.M '- :e mm?“ \ ‘. ‘ mu ‘3- Lax-A.“ mm.» . 3 'i f 'i :J..:_:.‘. :qu‘fi ‘3. ‘ 12‘“ £L-:;V.\E ”an,“ mm“, c: w. . 3-»4.“ C i‘ M. .. ‘ ‘21!“‘23 ”Hart: 3: . .7 “if“ ‘- ‘ ..\. x. ‘ 1“ J's-“v. n . ‘._. hymn.” «K I «am Us '35" A...“ .tz-fL. ‘21-. “h x..._ .u . x cm. ’ . m. a '1 .. 1: :fmu v3? . é a ",4 '3‘?» .. :. .,. SJ:}|:>.:.'u. -._‘.. .uu‘ . u . ‘ ,Ef‘r .w .=< 4.2}?!‘1' v. ... n; W. .. . Nauw...) ,«h... u... m. - A ‘ .. . . , 1 If 1c L“ _‘ \L1.\.‘ 4],], HEN-4,.“- ~ u . .r ”A _,.;; .;. o. ,.,., 3.. ~ 1" ‘ ‘ ‘ ‘ . . W Mr, is . 4. ‘ ' . ; r. 4.1 .;1..:‘.‘, ‘m’. “an. . -; ,3. 3., f u" ‘ ( .A. . .‘.“T.'.‘,‘£“‘ . . 4‘ A . ‘ v. 4 . ‘ "w. .. _ Van. .. I : iiiiiiiiiiiiii ~ This is to certify that the dissertation entitled BIORTHOGONAL SERIES SOLUTION OF STOKES FLOW PROBLEMS presented by Suhei] Khoury has been accepted towards fulfillment of the requirements for Ph.D. degree in Applied Mathematics ( x . Major professor g f Date 2/20/1994 MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771 LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. [4 DATE DUE DATE DUE DATE DUE a. MSU Is An Affirmative Action/Equal Opportunity Ins titution chlmMunS-DJ BIORTHOGONAL SERIES SOLUTION OF STOKES FLOW PROBLEMS By Suheil Khoury A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1994 ABSTRACT BIORTHOGONAL SERIES SOLUTION OF STOKES FLOW PROBLEMS BY Suheil Khoury In this thesis we developed an eigenfunction expansion method for solving prob- lems of Stokes flow , mainly in sectorial cavities and some other flow geometries arising in fluid dynamics which have never been studied before . The theory leads to the development of a set of eigenfunctions , adjoint eigenfunctions , biorthogo— nality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansions . The resulting infinite system of linear equations are then solved by truncation . These biorthogonality conditions are properties satisfied by the eigenfunctions and self-adjoint eigenfunctions , which enables us to compute the co- efficients of the eigenfunction expansion solution . We derived these biorthogonality conditions for a class of fourth-order boundary value problems with variable coeffi- cients that arise from separating variables of the governing Stokes equation . The method is applied to several creeping flow problems . One example is the slow , steady flow in a two-dimesional sectorial cavity . The fluid is set into motion by the uniform translation of a covering plate or belt . This problem has not been studied either analytically or numerically . In other examples , the matched eigenfunction expansion method is used for solving Stokes flow around a bend and through curved channels . The flow region is decom- posed into two simple subregions ; this enables the stream function to be represented by means of an expansion of Papkovich-Fadle eigenfunctions in each of these two i subregions . The coefficients in these expansions are determined by imposing weak C3 continuity of the stream function across subregion interfaces and then taking advantage of the biorthogonality conditions . Although the matching technique is done in a weak sense for solving Stokes flow in curved channels and the flow around a bend , we were able to couple the equations in such a way such that the resulting stream function we obtained is actually C3 continuous across the subregion interfaces . ACKNOWLEDGMENTS I wish to express my deep gratitude to Dr. C.Y. Wang , my thesis advisor , for suggesting the problem and for his helpful suggestions and patient guidance during the course of my research . Thank you to Dr. C.Y. Lo , Dr. M. Miklavcic , Dr. L. Sonneborn , and Dr. D. Yen for being willing to serve on my committee and for their patience and support . iii Contents LIST OF TABLES LIST OF FIGURES 1 Introduction 1.1 Literature review ............................. 1.2 Background ................................ 2 Biorthogonality Conditions 2.1 A Class of Fourth-Order Boundary Value Problems ......... 2.2 Stokes Flow in Sectorial and Rectangular Regions .......... 2.3 Axisymmetric Stokes Flow in Spherical and Toroidal Regions ..... 2.4 Discussion of other Flow Geometries .................. 3 Stokes Flow in a Sectorial Cavity 3.1 Formulation ................................ 3.2 Biorthogonal Series Solution ....................... 3.3 Numerical Results and Discussion .................... vi 14 ‘21 27 35 43 43 50 4 Stokes Flow around a Bend 66 4.1 Formulation ................................ 66 4.2 Matching Eigenfunction Expansions ................... 73 4.3 Biorthogonality Series Solution ..................... 81 4.4 Numerical Results and Discussion .................... 89 5 Stokes Flow in Curved Channels 113 5.1 Formulation ................................ 113 5.2 Matching Eigenfunction Expansions ................... 116 5.3 Biorthogonality Series Solution ..................... 121 5.4 Numerical Results and Discussion .................... 126 6 Discussion 142 A Eigenfunctions and Eigenvalues for Stokes Flow in sectorial Regions 146 A.1 Eigenfunctions ............................... 146 A2 Eigenvalues ................................ 148 A3 Discussion of Eigenvalues ........................ 151 B Axisymmetric Flow in a Torus 155 El Biharmonic equation in Toroidal Coordinates ............. 155 B2 Axisymmetric Stokes flow in a torus ................... 164 LIST OF REFERENCES 171 v List of Tables 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 4.4 4.7 5.1 5.2 5.3 Twenty first quadrant eigenvalues of (3.5) for a23 . ......... Error of the Papkovitch-Fadle series for N210 , a23 and a 2 Error of the Papkovitch-Fadle series for N215 , a23 and a 2 Twenty first quadrant eigenvalues of (3.5) for a22 . ......... Error of the Papkovitch-Fadle series for N210 , a22 Error of the Papkovitch-Fadle series for N212 , a22 and a 2 3. Thirteen first quadrant eigenvalues of (4.15) - (4.16) for a 2 3 . . . Twenty four first quadrant eigenvalues of (4.27) for a 2 3 . ..... Convergence of the Papkovitch-Fadle series for N224 , a 2 3 and 61 = g .................................... Convergence of the Papkovitch-Fadle series for N224 , a 2 3 and 61 = i} .................................... Twenty six first quadrant eigenvalues of (4.27) for a 2 2 . . . . . . . Convergence of the Papkovitch—Fadle series for N226 , a 2 2 and 61 = E .................................... 