dealt expressly with wave propagation in curved elastic tubes. we point out any similarities with our current efforts, as well as the diff- erences. we then close the discussion with some possibilities for future consideration. E‘ATIEWATICA 2.1 Introduction - m In this c if the th sis prc it}. equations w' tall. These eq Briers, Jr. [IE Trudi .ate systg Section Tissues, compre imilinear coc .fiéspecific 96 it stations, stations Pres :5 the SPECifi ah this disc Tu-L' "‘"thies for Final as . {“5th whi we firSt thr :3: ' "9“th at CHAPTER II MATHEMATICAL FORMULATION OF THE PROBLEM 2.1 Introduction In this chapter the complete mathematical formulation of the thesis problem is presented. We begin in section 2.2 with equations which describe the motion of a thin elastic shell. These equations were originally derived by J.L. Sanders, Jr. [18]. We formulate them in a curvilinear coordinate system erected on the middle surface of the shell. Section 2.3 contains the appropriate equations for a viscous, compressible fluid again written in terms of a curvilinear coordinate system. Section 2.4 is concerned with the specific geometry apprOpriate for final formulation of the equations. In section 2.5 the general shell and fluid equations presented in sections 2.2 and 2.3 are stated in terms of the specific geometry discussed in section 2.4. Concurrent with this discussion is the introduction of non-dimensional quantities for these equations. Finally in section 2.6 we apply the perturbation expansion which is utilized in the remainder of the thesis. The first three problems resulting from the expansion are presented at this point also. 24 1.: Field math -———-—"—" me} ll :2an of the sh We assume :: Cartesian coor 3.2.1) 2 in terms of th 2.2.2} ‘fle all quantit An orthoe ‘I'S’Jrface by d. é. 5. 7.2.4) ”led fro 4.2.3) E i 25 2.2 Fieldégguations for the Thin Shell gég;l_ Geometric Concepts we assume the existence of a geometry on the middle surface of the shell specified by coordinates 91,92 related to Cartesian coordinates by i i . (2.2.1) X — X (91: 62) 1 — 1:203 or in terms of the position vector E by (2.2.2) r = r(el.92) where all quantities with a bar denote vectors. An orthogonal triple of vectors is erected on the shell mid-surface by defining - _ -1_.§._ 2-_a§._.§.i e1 ' 0‘1 591 “1 ‘ 391 691 - -1. a' 2 a- a' (2.2.3) e = a -——- a = . .__. 2 2 662 2 662 692 en = el x e2. The element of arc length on the midsurface is given by 2 _ 2 2 2 2 (2.2.4) ds _ a1 del + a2 d92 . The principal radii of curvature, R1 and R2, can be obtained from 53 53 n _ -1 - n _ -1 - (2.2.5) 61 — alR1 e1 6 - “2R2 e2 . I? V V 5n We let 2 32:20:: along t ; is the thick“(3 ..- 9 3.“) < 1/2C .. RR _ scribed the 966 apations of moti . I I1 . Saniers . I I Lute.) ~ Sanders J teasmnptioné U Stra: surf per; ii) Norr mid 0th iii) Men for With r as ' “Ninthe dir We im :C%nents 26 we let 2 denote a thickness variable with direction along the normal en with -h/2'g z‘g h/2 where h is the thickness of the shell. We say a shell is thin if m§x(§) g 1/20 where R is a radius of curvature. Having described the geometry of the shell, we now proceed to the equations of motion for the shell. 2.2.2 Sanders' Thin Shell Theory - Membranefgguations Sanders linear theory is based on the following three assumptions: i) Straight fibers perpendicular to the middle surface of the shell before deformation remain perpendicular to the deformed middle surface. ii) Nbrmal stresses acting on planes parallel to the middle surface are neglected in comparison to other stresses. iii) Membrane forces are much larger than bending forces. With respect to our coordinates 91. we define components of the stress resultant tensor to be Nij where the stress vector is acting on the line 9i constant and in the direction of ej. we indicate the displacement vector 6 to have components tilet‘re body it 2.2.7) Denoting realty by p t‘: 3.2.8) (IZNll‘ 27 +U5 +UE (2.2.6) U = Ule1 2 2 3 n while the body force vector q has components uafl) a=q”)%+q”’g—q”’%. Denoting af/aej by f,j. af/at by f,t and tube density by p the membrane equations of motion are written (2.2.8) + (a2N11),1 + (alle),2 N12%,2 ’ N2202.1 (l) _ + alazq ‘ Phalaz Ul,tt (2.2-9) (alez),1 + (alsz).2 + N12%.1 ' N11a1.2 (2) _ + ala q — phala2 U2,tt -1 —1 _ (3) _ 11R1 + N22R2 )alaz alazq ’ phalaz U3,tt 2 (2.2.10) -(N The stress resultants Nij can be written in terms of strains 6.. , Ybung's Modulus E, and Poisson's ratio v. 1] as _. Eh , ‘V 1 = Eh (2.2. 2) N22 1 2 (622 + V611) -v EhE’ 1 = 12 (2.2. 3) N12 W The strains are then formulated in terms of displace- ments by _ -1 - -1 (2.2.14) 611 - a1 U1 1 + (alaz) UZOLL2 + UBRl 1.2.15) 622 2.2.16) 612 The 130““i 3; Ei = constan Sabstitut 1.2.14). (2.2.15) :1: the shell e 28 _ -1 -1 -1 — -1 (2.2.16) e12 - (Zaldz) {QZUZ'I + alUl,2 - UlOLl'2 — U202,1}' The boundary conditions for this theory require that for e. = constant on an edge of the membrane, N.. or U. 1 11 1 and Nij or Uj (i # j) be specified. Substitution of the strain displacement relations (2.2.14).(2.2.15) and (2.2.16) via the stress resultants Nij into the shell equations of motion (2.2.8). (2.2.9) and (2.2.16) produces three. coupled second order equations in terms of the three shell displacements and the three components of the body force q. The actual substitution described above will be presented latergonce non—dimensional variables and our particular geometry have been introduced. 1“...- II. I at... I " Fio‘d Squat; .H b v. I 2.3.1 9.5 I the space at: a three dime :xrhnates { C _ 5'4? between the trriinates give- :.3 .1) ‘13 the inverse 1.3.2) DenOting Mam) I t we de COntra‘ 29 2.3 Fieldigguations for the Fluid 2.3.1 Geometric Concepts The space occupied by the fluid will be described with a three dimensional set of orthogonal curvature coordinates (q?) j = 1,2,3. We assume there is a relation— ship between the curvilinear coordinates and Cartesian coordinates given by (2.3.1) x1 = x1(cpk) k,i = 1,2,3 with the inverse relationship given by (2.3.2) cpl = cpj(Xm) j,m = 1,2,3. Denoting the position vector to a point in the fluid by §(¢F) we define covariant base vectors gj by (20303) .9... = J3- j= 1.2.3 3 am and the element of arc length by (2.3.4) d32 = dfi - dfi. This in turn defines the covariant metric tensor gk‘ by (2.3.5) as.2 = g“ dcpk dq,‘ where now’ k,1 = 1,2,3. Indices repeated across the diagonal are summed over all values. Contravariant base vectors 51 can be derived by solving the equationsy (2.3.6) 9 - g. = 5. being th I a: be defin I. ..d’ ".138 : (2.3.9) here as i 1'19 Lei pre‘Ja N g _ u Lhe fl: 2795 are 30 5; being the Kronedker delta. Contravariant metric tensors can be defined by k‘-§k.§‘ orby cofactor of 9 (2.3.8) 9]“ = g 1“ (2.3.7) LG I where g = Igij" In order to formulate our equations we have need of the Christoffel symbols of the first and second kind defined respectively as ( 9) [130k] = 1/2(gij.j + gjk'i - gij,k) 2.3. m mn . {Id} = 9 [Wm] where as in section (2.2) f j denotes -—§£. This notation I am will prevail in the remainder of this work. Finally we require covariant derivatives of tensors Of the first and second orders, fl and flj. These deriva- tives are given respectively as i _ i i 11 f n. - f 'j + {nj}f (203010) i n i nnk + {nk}fnm - [mk}f n 2.3.2 Field4§guatigns of the Eluid The field equations which govern fluid motion at c(Dordinate q;1 and time t are written in terms of the Stress tensor tkm(¢},t), the rate of deformation tensor dlh(¢1.t). the velocity vector v(¢},t). the body force KC! f 5, gatity, an The figment o: :zstant an 2'3611) 52.3.12) 31 vector f(¢},t), the fluid pressure p(¢1,t), a scalar quantity, and the fluid density p1(¢1,t), also a scalar. The tensor component, tkm(¢§,t), denotes the component of the stress vector acting on the surface ¥F constant and in the direction of Sm at coordinate q} and time t. The component vk(¢i,t) of vector v(¢é,t) is that component acting in the direction of TF° The components fk and dkm are described in the same fashion as J“ and tkm. we are interested in those equations which constitute an apprOpriate continuum theory for the study of a compressible, viscous Newtonian fluid undergoing small amplitude motion. Those equations are conservation of mass: balance of momenta. both linear and angular; constitutive equations; equation of state and.kinematic equations. For a complete problem we require specification of boundary conditions on apprOpriate surfaces. we will list these equations and refer the reader to Eringen [19] for their derivation. They are Conservation of Mass k _ (2.3.11) p1,t + (pov )3k - 0 * Balance of Angular Momentum (2.3.12) t = t’ *The notation f t has been defined previously. I 1.3.13) 2.3.14) 2 must its: and 32 glance of Linear Momentum k n (2.3.13) t m;k + pogmn(fn-v 't) — o m .. 1,2,3. Constitutive Equations (Linear) k _ _ n k _ (2.3.14) t m — (p+)(d n)5 m + zudkm k,m — 1,2,3. ggpation of State 2 (2.3.15) p = plc0 . Kinematicfliations (Rate of deformation/velocity relations) k _ k kn i _ (2.3.16) 2d m - v ;m + g gimv :n k,m — 1,2,3. The constants p0 and c0 are the constant density and acoustic speed of the fluid while u and x are the shear and dilatational viscosities of the fluid respectively. To complete the boundary value problem we assume that the bounding surface S may be written as S = St U SV St ('1 Sv = (I where on St the fluid stresses are prescribed While on Sv the fluid velocities are known. Denoting the Surface coordinates by [mi] and the components of the Ontward normal to S by nk(cp:) we write . . k . . . . (2.3.17) tjk(cp:,t)n (mi) = (23%;) 3 = 1.2.3 for all m; on St and k i a: 1 (2.3.18) v (€93.19 = v (cps) k = 1,2,3 for all cp's on Sv. The t) and 37k are known on their respective surfaces . The nton, how at to to: 210 a EC: To is compo." :Ou q I “..an . 13.21: 33 These fluid equations completely describe the fluid motion, however other forms are often more useful in the work to follow. For this reason we rework these equations into a more desirable form. To begin we make some preliminary remarks concerning the components and base vectors utilized in the above formu- lations. Since the base vectors 51 are not necessarily unit vectors, we introduce physical components by normalizing the base vectors. We define v‘k) to be a physical component of 5 allowing 6 = 9* 5k to be written as G = v‘k) 5k where 6k = <'-;"k/'(gkk)1/2 (2.3.19) V“) = Vk(gkk)l/2 and the notation gk.k indicating no sum on the index ‘k. Physical components of the stress tensor tkm are given by _ ' 1/2 k (2.3.20) tO‘)(m) - WISE/913111) t m ° Enuations (2.3.13) now become /2 3 1 (k) (2.3.21) +1: (m)(g’-‘-]-‘-g‘l"i‘ _ tm) )-1/2 -1/2 [(gm) )‘1/2[ (9,“)1/2] 1,], (k) (9119;19-19 'm + pogkm[f(k)-v(k)'t]) = o m = 1,2,3. Another useful form is obtained when the stresses in the linear momentum equation (2.3.13) are eliminated by the constitutive equation (2.3.14) resulting in equations of rtien writtc =22 the Ravi 2.3.22) In‘ :7! V8 hi 2.3.23) The lies on ti 1.3024) El We (2.3 923530)] ( 7.3.25) 34 motion written in terms of the velocities. These equations are the Navier Stokes equations and have the form k ik n (2.3.22) (n+x)v {km.+ ugmng v :ik n - p'm + Pogm(f"-v 't) — o m _ 1,2,3. In vector notation, denoting the gradient operator by 3, we have for the Navier Stokes equations !&_ (2.3.23) (2u+;\)'v$('v°\7) - M?) x (3 x 3)) - vp + p0(f-v't) = o. ,1 The continuity equation or conservation of mass (2.3.11) takes on the form (2.3.24) pl't + pofim'r) = o. Eliminating p1 from (2.3.24) by the equation of state (2.3.15) and applying the divergence to the Navier Stokes equation (2.3.23) we obtain the pressure equation involving P alone and having the form 2 2 2 _. (2.3.25) (n+2u)v (P.t’ + poco V P ‘ p09.1:1: appropriat Mic ter injsa'tions. W1 flying i: ‘1) CIOss 5'29 shell ‘mCh is '35 Cartes (2.4.1) 35 3;2 Geometry of the System Having specified the equations of motion as well as the boundary conditions for both the membrane and the fluid it is necessary to specify the coordinate system appropriate to our problem and then determine the various metric tensors needed for the specific formulation of our equations. we view the prOblem as one composed of a fluid flowing in a curved membrane shell. The shell is circular in cross section of outer radius a+h and inner radius a. The shell itself is coiled on a larger circle of radius R which is constant. The relation between the tube and a set of Cartesian coordinates is shown in Figure 1a. A point in the fluid satisfies the following relationship (2.4.1) x = (R + r sin ¢)cos e y = (R + r sin n)sin e z = r cos n O g_r g.a o‘g 9.3 2w 0.3 (‘3 2n. we note that if R = 0 the coordinates are spherical Ixfler while if R approaches infinity the tube appears Straight. ‘We will utilize this last concept later in the discussion. E) ts; 36 m :xrdinate mmfi "1 ~07. Fr< 12.4.3) "i916 We h I I’M. I e 1 “L [2.2. 37 On the middle surface of the tube we establish a coordinate system as seen in Figure lb. Using (2.4.1), a point on this middle surface is given by (2.4.2) x = (R + a sin ¢)cos e y = (R + 2_sin ()sin 9 z =‘a cos n a=a+h/2 O__<_e_<_2'rr O_<_()_<_27r. From (2.2.3) to (2.2.5) we see C11 C12 Rl =1; R2 (R +‘a sin ¢)/sin n R +‘a sin n II In: (2.4.3) where we have identified 91 with n and 92 with 6. From (2.2.1) to (2.2.4) we find ll '1 (2.4.4) 911 922 = (R + r sin ()2 933 = 1 gkj=o R?!) where 1 corresponds to the y direction, 2 corresponds to the 9 direction and 3 corresponds to the r direction. a 5 ‘, ' 1.: F3315 — fl ‘6 I!“ v:“' u. but Ibe '\5: “A In: sigh: :L'oduce .5.-5) St} We 22.5.1) 333% we < l" I 2.3.2) E also (it 1.5.3) 38 _2L§ ‘gluid Shell Prdblem - Complete Equations Having found the metric tensors and principal radii of curvature we are now able to write the shell and fluid equations in a form appropriate to our geometry. We first introduce non-dimensional coordinates, velocities, displace- ments, stresses and pressure which will allow us to isolate an important parameter for our analysis. we set * (2.5.1) r* = r/a t = ubt where we define a frequency ”D by (2.5.2) mg = E/p_a_2(l-v2) . we also define 0* = CO/au’o l/TI = (HM/Poazub l/n1 = Wed-=12 (1b 12* = p/poazuf; (2.5.3) um = v(a)/aub ‘Wa = U(a)/é _ (a) 2 2 * _ 2 2 _ m = ph/poa 5 = a/R .g = g/R The ratio m. will prove to be a convenient parameter later in ‘the analysis. As before we identify subscripts 1,2,3 on field v’ariableswith n,e,r respectively. Hence £12 is replaced EL.“ I 99' J! O I I? isscribed i (til the s? isboiy fc :u; +Y nu! SWEE Ut: i: shell ( L“ 4) v In fitOUgh (2 mil (2 I2'50?) .Ls.e) “5.9) 39 by f *9. The fluid-shell interaction problem can now be described in terms of our specific geometry. We begin with the shell equations of motion in section 2.2 identifying the body force components q l" q 9' qr respectively with the v: S . S evaluated at r = l. fluid stresses 81")" re rr Utilizing (2.5.1) to (2.5.3) in (2.2.3) to (2.2.5) the shell equations of motion assume the form * * * ' N -N l+§31n¢ ‘W'l‘ l+_§.s1n(y a — + Ba? Sr.” ‘r*=l ’ w¢,t*t* * * (2.5.5) 23 COS WNW + N* + 11 Need; l+_§s1n¢ (”on 1+_§s1n¢ a — + 75; 5’9‘r*=1 - we'vt. , 'k . w, 1 + _§ sin ¢ mg rr r*=1 r,t*t* In like manner the stress—strain relations (2.2.6) through (2.2.8) and the strain-displacement relations (2.2.9) thirough (2.2.11) become (2-5.7) NW*= ew+v€ee (2-5-8) N9; = E99 + vew (2°5~9) Ne; =(1-\))eM (2°5°10) -w +w Ew' M r P.“- _. l2.5.11) Kill) We is fluid 2 536;; fo 2.5.13) 13.3.14) (m 7.5.13) 4O .2 . (2.5.11) e99 = T3373T3_W [we'e + wwcos n + wt s1n n] 2 5 12) 2 - + ‘3 [ - ( . . 66W —‘we'w ltfi sin W ww'e ‘we cos n]. We are now able to apply (2.5.1) through (2.5.3) to the fluid equations given in section 2.3. we assume there is rm>body force on the fluid. we have then Conservation of Mass 5 cos it _l_ L 1+5r*sTn .1. 11¢ + r* uh”) + 1+5r*sin e “9, 9 + ur,r* (2.5.13) 1 5 sin y l * + [--+ Jur = - P r* l+5r*sin W (c*)2 ,t* Balance of Angular Momentumg12.3.12) (2.5.14) SOLE3 = s a,B meaning n,9,r Bo Balance of Linear Momentum (2.3.13) 6 cos fly, 6 S¢¢,¢ + 1+6r*sin n Snn + 1+5r*sin ¢ Sye,e HAP, (2.5.15) + s * + [3;- + 14533.15” 5 cos [ -il+5r*sin * See — uw't* 2 5,cosg1 S + S + _+ g sin 1+5r*sin e we er,r* 1+5r*sin e] + “T S + * ‘ S = u * r elm) Mar 8111 s 96.6 e.t 41 ..L 5 Loos l) r* Sww + 1+5r-ksinwW Ser, 9 + 1+5r*sin ¢ Snr .1. .4. mini (2'5’17) + r* Sent: + Srr,r“' + [r* + l+5r*sin n] Srr 5 s1n l) __. u - 1+5r*sin n 599 r,t* Constitutive Equations [(2.3.14), (2.3.16)] (2.5.18a) SW = _p* + 35;;- [WNW ur] + 0}] - fill-)1) (2.5.18b) S99 = -p* + n1[l+ig;sin $1 [“909 + uw cos n + ur sin n] + (% --%:)D (2.5.18c) srr = -p* +l€1 ur,r* + (% --%:)D (2'5'18d) Swe = 71;; if; 11‘9”) + 1+6rfsin ( (uh: 9 ' “9 cos (0] (2.5,19e) Slr=-7%£[ul"r*+ *ur'w-fiuw] (2.5.18f) Ser=-T%I[ue,r* +1+5rfsin w (“me-“9 sin (0] (2.5.189) D=fi;[nine-Fur]+1+(5r12sin(I[magi-“Wm”W ‘ + + ur sin n] ur,r* It will be remembered that in addition to the above equations, we obtained the Navier Stokes equation (2.3.23) and the Pressure equation (2.3.25) through various manipulations. 1 01. fl (1.229313 Eve the 2.5.19; 15.20 42 Applying the same non-dimensional changes as before we have the Navier Stokes equation in component form. Navier Stokes Equation (2.3.23) 1+ QCOU l 1 FE (7:;77'iuloll ' “w + Zur.¢ r*(1+ar*§In l) [ulvl 62 2 + u ] + [-u cos n + u r (l+5r*sin ¢)2 W (:96 (2.5.19) - 2 cos n ue'9 - sin i cos n ur] ;L_ 5 sin W + [r* + l+5r*sin W] u¢,r* + u¢,r*r*} 1 * 1 p*,1|[t* wti. = _; [P '1‘ + — * 2 } I r o T] (C) 1 u . L008 l! ?E'[7f%%% + r*(l+5r*s1n w)“ e,¢ + 62 + (l+5r*sin ()2 [2 c°s s “l.e + “6.69 ' “9 (2.5.20) . _l_+ _£Lsin L + 2 s1n n ur,9] + [r*+ l+5r*sin VJU + ue,r*r*} * _ a * 1513.33: “e.t* ‘ (1+5r*sin () [P .e + n (c*,2 } .11 ..l... _ _ 5 cos l 111 ( *)2 [umw “r 2%,(1 + r*(1+5r*sm‘¢) [ “r 4. 62 - u + u - cos sin u ( W] (1+5r*sin ()2 [ r'99 l l l 2~5.21) _ . _ . 2 _};+ ‘gqsin ( 2 Sln n 119'e Sln w ur] + [r*+ 1+5r*sin ¢]ur * i - u — * + l p ,r*t* r,r*r* r,t* ’ P ,r* n (c*)2 Pi: 32.5.22) lfi'miary 2ieracti iii-fled Irv (:2 [15.23) 95.24) 43 Finally we express the pressure equation as Pressure Equation (2.3.25) {l+-——7[% +—]'_1]aa —.E""'_-}[ (13*)2p*0¢W+-};p*’r* (C ) r '1’ (205.22) + P* + 6—T [c:: w p V4- 6 * ,r*r* l+5r*s1n n (1+5r*sin l) p :99 + sin n p* r*]}= -?——;f p* t*t* To complete the boundary value problem we specify the boundary conditions appropriate to the complete fluid-shell interaction problem as well as some special cases to be examined for the fluid alone and for the shell alone. _§luid-Shell Interaction Boundary Conditions r =1 u¢=w¢'t* (2.5.23) u9 = we 12* I u _ — w r r,t* * * O p , u ,u ,u are finite. (2.5.24) r (I 9 r Fluid Boundary Conditions A~ 3191:: Shell 44 B. NO Shell (stress free) * r = l s = s = s = For both (A) and (B) (2.5.24) applies. Empty_$hell Boundary Condition Displacements have period Zn in ¢~ i ._- 1‘ 3:? (y .M'hi . Io JU . err. is has Its app-e a: . .. pertL .‘fi‘ssas 1'? ROM ." 5 it I. H Nib 45 2.6 Traveling_wave Solutions 2.6.1 Introductornyemafks A careful examination of the complete mathematical prOblem presented in section 2.5 reveals the enormous task one has in solving these equations in their most general form. The appearance of the parameters 6 and .Q suggest a method of perturbation for the dependent variables allowing us to pose a series of problems, each simpler than the original, yet contributing to the solution of the original problem. We recall that 6 and .5 are defined by 5 = a/R and '5 = (a+h/'2)!!-1 and that from the thin shell theory h/a < .05. Hence the ratios .5/5 and a/a are approximately 1. we also note that when 5 is small we have the situation of a slightly curved tube with R large. Conversely, the case of 5 large can be interpreted as R very small, yielding the description of a flow inside a sphere of radius nearly a. Our choice is to examine the situation of 5 small expanding all dependent variables in a perturbation about small 5. Once this decision is made two possibilities present themselves for our consideration. If our tube were a straight cylinder, then a dependent variable X would have a wave solution of the form '5 (here u u ;‘q l~‘~ 'u'. ye ”Mate 46 (2.6.1) X(r. (not) = X(r. ¢)exp[i(kC-wt)] Where r, WC represent cylindrical coordinates C being the coordinate along the axis of the cylinder. The parameters k and u) are the wave number and frequency respectively. Recognizing that in our curved tube Q could be taken to be Re, an appropriate form for us to consider for the field variable X is A (2 .6.2) X(r*.(). 