4 Convergence of the Papkovitch-Fadle series for N226 , a 2 2 and 61 =1 1 Twenty six first quadrant eigenvalues of (5.12) for r1 2 1 , 1‘2 2 2 . Convergence of the Papkovitch-Fadle series for N226 , r1 2 1 , r2 2 2 and 61 2 g ................................. Convergence of the Papkovitch-Fadle series for N226 , r1 2 1 , r2 2 2 and 61 2 g. ................................ vi 53 54 55 61 62 63 96 97 98 99 105 106 List of Figures 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.1 >—-* 3.12 4.1 4.2 Fluid fills the sectorial region V 2 {r,0 : 1 S r S a , —a S (9 S a} and is set into motion by the uniform translation of a covering plate or belt . .................................. Contour lines of the stream function (3.4) of the flow in the sectorial regionV2{r,0:1SrS3,—§S6S§}. .............. Enlarged view of the corner at (730) 2 (1, -—§) for the region in figure (3.2) showing the first corner eddy .................... Enlarged view of the corner at (r, 0) 2 (3, —§) for the region in figure (3.2) showing the first corner eddy .................... Contour lines of the stream function (3.4) of the flow in the sectorial regionV2{r.t9:1SrS3, —%S6S§} ............... Enlarged view of the bottom of the sectorial cavity in figure (3.5) . . . Enlarged View of the last edge eddy at the bottom of the region in figure (3.6) ................................. Enlarged View of the corner at (7', 0) 2 (3, —%) for the region in figure (3.5) showing the first corner eddy .................... Enlarged View of the corner at (r,9) 2 (1, —%) for the region in figure (3.5) showing the first corner eddy .................... Vorticityin V2{r,0:1SrS3,———Z—S0S—Z—}. .......... Vorticityin V={T,6:ISTS3,2%SQS%}. .......... Vorticityin V2{r,6:1SrS2,—§S6S%} ........... Vorticityin V2{T,0:1S7‘S2, —§SOS%} ........... Coordinate system of the channel with a bend . ............ Velocity profile u for a parabolic flow contributed by the \IIO(Y) term given in (4.10) for subregion (1) given in (4.86) ............. 44 56 57 58 59 59 60 60 64 64 65 65 67 100 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.1 O 4.1 )_n 4.1 [\D 4.13 4.14 4.15 4.1 0') 4.17 Contribution of the infinite series term in (4.10) on the velocity profile u given in (4.88) , for the flow in subregion (1) given in (4.86) , at different values of X ........................... Contribution of the infinite series term in (4.10) on the velocity profile 11 given in (4.88) , along the center-line Y 2 1 for the flow in subregion (1) given in (4.86) . ............................ Azimuthal velocity profile 129 for a circular flow contributed by the \Ilo(1‘) term given in (4.25) for subregion (II) given in (4.87) . Contribution of the infinite series term in (4.25) on the azimuthal ve- locity profile 129 . given in (4.89) , for the flow in subregion (II) given in (4.87) , at different angles 0 ...................... Contribution of the infinite series term in (4.25) on the azimuthal ve- locity profile 129 , given in (4.89) , along the center-line r 2 2 of the flow in subregion (II) given in (4.87) ................... Pressure at the center—line of the channel (i.e. where r 2 2) contributed by the infinite series term in (4.95) of subregion (1) given in (4.86) . . Pressure at the center—line of the channel (i.e. where r 2 2) contributed by the nonconstant infinite series term in (4.97) of subregion (II) given in (4.87) . ................................. Contribution of the infinite series term in (4.10) on the velocity profile u given in (4.88) , along the center-line Y 2 1 for the flow in subregion (1) given in (4.98) . ............................ Contribution of the infinite series term in (4.25) on the azimuthal ve- locity profile 129 , given in (4.89) . along the center-line r 2 2 of the flow in subregion (II) given in (4.99) ................... Velocity profile u for a parabolic flow contributed by the 9100/) term given in (4.10) for subregion (I) given in (4.100) . ........... Contribution of the infinite series term in (4.10) on the velocity profile u given in (4.88) , for the flow in subregion (1) given in (4.100) , at different values of X ............................ Contribution of the infinite series term in (4.10) on the velocity profile 11 given in (4.88) , along the center-line Y 2 % for the flow in subregion (1) given in (4.100) . ........................... Azimuthal velocity profile 129 for a circular flow contributed by the @100“) term given in (4.25) for subregion (II) given in (4.101) ..... Contribution of the infinite series term in (4.25) on the azimuthal ve- locity profile my , given in (4.89) , for the flow in subregion (ll) given in (4.101) . at different angles 0 ..................... Contribution of the infinite series term in (4.25) on the azimuthal ve- locity profile 129 , given in (4.89) , along the center-line 7‘ 2 g of the flow in subregion (II) given in (4.101) . ................. viii 101 101 102 102 103 103 104 104 108 108 109 110 110 4.18 4.19 4.20 4.21 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Contribution of the infinite series term in (4.10) on the velocity profile 11 given in (4.88) , along the center-line Y 2 $- for the flow in subregion (1) given in (4.102) . ........................... Contribution of the infinite series term in (4.25) on the azimuthal ve— locity profile '09 , given in (4.89) , along the center-line r 2 g of the flow in subregion (II) given in (4.103) . ................. Pressure at the center-line of the channel (i.e. where r 2 1.5) con— tributed by the infinite series term in (4.10) of subregion (I) given in (4.102) . .................................. Pressure at the center-line of the channel (i.e. where r 2 1.5) con- tributed by the infinite series term in (4.25) of subregion (II) given in (4.103) . ................................. One period of the priodically curved channel .............. Azimuthal velocity profile for a circular flow contributed by ‘Ilo(r) or {170(7") term given in (5.8) - (5.9) . ................... Contribution of the infinite series term in (5.8) on the azimuthal ve- locity profile of the flow at different values of the angle 6 in subregion (1) given in (5.53). ............................ Contribution of the infinite series term in (5.8) on the azimuthal ve- locity profile of the flow at different angles 0 in subregion (I) given in (5.53) close to the interface (i.e. when 0 2 g) . ............ Contribution of the infinite series term in (5.9) on the azimuthal ve- locity profile of the flow at different angles 0' in subregion (II) given in (5.54) ................................... Pressure contributed by the circular flows \Ilo(r) 85 {170(7") terms given in (5.8) & (5.9) respectively ....................... Pressure at the center-line of the channel (i.e. where r 2 1.5) con— tributed by the infinite series term in (5.8) of subregion (I) given in (5.53) .................................... Pressure at the center-line of the channel (i.e. where r, 2 1.5) con- tributed by the infinite series term in (5.9) of subregion (II) given in (5.54) .................................... Contribution of the infinite series term in (5.8) on the azimuthal ve- locity profile of the flow at different angles 0 in subregion (I) given in (5.63) .................................... Contribution of the infinite series term in (5.9) on the azimuthal ve- locity profile of the flow at different angles 0' in subregion (II) given in (5.64) . ................................. ix 111 112 136 137 137 138 138 139 139 140 5.11 Contribution of the infinite series term in (5.8) on the azimuthal ve- locity profile of the flow at different angles (9 in subregion (1) given in (5.65) .................................... 140 5.12 Contribution of the infinite series term in (5.9) on the azimuthal ve- locity profile of the flow at different angles 6' in subregion (II) given in (5.66) . ................................. 141 B.1 Toroidal coordinates in a meridian plane ................ 156 Chapter 1 Introduction 1 . 1 Literature review The basic equations of fluid mechanics are the Navier-Stokes equations . These equa- tions are derived from the conservation laws of mass , momentum and energy . In con- trast with the basic equations in many other branches of mathematical physics , those governing fluid motion are essentially nonlinear (more precisely , quasilinear) . Be- cause of their nonlinearity exact solutions are rare in any branch of fluid mechanics . Lighthill (1948) [31] has given a more or less exhaustive list of exact solutions for invis- cid compressible flow . Berker (1963) [3] has given a comprehensive review of such so- lutions for incompressible viscous flow . Most recently Wang (1989) [47] classified and gave a review of the existing exact solutions of the unsteady Navier-Stokes equations . Seeking analytic solutions of equations have a few disadvantages : 1) Some problems are unsolvable ; 2) Exact solutions may also be complicated or too tedious to obtain ; 3) The region or domain for solvable problems is limited to applicability ; and 4) Some equations may be explicitly solvable but their analytic representation may be unsuit- able for mathematical or physical interpretation or numerical evaluation [49] . The treatment in this thesis is based entirely on finding exact solutions in certain flow geometries for the linearized form of the equations of motion which results from omit- [\D ting the inertial terms from the Navier-Stokes equations , giving the so—called creeping motion or Stokes equations . This is done by assuming the Reynolds numbers are small (i.e. setting the Reynolds number to zero in the governing Navier-Stokes equa- tions) . This Reynolds number is a dimensionless parameter that arises in these equations which determines the relative importance of inertial and viscous effects : fluid density X speed x size R 2 Reynolds number 2 . . vzscoszty For small R (i.e. , slow velocity , large viscosity or small bodies) , the solution of the Stokes equations provide a good approximation to the solution of the Navier-Stokes equations . Creeping motion arises in many physical situations , such as flow in ducts and flow of a highly viscous fluid [19] . Other cases for which Stokes flow is valid include the slow sheet flow in curved , narrow channels , [48] describes the seepage flow through cracks and fissures of dams and pulmonary alveolar blood flow [17] . In some lubri- cation problems in the hydrodynamic theory of lubrication the Reynolds number is so small that viscosity dominates completely [30] . Creeping flow also arises in flow through porous media , see [2] and [13] . Civil engineers have long applied porous media theory to groundwater movement , see Bouwer [6] . In this thesis we developed a separation of variables theory for solving problems of Stokes flow in sectorial cavities and some other flow geometries . The theory leads to sets of eigenfunctions . self-adjoint eigenfunctions , biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion . This results in an infinite system of linear equations which is solved by truncation . This algorithm is illustrated by solving the problem of slow steady flow induced in a sectorial cavity by shearing the incompressible fluid on the top with a moving plate or belt . This problem has not been solved before either analytically or numerically . However , a similar problem which is the steady flow induced in a rectangular cavity by the uniform translation of a covering plate or belt was solved using the same algo- rithm we used for a sectorial cavity by Joseph and Sturges [26] . The biorthogonality condition they derived and then used to find the coefficients of the eigenfunction ex- pansion is restricted to rectangular cavities and thus it could not be applied to solve the flow in a sectorial cavity . The related special case of the rectangular cavity prob- lem was solved by numerical methods by Burggraf (1966) [7] and by Pan and Acrivos (1967) [37] . The analytic solution was found to be as accurate or more accurate than the finite difference solutions . The deficiences in the biorthogonality conditions derived by Joseph and Sturges [26] and others is that they are derived for that particular fourth-order boundary value problem which arises from separating variables of Stokes equation for that special re- gion they are dealing with . Besides most of the fourth-order boundary value problems they dealt with have constant coefficients . The biorthogonality conditions we derived are not restricted to one particular Stokes flow problem . In addition , the class of fourth-order boundary value problems we considered have variable coefficients . The techniques and algorithm we developed here were first introduced by R.C.T. Smith [44] in his solution of the biharmonic problem governing the bending of a semi- infinite strip clamped at its side and loaded at its top edge ; the solution is expressed as a biorthogonal series and the given data is expanded into a biorthogonal series . Smith also established conditions on the data sufficient to guarantee the convergence of the biorthogonal series . Smith’s conditions were too restictive in applications . Joseph [23] and Joseph and Sturges [26] established a much less restictive conditions suffice to guarantee convergence for all types of edge data which might be expected in applications . These conditions arise naturally in elasticity and in Stoke’s flow in cav- ities . Gregory (1980) [18] , Spence (1981) [45] and Joseph, Sturges & Warner (1982) [28] added new theorems to the theory of convergence and completeness of biorthog— onal series of biharmonic eigenfunctions . In the latter paper Joseph et al. proved convergence of biorthogonal series of biharmonic eigenfunctions using the method of Titchmarsh . Smith’s ideas were used by Joseph and Fosdick [25] in their study of a narrow gap approximation for secondary motions generated in the problem of the free surface on a liquid between cylinders rotating at different speeds . Joseph and Sturges [27] stud- ied the problem of the free surface on a liquid filling a rectangular trench heated from its side . In their paper they included a numerical analysis and they showed how the eigenfunction expansions should be used to compute solutions when the rectangular strip is not semi-infinite but has a solid bottom . The biorthogonal series expansion method was also used by Joseph [22] in his study of the free surface on the round edge of a flowing liquid filling a torsion flow viscometer . However , this is the first case where this type of eigenfunction expansion arises for a Stokes flow problem which is not biharmonic . Similar biorthogonal eigenfunction expansions and biorthogonality conditions are required for the axisymmetric Stokes flow problems in a wedge shaped trench studied by Liu and Joseph [33] , the axisym- metric Stokes flow in a cone studied by Liu and Joseph [34] and for the problem of Stokes flow in a trench between concentric cylinders studied by Y00 and Joseph [50] . The previous references are just a small sample of problems arising in Stokes flow and elasticity which can be solved in biorthogonal series of eigenfunctions generated by separating variables . A list of several other problems is given in Joseph, Sturges & Warner [28] . The use of the matching technique is motivated by the work of Trogdon and Joseph (1982) [46] in their study of plane flow of a liquid over a rectangular slot . They split the region into four subregions , this enables the stream function to be represented by means of an expansion of Papkovich-Fadle eigenfunctions which arise in a natural way using separation of variables in each of these four subregions . Imposing conti— nuity of velocities and stresses across the common boundaries along with satisfying the boundary conditions gave conditions which were inverted using biorthogonality conditions which are properties of the Papkovich-Fadle eigenfunctions . This led to an infinite system of linear equations which was then solved by successive truncations . Their biorthogonality conditions are restricted to rectangular regions and that is why they did not apply their matching technique to other flow regions . Phillips [38] used the same method for solving Stokes flow around a two dimensional constriction . The flow region is decomposed into a number of rectangular subre- gions . Within each subregion the stream function is represented by an expansion of eigenfunctions of the biharmonic operator . The coefficients in these expansions are determined by imposing 03 continuity of the stream function across subregion interfaces and then taking advantage of the biorthogonality conditions . A post- processing technique is also developed to improve the accuracy of the approximation around corners by determining the coefficients in the known locally convergent ex- pansions of the stream function there . This is necessary because of the singularities present at sharp corners . Very few terms are required in the singular expansion to obtain an improved approximation . Phillips [38] , however , obtained a stream func- tion which is C1 continuous across subregion interfaces while the second and third partial derivatives do not match across the interface . The deficiency in the solution he obtained is that these higher derivatives of the stream function are important in computing the pressure . Phillips & Davies (1988) [39] use the method of matched eigenfunction expansions to solve Poisson’s equation in a contaction region . They also describe how the approx- imation may be post-processed in the neighborhood of a reentrant corner singularity in order to obtain an improved and more rapidly converging representation since usu- ally near the singularity the expansions converge very slowly . The idea of matching eigenfunction expansions by decomposing the region has been done by different authors in various ways other than using the biorthogonality condi- tions . For example using inversion or collocation techniques . Dagan, Weinbaum & Pfeffer [10] used matching eigenfunction expansions to solve axisymmetric Stokes flow through a pore in a wall . Flow field is split into two regions : an infinite half-space outside the pore and a. cylindrical volume bounded by the walls of the