9.5.1:) = X((:.r*. a5)eXp[i(§-6fl - Bt)] Where s and B are the wave number and frequency respectively. A Since X depends on 5 which is small, it also seems appro- A priate to expand X as a series about 5 small, i.e. * * X(¢l ear Iblt) = [Xo(l’lr) + 6X1(Wor) +'°'}E (2.6.3) 9 2 E = exp{i[3 (30 + 5s1 + 5 32 +---) - at]] The form (2.6.3) yields as its lowest terms the f(Diem (2.6.1) while the form X(¢.e.r*.5.t) = {x;(¢.r*) + 5X;(¢.r*) +---ls:1 (2.6.4) E1 = exr>[i[e(sao+zss1 +~--) - Btll Yields as its lowest terms solutions apprOpriate to plane flow, i-e., flow independent of distance down the tube. In this 8ituation curvature effects are found in second or higher order terms. The form presented in (2.6.4) is utilized when one is concerned with the secondary flow patterns in the cross Section of the tube. 47 2.6.2 Perturbation Expansion in Small _5 We expand all dependent variables in an expansion suggested by (2.6.3) . We assume s=s +6sl+523 +... O 2 2 _ . O O l l 2 2 (wo'ua) — i[(wa.ua) + 5(wa,ua) + 5 (wa,ua) +---}E a meaning 4; or r _ o o 1 1 2 u2 (wanna) — {(We'ue) + 5(we.u e) + 6 (WE, ue) + }E * _ . o 1 2 2 Nona - J'{Ncm + 6NCIO. + 6 Nod +°'°)E . o 1 2 2 S = 1 S + S + S 4"... E 5: I g: 5 g: 5 g: } a meaning 5: or 9, g being 5.9 or r* * _ 0 1 22 New — [119er 5N61p + 5N e¢+°”}E So _ So 0 1 S2 (Saw W‘Hew'5)+5(swer)+°(sve r)+ )E * All fluid quantities such as uka are functions of ¢ and r While all shell variables are functions of ¢ only. It now remains to substitute (2.6.4) into the equation of section 2.5 collecting terms as coefficients of powers of 5 and equating these coefficients to zero. This allows us to define zero, first and second order problems which are to be solved in the following subsections. We will now present the equations in the order we intend to solve them. We will also drop the * from all variables realizing that all quantities are non-dimensional. 3x: 0 - O‘ ._..- {2.6.7' 48 To facilitate the presentation of these equations we choose to define the following differential Operators. The L operators are associated with fluid velocities while those designated with M Operate on fluid stresses. k k k 1 k k 1 k L1(u u u ) — E u W!) + sou 9 + E(ru r),r 1 a l a 2 (2.6.5)L = ——+- (r—- -s 2 r: N2 r ar 5r 0 1 L = L - _ 3 2 r2 k 1 k k k M1(s’;r,sw) = E[S(r + (”War + 3W4] k _ 1 k k (2.6.6) M2(S:W.Ser) — Emewaw + (r Sega] ”13(5".Sk.sk)=l[sk -s’;w+(rs") 1 ¢¢ rr r r¢,¢ rr ,r k = 0'1'2'3'... we also define __.211 RC — 1 15/6 [n + “1} (2.6.7) k1 = 1 - iB/TICZ 2.6.3 gero Order Problems (straight cylinder) Pressure Equation 2 (2.6.8) (L + L) (p ) = o 2 czkg o f .2.6.9) '2.6.1 49 Navier Stdkes Equations -%:[L3(u0¢) +-fi% ug'w} + iBui = (2.6.9) fili- L2 (1109) + iBqu = -sokipo %[L3(uor) - :2-2- now, (I + irsuor Continuity_Equation (2.6.10) Ll(u0¢.uoe,uor) =-:§; Constitutive Equations SoW = -po + (313-31? 1304-73;- 5099: -p0+ 01-1-7111) ”0 +712; SOrr = -P0 + (% -'%:) D'0 +'%i (2.6.11) 5°M = 711-1- [-souo¢ + % “09.)v} Sotr = 71;; (“01'1“ + 3'1: [“oru) ' Soar = 71%; [noevr - 30.10:} D0 = L1(uow,u0 ,uor) HIH 1190. (y klp0,r HIH M (1.6.1) SO Fluid Stress Equations of Motion 0 O O . O wr,S W) + 80$ )9 ifiu W o 0 o . o 0601 S I - S = '- (2 2) M2( 9)) 3 er) so 99 1511 e M (S 1 o o o ___. o M3(S “,5 ”,5 rr) + sos re’ iau r Shell Stress - Displacement Equations N0 w0 + w0 + vs W'0 W WU) 1' 0 (2.6.13) N0 s wo + (,{w0 + wo} 96 0 e ¢.¢ 0 -1- 0 - New—2(1v)[w 90W Sow“. Shell Equations of Motion 0 s N0 + NW)“ 0 9) ans ans I I U) £ Boundary Conditions at r = 1 A. Coupled Prdblem u°¢(¢,1) = -iBw (2.6.15) u°e(¢,1) = -iBw uor(¢,l) = -iBw ‘V" 51 If the fluid is inviscid then (2.6.15) becomes -i6w'O (w) oswszw 0 (2.6.16) u r(¢'1) r B. Rigid Shell (2.6.17) u0¢(¢,1) u09(¢,1) = uor(¢,1) = o 0 < w < 2v If the fluid is inviscid we replace (2.6.17) by (2.6.18) u°r((.,1) = o o 3 ¢ < 21r C. Stress Free Fluid (No tube) 2v V\ o _ O _ 0 _ (2.6.19) Sr¢()"l) —Sre(¢,l) —Srr(¢,l) —0 03¢ D. No Fluid Shell displacements are periodic in 5 with period 2v. Boundary Conditions at r = 0 All fluid variables are finite at r = 0. 2.6.4 First Order Problem Pressure Equation 2 (2.6.20) (1.2 + 2%.?“ = po{28081 - 226g sin w} 1 - E Poo) cos W - p0.r sin 5 f‘ f a ('DS‘ 52 Navier Stokesnguations 1 1. 2 1. . 1 1 2 -—-[L (u ) + ——-u } + ifiu = - k n1 3 5 r2 r,¢ W r 1 p1,¢ 1 2 O O . -.fi; {(2rsO u W + u )vr) Sin W + % cos ¢(u0¢'¢ + uor - 2rsouoe)} 1 1 . 1 __ _ 2 _ _1_ 2 0 (2.6.21) 71-; L2 (11 9) + 1E5u 9 — klsOpl n1[[2rso u 9 O O . l O _ Zsou r + u 6']:,] Sin W + E cos ¢[u 60¢ - 2rs uO ]} - rs p sin V o (1, o o 2 1 . l 2 [L (11 ) - u } + ifiu = k p 111 3 r :2- 5“) 1 1,r l 2 0 O 0 . -«fiI {[erO u r + u r,r - ZSOu ] Sin ¢ 1 O O + E COS 1)! [u ro‘)‘ " 11 $1] Continuity_Equation ' 1 1 1 _ £9 0 0 (2.6.22) L1(u ¢,u q,u r) — c2 p1 slu e + [rsou 9 O . O - u r] Sin W — u W cos W Constitutive Equations 1 1 1. 2 1 1 1 S = - + - --- D +-—— - u + u ] WW P1 (n n1) 1 “1 r { *0? r {2.5 53 1 l l 2 l O S = - + — - -—» +-—— + O O O . + u W cos W + (u r — rsou e)san w] 1 _ _ 1 __1_ .2. 1 S rr — p1 + (a n1) D1 + n1 u r,r 1 1. 1 l l 1 2.6.23) S = --[u + - u - u )] ( (r 711 (:r r‘ raw 1 1 __1_ _ _ o _1_ 1 o . S 9W - n1 [ sou Slu 9 + r u 90¢ + rsou s1n w - u0 cos )1 1 __1_ 1 1 . 8 er - ”1 {u BII + sou r u 6 Sln ¢ . O + (sl-rsO Sln ¢)u r} 1 1 D1 - L1(u will ecu r) Fluid Stress Equations of Motion 1 1 1 . 1 O O M S .S ) + s S = -1 u + S -S cos 1) wr ¢¢ O )9 B [ 99 1)} ¢ . O O 0 + s1n w {rsOS )6 - S Wr} - sls )9 (2.6.24) M2(Sl ,Sl ) - s S1 = -iBu1 - 280 cos W 95 er O 99 9 9) 0 O . 0 [25 at + rsoS ee)s:1n¢+ 81$ 96 1 1 1 1 __. 1 _ o M3(S rq',S ¢W'S rr) + SOS re — iflu r S Vr cos ¢ 0 o o . _ o + {rsos r9 + S 99 S rr} sin 5 318 re Shell 12.5.25) ((3 (2.6.26) 54 Shell Stress - Displacement _E‘guations _ 1 l _ . O 1 NJ'M—www+wr+()[(s]_5051n(1.)we+sowe O O . +W¢C°SW+Wr31nW (2.6.25) N166 = sowle + v{w1¢'¢ + wlr} + (sl-so sin (Owe O O . + w cos ((+- w r s1n q, l __ l _ 1 _ 1 _ _ . O N 61)- -2-(1 V)[w 80)) sow ‘)' (81 S0 811') (0w W ‘i - wO cos (3} ' 9 Shell Equations of Motion 1 1 1 1 2 1 O N + N + - s = - - N l WM) 50 9‘) 1“ r¢)r=1 f3 w ¢ 81 9‘) | o o o . - N - N O + N in 1) { (w 66} c S * S0 9) 8 W 1 1 l l 2 1 O 0 0 - N + — = — + (2 6 26) N “My 30 99 m S re)n=l 6 w 9 slN 69 _ 2N0” cos 5 - BONDBe sin (1. 1 1 1 _ _ 2 1 O . ..NW-fisrr)r=l—Bwr+N9931n¢ __.— M ’ I-uln' N “If 6 I!” A. Cougled I (2.6.27) Al] _— “1.5.!“ ._ SS Boundary Conditions at r = l A. Coupled Problem u1¢(¢,1) = -iBw1*(W) (2.6.27) ule(1).l) = -iBw19(¢) o _<_ (,3 21T u1r(¢.1) = -iBwlr(¢) If the fluid is inviscid (2.6.27) is replaced by (2.6.28) u1 (¢,1) = -iBw1 (1)) o < w < 211' r r -— - B. Rigid Shell (2.6.29) u1¢(¢,l) = ule(¢,l) = u1r(¢,l) = o o g (3 g 21r If the fluid is inviscid we replace (2.6.29) by (2.6.30) u1r(¢.1) = o o _g ()5 21r C. Stress Free Fluid (2 6 31) $1 ( 1) = s1 ( 1) = s1 ( 1) = o o < 2? ° ' rw )1“ re ‘1" rr 1)” S *— D. Noygluid Shell displacements are periodic in v with period 2w. Boundary Conditions at r = 0 A11 fluid variables are finite at r = O. For a: find 8an tie process eliminates 1 fluid stresi ;lacement rell shell and 11' (2.6.32) 56 .g;§;§ Second Order Problem For the second order problem it will not be necessary to find exact values of the pressure and velocity terms since the process for determining the second order correction eliminates these variables. We need to describe only the fluid stress equations of motion, the shell stress dis- placement relations, the stress equations of motion for the shell and the boundary conditions. Fluid Stress Equations of Motion (S2 2 ) + s 82 = -iBu2 - s 1 0 M1 11"5 w 0 (8 1 1S ()8 l 1 + cos ¢ {S W9 1 . 69 - S W} + Sln w [rsoS . 0 0 S ) + r cos W Sln W [S ¢W S 96} . 2 l 0 206.32 p - = '- ( ) M2(S 9W 8 6r SOS 69 ifiu 9 + 318 99 + 328 99 _ Zslwe cos w - sin y] 2819r + rsoslGe O 0 + rsls 96} + 2r cos 5 sin W S We . 2 O 2 O + + r sin 5 {ZS 6r r80 96] 2 2 2 2 _ . 2 1 M§(S wr,S W.S rr) + 308 re — -1Bu r - 815 6r . 1 2S 9r - S Wr cos w + Sin ¢ {rsoS (2.6.33) 57 + rs so + 81 - s1 } 1 8r 66 rr + r cos 5 sin 5 S0 + r sin2 5 [sorr 5r - s0 + rs 2 so } 98 0 6r Shell Stress - Displacement Relations N2 = w2 + wzr + V{cos 5(wl WOW sin 5) W) Wv) W . 1 O . 2 1 0 + Sln 5(w rdw r Sin 5) + sdw 9 + siw 9 + 52w 6 l O . O . 2 - (BOW 6 + 81w e)Sin 5 + sow 6 Sln 5) 2 _ 2 2 2 1 . (2 .6.33) N 99 — v(w 1)") + w r) + sow 6 + w e(sl-so Sin 5) O . . 2 + we(sz-sl Sln 5 + 30 Sin 5) 2 _ (1-5) 2 2 1 _ . N 9) 2 {w 9:) + SOW’¢ + w¢(s1 so Sln 5) O . . 2 +'w W s2-s1 s1n 5 + sO s1n 5) - cos 5(w1e -woe sin 5)} Shell Equations of Motion 2 s N2 + l S m 2 NW)" 0 95 _ 2 2 _ 1 r5‘r=1 “ —B w 5 SIN 95 O 1 1 - $2N 9) cos 5 {N WW N 09] + sin 5 [SON19¢+ [SlwsO sin HNOM) + sin 5 cos 5 [NoW — N099} 72.6.34) 112 2m 46.35) 2 N2 l 2 2 2 o. - +" =- (2 634) N 6):) s0 99 ms 9r|r=l B w e 1 O + SlN 69 + s2N 96 - 2N 6) cos 5 . l . O - Sln 5 {SON 98 + [sl-sO Sln 5]N 96} + 2 NO cos sin e») ‘” q’ 2 1 2 __2 2 . 1 - N (W - a S rr'r=l — B w r + Sin 5 {N 69 - sin 5 N096) Boundary Conditions at r = 1 A- Coupled Problem 2 t -iBw29(5) 0.3 ¢.S 2w u2¢(5.1) = -iaw (5) (2.6.35) u29(¢.l) -iBw2 (5) 2 u r(¢01) r If the fluid is inviscid (2.6.35) is replaced by (2.6.36) u2r(5.1) = 4th2 (5) o _<_ (3 zvr r 3. Rigid Shell (2.6.37) u2¢(5,1) = u28(5,1) = u2r(5,l) O 0 3 53 Zr If the fluid is inviscid (2.6.37) is replaced by (2 o6o38) “2:.(1101) = 0 O _<.. 1' < 277. 59 C. Stress Free Fluid (2.6.39) 5 (5,1) = s (5,1) = szrr(5.1) = o o _<_ 5 < 21) D. No Fluid Shell displacements are periodic in 5 ‘with period 2?. Boundary Conditions at r = O All fluid variables are finite at r = O. CURVATU 3.1 Introdu In 6 :V’md. 8188 if the me five. The N salution anc‘ m‘dimensic is the I; is the nOn-. (3.1.1) “We did 1 We set in “0t; 39mg n0 0‘ vibration CHAPTER III CURVATURE CORRECTION FOR THE EMPTY SHELL 3.1 Introduction In Chapter two we displayed those equations necessary for the complete description of a fluid flowing in a thin, curved, elastic membrane shell. The solutions we seek are of the form of an amplitude function multiplied by a traveling ‘wave. The wave portion of our solution is common to every solution and has the form exp i{396-1 - Bt] where B is a non-dimensional frequency, t is the non-dimensional time, 5 is the ratio aR-l, 9 is the angle down the tube and s is the non-dimensional wave number. we expanded s in the fashion of _ 2 (3.1.1) 5 — SO + 531 + 5 32 +... t as we did for all our variables except 6. We now assume that our fluid shell system.has been .set in motion somehow and now vibrates in some fashion, there being no outside forces interfering with the motion. Our task is to characterize this system by describing the modes of vibration. The modes are merely the various patterns which 60 —\ the vibratic =14 "or fluid It '1 is relation? fiber 5. iraiicate the vibrating an 0‘: fhlid , 'I Ember Const be utiliZEd 61 the vibration waves exhibit as they travel along the tube and/or fluid. It is our task to describe these modes by finding the relationship between the frequency, B and the wave number 3. Once this relationship is established we may then indicate the frequency at which the tube and/or fluid is vibrating and then know the wave number 3 corresponding to the frequency indicated. From these two values we can determine the speed of the waves traveling down the tube and/ or fluid. The relationship between the frequency and wave number constitutes a characterization of our system and can be utilized in other problems involving the same system. If we study the same model under the influence of outside forces during the vibration, we may utilize the modes to describe the outside force and assist in the solution of our new problem. Our immediate concern is to find a relationship between the wave number s and the frequency. we begin by finding the relationship between 3 and 6 following that O with the corrections to s namely 31 and 82. This is 00 the procedure to be utilized in the remainder of the thesis. we begin by considering the shell without the fluid, establishing the displacement equations of motion for the ezpty shell iaemine th 1'. arbitrary iirection. In t ”4 the funde ”5‘3! morior. kfdeen S 2‘.- ..e zero 0rd Second 0rde I ’R 0“” Wave r 62 empty shell in section 3.2. In the next section, 3.3, we determine the relationships between and 6 for s0'31'32 an arbitrary number of waves around the tube in the W direction. In the remainder of Chapter three we concentrate on the fundamental case of axisymmetric motion for the zero order motion with section 3.4 containing the relationship between s0 and B. In section 3.5 we discuss the effects of the first order corrections on the functions described in the zero order. Finally, in section 3.6 we establish the second order correction, s and relate it to the zero 2' order wave number so. ‘1 3.2 Shell Di; 1!) se :fmtion for (2.6.14), (2, . . i saved the s triers in egg; 10‘ 5338 the relations int «5 (2.6.34). ufferential F11 63 3.2 Shell Displacement Equations of Motion In section 2.6 we formulated the stress equations of motion for the shell in the zero, first and second orders, (2.6.14), (2.6.26) and (2.6.34) respectively. we also derived the stress displacement relations for the same three orders in equations (2.6.13), (2.6.25) and (2.6.33). we now make the substitution of apprOpriate stress displacement relations into the shell equations of motion (2.6.14), (2.6.26) and (2.6.34). To facilitate the notation we define three differential Operations on the displacement components as: k k k _ k 2 _1 2 _ k F1(w WM 9.))! r) -w WW + [B 5 30(1 VHW') + % so(l+v)wke'¢ + wkr,5 , F2(wk¢,wke,wkr) = _ % 30(1+v)5kw ( + {Bz-sngke (3.2.1) + %(l-v)wke'¢¢ - vsowkr . k k k __k _ k 2_ k F3(w V'w e.w r) — w W”) vsow 9 + (B l)w r , k = 0'1'2'3'...o we are now in a position to indicate the equations of motion for the shell, written in terms Of the shell displace- ments. It should be repeated that the thin shell conditions result in the shell displacements being independent Of r. Hence the wk 5 meaning 9,5,r, 'will be functions Of 5 g! only. LN F (3.2.2) F F fl P (3.23) F H.2‘4) 64 Zero Order Shell - Displacement Equations of Motion O O O 1 0 F (W ow lw ) = " "" S _ o o o O _ l 0 (3.2.2) F2(w WA)? e,w r) _ — a S ‘r=l , r9 0 o o 1 O F (W ow ow ) = + - S ‘ _ 0 3 5 e r m rr r—l First Order Shell - Displacement Equations Of Motion 1 1 1 l 1 l o Fl(w wyw e,w r) = - a Sr¢|r=1 - $1[§(1+v)w 9'5 (3.2.3) _ (I'V)3Owo5} + cos 5{%(3-v)sdw°e - w°5.5 - wor} + sin W[% 80(1+V)wo9:¢ +*w°w[v-sg(1-V)] - vwor,5}' 1 1 1 _ 1 1 1 o F2(W’¢,W'6,W’r) — - a Sr95r=1 + sl[§(l+v)w 5.5 + Zso'woe + vwor} + % cos (Iso(3-v)w°¢ (3.2.4) _ (1-V)w°e'¢} + % sin ((({2s0(1.--\))w°r 2 O o - (1-V-4SO)W 9 ' (1+V)80W t' W}: 1 1 1 _ 1 .1 o o F3(w w.W’exw r) — +.fi'srr‘r=1 + Vsl'we + vw V cos i (3.2.5) + sin ¢[SO(1-V)Woe + 2w°r + wan”). )3.2.6) (3.2.7) 65 Second Order Shell - Displacement Equations of MOtion 2 2 2 _ _ _1_ 2 _ .1. 1 F1(w’5fw e’w r) — m Sr5‘r=l Sl[2 (1+V)w 6v) _ so(l-v)wlw - .21- sl(l-v)w°w} - s2[% (1+v)W°e,5 - 30(1-v>w°¢) + cos )(§ (3-v)[sdwle + s1w° 1 _ wlw'w _ wlr} + sin 5[% (1+v)[SOwle,5 + Slwog,‘)] (3.2.6) + wlw[v-sg(l-v)] - Win.) - 25081(1‘V)"°5} . O O O + Sln W cos 5[w W'W - 80(3-V)W 9 +'w r} l . 2 O 2 ° + 5 Sin )[w W[3sO (l-v) - 2v] - so (1+v)w 9,5 + szor,5} + wow C052) . F(w2 w2 w2)'-152) 2 5' 9' r _ 5 r6 r=1 1 1 1 o 1 + 51)? (1+v)w W'W + ZSOW 8 + 81w 9 + VW r} + s2[% (1+v)w°¢ ) + 2sow° + wor) (3.2.7) + % cos 5{(3-v)[sow1¢ + slwowl - (1‘V)"18.¢} _ .2. gm 5((1+v)[sow 5'“, + slw 5'5] + [1 Mso]w e - asoslwoe - 2(1-v)[so‘w1r + Siwpr11 . 1 o O + COS l) Sln *[2 (l‘V)w 9") " 30(3-V)W 5} F) )328) 66 + .1- sin2 5(s0(1+v)w° 2 O 2 + [1-v+680]w WM) (l—\a)w°e c032): , NH" O - 280(2-v)w r] + 2 2 1 F3(w ¢,w e,w r SrrIr=1 + v[slw 8 + 32w 1 . 1 ' + vw W cos 5 + Sln 5((1-v)[sdw 9 + 31w 9 ll .5. BIH (3.2.8) + \J[2wlr + w1 w]] + (l-v)w° cos (i sin I) i)! W - sin25{vw° + sO(Z—\J)w0e - (1-2v)w°r}. lb!) 3.3 Co_rr_e_c_ In as well as for an arb; 1n SUbSEQUE fandamenta 1 We 12" in W ferric flm ? Jase Crit: displaceme] 67 3.3 Corrections for Arbitrary n In this section we deveIOp the frequency equation as well as the first and second order correction equations for an arbitrary number of vibration waves around the tube. In subsequent sections we analyze these results for the fundamental case Of axisymmetric flow in the zero order system. In (2.5.3) we introduced the non-dimensional parameter m = ph/boa, m being one-half the ratio of tube density per unit length to fluid density per unit length. If there is no fluid then poa = O and m-1 is zero. In this situation . k k k _ the fluid stresses Sr)’ Srq and Srr (k - O,l,2,...) evaluated on r = l have no influence on the shell displace- ments. The situation of mfl = O is the case we are now investigating. we refer to it as the empty tube case. we expect all shell displacements tO have period Zn in w and we further expect that in the case Of axisym- metric flow, displacement components are independent Of 5. These criteria suggest the following form for zero order displacements. It is o w’0 sin(n¢) 3 ll ‘) ‘1! (3.3.1) w°e = w°e cos(n5) n = 0.1.2.3.... wor = W0r cos(n)) For axisymmetric motion, we merely set n = O and the conditions mentioned previously are satisfied. i Subst .. . | farming sys .2 D - l 2 :1 order that fireTJire th E31511. It S. It alter the 'em‘inant 1 . h-i 68 Substitution Of (3.3.1) into (3.2.2) gives the following system in homogeneous form. It is 2 2 l 2 1 \ B - n - 5 so(l-v) - 5 nso(l+v) —n W 1 2 2 l 2 _ 5 nso(1+v) B - SO - 5 n (l-V) -VSO W0 -n -vs BZ—l w ‘ 0 (3.3.2) 0 = O O In order that we have non-trivial solutions for this system 'we require that the determinant of the coefficient matrix vanish. It should be noted that replacing n by -n does not alter the value Of the determinant. Vanishing Of the determinant leads to an equation of the form (3.3.3) 50 = f(5.v,n) which is known as the frequency equation. Given values Of the frequency, Poisson's ratio and n we can compute a value for so, the zero order wave number. In the general case we find that 30 can be Obtained from __ 4 1-v 2 2 22 2_ l _ 2 O - so (—-2 )(B -1+v) - B SOUB 1)§(3 v) + v (3.3.4) - n2(1-9)1 + 64(52-1) - n292[§(92-1-n2)(1-v) + 62]. We will re). retric mot; Ba '50 0111' pri: zero corre2) crder shell contain 5 right sides fCllOWlng f u) 69 ‘We*will return to this equation when we investigate axisym- metric motion for the zero order. Having found the zero order wave number, we proceed to our primary Objective: determination of the first non- zero correction for the wave number. we begin with the first order shell equations, (3.2.3) through (3.2.5), since they contain 81' the first correction to 3 Examination of the 0. right sides Of these three equations indicates that the following forms are needed for our first order terms. They are w1 = a cos(n+l)¢ +‘b cos(n-l)5 + c ¢ w W i + d sin(n-l)¢ + e cos n) + h sin n( ) W ¢ sin(n+l)¢ ‘w = a sin(n+1)5 + b sin(n-l)5 + c9 cos(n+l)5 (3.3.5) + de cos(n-l)5 + ee sin n5 + he cos n5 w = a sin(n+l)5 + b sin(n-1)5 + cr cos(n+l)¢ + d cos(n-l)5 + e sin n5 + hr cos n). These forms will allow for all possible trigonometric terms which might appear in the inhomogeneous terms Of (3.2.3) through (3.2.5). All coefficients of trigonometric terms are constants. Inserting (3.3.5) into (3.2.3) through (3.2.5) yields six systems of equations, three of which are homogeneous. The three homogeneous systems contain the C(o). 6“) and e(,) , ) coefficient. «efficient the three h )3-3.6) 7“» f? «ii .1 equen Where det (I): 7O coefficients separately as column vectors. If we denote the coefficient matrix of (3.3.2) by ”(n) then we may write the three homogeneous systems as CW\ 0 5 d) ’O \ M(n+l) c9 = O ”(n-l) d9 = 0 cr 0 dr 0 (3.3.6) ‘ 8W O M(-n) e9 = 0 er 0 Our frequency equation is merely the result Of det(M5n)) = 0 where det is the determinant. Since the determinants Of the first two matrices in (3.3.6) are not identical to that of "(n) Cramer's rule allows us to conclude that the c(.) and d(.) coefficients are all zero. In the third system of (3.3.6) we conclude that the e(.) coefficients are arbi- trary and set them to zero. The remaining systems are inhomogeneous and can be written as (3.3.7) M(_n_1) as = £9 M(-n+1) be = 9 )3.3 8 . ) {a 2.3.9) In (3.3 .7) 08 2 rd ( Caef \ ; tici e n t 71 and as h) k) (3.3.8) Mkn) h9 = 31 k9 hr r k) = % n(l+v)wpe + sO(l-V)WOw (3.3.9) k8 = % n(1+v)W‘°w + Zsowoe + v W0r _ o kr — v W 9 In (3.3.7) the inhomogeneous terms are all known in terms of zero order displacements. Again, the determinants Of the coefficient matrices are non-zero and so we can solve for the a(.) and b(.) coefficients by Cramer's rule. In (3.3.8) we cannot solve by Cramer's rule since the determinant is zero, this being our frequency equation. we adOpt a different approach and multiply both sides Of (3.3.8) by the row vector (w°¢.w°e.w°r) and take the transpose Of the left most pair of terms. Symmetry of M5 allows us to write the result as t w° h k wo‘) i) 0 0 0 V (3.3.10) MO!) 6 he = 31(W WW e,W r) ke w° h k where the quantity 1: system (3. S1 is zeif the right '9 continu but wi th a PD 81 = 0 We preseht in 72 where the superscript t indicates the transpose. The quantity being transposed is the zero order homogeneous system (3.3.2) and is zero. Our conclusion is then that 31 is zero as the product Of the row and column vectors on the right side of (3.3.9) is not zero. Knowing that 51 = O is Of modest importance so we continue with the same procedure to an analysis Of the second order equations, hOping to isolate 32 as we did 31' but with a non-trivial result. From (3.3.7) we can compute the first order dis- placements involving the coefficients a(.) and b(.). Since 31 = O 'we conclude that the h(.) coefficients are arbitrary and set them zero since the relation among them is already present in the zero order displacements. Having the first order terms computed we can now determine all the inhomogeneous terms in the second order shell equations, (3.2.6) through (3.2.8). For second order velocities we should have terms involving cos(n:2)¢, cos(nil)5, cos n5 as well as the sines Of these same arguments. This gives rise to ten systems Of equations, one-half of which are homogeneous. Our concern is only with those equations involving cos(n¢) or sin(n¢) since the second order correction, 32, appears only with these trig ) 5| ..2 b A ' Y 9 35' Ar cos I for these 5 fit") these We )3.3.11) Ashefore t “Zero. C This reqllir rewritten a rewrite all end COS * 73 these trigonometric functions. we designate these components Of lflz ‘by A sin n) + B) cos n5, these components of ¢ ) w2 by A cos n) + B sin n5 and these components of 102 9 9 9 r by Ar cos n5 + Br sin n5. Again, there are more components for these second order displacements but our concern is only 'with these six. We write our two systems as A F B 0 W W 2 (3.3.11) Mm) A9 = Fe (_m 89 - 0 Ar Fr Br 0 As before the B(.) coefficients are arbitrary and set equal to zero. Our next task is to determine the F(.) coefficients. This requires that products of the form cos 5 cos(nil)t be rewritten as % {cos(ni2)5 + cos n5). In fact we have to rewrite all such products, sin 5 sin(n:l)$, sin¢ cos(nil)t and cos V sin(n:l)¢, in a similar fashion so we can identify the coefficients of cos n5 or sin n5 in the inhomogeneous terms. we find then that F) n 82[§(1+v)w‘°e + 80(1-V)Wp¢} l l + 2[2 so[(3—v)(ae+be) + (l+v)(b9(n-l) - ae(n+1))] (3.3.12) + (n+1)a¢ + (n-1)bW - (ar+br) - v(ar¥br) (v-s§(1-v))(a -b¢)} V (3.3.13) (3.3.14) 74 "2| l _ 1 O O o 32{§ n(l+v)W w + 230W 9 + vw r} + %[(3-v)so(a¢+b ) + sO(l+v)[(n+l)aw - (n-1)bW] W 2 (3.3.13) - (l-v+4so)(aeébe) - (l-V)[(n+l)ae + (n-l)be] 1 — 2(l-v)so(ar-br)} + z[nso(l+v)w°¢ + 2(1-v+3s(2))w°e - 250(2-v)w°r} _ O l 1 Fr — vszw 9 + 5 v(a¢+bw) + 5 [v[(n-1)b¢ - (n+l)a¢] (3.3.14) + (l-v)so(a9-b6) + 2v(ar-br) _ vnwow - so(2--\))W°e + (1-2v)w°r}. The process used in determining s1 is repeated in this case and the result is = o . (3.3.15) 0 w 525 + WOBFO + w°rFr Since s2 is not a common factor as s1 was, it is not identically zero and we are able to write an exPression of the form (3.3.16) 32 = G(so,f3,v,n) which gives the second order correction to the zero order wave number so. w. this sect: in the 2e and procee assumptio. 75 we are now in a position to analyze the results of this section for the fundamental case Of axisymmetric flow in the zero order tube motion. For this case we set n = O and proceed to the next section. In all that follows, the assumption of axisymmetric zero order flow will be made. our empty equation ( to give (3.4.1) Se E“(nation a; arbitrary E 7 + . . I): Oint )3.4.2) 76 3.4 Axisymmetric Frequency Equation We are now considering axisymmetric vibration of our empty tube. We must therefore return to the frequency equation (3.3.4) and set n = O. The result can be factored to give 2 2 (3.4.1) (sgé-g-Y) - ansg - 221%}(62-1w2) = 0. - +v Setting the first term to zero gives us a frequency equation appropriate to torsion motion in which pr is arbitrary and W'oe and wor are zero. Setting the second factor zero is the frequency equation for axisymmetric flow. 0 O o I r O In this instance W l 18 zero as can be seen from setting n = O in the first equation of (3.3.2). Hence 2 1 2 (3.4.2) so = M79415) / B -l+v is the frequency equation for the axisymmetric vibration Of an empty, thin, elastic membrane. The equation (3.4.2) has been determined by Rubinow and Keller [5)] in a slightly different setting. The following analysis parallels theirs. For frequencies in the range (l-\22)1/2 5.6 < l s is pure imaginary and the waves do not propagate. For 0 frequencies outside this range the wave number so is real and the waves move down the tube. When v = O the frequency equation reduces to S0 = B. As large. At the value presented 77 2)1/2 becomes arbitrarily As B approaches (l-v s 0 large. At B = l the wave number is zero but soon approaches the value Of B as B increases. A graph Of (3.4.2) is presented in Figure 2. Two auxiliary functions which are Of benefit to our discussion are the phase velocity cP and group velocity cg. defined as (3.4.3) c B/Re(so) P (3.4.4) c g dB/d Re(so). The phase velocity gives an indication Of the speed Of a traveling wave as it travels down the tube. The group velocity denotes the speed at which the energy is prOpagated. It is the velocity at which a packet Of waves is prOpagated, all waves having similar wave length. [Kolsky, 20] we find that 2 2 (3.4.5) o = (5 '1“ ))1/2. 0 < f3 < (1-v2)1/2. P 52_1 1 < B and that it is undefined outside these frequency ranges. Differentiation of (3.4.2) with respect to Re 80 gives (3.4.6) = OP [1 + (J—V cg cp (l-B2 ) 22410 3.5.2) 205343 deUDnNuuh M02. Chum—w a... MNDGE. i=1 an. e and P C and P velocity aPproache velocity . Phase Velc asymptotic L4 :0 longi t. w° :- when (3.4.7) 79 1/2 This function is zero at B = (l-vz) and at B = 1 and has no poles for real values of B. Graphs of c and c9 are given in Figure 3. If v = 0 then both P c and c are 1. If the frequency is zero then the phase P 9 velocity is (1-v2)1/2, decaying to zero as the frequency 2)1/2. The same is true for the group approaches (l-v velocity. For' B ==l the group velocity is zero while the phase velocity is infinite. Both approach the value one asymptotically as the frequency increases beyond one. Let us now consider the ratio Of radial displacements to longitudinal displacement. Solving (3.3.2) for wp and 6 W0r ‘when WOW = O = n we find .3... .3... = __.—e Woe (B2-1)1/2(B2-l+v2)172 When the frequency is near (l-\22)1/2 or 1 we see that the displacement ratio becomes large indicating primarily radial motion, while for small or large frequencies the motion is mainly longitudinal, i.e. in the direction of 9 changing. The graph of (3.4.7) is presented in Figure 4 and demonstrates the behavior mentioned. All curves have a vertical asymptote at B = 1 while each has a separate asymptote at B = (l-\)2)1/2 which is just tO the left Of B = l. Hume. Elan N..? dmmw 4.qu 024. Alwefi ml: “Hanoi. QN no Q, m \ \\ / / \\ / / WV\ / / \ \ a I/ QN é: Se 55 (3.5.25 82 3.5 First Order Displacements In section 3.3 we demonstrated that for any value of n. 81 = 0, but 32 #'0. To determine the correction s 2 ‘we must first Obtain the first order displacements. we do so now. For axisymmetric flow we set n = O in (3.3.7) and arrive at one inhomogeneous system in cos 5 and sin V. It is \ . . 2 l 2 l B -1— 550 (1-5) 530(1+V) 1 av f5 1 2 2 l _ (3.5.1) 530(1+v) B -s0 - 5(l-v) -vso ae — f6 2 1 -\)80 B -1 ar fr From (3.3.2), with n = O, we write 0 _ 2 -l 0 We sz(B -50 ) Wk v # 0 (3.5.2) W0 = O V = O r Utilizing these relations we find _ O l 2 2 2 -1 f) -‘Wi[§ vso (3—V)(B -so ) - l}, _ l 2 2 2 -1 (3.5.3) fe - ngsou-v) - iszU-Mso )(B "'30 ) l. _ O 2 2 2 -1 fr - Wr \J[2+sO (l-V)(B -so ) 3, when v #'0 and when v = O _ 0 f — 3dee/2, _ 1 2 0 (3.5.4) f —- 2(430 -1)w9 . _ 0 fr - sowe -——.——_._ .— - (3.5.3) . by Crame of the c identica in the r They are (3.5.5) arbitrar L coeffici 83 Knowing the values of f , f9 and fr given in 4' (3.5.3). (3.5.4) we now solve (3.5.1) for a a and a W' 6 r by Cramer's rule. This procedure utilizes the determinant Of the coefficient matrix in (3.5.1). we find that the determinant does not vanish identically when S0 satisfies (3.4.2), however for v in the range 0 < v'g 3 - (8)1/2 there are two values Of the frequency, B, which cause the determinant to vanish. They are found as the solution to 2 (3.5.5) B = (v—l){v — 3 .1; [v2-6v+1]1/2}. NIH If we allow the frequencies determined by (3.5.5) tO occur, consistency requires that we set the zero order arbitrary constant, W2, equal to zero. In this situation the zero order system is replaced by the first order system as being fundamental. Thus, allowing the determinant Of the coefficient matrix in (3.5.1) to vanish is merely defining a frequency equation for the case of vibrations exhibiting one wave around the tube in the 5 direction. we thus restrict our frequencies to those for which the determinant of the coefficient matrix in (3.5.1) is non-zero. Denoting the cofactors of elements aij in the coefficient matrix Of (3.5.1) by Aij we write (3.5.6) where the and fr '21th ma 84 O Wr ~ ~ ~ 35 = BEE {A11f5 + Alzfe + Aler}' W: ~ ~ ~ (3.5.6) a6 = Fe-E {1511sz + A22f9 + A23fr), V 32’ 0 W3 ~ 2 N ar = Det [Al3f5 + A23f9 + A33fr]' where the tildas reflect division of the functions fl, f9 and fr by We and Det is the determinant of the coeffi- cient matrix Of (3.5.1). When v = O we designate the a , a and a by tildas and find that 3 9 r :5 = -wg [(62-1)(1+252)-13/B(2-Bz). (3.5.7) 59 = 2w? [64-462+2}/(2-62). ~ _ 0 2 2 ar — We (5 +2)/B(2-B ). Before we proceed to the determination Of the second order correction, 52, it is beneficial to examine the corrected displacement ratio, ((Wg + oar sin ¢)/(Wg + 5a sin ()I. 8 as a function of the frequency for fixed values of W. 6 and v. Specifically, we choose 5 = 30° and t = -60° where both angles are measured from the north pole. Remembering that 5 has value .15 to .3 in the human aorta, we choose for 6 the values .005, .05 and .5 as being represent is nearl; I by consic' M V‘Se I. )3 has the f (3.5.8) Zero for 39“ rati EmltiPle ratio bei Exhibits denominat designate the demon (3.5.9) 85 representative. Poisson's ratio for biological materials is nearly .5 but we consider v over the range [0,.5] by considering various sub-intervals for v. Case I. v = 0 When v = O we find the corrected displacement ratio has the fOllowing value (3.5.8) Gar Sln 5 = 1 6(oz+2)sin 5 “3+6ge sin 3 5 2'324‘26 sin 5534-462”) In the zero order case the displacement ratio is zero for all frequencies. However, for the corrected displace- ment ratio we see some new behavior. When 5 is an integer multiple of n the motion is completely longitudinal, the ratio being zero. However, for other values Of 5 the ratio exhibits both radial and longitudinal components. When the denominator is zero the motion is purely radial. If we designate 5 sin 5 by e in (3.5.8) we find the zeroes of the denominator at frequencies given by (3.5.9) 62 = (4e)'1(1+8e ¢_[1+3262]1/2}. The approximate values are given by 1/2 (3.5.10) 6 a ((1+4€)/2e) and 6 a (2-2e)1/2. When the sin 5 is negative the first root exists only for imaginary frequencies and hence the only real zero of the denominator occurs at frequencies slightly greater he \ es 87 1/2 . . . . . than 2 . If the Sln 5 18 positive, then one root is slightly less than 21/2 and the other is given by the first approximation in (3.5.10). The graphs Of the ratio in (3.5.8) are given for the two cases of 5 = -60° and 5 = 30° in Figures 5 and 6 respectively. In Figure 5 we see two frequency ranges for which the motion is primarily radial, one near B = 0 and the other between B = 21/2 and B = 2. At other frequencies especially those greater than B = 3, the displacement is nearly zero though not exactly zero. In these regions we can be satisfied with the straight tube model which predicts only longitudinal motion for v = 0. At the two frequencies mentioned above we see that there is a marked difference between the zero order theory and the corrected one. For frequencies less than 2 the 5 = .5 exhibits large quanti- tative differences from the smaller 5 values, even though the general behavior is the same. For frequencies greater than two the 5 = .5 curves are very close to the smaller 0 value curves. In Figure 6‘we see the effect Of the extra zero in the denominator of the displacement ratio. Now there are three verticle asymptotes for our curves. One is at B = O, 1/2 1/2 one is near B = 2 and the third is near B = {(l+4e)/2€) where e = 5 sin 5. For small 5 values this asymptote \N .\ ~N \\) \. 88 89 occurs at a rather large frequency. For larger values Of 5 the asymptote moves to the left approaching the frequency 1/2. 3 = 2 The change in abscissa at B = 2 was incorporated to show the radial motion present for B = O and B near 21/2. The curves for 5 = .005 and 5 = .05 are almost identical especially at the asymptotes. The 5 = .5 curves is again different from the other 5 values in degree though it too exhibit the three regions of radial motion. SO we have stated before near those frequencies at which the motion is primarily radial the zero order straight tube model is insufficient and higher order terms are required. For large frequencies the two models give essentially the same results. Case II. v # 0 From an examination of the displacement ratio using (3.5.6) we see that the ratio Of radial tO longitudinal dis- placement is like 8- and so at B = O and B = l the 0 ratio becomes unbounded indicating primarily radial motion. The same holds true for B = (l-v2)1/2. However, fOr large frequencies the corrected displacement ratio is like B'-1 and hence becomes nearly zero for large frequencies. Again we find that the motion is longitudinal at high frequencies with radial motion predominating at selected smaller frequencies. 91 In Figures 7 and 8‘we present graphs of the corrected displacement ratio for v = .25 and the two 5 values used previously, 5 = -30° and 5 = 60°. In Figures 9 and 10 the value of Poisson's ratio is increased to .5. In neither case do we intend these curves to be representative Of the entire spectrum Of 5 and v values. Instead, they are meant to shed some light on the nature of this corrected ratio Of di5placements. It should be noted that the zeroes Of the determinant (Det) are not the dominant influence in the ratio as the determinant is present in both numerator and denominator. In comparing Figures 7 and 9 we notice little difference between the two graphs. Both approach zero as the frequency grows large indicating longitudinal motion. For v = .25 there are two regions for 5 = .5 which are Off scale. Both are "parabolic” shaped, the first situated between B = l and B = 1.02, while the second Occurs between 1.02 and 1.28. These portions of the 5 = .5 curve in Figure 7 are another indication Of the differences between curves with 5 = .5 and those with lesser values Of 5. The radial motion exhibited near B = (l-\22)1/2 and B = l is predicted by the zero order displacement ratio. Hence, the straight tube model is a rather accurate representation of the curved tube except for very small frequencies. 92 ..u «O 73% 30 .Avsv e$73.. P232230. E QM 93 When 5 = -60° ‘we again notice little difference between v values of .25 and .5, as seen in Figures 8 and 10. Both curves exhibit a section for 5 = .5 ‘which is Off scale in the frequency range 0 < B < .3. FOr both of these cases the 5 = .005 curve rapidly approaches zero for frequencies larger than one. All curves approach zero for frequencies greater than two but for 5 = .05 and 5 = .5 we see one frequency region for each in which radial motion is predominate. These values are not found in the zero order case, indicating the unsuitability Of the straight tube model in which it is assumed that the flow is axially symmetric. Indeed, from an examination Of equations (3.5.1) and (3.3.2). one notices that if one were tO set n = -l in (3.3.2) we would Obtain the same matrix in (3.5.1) and (3.3.2). This suggests that the motion predicted by our corrected system (3.5.1) is essentially the same that one would Obtain in a straight tube model if one were to assume motion with one circumferential wave (1') = 1). In summary we might say that on the whole the straight tube model can be used in some curved tube situations if the ratio 5 = a/R is not too great and if certain select fre- quencies are avoided. In these last instances the need for higher order terms becomes apparent. new.” 13.).» m..n~ 7.51410 5.32.34)“ 9.73. F2220 GNP? Q; Q. ”mu—Drum». mu. .. 2 _ _ l 9 ._ _ _ QM - A-‘-. ——-¢.-.-_— 96 3.6 Second Order Correction - Axisymmetric Case we know from the first order displacements that they have the following form for axisymmetric flow up through the first order w = a cos ¢. w 5 w _ o 2 2 -1 . (3.6.1) we - Wr sz(B --sO ) + 5ae Sln w, v # 0, w = W0 + a sin W r r 6 r ' where a , a and a are given by (3.5.6). For v = 0 w e r we have w = 63' cos W, W W (3.6.2) we = w: + 555 sin w, v = o, wr = 5ar Sln W, where 3*. 39 and a} are given by (3.5.7). In either case we can proceed to the determination of the second order correction, 32, as a function of so. B, and v. For axisymmetric flow W0 = O, and the correction W can be found from (3.3.15). saesiil v = o. In this case W? = o and the correction can be found by setting Fe = 0 along with n, W% and W3. In addition, 80 = B, and so we Obtain (3.6.3) 32 - 1 {456 - 7B4 - 662 - 4]. v = o. - 46(2-fi2) 97 As 6 nears zero or the value 21/2 32 becomes unbounded but when the frequency grows large the correction behaves like B3 becoming larger than the zero order term it corrects. When the correction becomes larger than 5-2 we find we should include higher order terms. We propose to present the corrected phase velocity 8% which is given by ~_ 2-1 (3.6.5) cp — B(so+5 52) . It is instructive to compare this value to the zero order phase velocity which is 1 when v = O. for v # O we will also consider the ratio (so+5232)/so ‘which gives some idea of where the corrected wave number varies from the zero order wave number. When v = 0 this ratio is merely Ep-l and so it will not be discussed. We present the corrected phase velocity for v = 0 in Figure 11. The reader should note the change in scale which allows us to examine the region 0 < B < 2 in more detail. For the area designated by A in the lower left corner the curve for 5 = .005 is nearly a vertical line coinciding with the axis. In the upper portion B the curve for 5 = .5 is off scale though of the same shape as the other two curves. This is a further indication that 5 = .5 is much too large a parameter for our expansion. The curves 99 for 5 = .05 and 5 = .005 are almost always near one in value which is the zero order phase velocity. At selected frequencies the corrected phase velocity differs from one up to B = 2. After a frequency of two the corrected phase velocities approach zero. The curve for 5 = .005 will gradually become zero, but at much higher frequencies. It is in this region that the zero order predictions are in- adequate especially if 5 .OS or .5. However, for frequencies less than two, the straight tube model is quite accurate. Case II. 0 < v < 3 — (8)1/2 When v #'0 we have the correction coming from the expression _ 0 (3.6.5) 0 — w: Fe + Wf Fr which is merely (3.3.15) with W? = 0 for axisymmetric vibrations. In solving (3.6.5) for 52 we find that 32 has the following value (62"5‘02)2 2 2 2 -1 s2 = - 2 2 O {szo (B -so ) a‘ 4v sofi Wk 2 2 -l 2 + soae[l-v-V(B -sO ) (2sO +l-v)] 2 2 2 -1 0 (3.6.6) + v[2+sO (l-v)(B -so ) ]ar + Wr[1-2v + Vs (vs -4s +v)(Bz-s 2)-1 O O O O 2 2 2 2 2 -2 + v S0 (350 +l-V)(5 -30 ) 1} in which we have used (3.5.2). 100 For v in the range 0 < v < {3-(8)1/2} the determinant of the matrix in (3.5.1) has two real zeroes, the positive zeroes being near B = l and B = (2)1/2. Since the determinant appears in the denominator of the a , a and w 9 becomes unbounded at these ar terms. the correction, $2, . . -1 . zeroes. Fer s0 small, 32 is like 50 while for 50 large 32 acts like 503. Hence, at some point the correction 5232 becomes larger than the term it is correcting, so. At this point we must bring in higher order terms for additional correction. In Figure 12 we present the corrected phase velocity 8? = B/(so+5252). In the zero order the phase velocity was nearly one in value except in the neighborhood of B = (l-\)2)l/2 and B = 1. For the corrected phase velocity in the range 0 < v < 3 - (8)1/2 we see in addition to the two frequencies B = (1-\)2)1/2 and B = l, the corrected phase velocity differs from one for very small frequencies and also near two other frequencies, B «21.03 and B ~ 1.14. . The reader should note the change in scale at B = l and B = 1.04. This 'was done so that more of the behavior near B = 1 could be determined. In the regions marked A and B the curve for 5 = .005 is not plotted as it would appear only as a straight vertical line in this graph. In B, 5 = .05 is also not plotted for the same reason . (3 a We v Q 7.23.10 §>Q F34”? g fig 4.. Mud—35E. (N.. «.3 can} .rnx. n9. 23.. ,4: Q. A! _ — 4 mad. a 102 In Figure 13 we exhibit the behavior of the corrected wave number (30+5252)/so. Again note the change of scale at B = 1. It should be noted that the correction is nearly one with exceptions occurring in the neighborhoods of B = 0, B = (l-v2)1/2, B = l and the two zeroes of the determinant. In these regions one would expect that the zero order wave number alone is in adequate and higher order terms are required. In summary, we have shown that the corrected phase velocity agrees closely with the phase velocity predicted by the straight tube model in the regions 0 < B <./{-v2, 1 < B < 1.14, 1.15 < B. for 5 = .005, and .05. When 5 = .5 there is virtually no agreement at all. The corrected wave number and the straight tube wave number show good agreement in the frequency ranges 0 < B < l, 1.04 < B < 1.15, and 1.2 < B. In Figure 12 all curves eventually become zero while in the zero order situation the phase velocity approaches one for all v values as the frequency increases. Conversely, the corrected wave number goes to infinity as the frequency increases in Figure 13. For this reason the straight tube model is inadequate for large frequencies and as the corrected wave number gets larger, higher order terms would be needed to add to those already computed. 