pore . The solution they obtained led to Fourier . Fourier~Bessel and Dini series which were all invertible over the common matching interface . A system of linear equations was obtained which was solved by successive truncations . The same authors , in separate papers , used a boundary collocation technique to derive the infinite set of linear equations in the case where no inversion or orthogo— nality conditions were applicable for the common interface . In the first paper [11] they presented an infinite-series solution to the creeping-flow equations for the ax- isymmetric motion of a sphere of arbitrary size towards an orifice whose diameter is either larger or smaller than the sphere . In the second paper [12] they studied the general axisymmetric creeping motion of a spherical particle in a stagnation region of a finite planar surface . Collocation methods have been studied also by Phillips & Karageorghis [41] . They described a spectral element method for solving Stokes flow in rectangularly decom- posible domains . Flow region is divided into a number of rectangular subregions . The solution to the governing biharmonic equation for the stream function is represented by an expansion of Chebyshev polynomials in each subregion . The coefficients in these expansions are determined by collocating the differential equation and boundary conditions , and imposing C3 continuity across subregion interfaces . The same two authors , in a separate article [42] , described a spectral collocation method for solving the stream function formulation of the Stokes problem in a channel contraction . The flow region is decomposed into a number of conformal rectangular subregions . A collocation strategy is devised which ensures that the stream function and its normal derivative are continuous across subregion interfaces . In [40] they described a spec- tral collocation technique for Stokes flow in contraction geometries and unbounded domains . The previous three papers have references of a sample of other different spectral domain decomposition techniques that appeared in the literature . Most recently Poteete [43] used a matching technique to analytically solve seven Stokes flow problems whose domains can be divided into contiguous simple regions . In her work she used real eigenvalues and eigenfunctions whereas the Papkovich-Fadle eigenfunctions are complex-valued . This simplifies the computational work . Due to her choice of the eigenfunctions the boundary-collocation was not needed which usually requires long computations for its numerical evaluations . The purpose of this thesis is twofold . Firstly , to develop an eigenfunction expansion method for solving problems of Stokes flow , mainly in sectorial cavities and some other flow geometries bounded by curvilinear coordinate surfaces which have never been done before . The theory leads to the development of a set of eigenfunctions , self-adjoint eigenfunctions , biorthogonality conditions and an algorithm for the com— putation 0f the coefficients of the eigenfunction expansion . The resulting infinite system of linear equations can then be solved by truncation . Secondly , we consider more complex geometries composed of contiguous simple re— gions using a matched eigenfunction expansion method . The general solutions of the separate regions are expressed in terms of eigenfunction expansions and then matched across the common interface using the biorthogonality conditions we developed . Ex— amples for sectorial and other flow regions will be illustrated . 1.2 Background In this section we show how to develop Stokes flow equations using the basic equations of Fluid Mechanics , namely , Naviers—Stokes equations that are derived from the conservation laws of mass , momentum and energy . In the case of viscous incompressible homogeneous flow ,0 2 constant where p is the density , the complete set of Navier—Stokes equations reduce to an —+(u~V)U2f—le+1/Au (1.1) at p and where 1/ is the coefficient of kinematic viscosity y is the dynamic viscosity f is external body force vector These equations must be supplemented by some boundary conditions . For example u - n 2 0 , i.e. , fluid does not cross the boundary but may move tangentially to the boundary . For the Navier-Stoke’s equations , the extra term z/Vzu raises the number of derivatives of u involved from one to two . For both experimental and mathematical reasons , this is accompanied by an increase in the number of boundary conditions. For instance . on a solid wall at rest we add the condition that the tangential velocity also be zero ( the “no-slip condition” ) . so the full boundary conditions are simply u 2 0 on solid walls at rest. The mathematical need for extra boundary conditions hinges on their role in proving that the equations are well posed ; i.e. that a unique solution exists and depends continuously on the initial data. The physical need for the extra boundary conditions comes from simple experiments involving flow past a solid wall. For example, if dye is injected into flow down a pipe and is carefully watched near the boundary one sees that the velocity approaches zero at the boundary to a high degree of accuracy. Next we discuss some scaling properties of the Naviers-Stoke’s equations with the aim of introducing a parameter (the Reynolds number) that measures the effect of viscosity on the flow. For a given problem , let L be a characteristic length and U a characteristic velocity . These numbers are chosen in a somewhat arbitrary way . For example , if we consider flow past a sphere , L could be either the radius or the diameter of the sphere and U could be the magnitude of the fluid velocity at infinity . L and U are merely reasonable length and velocity scales typical of the flow at hand . Their choice also determines a time scale T 2 % Changing variables and introducing the following dimensionless quantities : u, _ u x] __ x t, _ t _ U — L a _ T where x 2 (:r./y,z) is some general coordinates . In terms of the new variables the equations of motion reduce to 81.1, + (u: v/)111 _ V VI I + I/ Alul (13) at _ LU p LU ' where p' = —-p pU2 v’-u’ = 0 (1.4) The previous