104 Case II. v = 3 - (§)l/2 When Poisson's ratio takes on the value 3 - (8)1/2, which is approximately .172, then the determinant (Det) has only one zero. We see then the situation of a more orderly behavior of the phase velocity and corrected wave number in the region of B = 1. From Figure 14 we see the behavior of the corrected phase velocity. In general the corrected phase velocity is near one for most values of the frequency, B. Again, for very small frequencies and for those near one. behavior foreign to the zero order phase velocity (Figure 3) is introduced. There are select values introduced for very limited frequency ranges. These are represented by the vertical asymptotes near B = 0, B = l and B = 1.1. The curve for 5 = .005 has not been plotted in the group of curves labeled A, B and C, nor has the curve for 5 = .05 for the C set. In each case these segments were off scale and they could not be accurately represented. Note also that there is a scale change at B = 1. As before we see that the 5 value of .5 gives results which vary significantly from smaller values of 5. This value gives a good indication of the general nature of the curves behavior and is included for this reason. In this case, it indicates that the corrected phase velocity approaches zero from above which is true regardless of the 5 value. This is contrary to the zero order phase velocity which approach- es one from above. 5“ u h 107 The corrected wave number presented in Figure 15 eXhibits the same general character found when v = .15. Agreement with the zero order wave number is good in the frequency range 0 < B < 1, 1 < B < 1.1 and 1.2 < B. The analysis for the corrected wave number (1+52s2/s0) follows that of the corrected phase velocity. There are portions of the curves in Figure 14 which have been omitted because of their existence in select frequency range nature. They correspond to the A, B and C portions of Figure 17 and are mostly off scale. Case III. v) 3 - (8)1/2 For v > 3 - (8)1/2 'we have a less segmented behavior for the correction, the corrected phase velocity and corrected wave number. Since the determinant does not have a real zero we do not obtain the great variation in behavior for our variables for frequencies between 1 and./§. The rapidly changing nature of the curve present near B = 0 is still apparent for v > 3 -./§. Except for 5 = .5 the behavior of all curves settles down rather quickly and follows that of the zero order system rather closely. This feature indicates that the straight tube model is farily accurate except for frequencies near zero. Here we find rather large values for our curves indicating use of higher order terms is required. ——III 109 fl..nu 22.910 gfl M5232 mks? DUPQNNMQV k... HMDJE. _ 110 The differences between the straight tube and curved tube expressions would become significant if large order scale factors were brought into play, thus accenting the differences. Hewever, for frequency values less than two and away from zero the straight tube and curved tube models have good agreement. For 5 = .5 we notice a strange behavior for B near (l-v2)1/2. There is a vertical asymptote not present for the smaller values of 5 which is further proof of the unacceptable size of 5 at this value. It should also be noted that in Figure 19 the .005 curve for the a section of the curve is omitted due to its existence for only small frequencies. In both Figure 19 and Figure 20 the curves for 5 = .005 and 5 = .05 are nearly identical for B values larger than one. Although the curve for 5 = .5 differs greatly from the other two curves, it does indicate the general behavior of the curves for B > 1. we see that as the frequency becomes very large the phase velocity approaches zero and consequently the wave number grows large. In conclusion we can say that except for narrow regions in the frequency spectrum, the corrected variables we've observed vary only slightly from the straight tube variables in the frequency range we've been considering. From our various figures we can conclude that the discrepency between the straight and curved models is slight until large frequencies are Obtained. This information results from observing the lll behavior of 5 = .5 curves which are indicative of the general trend of the functions we are graphing. It is in these larger frequency ranges that one should resort to the curved tube mdel to gain some appreciation of the behavior of our tube. CHAPTER IV VIBRATIONS OF THE FLUID 4.1 Introduction In Chapter three we investigated the free vibrations of an empty curved shell. We found and thoroughly examined a frequency equation relating s to B as well as the O correction expressions for 5 namely 31 and s 0' 2' We are now going to undertake a similar process for two prOblems in which the fluid is the primary factor. we begin in section 4.2 by solving the zero order pressure and Navier-Stokes equations. Section 4.3 contains a discussion which develops the necessary equations to find s1 and 32 in the general situation. In section 4.4 we analyze the fundamental prOblems of axisymmetric, inviscid flow in the zero order system by determining two frequency equations and investigating their properties. Section 4.5 contains the first order terms associated with our fundamental prOblems and it develops the effect of these terms on zero order quantities. Finally in section 4.6 'we find the second order correction for our two prOblems and make some comments on its effects and behavior. 112 113 4.2 Z250 Order Pressure and Velocity Equations 4.2.1 Pressure Equation Examination of the Navier-StOkes Equations (2.5.19) through (2.5.21) indicates the presence of the pressure on the right side of all equations. This requires our determining the pressure explicitly before we proceed to the velocity equations. We do this by solving the pressure equation (2.3.25). We begin the zero order problem then with the zero order pressure equation (2.6.8) written as 2 l 1 2 -—— p + —(rp ) - (s - )p = O ( r2 0,¢¢ r 0,r ,r 0 c2k2 0 4.2.1) 2 l l . 2 [11 n1] B/ We expect that our solutions all have period 2? in W and so we separate variables for pO as (4.2.2) p0(¢,r) = f(r)cos n) n = 0,1,2,3,... Use of (4.2.2) in (4.2.1) leads to the following equation for f(r) where the primes denote ordinary derivatives with respect to r. 2 2 (4.2.3) f”(r) + l f'(r) - (9— + s 2 _ Ji—nzm = o, r 2 0 2 2 r c k 2 2 2 Denoting s - by a we can write as 0 C2k2 our solution to (4.2.3), (4.2.4) where A a Hm) are n second kind Knim) 18 us to write (4.2.5) when n = in the V fulfill th In with l‘eSpe while . In (5 functiOnS. written 0U 1 114 (4.2.4) f(r) = AIn(ra) + BKn(ra) where A and B are arbitrary constants and In(ra) and Kn(ra) are the modified Bessel functions of the first and second kinds respectively of order n. [Watson, 21]. since Kn(ra) is not finite when r = 0 we set B = 0 allowing us to write (4.2.5) po(w,r) = AIn(ra)cos nw. 4.2.2 Navier-Stokes Equations we now turn our attention to the zero order Navier- Stdkes equations (2.6.9). we separate variables as follows ui(¢.r) = 33(r)sin n) + 32(r)cos n) (4.2.6) ug(w,r) = 53(r)cos nw + 3%(r)sin nw ug(w,r) = 52(r)cos nw + 92(r)sin nw When n = 0 we have axisymmetric flow and there is no velocity in the W direction. Hence 90(r) must be zero in order to 4: fulfill this condition. In what follows the primes indicate differentiation with respect to the argument. Hence f’(r) indicates g%-f(r) . , d . while In(ra) means 57:57'In(ra)' we now drop the tildas from all variables as well as delete the arguments of the functions. All functions have r as the argument except when written out. In results ir. T‘r. (4.2.7) Hhi Ie the (4.2.8) 115 Insertion of (4.2.5) and (4.2.6) into (2.6.9) results in the following systems of equations. The first system is 2nr vr — 0, -1 O, ’ 2 -2 2 . O _ (4.2.7) r (rve ) — [n r + 30 - 1B'n1}v9 — 0 "1 O"_ 2 -2 2_. O: r (rvr ) {(n +1)r + 30 1B'nl}vr 0. while the second system assumes the form'below, -1 O; p 2 -2 2 o 0 -2 O r (ruw ) - [(n +1)r + 30 — an1}uW - 2nr ur -1 I 2 = - Anr n1{l-1B/nc )In(ra), -1 O ’ ' 2 -2 2 0 0 (4.2.8) r (rue ) - [n r +50 -1B'n1}u9 _ _- 2 r (rur ) - [(n +1)r +so -1Bn1}ur + 2nr “W = Aan1{1-iB/nc2}I;(ra). The first equation of (4.2.7) indicates v3 = 0 unless n = 0. If n = O, the last equation of (4.2.7) . o_ 2_ 2_. yields vr — A1I1(rY) + B1K1(rY) where Y — so 1Bn1. and, A1 and B1 are arbitrary constants. Finiteness at r = 0 requires B1 = 0. [Note: The argument of finite- ness for all variables at r = 0 ‘will automatically be made in the remainder of this work. Hence only modified Bessel functions of the first kind, In(ra), or In(rY) ‘will be given in our solutions.] From the second equation of (4.2.7) we write (4.2.9) 'I‘) first and indePende: as (4.2.10) 116 we'write (4.2.9) vg(r) = A21n(rv). The system (4.2.8) exhibits coupling between the first and third equations while the second is completely independent. The coupled pair of equations can be written as 0 L33 -2n uw 2 ~ann(ra) (4.2.10) 0 = Anl{1-iB/nc } 2 , -2n L33 ur ar In(ra) _ .2. _d_ _ 2 2 2 where L33 — r dr(r dr) (n +l+r Y ). Since L33 and Zn are operators which commute we apply L33 to the first equation and Zn to the second and subtract. The result is (4.2.11) {(L33)2 — 4n2}u$ = -An(a2-Y2)r3fl1[i-iB/nc2}1n(ra) The Operator in (4.2.11) can be factored as [L33-2n}[L33+2n}. The homogeneous solutions for the first portion of the Operator are I (rv) and K (rY). while the homogeneous solutions n+1 for the second part of the Operator are In_1(rY) and Kn_1(rY). n+1 we utilize these four functions through variation of parameters to find a particular solution for (4.2.11). we designate the particular solution by u$p(r) and find 117 u3p(r) = % An1{1-iB/:c2](a2-Y2){In_1(rY)I2r21n(ra)Kn_l(ry)dr - K-n _1_(r)()‘for2 In (rc1)In _1(rY)dr r 2 (4.2.12) - In+1(rv)jor In(ra)Kn+1(rY)dr r 2 + Kn+l(rY)IOr In(ra)In+l(rY)dr} -1 = -Ar'1n(az-v2) n1{1-iB/nc2)xn(ra) we finally have as our general solution to (4.2.11) the velocity 0 uw(r) = A3In+l(rY) + A4In_1(rY) (4.2.13) _ — Anr'1 n1{1-iB/nc2}(a2-Y2) l In(ra) where A3 and A4 are arbitrary constants. Substitution of (4.2.13) into the first equation of (4.2.10) gives us ug(r) as u0(r) = A 1 (rY) - A 1 (rY) r 3 n+1 4 n-l (4.2.14) + Aanl[1-iB/nc2 )(a 2"14(2) Ig(ra) The second equation of (4.2.8) can be solved directly by variation of parameters. we have then -1 (4.2.15) ug(r) = A51n(ry) - Ason1[1-iB/nc2](a2-Y2) In(ra) where A5 is arbitrary. We and pressun the presend tions prob} will always the Others 118 4.2.3 ‘cOntinuity Equation we now have solutions for the velocity components and pressure for the zero order. Quick observation reveals the presence of six arbitrary constants. In a free vibra- tions problem in which we search for vibration modes there will always be one arbitrary constant. So we must determine the others remaining after designating A to be arbitrary. It should be noted that we have solved the pressure equation and the Navier-Stokes equations without dealing 'with the continuity equation directly. Our first step in determining arbitrary constants will be to require that all velocities satisfy the zero order continuity equation (2.6.10). This gives rise to a pair of equations -1 O O -l O , _ . 2 (4.2.16) nr uw + soue + r (rur) — (la/c )AIn(ra) o —1 o , _ (4.2.17) sOve + r (rvr) — 0. When n # 0 then v2 5 O and by (4.2.9) and (4.2.17) we find A = 0. If n = 0 then by (4.2.6) the v components 2 disappear and (4.2.17) is satisfied identically. substitution of (4.2.13) through (4.2.15) into (4.2.16) reveals that _ —1 The three zero order velocities and pressure are given by (4.2.18) whe If the flu: (4.2.18) 1 (41.19) 119 po(¢,r) = AIn(ra)cos nW O AMnln(ra) . u¢(¢,r) = [A3In+1(rY) + A4In_1(rY) - r(a2-Y2) } s1n n). AMs I (ra) O Y O n (4.2.18) u (1,r) = {——-(A -A )I (rY) - ] cos n) 9 s0 3 4 n (Oz-Y2) O Abnlrflra) ur(¢,r) = {A3In+1(rY) - A4In_1(rY) + w} cos I“): where M = n1[l-iB/nc2}. If the fluid is inviscid then n ’1 = n‘1 = o, k2 = 1 and 1 (4.2.18) reduces to P0(r.¢) AIn(ra)cos n) . uo(¢.r) = {-AnIn(ra)/ri8} sin n). (4.2.19) 0 ue(w,r) = {-AsOIn(ra)/iB] cos n). ug(w.r) {AaI£(ra)/i5} cos n) . We Should 120 4.3 Discussion for Arbitrary n As in Chapter three we present the procedures for determining the frequency equation and the higher order corrections for arbitrary n. In subsequent sections the analysis of these equations is restricted to the fundamental case of inviscid, axisymmetric flow. For the present, we ‘will make no such assumptions while proceeding to detail the process for solving our prOblem. 4.3.1 Two Frequency Equations The primary concern of this chapter is to discuss two problems which feature the fluid component of our model. In the first of these prOblems we postulate that the fluid is flowing in a tube which exhibits no movement of its own. we will refer to this as the "rigid tube" case. In the second prdblem we state that the fluid is flowing in a curved path without benefit of a container. we designate this situation as being "stress free" since no stresses are exerted on the boundary of the fluid. In the rigid tube case we say that at the boundary of the fluid, which is the tube. the fluid velocities are all zero. Since the tube is not moving it is reasonable to expect that the fluid immediately next to the tube is also stationary. we should note that for inviscid flows there can be slippage at the wall so that only the radial velocity is zero. Tr I= 1. gi arbitrary is written (4.3.1) from (4.2 121 The vanishing of all three velocities at the boundary. r = 1. gives us three homogeneous equations in the three arbitrary constants A,A3 and A4. This system of equations is written MnIn(a) . In+1(Y) In-1(Y) ' (0242; A3 0 _ YI (Y) y: (Y) Ms I (a) 3 O O (a -Y ) I , l MaIn(a) ( / In+l(Y) 4.1-1”) W A \0 / for viscous fluids. For inviscid fluids we set ug(l) = 0 from (4.2.19) to get (4.3.2) aIg(a)/i8 = O. In the viscous case we seek non—trivial solutions and so require that the determinant of the coefficient matrix in (4.3.1) vanish. Since both y and a involve s and B O the vanishing of the determinant leads to a complex transcendental relationship between 50 and B which we have called the frequency equation. In general it is quite difficult to find all the possible relationships between 3 and B analytically. 0 Normally, assumptions are made which simplify the Bessel functions using only one or two terms of an expansion. Doing this reduces the frequency equation from a transcendental equation to an algebraic one. Ouz equations 1 (4.3.2) cor This equati for S an 0 For set all flu to zero. .1 to (2.6.11) flnd after (4.33) -+l (4.3.4) 122 Our concern, however, is to present the various equations for the general case. For inviscid flow equation (4.3.2) contains the relationship between s and B. O This equation is simpler and a relationship can be established for s0 and 6 once the zeroes of 15(a) are known. For the stress free case we find that we must set all fluid stresses acting at the fluid boundary equal 0 O O . to zero. These stresses are Sr¢ . Sre and Srr' Returning to (2.6.11) and inserting (4.2.18) in those equations, we find after many algebraic manipulations that at r = l 1 2mm -{A YI (v) +Av1 (y) .- n 3 n+2 4 n—2 2 2 (4.3.3) 1 (a “Y ) [(n-1)In(a) + aIn+1(a)]} = O. —1 Eli-[151380 [manm + (Y2‘302)In+1(Y)] 2AMs (4.3.4) + A4so'1[ny1n(v) - (Y2-802)In_1(Y)] - ——2-9-—a1r;(a) (0 -Y2) = 0' 2A 2A . 3 4 I. 1 1 (4.3.5) + .E-g-M—E-i- {(nzmz-nnnm) - aIn+1(a)] = 0. a -Y when the fluid is viscous. If the fluid is inviscid. then only the normal fluid stress, Sgr(l), is zero, and we have then (4.3.6) AIn(a) = O. 123 As in the system (4.3.1) we require the determinant of the coefficient matrix for the undetermined constants, A,A3 and A4, to vanish so as to have non-trivial solutions. As before, the viscous case leads to a complex, transcendental equation involving s0 and B which in general is difficult to solve. For the inviscid case we merely prescribe the value of n we are considering and then we are able to find the zeroes of In(a). From this the frequency equation is written as (4 3 7) s 2 - 23-- 22 ° ' O _ 2 nm ’ c where Jh(znm) = O, m = l,2,3,4,..., and Jn is the Bessel function of the first kind of order n. 4.3.2 First Order Correction After determining the frequency equation apprOpriate to the zero order fluid equations, i.e., the straight cylinder case, we attempt now to find the first order correction, 51' We begin with the first order written in terms of the stresses rather than the Navier-Stokes equations. The proce- dure is straightforward. we multiply the stress equations by the zero order fluid velocities and integrate over the cross- sectional area occupied by the fluid. By an integration by parts we are able to eliminate the first order stresses,. velocities and pressure in the region except for boundary integrals which will involve these terms. In this way we can isolate the contribution due to sl. 124 From (2.6.2A) we write the first order stress equations of motion for the fluid without benefit of the operators ‘Ml' M and M3. They are 2 -18 l 1 _1S 1 -l l __ . l SW1 W + SOS¢9 + r Swr + r (raw),r — ifiuw (4.3.8) 0 0 . O O _ 31519 + cos 11599-SWW1 + Sln 1[rSOS¢e-Swr1 -1S 1 -1 1 1 _ . 1 0 89¢ W + r (r891),r - sOS99 — -1Bue + 31886 (4.3.9) 0 O O — 289$ cos W - sin Wi236r+r803991 -l 181 l 1 _ . 1 (Sr) W_ 51)} + r1(rSrr)’r + Sosre — -16ur (4.3.10) 0 O . O 0 O - Slsrg - SWr cos W + s1n ¢[rsoSre+See-Srr} we multiply (4.3.8) by ui, (4.3.10) by u: and add all three together while integrating (4.3.9) by u: and over the cross section of the tube. The result is an expression of the form S +u S 1 0 +110 1 o 1 2W 1 [u 81 WW “9 gw r r¢|o dr 2w 1 + I r{u 031 +u081r +uOS1 d) 0 ¢ ¢r r rr'o 2w 1 -1 1 u0 1 O 1 O ' 10 I0 r [51)“ i i “Sweue.)+sir rvw )11111 S1 O l O - I: H1018Wru1'r +S 9ru9,r+srru r r)rdrd1) —uOS1 ) + s (u 0S 1 -u081 +u 01S )]rdrd¢ 2w 1 -1 O 1 (4.3.11)+ 10 I0{r (uWSWr r11 0 $819 9 99 r 9r 125 2w M0 1 O + 15 I; I? ue ue+urur1]rdrd¢ + f:wfl{sl -rsO sin ¢}[uOS O -uOSO+ OSO )rdrd) o i we 9 99+ urs er 2w 0 O O O O O + 10 I: sin 1{u¢S¢r+2ueser+ur(Srr'sgg)errd1 2w 1 O 0 SO 0 O uOS O + cos w u S e) + 2n 8 + rdrd) = O I Io { 1‘ 11899 e e) “r Srv] For the general case ufi is odd in ¢ (sin nW) while ug, u: and p0 involve cos n1 and are even functions of W. Examination of the fluid stresses reveals that S0 . SO . S0 0 ii 99 rr and S are even functions of W ‘which 80 and S0 are 9r 9) r1 odd functions of w. Utilizing the fact that the integral of an odd function over its period gives zero, the last two inte- grals in (4.3.11) are zero as well as the term involving sor sin 1 in the third from the last integral in (4.3.11). Our purpose is to eliminate all first order terms except s1 by replacing them with known expressions involving zero order terms. To do so we replace the first order stresses by the first order constitutive equations (2.6.23) and integrate by parts any derivatives of first order velocities. The resulting expressions are written so as to have the first order fluid velocities multiplied by the zero order Navier- Stokes equations which are zero. We finally arrange the equation in the form 126 N 01 01_ u10 10 101 r u s +u Bes rr+u Sr 5 u s -u 3 d1 10 [wiir u¢¢r99rrrrio 1 01 1 1o 10 1021T + +u S +u S S S u S -u S d 10% *1 9) r) i i W) 9 91 r r) 0 27T1. 2 -1 1 1 + 10 IO{1BpOp1/c -pO[r (u¢'¢+[rur]' ) + sou (4.3.12) + (3 -rs sin MuO + u0 cos W + u0 sin ¢]]rdrd¢ l 0 e V r 150 o o o _ I: VI 9{(s l-rsO sin Wue + uw cos W + ur sin ¢]rdrd¢ + 51 I: ”f0 [2u W OSew-uOSOe +2uOSO r)rdrdw= Due to the periodicity of all variables in ¢, the second integral of (4.3.12) is zero. In the third integral the terms in brackets, multiplying pb, constitute the first order continuity equation (2.6.22) and can be replaced by inl/c2 thus making the entire integral zero. Finally in the fourth integral the only non-zero term multiplies 51 so that (4.3.12) assumes the form 2? l i r{uo l O 1 O 1 1 0 l O 1 0 ‘ O wS¢r+ueser+ ursrr-uys¢r-9 u S er-ursrr o d) (4.3.13) 2 1 W 08 O 0 O O + 231 f0 IO[UWS GW-uMSe+u%Sr1rdrd1= The first integral in the above equation will make some non-trivial contribution to the general interaction problem. However, for the rigid tube all velocities are zero at r = l and for the stress free case all stresses are zero at r = 1. In either of these special cases the first integral 127 is zero. Since the second integral is not zero, we conclude that = 0 s1 in the stress free and rigid tube cases. As in the empty tube problem we are not satisfied ‘with this result so we proceed to a determination of the next correction, 52. This procedure involves second order equations which require in turn the solution of the first order fluid velocities and pressure. 4.3.3 First Order Terms for Arbitrary_ n we begin to determine the first order velocity fluid by seeking the solution of the first order pressure equation which we repeat here for convenience. we have that pl satisfies -2 _1 2 2 2 2 r p1.W + r (rp1,r),r - (so -B /c‘k )pl (4.3.14) _ -l . 2 — -r PO'W cos W - Sln W[po'r+2rso po] 2 with k defined by 04.2.1)and with d2 defined by a2 = (soz-dz/ezkz). The left hand side of (4.3.14) is of the same form as the left side of the zero order pressure equation (4.2.1). Substituting for pb(r,¢) from the (4.2.5) the inhomogeneous terms may be written . 1 -1 2 ' _ 1 , A 31n(n+1)¢[§ r nIn(rc)-rsO In(ra) 2 aIn(ra)] (4 3 15)+ A sin(n—1)W[} r-lnI (ra)+rs 21 (ra) + 1 aI’(ra)} ' ' 2 n O n 2 n + 2AsoslIn(ra)cos nw 128 Assuming that the first order pressure has the form P1(r.