equations are the Navier-Stokes equations in dimensionless variables . The Reynolds number R is defined to be the dimensionless number 3:2 I/ We call % A'u' the difiusion or dissipation term and (u, - VI) 11' the inertia or convection term . The equations say that u is convected subject to pressure forces and , at the same time is dissipated . We shall only be interested in cases where R is very small . For steady motion and omission of inertia terms (i.e. setting R 2 0 ) reduce equations (1.3)-(1.4) to —V’p’ + V"u’ = 0 (1.5) v’.u’ : 0 (1.6) In this thesis we distinguish the following two cases , (i) Two-dimensional flows : In bounded systems , where a solution is obtainable , the flow pattern is the same in all planes , say x-y planes . Thus we may write the velocity vector 11 : (“($vylav($3y)70) 11 We introduce the concept of stream function \I’(a:, y) . The component velocities are related to the stream function by the expressions .1...) z _ (may) = —%:-I The continuity equation (1.6) g+gzo is automatically satisfied by any choice of \II . Eliminating the pressure from the resulting Navier-Stokes equations (1.5) , see [19] , we get the following biharmonic equation : v41; 2 0 (1-7) The use of the stream function greatly simplifies the solution of all two-dimensional problems ; the solutions of the equations of motion is reduced to the search of a single scalar function . In cylindrical coordinates , the velocity components and the stream function \II are related by _ — — _1 — v9 _ , vr _ where v, and v9 are the radial and peripheral velocity components respectively . The continuity equation in cylindrical coordinates , namely , 18 16129 WW + 755 = 0 12 is satisfied by any choice of \II (ii) Axisymmetric flows : In the general case of three-dimensional motions it is not possible to unify the previous method of approach to the solution of the Navier-Stokes equations to the search of a single scalar function . There exist , however , a number of classes of three-dimensional flows which can still be uniquely characterized by means of a single scalar function . Each of these involves a certain symmetry in the flow pattern . The most important is flow past a body of revolution , parallel to its symmetry axis . Such motions are termed axisymmetric . They are characterized by the existence of a stream function . In this case the resulting equation , see [19] and [30] , is given by EW = 0 (1.8) where , for instance , in cylindrical coordinates , 32 18 a? 2 E4 = (a: — m: + 52—2) (1'9) and , in spherical coordinates , 82 1 82 cot9 (9 2 4 _ __ _ —— E _ (0r2 + r2 692 r2 30) (1.10) We note here that the E“ operator is slighly different than the biharmonic operator V4 , for example , by comparing them in cylindrical coordinates : 82 13 82 2 V4 2 (5:2- + F5; + 27);) (1’11) 13 From (1.9) and (1.11) we see that the only difference are the signs of the middle terms . Because of this similarity of the two operators the well-known theory of solutions of Laplace’s equation , V2 \IJ 2 0 , in various system of coordinates is of important value in guiding one to the corresponding solutions of the equation E2 ‘II 2 0 and , ultimately , to the more complicated related equation E4 \I' 2 0 [l9] . For instance , in Appendix B , we use seperation of variables to solve the biharmonic equation in toroidal coordinates and this guided us to solve the equation E4 \I/ 2 0 using the same coordinates . The latter equation represents axisymmetric Stokes flow in a torus . Complex Transfoms , integral representations , and separation of variables can be employed to find analytic solutions to Stokes equations given in (1.7) - (1.8) . In our thesis we use separation of variables and we take advantage of the biorthogonality conditions to compute the coefficients of the eigenfunction expansion solution for the stream function . This is restricted to very simple geometries . For more complex geometries whose domains can be decomposed into simple contiguous regions , each sub-region is solved by separation of variables then we use a matching technique across the common interface to compute the coefficients . Chapter 2 Biorthogonality Conditions 2.1 A Class of Fourth-Order Boundary Value Problems In this chapter we derive a biorthogonality property satisfied by the eigenfunctions and adjoint eigenfunctions of a fourth-order differential equation written in self-adjoint form given by I (Pom y m)" + (P1(r;a)y'(r)) + awe) ya) = 0 (2.1) T E [T17 T2] with certain restrictions imposed on the coefficients as described in theorem .2 . The boundary conditions are given by y(T1) == 9(T2) == y (T1) == y (T2) == 0 (2-2) We need to write equation (2.1) as a 2 X 2 system . First consider the following fourth-order equation in self—adjoint form II I (7)) + (at) y'm) + a3(r>y = 0 (2.3) H (01(r) y We have the following theorem 15 Theorem .1 Let a1(r), a;(r), a3(r) be continuous and a1(r) 76 0 on r1 S r S r2 then equation 2.3 can be written in the form ll 2 + P(T)Z = <10")w (2.4) w,l + p(r) w 2 s(r) 2 where p,q, and s are continuous on r1 S r S r2 with q(r) 2 0 and p/q of class 0(2) . Conversely, ifp,q, and s are continuous on r1 S r S r2 with q(r) 2 0 and p/q of class 0(2) then system 2.4 is equivalent to equation 2.3 when a1(r) 2 0 and a1(r),ag(r), and a3(r) are continuous on r1 S r S r2 . Proof: Let (11:1. 