¢) = f1(r)cos nw + f2(r)sin(n+l)¢ (4.3.16) . + f3 (r) sin (n-l) w, as its particular solution, we substitute (4.3.16) into (4.3.14) and arrive at three sets of equations. Designating an operator Lm to have the form _ -1 _c'1_d __ 2 -2 (4.3.17) L — r --(r dr) - {mr m +a2] m an integer we can write the three equations as (4.3.18) Ln{fl(r)] = 2A30511n(ra) (4.3.19) Ln+1{f2 (r)] = - % A[aIn+1(ra)+2rsozln(ra)} (4.3.20) Ln_l{f3(r)} = %‘A{aIn_1(ra)+2rsozln(ra)} To find the general solution for each of these equations we combine the homogeneous solution with a parti- cular solution which can be determined through the variation of parameters. The particular solutions are f1(r) = Cn1n(ra) + Asoslra 1In+ 1(rd) f2(r) = Dn+lln+1(ra)- % As ozrza-1{In+1(ra) (4.3.21) + 2n(ra)-1In(ra)} _ 1 2 2 -1 f3(r) — Dn_lIn_1(ra) + 4 Aso r a [In_1(ra) - 2n(ra)—11n(ra)] 129 For inviscid flows we set k2 = 1 changing the value of a2 to (sOZ-Bzc-Z). To solve for the first order velocities we consider the first order fluid equations given in (2.6.21) and examine the inhomogeneous terms which are written in terms of the known zero order velocities and the first order pressure which is known up to the arbitrary constants Cn.C and n+1 Dn-l' plus the arbitrary constants arising from the homo- geneous solutions, namely DnIn(ra)sin nW, Cn+11n+l(ra)cos(n+l)w, c 1(ra)cos(n-l)¢. n-lIn- Knowing the first order pressure we can now establish the procedure for determining the first order velocities. The inhomogeneous terms for our system are composed of the first order pressure, its derivatives, the zero order velocities and pressure and their derivatives. Denoting the partial differential operators to be used by 2 2 L =r2L-+r-a-+-h—--(l+r2Y2) ll 2 ar 2 5r 6‘) 2 2 (4.3.22) L22 = r2 §—§-+ r-§% +-h—§ - rzyz er a1: = .JL L13 2 a) we write the first order system as 1 L L w F (4.3.23) 11 13 i = 1 ’L13 L11 ‘wr F2 130 (4.3.24) L w: = P where _ . 2 o 2 o 0 F1 — rn1(l+iB/nc )p1'¢ - [rww'w-Zr sowe+rwr]cos W - sin W[2r3s 2wO+r2wO ] O W ¢,r (4.3.25) F2 = r2n1(1+iB/nc2)p1'r + COS W[rW$-rwg'w] + sin W[ZIZSOW%-2r3SOZWg-rzwg'r] F3 = —r2n1(1+iS/nc2)sopl + COS n[2rzsdw$-rw:] + sin ¢[2rzsdwg-r2w:'r-2r3sozwg . The inhomogeneous terms contained in the functions F F and F are composed of the trigonometric terms 1' 2 3 cos(n i_1)¢, sin(n i_1)¢, cos nw and sin n). we must therefore write the first order velocities in such a fashion as to give us all the trigonometric possibilities. The result is six systems of coupled second order ordinary differential equations containing both the wl and w1 components. In w r addition there are six second order equations involving only the w: velocity components. One half of these equations have inhomogeneous terms which are multiplied by the three arbitrary constants found in pl(r.¢). The other equations involve these arbitrary constants with other known constants which make it possible to determine some of the arbitrary constants through the boundary conditions. 131 'We will see later in the next section that only certain trigonometric functions give a non-zero value when integrated in the correction equation. we will see that only the cos m) portions of wk and the sin m¢ portions of ‘w1.‘w1 and p1 give a non-zero value for the correction. 9 r Here we allow m = (n-l), n or (n+1). In each system of coupled equations we find the same pattern as in the zero order system. By operating on one equation of the system with a differential operator and subtracting from it the other equations multiplied by a constant we arrive at a fourth order system which can be solved just as in section 4.2. The actual steps will not be produced here as they are cumbersome though of the same nature as those in section 4.2. 4.3.4 Second Order Correction To find an expression for 32 we begin with the second order stress equations of motion (2.6.32) for the fluid. They are S2 S2 - -iBu2 - s S0 fiiswrs ww ¢+(rswr :r] + 303 we ' v 2 we 81 1 + cos W(S: MW} + sin $[rsOS )9 -S *r} + r cos W sin ¢[S¢¢-S SO 9} + r sin2V[Szr-rs O 8*8) (4.3.26) —12 2 2= 2 o {$9¢.¢+(rser)r ,r ] - sOS99 —i6ue+ $2899 1 l — 288$ cos W - sin $[286r +rsoSL } + 2r cos w sin V S0 + r sin2){280 +rs 280 I *9 er 0 132 -1 2 2 2 _ . 2 {Srw.¢—S¢¢+(rsrr) ,r} + SOSr r9 — 1f3ur O l 1 1 1 _ SZSer - SWr cos W + sin W[rsOSer+See-Srr} + rsir cos W sin W + r sinZMSO M—S:e+rs 2 S0 i It should be noted that in the three preceding equations we have set 31 = O. The remaining procedure is exactly the same as that used in determining s1. Rather than dup1icate the derivation, the final expression is presented below. It is —u 1rdr O 2 O 2 u2 SO 2 SO I o{“¢s¢¢+uesnw+“r Sr r1 ”w 11 e Sew “r zsrw‘o 2w 1 f r{u° 32 mos2 r+u°s2 r-uzs0 -uzsO -uzs° o 1 1r 1 «r 9 er I rr‘o d' 2v 1 + 232$ I {uosO -uOSO +uOSo r)rdrdw O 0 w 91 9 99 cos ¢{u S +u (S -S )+2u S +u Sr -2u Se e)rdrdw ZWII 1 o o 1 1 o 1 o 1 1 0 IO 0 e 91 w vw ee 9 91 r r1 #9 2 1 - I ”I r cos w sin W[3u OSO +u 0(So -280 e)+uOSO o 0 SW W ¢¢ I IV uOS u¢ See}rdrdW (4.3.27) 2w 1 . O 1 1 + (0 f0 s1n w[u¢(s: r -rsos¢9)+uO (“08994-289r ) O 1 1 1 1 O O + ur(Sr r 868- rsosre) -rsoutse¢+u: (So Br+2r80899) :(rs 50 +250 e)}rdrd¢ 0 9r 2w 1 + I I sin 21(uw O(2r2s So- Sor) -u 0(3r2 8 So +3rSOr ) o 0 0 H1 1 0 ea er 2 o _ S -rSrr+3rSBe))rdrdV - O. + ur(2r sO er 133 Since all variables have period 2v in W the first integral of (4.3.27) is zero. Evaluation of the remaining integrals is simplified if we note that all the integrals involve products of trigonometric functions. Hence, only integrals involving coszmw or sinzmw, m an 2v integer, are non-zero. In fact the value of I cosZdew 2v or I sinszdw is W unless m = O in which case the O integrals have value 2F and zero respectively. we can 1 l . S V 9* and Sit are non—zero when integrated over W. In a similar conclude that only the cos(n i_l)¢ portions of u fashion only the sin(n i_1)¢ terms in the remaining velocities and stresses are non-zero when integrated. Since each of the first order terms contributes two terms which will be non-zero when integrated, it is more convenient to leave first order terms as they are and comment only that the fifth integral in (4.3.27) is also zero since the trigonometric functions for each term can be written as % [cosz(n-1)¢ - cosz(n+l)$} which is zero when integrated on ¢ over 0 g.¢ g_2w. we finally write the correction as O 0 u S +u S rdrd Riff: uwseweeerer] “ 02 02 20 20 201 x {f:v r[u% $r+u eSer+ursrr- uws¢r-ueser- rSrr‘o dt 81 O 1 O l 1 O +I: Ff: cos ¢[uesew+ug(swwfl err)+2ueSe¢+u S rt —2utseg]rdrd$ 134 N 71' (4.3.28) + I I 1 1 )+uO (rs S +28 ) s1 5 Sin ““3 ( "rs e 0 ea er yr 0 o ,. o :4 1 W l l 1501 O 0 Sr r-S —r S -r u +u S +2r S ( 99 SO re) 80 $5 aw 9( 9r 80 99) 0 BW Hal-0 HGO l u(rs OS9r +23: e)]rdrd¢ + I:Wfo sin2¢{u:(2rzsos O O 2 O O O 2 0 O - rswr)-ue(3r sosee+3rser)+ur(2r SOser-rsrr + 3rsze)}rdrd¢. In the rigid tube case the first integral in the numerator of (4.3.28) is zero since all velocities are zero at r = 1, while in the stress free case this same integral vanishes since the stresses do. For inviscid flows all the shear stresses are identically zero and the normal stresses can be replaced by the negative of the pressure for that order. Hence, Sk = Sk = Sk = O for all k and 9 r 9r k k k W W S rr = S 96 = S WW = -pk [Note that in this case, the integral vanishes.] Now that we have determined the correction equation for the general case we turn to the discussion of the two cases we are most interested in, the stress free and rigid tube cases. 135 4.4 Frequency Equation For Inviscid Axisymmetric Flows Having seen the general procedure for determining the pressures and velocities required to find the correction s we propose to examine the situations of primary interest 2' in this chapter. These situations are the stress free and rigid tube cases of an inviscid fluid exhibiting axisymmetric motion in the zero order. Since the fluid is inviscid the non-dimensional viscosities. l/fl and l/nl, are taken to be zero. The axisymmetric condition allows us to set n = O in our zero order system. From (4.2.19) we see that p0(r) = AIO(ra) ugh) = —AsOIO(ra)/i6 (4.4.1) 0 O . u¢(r) = 0 ur(r) = Ad11(ra)/&B For the rigid tube boundary condition, ug(l) = 0 so (4.4.2) a11(a) = O. Designating the zeroes of J1(z) by 21m m = 1,2,3.... we find that a = --iz1 is a root of (4.4.2). From the definition m of a in (4.2.3) we see that the frequency equation correspon- ding to (4.4.2) is written 2_22 2 (4.4.3) 30 — 6 /c - 21m. 136 For the stress free boundary we have (4.3.6) which is (4.4.4) IO(0L) = O. Designating the zeroes of Jb(z) by 20m m = 1.2.3,... we find the frequency equation for (4.4.4) to be 2=62/c2- 2 . (4.4.5) 20m So Since the zeroes of both J1(z) and Jb(z) are well known we are able to compute s easily in either case. If 0 the ratio B/c is less than the zero 21m or 20m ‘we have no prOpagation as S0 is imaginary. However, for frequencies greater than cz1m or czOm 'we have propagation of waves in the 6 direction. As the frequency increases the value of 80 approaches B/c asymptotically from below. This is true regardless of the boundary condition. To allow for the many possible values of the acoustic speed, c, we present the graph of the zero order wave number as a function of B/c for the first few modes of both cases. This is done in Figure 18. Recalling the definition of the phase velocity cp and group velocity cg. as, c9 = dB/d Re 80. (4.4.6) fig wwaPuE AllUMmunrflnmuukfl Avg M9099. Giro Uflfirfliflr ..u\a fl.) 94. .0.» M923» 139 we find immediately C (4.4.7) fl = (l-czzimB-2)-l/2 and C C (4.4.8) —-‘l = —— C Cp for the rigid tube case. For stress free conditions we insert 20m for 21m and then perform the needed compu- tations. From these equations we observe that for B < czOm or B < cz1m there is no propagation in the respective cases. At B = czOm or B = cz1m the value of s0 and cg/c is zero while cp/c is unbounded at the same value. As 6 increases 50 approaches B/c and both the group and phase velocity approach C in value. The graphs of cp/c and cg/c are given in Figure 19 for three modes. We should note that the first zero in the series 21m 18 O and when 21m = O we have so = B/c so that cp/c = cg/c = 1. This can also be seen in Figure 19. Due to the interlacing of the zeroes of Jb(z) and Jl(z) we can easily assess the behavior of the velocity ratio (4.4.9). For example, consider the rigid tube case which corresponds to a = -iz1m where m is the mth zero. Since the numerator will vanish at r = 1, it will also vanish at positions rén) < 1, corresponding to (4.4.9) rémz1m = z1n n = 1,2,...,m—l. 140 Furthermore, at a zero of Jl(z), Jo(z) has a maximum or minimum value there and is nonzero. Thus at these points, (4.4.9) . the ratio of velocities is longitudinal. Conti- nuing, since the zeroes of J1(z) and Jo(z) are inter- laced, we find that the denominator vanishes m-l times, or, between the locations (4.4.9) and, hence the motion is radial at these stations. We note that the behavior just described for the rigid tube case holds as well for the stress free case if one interchanges z1m w1th 20m in (4.4.9)- As in the case of the empty tube, we find it instructive to examine the ratio of velocities as r increases from the center to the outer edge of the fluid. In the shell we considered a displacement ratio over a certain frequency range. For the fluid we have the additional variation of the variable r. We write uO(r) a11(ra) (4°4°10) 5 = 2 2 21/2 ue(r) (5 /b +a ) Io(ra) where a = 'izlm for rigid tube flow and a = -1zOm for stress free flow. When we have the rigid tube case the numerator will be zero at r = 1 while the denominator will be zero at some intermediate value of r. Just the Opposite will occur in the stress free situation when the denominator is zero at r = 1 and the numerator is zero someWhere else. 141 All of this takes place for some fixed value of B,c and or 2 When 6 = cz or cz then the denominator 21m 0m' 1m Om is zero regardless of value of r. Graphs of the velocity ratio for various values of B,c and zjm, j = 0,1 are presented in Figures 20 and 21 for r between 0 and 1. We see that in the rigid tube case for the second mode r 2 2.40/3.83 is the value at which the ratio becomes unbounded, indicating radial motion. For small values of B/c the velocity ratio is nearly always large. However, as the value of B/c increases the motion is mostly longitudinal except near the value of r mentioned beforehand. As we consider higher modes for the rigid tube case, we find there are more values of r, O g_r 3.1 such that rz1m is equal to a zero of the denominator, namely 20k k = 1,2,...,m—l. In Figure 20 a we indicate the behavior of the velocity ratio for the second mode = 3.83171. [Note: "312 We are considering 211 = O and for this value the velocity ratio is always zero.] In Figure 20 b ‘we exhibit the velocity ratio for the next mode, 213 = 7.01559. Here we see the presence of two regions exhibiting primarily radial motion. From this it is easy to imagine that for very large modes the velocity ratio would be a series of verticle "spikes" indi- cating radial motion nearly everywhere. There are also (m—2) zeroes since these are (m-l) places where rz1m = zlj 142 Zero Order Velocity Ratio for the Rigid Tube..=5 Values of B/c are Indicated Zim=3.8317 .25 2:u\“.55ugifi=fl7’ Zero Order Velocity Ratio for the Rigid Tube Case '25 Values of fi/c Indicated Zim=7.01559 ‘wai-z *zb ~55 Z ‘K‘ 7. A‘SC‘ 143 j = l,2,...,m—l. Hence there are (m-l) + l = m places where the velocity ratio is zero and m-l places where it is infinity for the rigid tube case. In general we might say that the motion is more often longitudinal, but that at higher modes this motion is fragmented by regions of radial motion. In the stress free case we see that at r = 1 the denominator of the velocity ratio is zero, it being a multiple of the pressure. At r = O the ratio is zero as in the rigid tube case. If we consider a higher mode, say 2 we have (m—l) points at which the denominator is zero Om' in addition to the value r = 0. There are also (m—l) points at which the numerator is zero in addition to r = 0. Hence for 20m, we find that in the stress free case there are as many points where the motion is purely radial as there are places where it is purely longitudinal. Thus we find that the motion becomes more longitudinal as the frequency increases. This suggests that at very high frequencies the motion is nearly longitudinal except at a finite number of discrete points, these points being the zeroes of the denominator. 144 Zero Order Velocity Ratio for the Stress Free .55 Case. Values of B/c are Indicated. 145 4.5 Determination of First Order_§unctions In 4.3 we indicated how one proceeds to the deter- mination of the pressure and velocities of the first order system which are needed in the second order correction equation. In the case of inviscid axisymmetric flow we set n = O, % = ?%-= O, and k2 = 1. In this case, by (4.3.16) 1 and (4.3.21) we find P1(r:W) = COIO(ra) + 11(ra)[C cos V + D sin V} (4.5.1) - l A sin ¢IrstZa-1 2 11(ra) + rIo(ra)} where C = C1-C_1 and D = Dl-D-l' Returning now to (2.6.21) and inserting (4.5.1) as well as the inviscid assumptions mentioned above we find 1 _ . u‘b — p1,¢/r1B (4.5.2) u: = so(rp0 sin W - p1)/iB u: = p1,r/iB we now proceed to the boundary conditions noting that the continuity equation (2.6.22) is satisfied identically. For the rigid tube case, (2.6.30), the radial velocity component must vanish at r = 1. Hence, by (4.5.2), (4.5.3) p1(r.‘))' = O. r'r=1 146 and inserting (4.5.1) we obtain, f _ Codll(a) - 0, (4.5.4) Ca1i(a) = o. Dali(a) - % A[(2302d-1+a)11(a) + soin(a) + Io(a)] L =0. Recall now that a must satisfy (4.4.2), aIl(a) = 0. Although a = O is a root satisfying (4.4.2). it is easy to see that in this case that (4.4.1), (4.5.1) and (4.5.4) would lead to the conclusion u0 = u0 = u0 = p0 s O and our first order system would become the fundamental system. Hence we conclude a # O, and proceed to the non- zero roots of (4.4.2), i.e., Il(a) = O, a = -121m. From (4.5.4) to (4.5.6) we find C arbitrary, 0 (4.5.5) C = O: and _ l -1 2 (4.5.6) D - 5 Ad [80 +1]. Knowing the value of the arbitrary constants we are now in a position to completely describe the first order pressure and velocities for the rigid tube case. 147 For stress free flow (2.6.19) requires p1(l,¢) = 0. Inserting (4.5.1) gives ’ COIO(a) = 0 (4.5.7) < CIl(d) = O 1 2 l K{D - 5 Aso /d}Il(a) - 5 AIO(a) — o where now a satisfies (4.4.4), 10(a) = 0. As before we find CO is arbitrary, that C = O, and _ 1 2 (4.5.8) D — 5 A80 /C1. In summary we have, for zero and first order, po(r) = AI0(ra) u:(r) = 0 (4.5.9) 0 O ue(r) = -AsOIO(ra)/iB ur(r) = AdIl(ra)/ifi p (r w) = l sin ([(ZDd-Arzs 2)I (ra)/c - ArI (ra)} 1 ' 2 O 1 O ut(r.¢) (;%§) % cos ¢[(2Da-Arzsoz)11(ra)/b - ArIO(ra)} (4.5.10) u:(r,W) = (§%0 % 50 cos ({3Ar10(ra) — (2Da-Arzsoz) 11(ra)/o] u:(r,¢) = % sin ((10(ra)[2na - A(r2302+l)] - 11(ra)[2Da + Ar2(sOZ—zz)]/ra}/i5 where Da = % A302 for stress free flow, Da = % A(302+1) for rigid tube flow and z is a zero of J1(z) or Jb(z) depending on the boundary conditions. Note that since CO is arbitrary it has been incorporated into the arbitrary constant A. 148 Let us now consider the first order functions on the velocity ratio for various modes and frequencies within the two cases. we write the ratio as ug(r) + 6n:(¢.r) O 1 ue(r) + bueWm) and examine it for the cases presented in Figures 22 through 29. These curves are not meant to be an exhaustive display of the velocity ratio's behavior, but only indicative of the behavior as various parameters are changed. The graphs are given as r varies from O to 1 each graph being for a fixed frequency and mode. In Figures 22 through 25 we present the stress free condition. The addition of the lower case "a" to a figure number will denote the upper graph for each figure will the lower case "b" will denote the lower graph. In the zero order stress free case for the first mode (201 = 2.4048) the velocity ratio is zero at r = O gradually growing large as the boundary is approached. This indicates that the motion becomes radial as we approach the boundary. Increasing the frequency merely delayed the onset of the radial motion to larger r values. [See Figure 21a]. Examination of Figures 22a and b reveals the same behavior ‘with the addition of their being more radial motion at r = 0 149 Corrected Velocity Ratio: (=30° L4> 6:,005(...) &=.05(-——) 6=.5(- -) 150 in the corrected ratio. For r in the range .3 < r < 1.0 the zero order and first order ratios are quite close. Near the center we see that the first order ratio displays behavior not found in the zero order, indicating a need for higher order terms. We also see a greater variation for the largest value of 5 indicating even more descrepency between the straight tube model and those which are highly curved. In Figure 21 b we showed the behavior for the zero order ratio for the next mode (20 = 5.52007). The zero order indicated two regions which contained primarily radial motion. Examination of Figures 23a and b reveals the same behavior at approximately the same r values, .5 and 1. As in the zero order case the corrected ratios tended to become unbounded more abruptly as the frequency increased for a given mode. Again the 5 = .5 curves showed their variance with the zero order and smaller value delta curves near r = 0. When we consider the case of W = -60° 'we find that there is more variation from the zero order cases than when w = 30°. In the smaller frequencies, Figure 24 a, the 5 =’.5 curve is the only one to offer significant difference from the zero order curve in Figure 21 a. However, increasing the frequency, as in Figure 24 b, caused the 5 = .05 to adOpt the pattern of the larger 5 value, exhibiting a new region of radial motion for r < .4. 151 Corrected Velocity Ratio: Stress Free #30" Lb z=5.52007 6=.005(---) 5=.OS(———J b=.5(--) .5 We - usage? 152 153 When we increase the mode to 20 = 5.52007 as in Figures 21 a, 25 a and 25 b we see the presence of two regions of radial motion. When ¢ = 30° the corrected ratio is nearly the same as the zero order. However, for w = -60° we see new behavior especially for 5 = .5. In Figure 25 a, the higher frequency curve, the 5 = .5 curve introduces a third region of radial motion near r = .1. This behavior is not found at all in the zero order curves of Figure 21 b. Hence, when w = -60° we find that the zero order model is less suitable for curved flow than when t = 30°. For the value of 6 = .05 the curves of t = -60° in the corrected ratio are differing from the zero order more so than those of W = 30° do. FOr the rigid tube case the w = 30° curves of Figures 26 and 27 do not differ substantially from the zero order system in Figure 20. Both curves tend to reach points of radial motion more abruptly as the frequency is increased. Both systems introduce new regions of radial motion as higher order zeroes are used. Only the 5 = .5 curves show marked differences from the zero order motion, these differences being present primarily for small values of r. Again, this merely suggests the unsuitability of the straight tube axially- symmetric model for tightly curved shells. 154 Corrected Velocity Ratio: Stress F400 Le. II -| 3' I I I I I I I z=5.52007 '\u 6:.005(o o o) 6=.05(-—-) 6=.5(--) 155 Corrected Velocity Ratio: Rigid Tube #30. Io zl=3.83171 6=.005 (. . .) 6=.05 (——-) 6=.s (--) 156 157 When I = -60° the 5 = .5 curve for small frequencies. Figure 28 a, differs remarkably from the zero order case, Figure 20 a. When the frequency is increased as in Figure 28 b, then even the 5 = .05 curve has altered its behavior significantly from the zero order system. Only 5 = .005 retains the true behavior of the zero order system. As the mode is increased as in Figure 20 b and Figure 29, the 5 = .5 curve is quite different from the zero order system. In this case of the higher mode number, the increase in frequency does not alter the 5 = .05 curve of Figure 29 b nearly as much as increasing the frequency did in 28 b, the lower mode. However, other data suggest that as the higher frequencies are reached for the second mode of Figure 29 the curve for 5 = .05 ‘will change from the zero order case in greater degree. Our conclusions from all of this indicate that the zero order model is a reasonable representation for the curved tube on the outside of the tube and for small values of 5. However, on the inside the straight tube becomes unsatisfactory for all but the smallest 5 values and for all but the lowest frequencies. It is in this region that one must necessarily include higher order terms if the true velocity pattern is to be seen. 158 Corrected Velocity Ratio: Rigid Tube LA) i=~60° z=3.83l7 6=.005(o--) 6:.05(-——) 5=-5 (--) .5 39/4. ‘ 5.52M? $4135. ‘25 .5 We. . l5.&3|'7 159 Corrected Velocity Ratio for the Rigid Tube LA) Case t=60° Two Modes Two Frequen- cies for each mode CL Delta values indicated on z=7.01559 4.6 ZEI‘O are é MESH we kr thes§ (4.6. In (4 beth 1 Only . U W 160 4.6 §econd Order Correction we turn our attention to the correction equation (4.3.28) for 52 developed in section 4.3. For inviscid k k k flows, the shear stresses. S rW' S re and S *9 are all zero since they depend on viscosity. The normal stresses are all that remain, being equal to the negative of the pressure for their respective order. For axisymmetric flows we know that now = 0 = uoe'* = uor,¢ = po'¢. Utilizing these facts in (4.3.28) we see that now 82 has the value 1 2w 1 0 O -1 2V 1 l O s = - - - u S rdrd cos -2u S 2 ZIIO I0 9 99 I} {I0 jo( II w 69] . 0 l l O 1 0 (4.6.1) + Sln ([rsou BS 88 + 2rsou GS 99 - 2u rS ee] . 2 0 2 O 0 0 O + Sin w[—u e3r 50S 99 + ru r(38 ee-S rr)errdI). In (4.3.28)‘we have set the boundary integral zero since both stress free and rigid tube cases lead to this result. Only the cos w terms of ul¢ and the sin V portions of u1 ,u1 and S1 will make a non-zero contribution to the 9 r 99 total. Our first task is to perform the integration required in (4.6.1). In the first integral the V integration gives 2? as the value while in all of the others the result is v since we have non-zero results from the sinzt or coszt 'terms. In order to integrate the remaining terms which are fun (4.6 Afte (4.6. >13 II (4.6.4 where 161 functions of r we require the following integrals I: rIg(ra)dr = % r2{Ig(ra) - Ii(ra)} fr rZIO(ra)Il(rd)dr = % r21i(ra)/d 0 (4.6.2) I: rBIg(ra)dr = r4{315(rd) - 21i(rd) - I§(ra)}/12 fr r31i(rd)dr r4{Ii(rd) - I§(ra)}/6 Ir r411(ra)IO(ra)dr = r4[21§(ra) + I§(ra)}/6a After utilizing these results in (4.6.1) and dividing all terms by wAZ/iB ‘we find 2 153 _ 1 2 2 -l 2 Da _ _ O 2 2 3s 8 2 Da 0 2 5 l. 0 1 1 A 202 O 4 (1: 2a.: 2 2 So 1 2 5 2 + Z 12(a)[§ _ 23130] In the rigid tube case we set 11(a) = O and Da _ 1 2 . _ . . 7?" §(so +1). we also flnd 12(a) - 10(a) glVlng 2 4 S = — l {380 + l. 4" fig") 2 25 4 2 2 0 4a (4.6.4) _ l 4 2 2 0 1m Where as before Jl(zlm) = 0 m = l,2,°°-. 348 (4.6 Situ free reSp 0rde bec01 for I norm; the I quite and 1 162 For the stress free case we set 10(0) = O and 22-= l s 2 we find I (a) = 21 (a)/c This allows us .A 2 0 ° 2 1 ° to write _ 1 4 2 2 2 2 52 — g;—;-—Zj{so [zom +4] + 330 20m (2 20m ) (4.6.5) 0 0m + 22 4) 0m We can see from (4.6.4) and (4.6.5) that for the situation of small values of 30 free corrections behave like - 1/4 80-1 and + 1/4 80- the rigid tube and stress 1 respectively. For large values of s 3 0 . become larger than s 0 they are both of order 3 At some point then the value of 52s2 ‘will 0 indicating that we need to consider some higher order terms in the expansion. The same is true for values of 30 which are very small. In Figure 30 we show the corrected phase velocity, normalized by the non-dimensional acoustic speed, fOr both the rigid tube and stress free cases. Their behaviors are 0 quite similar, both being zero at the point where 80 and both indicating an asymptote very near this same point. After this initial behavior the phase velocity quickly approaches one. Eventually it becomes zero with large values of B/c. For 5 = .5 we see that in both cases the phase velocity becomes zero much faster than for smaller values of 5. For small delta the straight tube is an accurate nun..- gingham. in. 459mg .52.». AIIWMAmfiu MEDVQJB. “9.1.344”; W93...» fig )I'e )IO te +5 IE CC 164 representation for the curved tube since its phase velocity approaches one from.above. Only when the frequency becomes quite large or the value of 5 grows large does the straight tube mode cease to be an effective predictor of phase velocity. The curves for the corrected wave number, (so+5232)/s0, have been omitted since they are essentially the reflection of the phase velocity about the line 85/8 = l. The correspondence is not exact but the behavior can be determined by noting that as the phase velocity approaches zero the wave number becomes infinite and as the phase velocity becomes unbounded the wave number approaches zero. CHAPTER V GENERAL INTERACTION PROBLEM 5.1 Introduction We have presented analyses dealing with the vibration of the empty tube in Chapter 3 and with the vibration of the fluid in Chapter 4. It is now our task to analyze the inter— action problem in which the fluid and membrane are allowed to vibrate together. In section 5.2 we begin by establishing the frequency equation for an arbitrary number, n, of waves around the tube. we also discuss the correction equations apprOpriate and s to Section 5.3 deals with the frequency equation 51 2. appropriate to the fundamental case of axisymmetric inviscid flow which is the concern of following sections. Section 5.4 contains the determination of first order velocity and displacement components for the fluid and shell respectively. As before we examine their effect on correspond- ing zero order terms. Finally in section 5.5 we examine the correction, s and determine its effects on the zero order 2' wave number so. 165 166 _543 Discussion for Arbitrary n In section 4.2 we established the zero order values for the fluid pressure and velocities for the viscous fluid flowing with n waves in the ¢ direction. They are given by (4.2.18). Knowing the fluid velocities it is possible to write out the fluid stresses which are inhomogeneous terms of the shell equations. The shell displacements can be written from (3.2.2) as 62_n2_ % 302(l-V) _ % nso(l+v)-n -n W? - % nso(1+v) Bz—soz- % n2(l-v) -vso W2 -n -VSO 52-1 W2 (5.2.1) g3. = —$ ggr -§Er r=1 where the tildas reflect division by the apprOpriate trigono- metric functions and where the three shell displacements have been written as (5.2.2) w? = W? sin nw w% = w° cos m) w: = w: cos n) 8 m(= is (5. (5.2 Wher in ( the fluid deriv (5.2. InSer WhEre 167 Unlike the empty tube case the mass ratio m(= ph/poa) is no longer zero and a solution of (5.2.1) is given by the three relationships 0 _ 1 go “0 (5.2.3) WW — - a (net) 1(A118 rI + Alzsre’ A13 Srr}r_ o _ 1 So ~o “0 (5.2.4) We — " El (Det)_ 1.8[A12 1") + A zzsre - A23Srrlr=1 o _ 1 -1 “o Mo “0 (5°2°5) wr ’ ' a (pet) [A13Sr¢ + A23Sre ' A33Srr}r=l where Det represents the determinant of the coefficient matrix in (5.2.1) and the Aij are cofactors of elements aij in the same coefficient matrix. 5.2.1 Frequency Equation The boundary conditions require that at r = l the fluid velocities match the shell velocities which are time derivatives of shell displacements. we see then 0 0 (5.2.6) ..iawg = u§(r=1 g = I.e.r Inserting the fluid velocities (4.2.18) into the fluid constitutive equations (2.6.11) we find from (5.2.6) that C11 C12 C13 A3 0 (5.2.7) c21 c22 c23 A4 = 0 C31 C32 C33 A 0 where 168 c -- 2 -1-[A +2(Y) +A ‘1[v21'(v) 2:: (V) 11 - m Det n1 llYI 12 50 n - SO n+1 ] " 2A13YII:+1(Y)} ' In+1W)’ C _ 2 ‘1 (A. (Y) - A s -1 YZI'(Y) - 21 ( ) 12 _ - m Det n1 11YI n 12 0 [ n SO n-l Y ] + 2A13Y1n _-1(Y)] I__n _lH). _ M {-321 C13 — - a2 -Y2)m Det n1 [ 2nA11[Q'In (a) - In(a) 2 . -20IsOA121n(o.)} + 595?: {A13[l + $9— ($1 - 7‘11.) ]In(o) C 2 . "“15“” _ A13MO' In(a)] — T?— . a — 2 _ -1 . 2 C21 ’ " 31%‘1375? [A12 YIn+2(Y) + A22so [YZIn‘Y’ ’ so In+l(Y)] (5.2.8) + 2A23YII;+1(Y)} — Yso'11n(Y). 2 _ 2 C22 -- - TEE—DE {A12Y1n (Y) - SO 1A2112[Y21 (Y) - so In_1(Y)] + 2A23Y1n _1(Y)} + Yso 1In(Y), 2 C = - [- (aI'(a) - I (a)) 23 m Det n1(: %_Y2) nA12 n n + so A221n(a)] +A2 31 n(a)[l +l9 (. -—)] C2 _ MoII”(oI)A (a Z-Vz)A 2 _ -1 . 2 s31 ' " fia—Ifis—t23 ”‘13“ +2”) + so A23H21n”) ‘ so In+1(Y)] - 2A33Y1n+l(Y)} - In+1(y)' 169 2 _ Q -l 2 o 2 C32 “ - mnl Det {A13Y1n(y) - So A23[Y In(Y) ’ so In_1(Y)] + 2YA33In_1(Y)} + In_1(Y). 2 _ _é___, 2M 0 C — - [- (a1 (a) - I (a)) 33 m Det n1(a2-Y2)[ nA13 n n + sO A2311. ’(a)] + A 33[(1 + :13 [% - all-mum) _ Ma213(a) Maln (a)] 77—1“ *72“ In order that we have non-trivial solutions for our prOblem we require that the determinant of the coefficient matrix in (5.2.7) vanish. This defines an implicit relation— ship between s and fl which we designate as the frequency 0 equation. 5.2.2 First Order Correction In Chapter 4 we found the first order equation for s1 and were able to verify that in the stress free and rigid tube cases. 31 = 0. We now consider this equation (4.3.13) for the interaction prdblem. The line integral of (4.3.13) is repeated here as it is no longer zero. It is = 2W 0 1 O 1 ul 50 I0 u$S fr + ueser + urSrr - uws #r (5.2.9) 1 O 1 O uGser - ursrr}dw where all arguments are evaluated at r = l. The first order fluid stresses at r = 1 may be replaced by the first order Shel zero (2.6 the byt Noti over (5.2 170 shell terms with which they appear, (2.6.26). we replace zero order fluid stresses in a similar fashion using (2.6.14) and replace the fluid velocities at r = l by the shell displacements. Shell stresses are then replaced by the shell displacements using (2.6.13) and (2.6.25). NOting that the integral of an odd function in W is zero over the interval we are considering we write _ . 27 1 WO 0 ON 0 O 0 I — ifimsljo {— wwNow- 'woN 69 + vwvvo'o 1 (1_ 02 - -2- so \a)(wo)2 sows) - -(1- -v)w9w° (W 00 . 21r 1wo WW01 (5.2.10) - wo}d(: - 16m Io {w (W w'w- wwo t W l 0 WO 1 l -1-(1 v)(wow (MW- we 9 W) +5 so (1+v)(w°w owo O 1wowo) ‘wwwe’.w"" ‘ MW Examination of (5.2.10) indicates that the second integral of (5.2.10) contains two basic terms. The first has the form 2w (5-2-11> 12 = Io Ift¢)g,¢¢ - g<¢>f,*,3dw which when integrated twice by parts in the first term yields 2n = {f(WJW) - 9H)f(¢) \ 0‘” I“, o w + I: {9(¢)f(¢)'¢o - g(¢)f(¢)'**}dg, ide Usir the 171 The first term is zero by periodicity and the integral is identically zero. The second term to be found in (5.2.10) is 2 (5.2.12) I3 = In [:2ngan = f(WsW) I07r = 0 O by periodicity. Hence (5.2.10) reduces to _ . 2v 1 o o O o O 0 I — lamsl‘ro {'2' W¢N9¢ - WQNGB + WVWGM) 1 O 2 O 2 1 0 0 (5.2.13) _ 2 30(1-V) (ww) - 50(We) " '2' (l-V)Weww'¢ O O _ wrwe}dw. Using (5.2.12) as the boundary integral in (4.3.13) we find the equation (4.3.13) can be written as since F(sO,B) g 0, we conclude that = 0. 31 If we are to have any sort of correction we must find 32 and this requires the solution to first order velocities and displacements. The first order pressure and velocities can be found as in section 4.3. Using the boundary conditions we match first order fluid velocities with first order shell velocities at r = l. The result is a determination of the arbitrary constants acquired in solving the first order problem. This having been done our next step is the solution of the second order correction. 172 5.2.3 Second Order Correction for Arbitrary n The second order correction equation for 82. (4.3.27) developed for the fluid alone must be modified to account for the fluid-shell interaction. we begin with the boundary terms in (4.3.26) and repeat here those integrals which are not identically zero at r = 1. due to periodicity in W- The integrals just mentioned, in (4.3.26). are designated as _21r 02 02 02 20 11 — f6 [uwsrw + uGSrG + ursrr - “vsrt 20 20 - ueSre — ursrr]d¢ where the integrand has argument r = l. we eliminate the fluid stresses by means of (2.6.14) and (2.6.34), and we eliminate the fluid velocities by the boundary conditions (2.6.15), (2.6.35). After some simple algebraic manipulations it is easy to show that only certain components of the integrand will make a non-zero contribution. we designate by tilda, ~. the sin(n+1)¢ components of Ni) and Nte and the cos(n+l)¢ term of Nt)’ we use a hat. A, to designate the sin(n-l)w portions of Ni) and Née as well as the cos(n-l)t term in NGW' Denoting the constant portion of the zero order terms by * ‘we write 173 ‘ . 2" oo 11 = leSZIO [WWNGW - we Nee}d¢ 2 + iBm‘fow {w3[Ni¢.) + 3 CNN] wO 2 W2 0 0 2 2 +w9[N8¢.¢—SON66] - wrN¢¢+ WINWW 2 O W2 NO 0 ' ww[N)).w + S oNew] e[N e) w SoNee]}dN (5.2.14) "1 A . ~1N1 -1Bm H[W[NW+NW-Nee- N66 A “’1 l * O + SONNGN - New) + 50 Ne w] A O O “1A1 ”l 1 + w9[s (Nee - N69 + Nee) - 2m“ + N9*)] .1 A + wr 0[ *N0 N N19]} The second order shell terms are now eliminated by means of (2.6.33) and we finally achieve _. 2” quoo 00 Il — 16m52f6{w¢N 9W - weNeehiw A ~ A +iBm12T wO[-nv(\::'+wt+w +w1+w O "1 + 252We - so('we - w 9) + dee) 1 *1 A1 0 o ~1 (5.2.15) + 5 sO(l-v)(so[w¢ - WW + WV] + 282W+ - +tw t)} W 1 Al ~1 O + ifim - we{§ (1-v)[nso(w* -‘w*) + 2n82W~ + nsowg —n(G1-$1)]+sz(&’1-$1+wh+2ssw°) 9 9 O 9 9 . 1r 0 ~1 A1 ”1 Al -1f3m§WV{w¢+w¢+wr_ r 9 - so(w9 - W’e) 174 ‘ 0 A1 -wr+(232 +so)w 3% H[W[N*:+ NW ~1 A1 A1 _Nee-Nee+so(Né¢- New) 3*] ~ A ~ A +w %[s 0*(Nee- Nte-i-Nte) -2(:C1;:+Nle¢)] ~ A + Wg[*N%e' N199 1]} . 277 o 2 2 o ' 13me “’me " ”w” WNW + 5 ‘1‘”) [wee w _ wnge'w] + 1 (1w) [w 29 - Wew¢]'¢ 2 0 WoW 2 + (waw- “W W}d N. we recognize the integrand of the last integral above is zero due to terms of the form found in (5.2.11) and (5.2.12). Coupling this equation, which has no second order terms except 32, with that develoPed in section 4.3, for the fluid we have an equation for the second order correction in terms of known zero and first order quantities. 175 5.3 pgrequencyEguation for Inviscid. Axisymmetric Flow we repeat. for convenience, from (4.4.1), the pres- sure and velocities in the inviscid, axisymmetric case: p0(r) = AIO(ra). u3(r) = 0, (5.3.1) u%(r) = -AsOIO(rd)/iB, ug(r) = Aa11(ra)/iB, a2 = s: - BZ/cz, where A is arbitrary. In the inviscid case only the normal fluid stresses are non-zero and hence the only body force on the shell is Sorr r=1 = -po(l). The shell displacements then satisfy the following (5.3.2) [52 - %{1-\flsg]w% = o to HOG: 2 2 (5.3.3) 5 ~30 -\so 0 3‘? 10(a) 2 -\)so B -l w 1 Solving for displacements we see that for axisymmetric flow w?” = 0 and w% = -szAIO(a)m-1{(l-vz-Bfi)sg - 52(1-52)}'1 (5.3.4) w: = -(Bz-sg)AI (a)m'1[(l-V2—Bz)sg - 62(1-BZ)}-1 0 where m = ph/poa. In the inviscid prOblem we match velocities in the radial direction giving us the frequency equation for the interaction problem. It is 176 2 (5.3.5) m11(a)[(BZ-1+v2)sg—fi2 (62-1)} —62(a -s§>10(a) = o This equation has been determined by Rubinow and Keller [9 ] in a different situation. As before we utilize portions of their analysis as the basis for our discussion. Our first task is to recognize those portions of (5.3.5) which have appeared before in special cases. We shall refer to those modes which reduce to the empty tube frequency equation (3.4.2) as §2bg_mggg§. Those modes which reduce to either of the two modes associated with the fluid will be called acoustic modes. due to their dependence on the non-dimensional acoustic speed. When m approaches zero we may interpret this as meaning that tube density is small as in the stress free case. In fact setting m = 0 produces three possible equations: 10(d) = 0, s0 = B or B = O. The first equation is the stress free boundary condition (4.4.4). However, the next mode is a new mode which results from the interaction. It is the tube mode (3.4.2) when x»: O. The last equation, B = O, is of no interest since 30 is undefined. Hence letting m approach zero gives us a mode which we recognized from'before. the stress free mode. It also produced a new mode, 3 = 6, not normally associated with small tube 0 density. If we let m grow unbounded in (5.3.5) we find that a11(d) = O or 38 = 62(l-Bz)(l-\?-Bz)-1. The first mode 177 is the rigid tube mode (4.4.2) which we could associate with ph growing large (and hence m also). The second mode we found to be the empty tube mode (3.4.2) which we could assoc- iate with poa becoming small (and hence m ‘becoming 1 large.) In this situation setting m7 = 0 results in two modes, both recognizable from before. 5.3.1 Frequency Equation for v = 0 If we set \)= O in (5.3.5) we see that the equation can be written as (5.3.6) (Bz-sg)[mall(a)(32-l) + 6210(a)} = 0 we see that 50 = B is a frequency equation for all values of m ‘when \)= O (and for all values of \9 when m.= O as in (5.3.5)). Let us now examine the term in.braces in (5.3.5) utilizing an interesting and informative approach of Rubinow and Keller [ 9 ]. Case I: d2 < O We set a ix where x is real. we then write J (x) 2 1 _ (5.3.7) -XW—;‘—(1§;-B—2-)- . If we differentiate the left side of (5.3.6) with respect to x 'we find 2 J (x) J (8) 1 —x l + 1 (5.3.8) 3- —x—— = JO(X) J30” dx 178 Hence for positive x the left side of (5.3.7) is always a decreasing function of x. If B > 1 then the right side of (5.3.7) is an increasing function of m. If 6 < 1 then the same expression is decreasing in m. Since the left side of (5.3.7) exhibits the same behavior as the right side we can say that for B > 1, x is a decreasing function of m and for B < 1, x is an increasing function of m. Since (5.3.9) 38 = Bz/c2 — x2 we have that for real values of so. 30 is a decreasing function of m when 6 < l and ana increasing function of m 'when 6 > 1. If 50 is imaginary the conclusions are just the Opposite. Knowing the qualitative behavior of so we are now in a position to understand the behavior of the compu- tations we are about to make. we know how the acoustic modes behave in the extremes of m = O and m = o. Hence ‘we expand in a MacLauren series about 5 small where e = m or m-l. We say + e 95- + . . . (5.3.10) 8 dc|€=0 o = So )e=o ‘we see then that when m is small we have _ n)1/2 22 /2 (5.3.11) so _ (a 22-/c +szn (52 -"’1)/5 (BM/c011} where JO(zOn) = O n=l,2,3." If we rewrite the term in braces of (5.3.6) so that e is m-1 then we find 179 (53°12) S = (132/3413”” + Bz/Imu-aznez/cZ-zl 2W2} 0 n where J = O. n = 1,2,3. 1(zln) We plot the frequency equation for these acoustic modes by using (5.3.11) and (5.3.12). we know that the curves for m = O or l/m = O are the extremes for any intermediate values and we know that at B = 1 there is a change in s from a decreasing function in m for B < l. O to an increasing function of m for B > 1. These curves are presented in Figure 31 for c = 0.1. These curves are the lower set of curves which exhibit an abrupt change at B = 1. Real values of sO dictate that B 2 cz1n or B 2 czOn depending on the mode. Hence if c is large enough, cz or cz will be larger than one and In On the acoustic curves will start further to the right, dis- playing no switch from decreasing to increasing functions of m. In fact, if the fluid is incompressible (c very large) the acoustic modes exist only for extremely high frequencies. If we examine the phase velocity cp = B/So in Figure 32 we see the obvious sort of behavior for the curves near B = 1. These curves are plotted for c = $1 and if c is large enough the acoustic modes exist only when 6 > 1 and hence do not exhibit the switch from increasing to decreasing functions of m. [Note: Since cp = B/so. c is P increasing in m when s is decreasing and vice versa.] 0 «Xian @dfl Amman). MUD? thfiu 1.0..3 0.33.. 359644 A VA K153? gfl A: ..ud.nflnn~ iéfiwg g , $44. .0¢1MUD{£. 182 Case 11 a2 > 0 (Tube Modes) If we return to the expression in braces in (5.3.6) we may rewrite it as 2 2 (5.3.13) maIl(d)(l-B ) = B Io(d) or as (5.3.14) m(l—BZ) = 10‘0" B2 dIl(a) When a is real the right side of (5.3.13) is a positive decreasing function of a which behaves like 2 2/a for small a, i.e. 50 near B/c, and like l/d for a large. If we replace the right side of (5.3.14) by 2/a2 then we see for small a that (5.3.15) .3- 2_2—5. .25 m(l-B ) c while for a large we find 2 2 52 1 (5.3.16) 80 = B 2 2 2 + -3- . In (1-5 ) c In plotting the curve for this mode we utilized (5.3.15) until B2 = 2m(2m-)-l)-1 at.which point.both repre- sentations are equal. Then we plotted (5.3.16) for the re- mainder of the B values. Since the right side of (5.3.14) is positive. B is restricted to values less than or equal to l. we also see that the left side of (5.3.13) is increasing in m (B _<_ l) and that the right side of the Same equation is decreasing in (1. Hence a must be 183 decreasing in m and therefore sg(= a2+B2/c2) is also decreasing in m. The curves for this mode are plotted in Figure 31, and are labeled with t. We see that the first acoustic modes are close to the m large tube modes for B < 1. When the acoustic modes exist only for B > 1 then there is no similarity. The phase velocity associated with the mode of (5.3.15) and (5.3.16) is given in Figure 32 and it exhibits the behavior we expect from looking at the frequency equation. If one returns all variables Undimensional form in (5.3.15) and sets B = 0, the resulting phase velocity is that of Korteweg mentioned in the introduction as c At B = O 1. the phase velocity for this mode has value fch/[2c2+m])l/2. If the acoustic speed is small then the phase velocity starts near zero and becomes zero at B = 1. If the fluid is nearly incompressible. c ~ m, then the phase velocity begins near (m/2)1/2. If instead we allow m to become large then the phase velocity starts near the acoustic speed c. Having examined the two acoustic modes (5.3.11) and (5.3.12) as well as the tube mode given by (5.3.15) and (5.3.16) we examine the displacement and velocity ratios associated with these modes. From (5.3.1) we have (5.3.17) (ug(r)/u%(r)) = )-aIl(ra)/s010(ra)| . 