02 =2?!) and as = lP/ql" + [192/91— 3 Substitution into equation (2.3) gives ly"/ql" + Qipyi/ql' + yIP/ql" + ylpz/ql — 8y = 0 or can be rewritten as [yu/q + py/ql" + pix/"M + py/ql = 61/ Setting we get system (2.4) . To prove the converse we divide the first equation of system (2.4) by q and compute 16 w” from the resulting equation . Upon substituting this value into the second equation of system (2.4) and simplifying one obtains equation (2.3) . The following theorem .2 gives the biorthogonality property for the boundary value problem given in equations (2.1) and (2.2) with some conditions imposed on the coefficients . Theorem .2 (Biorthogonality Condition) Consider the boundary value problem given in 2.1 and 2.2 where P0(r) , P1"(r; a) , P2(r; 01) are continuous and Po(r) 2 0 on r1 S r S r2 . P. in equation 2.] is a polynomial of degree at most i in the parameter a ,in particular, let P1(r; a) 2 pn(r) a + p12(r) , and we require P12(r; a) — 4P0(r)P2(r; a) 2 p31(r) a + p32(r) where Piif”) ‘l' Piaf?) 7% 0 Then with P; defined by .2 «4%) P" = / MW) . Wm] B dr " WW) we have the following Biorthogonality condition : .2 We) / [aimlm , #2in B(r) air = P; 6,... (2.5) " 3%) where 67,.” is the Kronecker’s delta, _lP11(T] 0 2 P0(T) 3(7‘) = ipidfl + $333]; ‘lplollrl 17 with ¢1”’(r) = yn(r) ¢§”’(r) = Pom yxm + %P1(r;an>yn (2.6) ’U} U) where _lP12 7‘ l 2P0(r) Po(r) Ah") = 1 " 1 (r 1 1') 170 (7) BO“) = -;-p13 + in; -%'}%;(I) and z = y 18 Clearly the condition pf1(r) + p§1(r) 54$ 0 ensures that the matrix B(r) is not identically zero . We note that system (2.6) is linear in the parameter a which resulted from the fact that s(r) given in equation (2.4) has the form 1 u 1 5(T) : 5P1 +m{P12—4P0P2} which is linear in the parameter (1 since in theorem .2 we require that the term (Pf — 4P0 P2) be linear in a ; so the term P2 which is quadratic in a cancels out . Thus , the latter condition is crucial for obtaining a linear system . We can rewrite system (2.6) as (2.8) with the following boundary conditions which follow from equations (2.2) & (2.7) 2m) 2 2(r2) = z’m) = z’m) = 0 (2.9) 19 with boundary conditions s2(r1) = 52(r2) = s;(r1) = s;(r2) = O (2.11) Upon setting s2 = y the 2 X 2 system given in (2.10) with boundary condi- tions (2.11) reduces to the single fourth-order equation given in (2.1) with boundary conditions (2.2) . Expanding equation (2.10) we find I 51: POUR/I + %P1(r;a)y 322:!) System (2.10)-(2.11) is actually a self-adjoint system to (2.8)-(2.9) and that follows from the fact that the two diagonal elements of the matrix (A(r) + aB(r)) are identical . The eigenvalues for the boundary value problem (2.8) - (2.9) are identical with those for the self-adjoint boundary value problem given in (2.10) — (2.11) . Let xn be an eigenfunction of the system given in (2.8) - (2.9) corresponding to the eigenvalue an , and 5m be an eigenfunction of the self-adjoint system given in (2.10)-(2.11) corresponding to the eigenvalue am , so we have x1n = (A(r) + anB(r))xn (2.13) 20 and (2.14) Multiplying equation (2.13) from the left by 5m and equation (2.14) from the right by xn then subtracting both equations we get 01' but since -3- - (3» , i ( -) wn and from the boundary conditions (2.9) 81, (2.11) we have (2.15) (2.17) 21 then using equations (2.7) & (2.12) and the assumption that the eigenvalues are simple we get the biorthogonality condition given by equation (2.5) . 2.2 Stokes Flow in Sectorial and Rectangular Regions We consider here finding the biorthogonality condition for two 2-dimensional Stoke’s flow problems by direct application of theorem .2 . The first example is that of the flow in the two-dimensional sectorial region V = {r,9 : 0(-) since where [2x2 is the Identity matrix . The eigenfunctions satisfy (2.49) 43%) = (32 — 1)2T;.'< ) + [302—5) — 3(32_1)] T39) and the adjoint eigenfunctions satisfy Ms) = (32-1)2T,;;(3) + [W—o) — 3(32_1)] 113(3) (2.50) where 2.4 Discussion of other Flow Geometries We consider two other flow geometries where the hypothesis of theorem .2 is satisfied thus the biorthogonality condition can be easily derived by direct application of the theorem ; we then explain why the hypothesis of theorem .2 is satisfied by several Stoke’s flow geometries of simple configuration . The first example is that of the flow in wedge-shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge , see Liu and Joseph (1977) [33] . They studied the motion and the shape of the free surface in a wedge-shaped cavity heated from its side . The region is given by They reformulated the problem , and after changing variables they had the following edge problem in V 2 62 16 182 2 V4‘I/(r,6) = ( They require the velocity to vanish on the rigid boundary radial lines 6 = i 61 so @(T,i91) = “'88—‘30",i91) = 0 (2.52) Other edge data are prescribed at r = r1 , r2 as shown by Liu and Joseph (1977) [33] . We note here that the radial lines of the sectorial cavity are rigid while we have 36 a free surface along r =: r1 . The problem we are considering in chapter 3 which is slow steady flow induced in a sectorial cavity by shearing the incompressible fluid on the top with a moving plate or belt is different . In the latter 2 however 2 the concentric circles of the sectorial cavity are rigid while the radial line 6 = 61 is set into motion . In the first problem we seek a solution in 6 direction while in the second we require a solution in r direction . Separable solutions of ( 2.51 ) & (2.52 ) in the form ‘I/(r2 6) ~ {r’l 2 rz'A} y(6) exist 2 see [33] 2 when y(6) satifies the following equation y(4) + [)2 + (A _ 2)2] 31(2) + 120‘ _ 2)? y = 0 (2.53) and the boundary conditions y(igll = l/(igil = 0 (2-54) Comparing with theorem .2 we require an # am and we have P0(6) =1 ; P1(6;a) = 2(a+1) ; P2(6;a) = (a— 1)2 where a = (/\—l)2 37 so the Pi’s are polynomials of degree i in a and 1312(6; oz) — 4Po(6)P2(6; a) = 160 Hypothesis of theorem .2 is thus satisfied when (An — 1)2 96 (Am — 1)2 . The biorthog- onality condition can thus be easily derived . The second example is that of Stoke’s flow in annular trenches bounded by horizontal parallel planes and concentric vertical cylinders 2 see Jung and Joseph (1978) [29] . The region is given by V = {r22 : 0 O , the first quadrant roots and are arranged in a sequence corresponding to increasing size of their real part and define A_n:Xn (n=1,2,3....) where the overbar designates complex conjugate. Then «st-"m = WW) (n = 1 . 2. 3. ...) is the eigenfunction corresponding to the eigenvalue A-.. . Since the edge data given in (3.3) are real then 3.2 Biorthogonal Series Solution The coefficients En and Fn must be selected to satisfy (3.3) ; that is, co '(n)(7.) _1 = g [En/\ncosAna —F.A,. sin Ana] 3‘? (3.6) 0 = :0 [En sin Ana + Fn cos Ana] ¢[n)(r) (3.7) 0 = i [EnAn cos Ana +F,.A.. sin Ana] WM.) (3.8) 0 = i: [Fn cos Ana — En sin Ana] o[nl(r) (3.9) —00 46 We can find the coefficients En and Fn by application of the biorthogonality condition for a sectorial cavity (See Section 2.2) to the edge data given in equations (3.6 - 3.9) . With P7: defined by (n) a () < ) 1 (r) P; = / Wm , WWW) dr ¢é”’(r) then the biorthogonality condition is given by . ¢l”’(r) / [aimlm , tgmlm B(r) dr = P; 6.... (3.10) 1 [(nl . $2 (7) where «5%) = Wynn.) — g: 3%) (3.11) Wm = We) and WW) — ¢ here ,un = 1 + 2Ai and 2—1. 