184 Case I Acoustic Modes The velocity ratio (5.3.17) for the acoustic mOdes is plotted in Figures 38 and 40. The curves labeled 5 = O in these figures represent the behavior of (5.3.17). When m = 100 (Figure 38) we are near the rigid tube mode and find that the 6 = 0 curve of Figure 38 is very nearly the same as the rigid tube case of Figure 26. Increasing the mode number to the second mode for the interaction gives us similar behavior to the second mode of the rigid tube case. i.e. the presence of a second region of radial motion as in Figure 27. Thus examination of the interaction prOblem near the rigid tube case indicates that there is little change from the rigid tube itself. The acoustic mode for m small given in Figure 40 eXhibits behavior nearly the same as the stress free mode of Figure.22. As for the stress free case further computations for the interaction prOblem show that increasing the order of the mode increases the number of regions of radial motion. In examining the displacement ratio (wg/w31 for these two acoustic modes we see from (5.3.4) that ‘wo = O 9 and the tube dispacements are purely radial. Case 11 Tube Modes Having examined the acoustic modes and their effect on the velocity and displacement ratios we turn to the tube nodes associated with v = O. The first mode is so = B 185 2). and the argument of the Bessel functions is d2 = B2(l—c- 'we note that if c > 1, then a is real and 10(a), 11(a) are real. However if c < 1, then Io(a), 11(c) must be replaced by Jb(a), Jl(a) which are, of course, oscillatory. Hence if c < 1 then the v locity ratio (5.3.17) will have zeroes when rd = z1n and will be unbounded when ra = 20“. Therefore for any fixed c < 1 increasing B increases la) and hence rd is equal to more of the zeroes z1n or 2 On as r increases from zero to one. 0 For 30 = B we find that the displacement wr is determined in (5.3.4) but that ‘w% is arbitrary. Examination of the next order displacement equations indicates that both w% and the arbitrary constant A are present. Since A is the arbitrary constant for this problem,'we set 'w% = O Obtaining only radial motion in the shell. Finally we examine the ratios for the tube mode given in (5.3.15) and (5.3.16). we approximate the modified Bessel functions for re large and re small finding that the velocity ratio is like I-raz/Zsol for (re) small and it is like i-a/501 for (re) large. The velocity ratio is then zero at the center of the tube growing to some value )-a/so| at an intermediate point. we note that since so # B (from (5.3.15) or (5.3.16)) we find that (5.3.4) yields 'w% E 0 so that the motion is purely radial in the tube. 186 §;§;3_ Poisson's Ratio is Non-Zero To examine the frequency equation when Poisson's ratio is non-zero we expand so in a series about m small or m"1 small using (5.3.10). In this section only we shall denote 80 as a function of m by the symbol g0 and we shall reserve the symbol S0 to designate the value of so corres- ponding to m = 0 or m-1 = 0 where appropriate. For m small we find from (5.3.5) and (5.3.10) for the acoustic mode 10(d) = 0. we have m zOn2[(1-V2-32)Sg-Bz(1-52)3 (5.3.18) E’ = s + O 0 2 2 2 B (B -so)sO where 2 2 2 2 _ s0 2 B /c -zOn , J0(20n) - 0. The expansion about the tube mode. 30 = B, is from (5.3.5) and (5.3.10) ~ _ 2 (5.3.19) 50 — B-mv a11(a)/2B10(a) where a2 = B2(l-l/c2) For m large we have the acoustic mode, I1(d) = 0. and we obtain 62(s3-62) (5.3.20) §' = s + ° ° msoi(1-v2-62)s§-62(1-62)) Where 187 2 _ 2 2 2 _ so — B /b -zln , J1(zln) — 0. For m large we have the tube mode 2 2 2 ~ 3 (S -B )I ((1) (5.3.21) 5 ° ° = s + 0 2mso(1-v2-B2)a11(d) where s3 = 62(1-BZ)(1-v2-62)'1 and a2 = sz-B2/b2. 0 We now analyse the behavior of go for the cases we have given. Case 1. Acoustic Modes (5.3.18) and (5.3.20). For v = O we found a change in the behavior of the acoustic modes as B changed from less than one to greater then one. we expect a similar behavior for §' 0 when v #’0. ‘We write the acoustic mode (5.3.18) for m small as 2 2 2 2 ~ _ mZOn (l-v -B ) so 1_ 2 (5.3.22) 80 - so - 2 2 -§-- -—-§i—§' 80(80 -fi ) B l-v -B 2 _ 2 2 2 _ _ ._. where so - B /c -20n, Jo(20n) — 0 n — 1,2. . In what ~ follows we will assess the behavior of 80 given in (5.3.22) as a function of m. i.e., we will determine the regions in which go is an increasing (or decreasing) function of m. For this purpose we examine the sign of the three terms which form the coefficient of m in (5.3.22) - the curly bracket term, 188 Starting with the curly bracket term in (5.3.22) and noting that for c = 0.1, czOn < 1 for n=l,2,3, define Y1 83/62 . (5.3.23) y2 = (1-B2)(1-v2-Bz)'1. The functions y1.y2 are given in Figure 33 and we note that yl-y2 can be > O, = O, or < 0 depending upon the value of czOn. If czOn < l, as in Figure 33a It is easy to show that y1 = Y2 at two frequencies, say a1 and a2. It is found that yl-y2 > 0 for a1 (l-\32)l/2 and y1--y2 is never zero fOr real values of B. Hence for czo3 < B < b1 189 J_ I Represen- 6" _—-H_—— _- tation of 3" I 3) possible inter- I action of com- ponents in frequency equation I (5 .3 . 20). Func- I 31, tions y1.y2 are defined in l (5.3.21). Third I o possibility C?- 04 (1; places b1 | 5‘ t 0 left CL. of B=l in e V | '9 32b. Curves (P0); are not to scale. Q «>9 "7~ Eaves: ‘35 190 ~ so is decreasing in m and increasing in m for B >‘b. Finally for n‘Z 4 CZOn > 1 for c = .l and we find a1 < CZOn < a2 < b1 as in Figure 33b. Hence so is decreasing in m for a2 < B < b1 and increasing for CZOn < B < a2 and for B > bl' Let us now rewrite the acoustic mode, 11(a) = O. for m large, (5.3.19) in the form (56‘52) $6 1 -1 s + ——_—_:L 2 U) II o o msO(l-v2-BZ) 62 l-vZ—Bz (5.3.25) 2 2 - E..- 2 _ s0 — C2 zln ' Jl(zln) — 0' It is easy to note the similarity between (5.3.22) and (5.3.25), the major difference being the root z1n rather than zOn' For the smallgst root, 210 = 0, s0 is decrea31ng in In 'when b2 < l + and increasing when the inequality v c2-1 is reversed. Since the roots z1n satisfy the inequality cz1n < 1, if n=l,2, the behavior of so given in (5.3.25) is the same as that indicated by (5.3.22) if z1n replaces 2 there. and a1,a2,b retain their meaning with respect On 1 to the functions 33 - B2 and yl,y2 defined in (5.3.23). When n 2_3, we find that in the intervals czlnb1 §' is an increasing function of m and is a O decreasing function of In when a2 a2. When czjn > 1, j=O,l, then ~ so is increasing in m for B > b1 and decreasing in the interval czjn < B < b1, 3:0,1. We present the graphs of the frequency equation for m large, (5.3.25), in Figure 34w Only one situation in which the so changes from decreasing to an increasing function of m is presented. This is due to the fact that the other changes occur very close together on the frequencyaxis.. we have designated these regions by an asterisk, *, and give as an example of the behavior of so in these regions the curves in Figure 35. This set of curves shows the behavior of the second acoustic mode for m large near the starting Here we see the function s is a decreasing 12' 0 function of m for c212<a1 and increasing in frequency cz m elsewhere. éflwfiukgég :04qu 14mm. 243mm,. 39g... wind “5.11%.... 030191.346: 4.04.44. .Tflog 194 Case II. Tube Modes N we have examined the behavior of so for the acoustic modes, noting their similarities. We now turn our attention to the tube modes (5.3.19) and (5.3.21). In all that follows z1n ‘Wlll be a zero of J1(z) while ZOn Will be a zero of Jo(z). Examination of (5.3.19) indicates that for c > 1 the modified Bessel functions, Io(a) and 11(d), have real argument and are themselves real. We find then that for c > 1 so hand c < l the arguments of the Bessel functions are imag- is a decreasing function of m. If, 1 . In either case d2 is positive in only a limited frequency range so that in general the modified Bessel functions have imaginary argument and therefore behave like 2 . 2 ~ Jo and J1. When B is near (1-v ), so and so both become unbounded. When B approaches 1 then so is near zero and g0 is arbitrarily large. When a2 = -zln2 then go, given by (5.3.21), becomes unbounded while when 2 _ 2 ~ _ . . . d - -20n then so - B. Utilizing the values of so and a2 given with (5.3.21) we find that there are two possible . 2 _ 2 2 _ _ 2 frequencies when a - —z1n or a — zOn . 196 The graph of s is presented in Figure 36 as the 0 solid line for one value of m, namely 100. Defining 2 _ 2 . do — -d we find from (5.3.21) that for zln<:a0<:zO(n+l)’ s0 is increasing in m while it is decreasing in m for other values of do. Hence, if one is to determine the behavior of so as given by (5.3.21) for some m large other than _m“ 100 , one would have to find the relationship of the frequency to the zeroes and determine then whether the curve were above or below that given for m = 100. The phase velocities for these two modes, (5.3.19) and (5.3.21) are given in Figure 37 over a larger frequency range. we must indicate that no comparison between the m large and m small curves regarding increasing or decreasing prOperties has been made. Let us now consider the velocity ratio, ug/ug, and the displacement ratio, wg/wg, obtained from (5.3.1) and (5.3.4). respectively. The behavior of these ratios is determined by the solution so of the frequency equation (5.3.5). To show how the various modes, acoustic and tube, alter the amplitude ratios we present the ratios using 3' found in (5.3.18) to (5.3.21). 0 The velocity ratio for the acoustic mode, (5.3.20), is presented in Figures 38 and 39, and for the mode (5.3.18), in Figures 40 and 41. The apprOpriate curves are the 6==O curves in each figure. 9. 90 ON. 9 .190 All ,0... .2 nlv 0.9.. .2 9 mdnw 3942 U93. 192g? 4 n. 198 . .nu 99m AMWV/M «W. A)? 1N. 930.2 .MDDr dam EH3 H.923». new» Must.» 199 A comparison of Figure 38 with 39 or 40 with 41 shows that the effect of V is insignificant for these modes. Furthermore, comparing the velocity ratio found here with the ratio found for the rigid tube or stress free tube (in Chapter 4) shows that the fluid behavior is only slightly altered. Hence, the elasticity of the tube wall is insignificant if the motion of the fluid is governed by an acoustic mode. Let us now consider the velocity ratio for the tube modes, (5.3.19), (5.3.21). The ratio is presented in Figures 42 through 44. The velocity ratio for this case is now almost a linear function of the radial dis- tance becoming very radial in character at the edge of the tube. Note that the two expansions in m yield almost the same ratio values. for 6==0. The graphs of the displacement ratios are given in Figures 45 through 48 where again the 6 = 0 curves are the zero order curves. In Figures 45 and 46 we present the first two displacement ratios for the acoustic modes. We see that for both modes the displacement ratio grows larger as the frequency is increased, indicating that the displacements in the tube become more radial at higher frequencies when the frequency relation is near the acoustic modes. In Figure 47 we present the displacement ratio for the tube mode (5.3.19). This ratio exhibits behavior 200 radically different from the empty tube mode and demon- strates the effect of the compressibility of the fluid. O The displacement ratio wS/we Obtained from (5.3.4) is given by 2 ~2 B - so VSO Since c = 0.1, (5.3.19) becomes mvzzJ (z) S' = B + 1 O 2BJo(z) if we use 2 = id there. Thus when J1(z) = O, z = zim' the ratio is zero whereas when Jo(z) = 0, the ratio is infinite. It is apparent that for c > 1, the behavior is altered since Il(d) and Io(d) do not vanish. When we examine the displacement ratio for the tube mode associated with the empty tube case (5.3.21) we find the displacement ratio behaving nearly identically to the displacement ratio for the empty tube. Figure 48 gives the displacement ratio for the interaction prOblem, near the empty tube mode. There is no ratio given for (l—\)2)1/2 g B < 1. Near these values of B[=(l—\)2)1/2 and l] the displacement ratio indicates primarily radial motion, while for other values of B the motion is longitudinal. 201 §;g_ First Order PressureL_Velocities and Displacements From (4.5.1) and (4.5.2) we have the first order pressure and velocities, determined up to their arbitrary constants. The first order pressure influences the first order displacements so we write our displacements in the shell as w1 = a cos) + b sin) + c ‘1' ‘) )1) 1) (5.4.1) w1 = a sin) + b cos) + c 9 8 6 9 w1 = a sin) + b cos) + c r r r r If we designate the matrix of (5.2.1) by ”4(n) then the first order displacement system, upon using (5.4.1), (4.5.1) and (4.5.2), becomes (5.4.2) 'bw o M(_1) ‘be 0 . (5.4.3) f) M(+1) = fe . r fr (5.4.4) [52 - %sg(l-v)}c¢ = o, 2 2 (5.4.5) B -sO —vso ce 1 O 2 _ - C010 (0') 1 o -vso B -1 cr where 202 f) = 550(k)”): " "(13 ' (5.4.6) f9 = -%(l-v+4sg)w% + 50(1—v)wg . fr = 2vwg - fiIDIl(a) - %A[sga-lll(a) + IO(G)]) + so(l-v)w% . The constants Co. D and C are to be determined by matching the radial velocities at r = 1. Hence, we have, in addition to (5.4.2) to (5.4.6), the boundary conditions -chr = coa11(a)/iB. -iBbr CaIi(d)/iB, (5.4.7) . I 1 2"]. 2| -1631' {Dd1103)— §A[ZSOQ I1(d) + soll(a) + Io(d) + aIl(a)]}/iB. The procedure is rather straightforward. Recognizing that so is related to B by means of (5.3.5), the frequency equation, we have the determinant M(_1) is non zero so that (5.4.8) b = b = b = 0. Since the determinant of (5.4.3) is the same as the determinant of (5.4.2) we can solve for aw. a9. and ar. Finally,(5.4.4) and (5.4.5) yield cw. 6 c and c r 203 Substituting cr into the first of (5.4.7) reveals Co is arbitrary provided (5.3.5) is satisfied. We take (5.4.9) C = c = 0. The second of (5.4.7), using (5.4.8), yields (5.4.10) C = 0. A solution ar of (5.4.3) involves D, and so the third of (5.4.7) may be used to determine D. we write this symbolically, n[aIO(a) - 11(4) + m‘16243311(a)/bet) i = Det{A13f) + A 23£8 + 1133“!va + so (1- v-)wo (5.4.11) 1 -l 2 -l +-2-m A(soa 11(0)) + Io(a))]] l -l 2 2 2 1 + §A{a (Zso+d )Il(d) + soIl(a) + 10(0)} where Det is the determinant of M(+1)' Aij is the cofactor of the element a.. in M , and the 13 (+1) f)' fo, fr are defined in (5.4.6). Incorporating all of this, (5.4.8) to (5.4.11), determines the first order fluid velocities, pressure and shell displacements subject to the relation (5.3.5) Let us now examine the corrected velocity and displacement ratios, )ur/uo) and )wr/wol, respectively, Velocity Ratios mmloo Acoustic Mode #11. v=0. &=O(-—) 5=.05 (r-) ' 6=.5(---) v=o z=3.832 B/c=8.832 c=.l 204 205 Velocity Ratios m=100 Acoustic Mode 232» v=.5 6=O(-—) 6=.OS(--) 6=.5(°'-) z=3.832 B/c=8.832 IZ> 206 Velocity Ratios m=.01 v=o z=2.4048 c=-1 LO B/c=7.4OS 6=0(-—) 5=.05(--) 5:,5(...) First Mode Acoustic 207 Velocity Ratios m=.01 v=.5 z=2.4048 C=.1 \‘Q B/c=7.405 Acoustic First Mode 5=0(-) &=.05(--) 4:.5(- --) 208 for the effect created by 6, Poisson's ratio v, and the polar angle ). We concentrate our attention on v = 0.5 as representative of biological materials and we allow 6 to be 0.005, 0.05 or 0.5 again banding the situation presented in the aortic arch. Two angles, ) = 30°, ) = —60°, have been chosen so that their effect can be I' measured as well. The graphs of the velocity ratio for the different i modes of motion are presented in Figures 38 to 43, and u the displacement ratio is graphed in Figures 44 to 48. Case I Acoustic Modes We first examine the corrected ratios for the acoustic modes. The acoustic mode for m small is given by (5.3.11) for v = 0‘ and by (5.3.18) for v #'0 and the velocity ratio evaluated for this mode is graphed in Figures 40 and 41 respectively. First, um: notice that changing v from 0.0 to 0.5 does not alter the behavior significantly. Second, Mme see that the behavior of the zero order case (6 = 0) and the corrected value for 6 = .05 are quite close, with the 6 = .5 curve showing significant differences from the lower 6 values, especially for ) = -60°. The behavior of this ratio is remarkably similar to the stress free velocity ratio presented in Figures 22 through 25. When we consider the acoustic mode for m large given by (5.3.12) for v = 0 and by (5.3.20) for v # O 209 we find the corrected velocity ratios given in Figures 38 and 39. As before, little change is introduced by allowing v to be 0.0 or 0.5; the 6 = .05 curve is similar to the zero order case (6 = 0); as we change angle we can expect to introduce even more significant deviation of the 6 = .5 curve from the zero order curve. ! The displacement ratios corresponding to these acoustic modes are given in Figures 45 and 46 for v = .5. 3.» 1' In general the corrected ratios follow the zero order El ratios rather closely except for the asymptote in the m = .01 curve. In general the motion in the tube becomes more radial as higher frequencies are encountered. For v = 0 the corrected ratio is not given. In the zero order the motion is purely radial for these modes when v = 0. The corrected ratios are quite large indi- cating generally radial motion. Case II Tube Modes The tube mode for m small is given by the so expansion (5.3.19) and the velocity ratio for this mode is graphed in Figure 42. Note that the 6 = 0 curve in this figure is the plot of the straight tube velocity ratio and we have already remarked upon its increasing radial character. The character of the ratio for 6 = 0.005 and 0.05 is also radial from the point r = 0.2 to r = 1 and is more radial than the straight tube case. However 210 Velocity Ratios Tube Mode m=.01 v=.5 B/c=l. c=.l =O(—-) 6=.05(——) 6=.5(--o) Eh 6=.5 off scale Ei 211 when 6 = 0.5, the ratio, apparently, reverses the trend for small 6 and goes from predominantly radial at r ==1. to almost longitudinal there. It is felt that this is probably a spurious result due to the size of 6. If we examine the velocity ratios for the m large tube modes given by (5.3.21) given in Figures 43 and 44 we see that for v = 0 all 6 values are rather close, the velocities being mostly longitudinal near the center and more radial near the outer edge. When v = 0 ‘we have essentially that so = B. For v = .5 the 6 = .05 curves in Figure 44 behave much differently from the 6 s 0 curves, indicating further restrictions on the use of the straight tube to model the curved tube for this flow. To examine the displacement ratios for these two tube modes we turn to Figures 47 and 48. Figure 47 represents the m small mode displacement ratio plotted for v = 0.5. The figure contains the results for 6 = 0 and 6 = 0.05 and they are almost identical. Plotted against B we note several regions of radial motions and this is due primarily to the compressibility of the fluid. If c > 1.0, then So and the regions of radial motion do not exist. of (5.3.19) does not become infinite 212 Velocity Ratios. 6=.005 curves are nearly equal to 6=.05 curves. £5£> 6=.OS off scale 213 Velocity Ratios. Tube Mode. m=100 v=.5 £5. B/C=l. c=.l 6=O(-) 6=.005(--—) =.05(--) 6=.5(-") 6 values not shown are off scale. 214 Figure 48 gives a representation of the displacement ratio for the tube mode of m large (5.3.21). This mode is nearly the same as the empty tube mode so we expect that the displacement ratio will behave like the empty tube displacement ratio examined previously. Note that the ratio evaluated at ) = 30° and at ) = —60° is given on the same graph and indicates little sensitivity to the ) location. 216 III Alla 09 12 <3 [human OMMN 9, 4.0 nlu 3:52 224190 9. «G. 3-1.... nus/233 emu 950.2sz; stag a; n. 219 5.5 Second Order Correction In section 5.2.3 we gave, in integral form, the expression that allows us to determine the s2 correction to the wave number. Equation (5.2.15) when substituted into (4.3.27) and (4.3.28) yields the equation for 32. For the inviscid fluid this equation is simplified since the shear stresses Sr) and Sro vanish. Further, in the case of zero order axisymmetric flow, u? = n? W = u: ) = 0. Employing these restrictions we obtain 2 O O O 2 2 s2[—4mB [sowo + vwr]wo + 2A so[Io(a) 2 0 - ao[ (2so + l-v)w 2 2 O 11(Q)]} = mg [zsoawwe e O 0 0 so(l-v)wr] + ar[so(l—v)wo + 2Wr] o 2 V o o 0 (5.5.1) + wo[ (350 + l- )we - so(2-\J)wr + so(\)—2)wo] o o 2 2 D01. 153(2) + wrwo(l-2V)) + A {10(0) [T - 1 - 8 ] 2 2 33 s 2 Da 0 l o 1 2 +11(O"[A(22‘1)'2‘ 2(1+2So) a a 2 2 S S + (5/4)sg] + I§(a)-49-[% - 423]} a where a2 = S: — B2/c2 .and w‘ 220 O we = -szAIO(a)m 1{(1-v2-Bz)s§ - 52(1-32H'1 (5.5.2) 0 2 2 -1 2 2 2 2 2 - wr = ..(s -so)AIo(a)m [(l-v -6 )s0 - B (1-6 )1 1 The aw, ao and ar terms are those of the first order I shell displacements which are solutions of (5.4.3). _- Eventually we are able to write the entire expression in terms of the modified Bessel functions 10(a) and 11(a). .fi we should note that m = ph/poa where ph is associated with the tube and poa is connected with the fluid. When there is no tube, ph is zero, and since there are no shell displacements we interpret (5.5.2) so as to have w% = O = w?. This necessarily means that Io(a) = O, which is the stress free boundary condition. Examination of (5.5.1) shows that we arrive at the stress . . 0 _ 0 _ _ 2g_= l 2 free correction if we — wr — Io (a) — O and A 530. -1 For the case when poa = 0 'we see that m = 0 and so if we interpret (5.5.2) so that {(l-vz-B2)sg - B2(1-B2)} 4»0 as 111"1 4’0 then ‘W3 'WE are not zero. Understanding that A must.be zero if and there is no fluid we find that we have [(l-vz-B2)sg - B2(l-B2)} = 0 as the frequency equation and we have found the correction for the empty tube. 221 Finally if ph grows infinite in size we interpret l 0 and w0 = w0 = 0. e r Understanding that I1(a) = 0 ‘we find the correction for this case as the rigid tube. Then m- the rigid tube. Hence the correction (5.5.1) contains within it F all the special cases we have been considering and is therefore a composite of all those situations we considered earlier. To measure the effect of 32 we now concentrate our attention on the phase velocity (5.5.3) o - ———é———-. Equation (5.5.3) has been evaluated for the four main modes considered previously. For example, using the acoustic mode for m small, i.e., (5.3.18), we obtain the corresponding value for s from (5.5.1). New choosing a 2 6 value and a v value, we evaluate (5.5.3) versus B where we have standardized c at 0.1. Our results are given in Figures 49 through 53. Case I Acoustic Modes In Figure 49 we present the phase velocity (5.5.3) in which we have used the acoustic mode expansion (5.3.18) for so for m = 0.01. we have plotted the curves for (5.5.3) for the first two zeros of Jo(20n); n =l,2 and labeled the curves corresponding to n as l or 2. 223 The curves for 6 = 0.005 and 0.05 were extremely close together and their distinction on the plot was not possible. Thus we show only the 6 = 0.05 curves as the solid lines. A comparison of the phase velocity obtained from (5.5.3) and that corresponding to 6 = 0 (previously plotted in Figure 31) shows extremely good agreement over the interval in B to about B = 1.5 at which point (5.5.3) starts to rise monotonically. The agreement range 1 between the 6 = 0 cp and the corrected cp expands in B to 2.0 for the second mode n = 2, and it is expected that a larger interval of agreement would be Obtained for higher modes. For the sake of comparison we include the results predicted for 6 = 0.5 and for this value of 6 ‘we find no agreement with the straight tube case (6 = 0). Let us now consider the phase velocity for the second acoustic mode using (5.3.20) and n = 1,2. This case has been plotted in Figure 50 wherein the two modes corresponding to n = 1,2 are labeled as such and where 6 = 0.05 is the solid curve. we note that the two 6 curves for n = l are close until B reaches 1.1 and the 6 curves for n = 2 agree until B is near 2.0. However, and more importantly, the agreement between the 6 = 0.0 curves of Figure 31 (m 100, there) and those in Figure 50 is very good for B nearly 2.0. 0:109:49 01:094. an» 7914 999.452 9342 9. «0 4.47.2 9002 4?? 06.39.97 .9926. fig 49... 9959* 4.9 9.. 4., 9. / // / / / / / — / v2 / 225 Case II Tube Modes The phase velocity of the straight cylinder interaction problem is given by the tube mode so = B or, cp = l, in the case where Poisson's ratio is 0.0. For v nonzero, however, we have found that cp, predicted by (5.3.19) and (5.3.21), exhibits asymptotes due to the k... compressibility of the fluid. Figure 37 is a graph of c for v = 0.5 and for m large and small, 100.0, P 0.01, respectively. While the m small mode exhibits (‘4: the general behavior pattern, cp 1.0, except for the asymptotes, the cp curve for m = 100.0 shows a cutoff in the interval \/l-v2 < B < l. The character of this mode is much more akin to the empty tube mode previously examined. Let us now examine the effect of curvature on cp by inserting (5.3.19) or (5.3.21) into (5.3.3) for v = 0.5. The results are illustrated in Figure 51 for the value 6 = 0.05. A direct comparison with the zero order empty tube mode, m = a, Figure 3, shows remarkable agreement with the cP value Obtained from (5.5.3) using (5.3.21) with m = 100.0. The agreement is good over the entire frequency range B > 0, except near B = 0. In the interval 0 g_B g_0.02, the cp obtained from (5.5.3) becomes negative and approaches zero through negative values due 226 AN; 227 to the asymptote at about 0.02. The corrected velocity differs from the velocity, Figure 37, in that the com- pressibility of the fluid is suppressed now when m is large. Turning our attention to the m. small case we note good agreement with the straight cylinder phase “46 velocity plotted in Figure 37. Compressibility of the __.“ |_. fluid is responsible for the asymptotes and in general 5;? cp is near 1.0 in value as in Figure 37. We present further results concerning the m small mode in Figures 52 and 53. The pertinent data for these cases is v = 0.0 and 6 = 0.05 (Figure 52) and 6 = 0.005 (Figure 53). Drastic behavior changes in cp are found in these cases and especially in the 6 = 0.05 curve. The phase velocity is influenced primarily by the fluid compressibility especially in the neighborhood of B = 1. For the smaller curvature, cp is again near 1.0. Ill-Eb 4.64 8.2.9 ME» 9qu 09.9.de 0929.90 94.....N ,4.n.2 4....0 9932 993. c.8449) Rosa. 9»? «9... 9&1.qu 4., m. , 1.3 5 / CHAPTER VI COMMENTS AND CONSIDERATIONS 6.1 Introduction In this chapter we conclude the present work with some comments on what has been presented in the preceeding chapters. In section 6.2 we consider the situation in which the mass parameter, m, is the same size as the perturbation parameter 6. In section 6.3 we present sane restricted results and make a comparison to available experimental data. Finally in section 6.4 we summarize the present contributions and indicate what further studies are contemplated. We are also able to make some statements regarding areas of analysis which must be mastered before further generalizations can be contemplated. 230 231 §;g_ Comments on the Size 951 m Implicit in the development of all the equations in Chapter II is the assumption that the mass parameter, m, is of order 1 in comparison to the perturbation parameter, 6. We recall that m = ph/boa and that in biological cases p/bo is nearly one. We also note that h/a is required to be small in the thin shell theory so that it is possible for m to be small. we therefore choose to consider m = Q6/boa where Q and (poa) are of order one compared to 6. If we examine typical terms from the shell equations (2.2.8) we find that there are three basic terms written as 2 2 2 2 2 2 2 phé-uszB' poa éfibSdB' ph§_bi w where we have non-dimensionalized using (2.5.3) and set qg = Sr§)r=l' Here N35 is a typical shell stress resultant, S is a typical fluid stress evaluated at r = 1 and w GB is a typical shell displacement. If we replace ph by Q6 then the zero order shell equations are reduced to setting the fluid stresses (at r = 1) equal to zero. we see then that if we are to consider the shell displacements of order k (k21) then the fluid stresses are of order (k+l) in these equations. For the zero order inviscid fluid the axisymmetric velocities and pressure are given in (4.4.1). Setting the fluid stress equal to zero at r=1 gives 232 (6.2.1) sgr(1) = -po(l) = AIo(d) = o where a = $3 - Bz/c2 We proceed to the first correction equation (4.3.13) l written as follows in the inviscid case. '41 2w 1 0 0 - (6.2.2) J‘ouo(1)slr(1)<1)=251 jojouo(r)soo(r)rdrd). 9.3 We replace s:r(l) by the remainder of the first order shell equation (3.2.2). i.e., Q 1 _ _ _ O _ O S r(l) — +5-63{(B21)w0 W4“) vsowo}. Since only radial velocities must match at the boundary we assume that w? and w% are not necessarily axisymmetric as are their counterparts in the velocity field of the fluid. We assume w? = aocos) + bwsin) + co 0 _ . (6.2.3) wo —- aoSin) + bocos) + ce 0 w = c r r where the a's, b's and c's are constants. The correction is written as 233 AaI (a) Q 1 2 O O " “TB" 90a IONS ’1) ”r " WM (6.2-4) —vso[aosin) + b ocos) + co]}d) 2 —josol O(ra)rdr. _4Ws 1A The integral of fiw? W is zero by periodicity as is true of the sin) and cos) terms. Hence we have AaIl(a) Q 2 " 1B poa [(B ‘1)Cr - V8009] (6.2.5) S0 S1 A2 = 16 {10 (a) - 12 1(61)}. we must solve for cr and co if we are to evaluate the correction. The equations for the zero order shell displacements involve p1(l). we find the first order pressure from equation (2.6.20) and determine P1(r.)) = C0 10(ra) + I l(rd)[C cos) + D sin)} (6.2.6) - %A sin){rzsga1 11(rd) + rIo(ra)) -1 + Asosld rIl(ra). We should note that for our earlier work in chapter IV 81 = 0 so that the last term in (6.2.6) was not present. [cf. (4.5.1)]. L. 234 Substitution of (6.2.3) into the first order shell equations (3.2.3) through (3.2.5) gives 62-1-%s§(1-v) %_-So(l+v) a ¢ 0 (6.2.7) = -so(1+v) Bz-sg~%(l-v) a9 0 52-1-%sg(1—v) - -%-so(1+v) 19‘y 0 (6.2.8) = -—SO(1+\)) 62-53%(1-V) be 0 2 1 2 _ (6.2.9) [B - —so(l-v)}cw — O 2 B - s0 0 poa Asos1 (6.2.10) = - ~6—' a 11(a) -v50 1 _ pOa -b¢ - vs Obe — - Q CI 1(a) (6.2.11) P _ 0a 1 2 a11’ - vsoae — - Q “[D- EASO/o}11(a) By (6.2.1) the frequency equation is 10(a) = O, i.e., J0(ic) = 0. Since the determinants of the coefficient matrices in (6.2.7) through (6.2.10) are not identically b b and c are all V 9 W zero we can conclude aw, as, zero and that 235 2 c _ - Avsosll}}g)poa 6 — aQ Det ' (6.2.12) 2 2 C = — A(B -so)sos}11(a)p0a r aQ Det ' _ 2 2 2 2 2 Det - (B -so) (6 -1) v so . . “~n From (6.2.11) we find C = O and D = %Asg/o which is the value of D for the stress free condition. Matching radial velocities we find that i1 and therefore (6.2.13) - “291)“ we have then found 31 by matching velocities, but we also have the correction equation (6.2.5). Substitution of (6.2.13) into (6.2.5) satisfies the latter relationship exactly. When m = 0, as in the stress free case, we interpret this to mean that Q is zero and hence = O. 31 Therefore the situation in which m is small reduces to the case m = O with s1 becoming zero just as in the stress free case. 236 Use of (6.2.13) in (6.2.12) leads to the following relations for the zero order shell displacements: w? = 0, (6.2.14) w% = Avsoa11(a)/BZ(B2-Sé)o w: = AaI1(a)/BZ. We are also able to write P1(r.W) = C I (ra) + 1A sin([s 2 -1 2 O O 2 a (l-r )Il(rd) O a(Det)rIl(ra) - rI (ra)}——A—Q- v 11 O pOa 62(52—53) For inviscid flow we have from (4.5.2) that Ufi(rl ‘1‘) = Pl'w/rifi: (6.2.15) ué(r,¢) = so[rpo sinw-p1}/iB. 1 . ur(ro W) = p1,r/IB ° We see that u; does not depend on Q and hence 31 since the derivative with respect to ¢ eliminates the Q term from p1(r, w) . One can note then that the first order correction has no influence on the circumferential velocity, its effects being felt in the pressure and other velocities. 237 §;§_ Comparison with Experimental work There has been some experimental work reported in the literature regarding the wave prOpagation char- acteristics of fluid filled curved elastic tubes. This 'work was performed at Stanford University and is summar- ized in a report by Anliker and Van Buskirk [22]. In general a fluid filled flexible tube was coiled in a semi-circle and suspended by flexible strips from another rigid tube. A disc was inserted in one end of the tube and vibrated in a controlled manner. Three separate types of waves were generated during the course of the experiments. The three waves were denoted by the terms axial, pressure and flexural respectively. Daras, in his thesis, compares his analytical results with those reported by Anliker and Van Buskirk. All investigations assume an incompressible fluid so that 2 a = sg-Bz/c2 reduces to a = so. Case I. Flexural Waves (one wave around the tube) Daras assumed in this case that B < < l and that the zero order terms have the form 0 _ O . O = O = ww — W¢51n¢, we wgcosx), wr wgcost (6.3.1) po(r,¢) = A11(rso)cos¢ 238 Inserting these expressions into the shell equations (5.2.1) with n = 1 results in a system of equations in w0 w0 w? and A. If we ignore the 62 terms we WI 8' find 1 2 l O -l--2—sO (l-v) #580 (1+v) -1 WV 0 A 1 2 1 _ A “1 (6.3.2) —§so(l+v) -s07§(1-v) -vso wg —-m;1(so) o _ l _ V30 _1 w: 1 J“— E Solving for w: and matching velocities at r = 1 gives 1.18 (6.3.3) -ing = ug(l) = p°(r,(),r/16|r=l . The result of (6.3.3) is a frequency equation of the form 2 4 2 _ 2 5 ' (6.3.4) B Il(so){so+250+l] — m(l-v )soll(so) If we make the further restriction that 30 is small, writing 11(30) arsO/Z and 10(50) 2 1, then the result takes the form (6.3.5) sg{Bz-m(l-v2)} + 26288 + 62 = 0. We present the graph of the phase velocity, cp(=B/so), in Figure 54. we converted the data of the Stanford experiments into our notation using h/a = .0967, p = 1, Q) = 1.248 and v = 0 or v = .5. These curves compare well with the experimental data presented in the 239 ._. nee. one. 3.6.2 20.1.33 T1 m. us I o us 565 32.; z 33; .35me 3.8.; 82.6 .3. .oz 952... mac. 0.. °s m cm. 240 Stanford reports and indicated by crosses on the graph. Daras solved (6.3.5) by the quadratic formula, expanding the denominator in a geometric series. Taking the lowest order term he finds (6.3.6) 33 = B[m(l-v2)}_1/2 Computing cp from this value also gives reasonable agreement with experimental data. Case II. Axial waves. The second situation Daras considers is that of axisymmetric axial waves. In this situation the flow in the zero order system is assumed to be inviscid and axisymmetric. Daras set w0 = 0, WC = 0 and w0 = 0. (l; ru) 60* If one makes a long wave length assumption, i.e., so is small and m < < 1, then the zero order shell equations take the form 62-52 -vs w0 0 0 O 9 A ~vso B -l wr l Daras notes that under these conditions Anliker and Maxwell [23] have found w: small for small B. Setting w: = 0 forces us to conclude that 30 = B 0 _ we - A/vsom where we have replaced 10(30) by 1. 241 If we now pursue this prOblem further by searching for the correction s we find that the first order system 2 assumes the form 3 2 3 -l-+Zso 250 1 aw f) 3 1 _ (6.3.8) 7150 -4 —sO/2 ae - fe ' “‘1 2 l —sO/2 sO-l ar fr .61I where we have utilized the fact that i] i 1 _ 1 _ . 1 _ . _ WW — awcos), we — a951n), 'wr ar51n), v .5, and f) = 5A/2m. _ 2 (6.3.9) fe — -A(l+8sO)/2som, Ds _ o A 2 fr — A/m - [ 2 - 2[so/2+l]]/m. Using Cramer's rule we solve for ar and set it equal to fii(l)/B2, with the tilda denoting the sin) portion of ~1 u . r If we keep only those terms which involve m-1 we find that 0 'q 5 .3..- __. and from that we find pl(r,)) = 3Ar sin). 242 This value for p1 corresponds to that found by Daras. S . Solving for ar by setting a = -l‘£- we r 62 we find for the lowest order terms in so that ar = 3A/B2 which also corresponds to the value obtained by Daras. Returning to the shell equations of first order ILfi we find by Cramer's rule that a =35- ' 6 mfi ._ which agrees with Daras. Finally we solve for a Cramer's rule again, obtaining a -17A ) — 8m ' . 2 . . . _ 61) Daras Obtains 3A/B 'Wthh he finds by setting at - 2. B i.e. from matching velocities in the ) direction. This procedure is not applicable in the inviscid fluid since there are no shear stresses in the fluid and hence no coupling of the fluid and shell velocities in the circum- ferential direction. Utilizing the values we have found for a a ar, t. 90 O by taking we and D we examine (5.5.1) and find 52 those terms which are of lowest order in B. ‘We find (6.3.10) 82 = -3m/lGB so that the corrected phase velocity is given by 243 (6.3.11) cp = {1 — 3m62/l6BZ}-1 Daras finds a different value for cp although it is of the same form. He expands the right side in a geometric series which is not correct as the expression 3m62/1662 is not less than one for B near zero. In fact there is a vertical asymptote for B = 6V/3m74. The graph of cp is given in Figure 55. We have utilized the following values for the parameters appearing in (6.3.11): m = 0.075, 52 = 0.01 and 0.0 < B < 0.1. The agreement between our theoretical values and the experimental results are quite good. Case III Pressure waves In this case v = .5 and the acceleration terms are neglected in comparison to other terms. As in the axial waves the flow is inviscid, incompressible and axisymmetric in the zero order. The wave lengths are long so that 50 is small. The shell terms are written as 2 0 --s0 -vso W6 A 0 (6.3.12) 0 = "H 10(50) I -vs0 -l wr 1 while the fluid velocity in the radial direction is u: = soIl(sO)/i6 , and the boundary condition takes the form . 0 _ 0 -1f3wr - ur . 244 mbd.I/A ““03“” 970. (W 7.094434% flux/14.), 1,1Xxx CV53)? U143». 9% mm ¥§ .. 8. 245 Solving for ‘WS and satisfying the boundary condition we find so = 6/8/3m, w0=4§ W0=_2A [3m w0=0 r 3m' 9 3mB £3 ' ) ' The first order displacement equations satisfy I. 1 2 3 “ ‘1 ’ 250 280 1 a) f) 3 2 1 _ (6.3.13) 7160 --s.O — 4 -sO/2 a9 — f6 J; 1 - 0/2 1 r fr where the first order displacements are the same as those in the axial waves, and W w? 353 1 (6.3.14) f9 = g—[— + g} 0 2 Ds _ 3 0 l_ 0 _ §_ 2 fr " Zwr ' m[ 2 2(30/2+1H Solving for ar and satisfying the boundary conditions DsO we find A = -l and that _ 2 ar — -A/B (6.3.15) a9 = -5A/3som a = -A/msg . 246 As before we have some disagreement with Daras due to his use of boundary conditions to determine other than the radial velocity. Utilizing these values for the zero and first order velocities we find from (5.5.1) that keeping only terms of order (ms(2))"1 we find (6.3.16) 32 = 3/l6sO and the corrected phase velocity is given by (6.3.17) c = «gm-[1 + 2 P 8 1266 The corrected phase velocity is plotted in Figure 56 and the comparison with experimental data is rather close. All of the above cases have shown close agreement between theoretical and experimentally predicted phase velocities. The results plotted by Daras are different analytically from those presented here yet they also agree fairly well with experimental data. As can be seen the experimental data exists only for a limited frequency range and the values given are essentially the first term of the phase velocity. Further experimental work needs to be done on the very small frequencies in order to determine the exact behavior in this region of small frequencies. 248 6.4 Final Considerations We are now in a position to reflect on what has gone before in this treatise. We established the equations for a fluid flowing in a curved membrane shell. We attacked the problem through the use of a power series expansion in a small parameter for all our variables. The result of this procedure was a series of problems starting with a prOblem apprOpriate to a straight tube flow. we analysed the frequency equation resulting from the straight tube prOblem and then proceeded to find corrections for our zero order variables. We found for a particular variable, the wave number, that the first correction was zero and that the second order correction was required. We interpreted this correction as accounting for curvature effects not found in the straight tube. Various ratios of velocities and displacements were calculated and these values com- pared to those of the straight tube. we found some regions where there were differences between the straight and curved tubes. In other frequency ranges the straight and curved results are nearly identical. ‘We have also found that.when the parameter m is of order 5, then the first correction. U: the zero order wave number is not zero. This fact has not.been noticed by previous investigators and therefore deserves fluflmrshfly. 249 Finally we should comment on the basic prOblem present in such an analysis as we have presented. The fundamental prOblem is analysis of the frequency or dispersion relation deve10ped for the zero order system. This equation is transcendental in nature and in general quite difficult to solve. A recent paper by Scarton and I Rouleau [24] presents a computer analysis of a frequency -—m equation similar to ours. In this analysis the complex roots are sketched. Unfortunately it is difficult to find ,: an analytic relationship among these roots which could be gJ used in computing corrections for lower order terms. In conjunction with the last comments we should point out that there are further problems which require solution. For example the bending terms might be intro- duced into the shell equations or the shell might be described as anisotropic. Another problem would involve the introduction of viscous terms into the solutions. we have pmesented the solutions but the complete analysis of the frequency equation for these more general models is lacking. What we have been able to do is solve a basic problem, fundamental to the field of biological fluid flows. we have analysed the zero order situation and then pro- ceeded to write an expression which accounts for curvature corrections. In addition to demonstrating the existence of such a correction we have also computed it and shown its effect on lower order terms. BIBLIOGRAPHY 10. ll. 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