0 N ‘10) 0-) t0| .3... to 47 To prepare for the application of the biorthogonality condition we substitute n 27‘ I” n 27' n Wm=7wNm—7§m given by equation (3.11) into equations (3.6) and (3.8) so we get , °° 2 n 2 {lman—WWH ab #11 71 Z [ERAn cos Ana — FnAn sin Ana] (3.13) _1: 0 = ‘2 [En sin Ana +Fn cos Anal 9'51”)(7‘) K) _"'_ ¢(")(r)} (3.14) 2 n 00 9‘ II 0 = 2 IE”)... cos Ana +Fn/\n sin Anal {i 410% )— _00 [an (3.15) 2 [—En sin Ana + Fn cos Ana] ¢[n)(r) —OO O = Coupling equations (3.12) and (3.13) then rearranging terms we get 961 (7“) oo 0 —— Z [En sin Ana + Fn cos Ana] (2”)(7‘) —00 2 [En sin Ana + Fn cos Ana] WW) —OO 00 9 +2 33—)... [En cosAna — Fn sin )‘nal ( ”151 -00 #71 Similarly coupling equations (3.14) and (3.15) then rearranging terms we get 48 °° ¢in)( ) oo 0 Z[_En sin Ana+Fn cos Ana] — :[—En sin Ana+Fn cos Ana] _oo ¢(2n)(r) -00 £100“) 00 2 0 0 +2 —An [En cos Ana + Fn sin Ana] ( ) = ( ) (3,17) 7‘45 (7‘ ) -°° " 1"”’§"’ 0 Now applying the operator /“ 118"” ()r Lit-3W (018(1) :11 to both sides of (3.16) and (3.17) then making use of the biorthogonality condition and letting . 0 g... = / [¢{m’(r) , 8.ng B(r) ] dr (3.18) 1 —1 a 0 v... = /1 MW). wlm)(r)18(r) ] dr (3.19) 86(7) 0 Wm: / [wlmlfl /§""(r)] B(r) ., ] dr (3 20) r¢.‘"’( ) W ) we get '2 gm =(Pm sinA Ma)E + (Pm cosAma) Fm +:( (—An 14/”... cosA a -an sin Ancr)En _oo “77. OO 2 _ Z( (—An VansinAn a +V mncos Ana) Pu (3,21) _00 it". 49 °° 9 0 = (—P,’,“1 sin Ama)Em + (P; cos Ama)Fm +2: (LAnVan cos Ana +an sin Ana) En ._00 TI. 0° 2 + Z (—Anl/an sin Ana — an cos Ana) Fn (3,22) -00 Ian To rewrite the infinite system of equations (3.21) and (3.22) in matrix form let D = diagm(P;) ; L = diagm(2Am/,am) H = diagm(sin Ama) and K = diagm(cos Ama) then (3.21) in matrix form becomes (DH + LKW — HV)E + (BK — LHl/V — KV)F = G (3.23) and similarly (3.22) becomes (LKl/V + HV — DH)E + (BK + LHl/V —— KV)F = O (3.24) We can eliminate E from (3.23) and (3.24) to obtain [(DK — LHl/V — KV) — (DH + LKl/V — HV)Q"IR]F = G (3.25) provided that the inverse of Q exists . Here Q = LKDV+HV—DH R 2 DK + LHW — KV Equation (3.25) represents an infinite system of equations for the components of the vector F. From (3.19) and (3.20) the elements of the infinite matrices V and W are given as : 50 ¢(m)( (n) v... #01212 M52 (1)31 (326) T1. 1 m n Wm = L —¢< )(r )¢1 (r )dr (3.27) 4 1 7‘3 From (3.10) and (3.18) the components of the vectors G and D are given as : Pi: [[5,sz r1)¢ >(r) +(2%¢1m’(r) +9-1—.1()) 308(1)]. (328) — Him )dr (3.29) 917‘? 3.3 Numerical Results and Discussion The first example we consider is flow in the region V={r,6:l§r§3,— S63 } 7r 4 Al>l The complex eigenvalues given in (3.5) were found numerically correct to eight deci- mal places using a subroutine from the IMSL Library based on Muller’s method with deflation (See Table 3.1) . A discussion of these eigenvalues is given in Appendix A . Equations (3.21) and (3.22) form an infinite system of equations to be solved for the coefficients En and Fn , n : :l:1, :l:2, . To solve this system of equations we must truncate the infinite sums appearing in (3.21) and (3.22) to finite ones . We do this by replacing the lower and upper limits of summations by —N and N , respectively. The truncated systems are then solved using a subroutine from the LINPACK Library 51 based on Gaussian elimination with partial pivoting to compute the LU factorization of the complex matrix and then solves it . The coefficients of the finite system which are given as integrals are evaluated exactly using MATHEMATICA . The convergence of the solution of the truncated equations was then checked numer- ically . Table (3.2) shows the edge data for truncation number N=10 at 20 mesh points and clearly the matching is not quite good . Convergence is reached when N is further increased to 15 where the solution matches the edge data quite well at the top and very well at the bottom of the cavity (see Table 3.3) . The solution remains unchanged if N is larger than 15 . The solution in the interior of the cavity consists of edge eddies ; contour lines of the stream function (3.4) are plotted in figure (3.2) . In figures (3.3) and (3.4) we have an enlarged view of the two corner regions at the bottom of the cavity so as to emphasize the existence of corner eddies of the type discussed by Moffatt (1964) [35] ; so for this case we have one edge eddy and two corner eddies at the bottom of the cavity . The second example we consider is flow in the region / V={r,9:1§r§3,—‘ 393.} h tvl‘l [VI-l so we increased the angle in example one from to g thus we have here a longer Ala cavity . We get three edge eddies and two corner eddies ; so the longer the cavity the more edge eddies we have . Contour lines of the stream function (3.4) are plotted in figure (3.5) . In figures (3.6) and (3.7) we have an enlarged view of the bottom region of the cavity . Figures (3.8) and (3.9) give an enlarged view of the two corner regions at the bottom of the cavity . The third example we consider is the flow in the region I V={r,9:l_<_r_§2,—— 39$ } $44 Ma We reduced the width of the cavity which is three in example one to a=2 . Comparing tables (3.2) and (3.3) with tables (3.5) and (3.6) , respectively , we notice that the wider the cavity the slower the convergence is . For N =10 for both cases we notice that the edge data matches much better for the cavity with a=2 than the one with a=3 . For a=2 we need only N=12 terms to get convergence compared with N=15 terms when the width of the cavity is a=3 . From the previous examples we notice the increase in the number of edge eddies when the length of the cavity increases and a decrease in the rate of convergence as the width of the cavity increases . The vorticity is defined as 1 a 1 8v, (2 = ;‘a‘;('rv€) — ; 89 (3.30) with the velocity components given by 8&1 —18\II ”9 = , 1.: 7% E where vr and v9 are the radial and peripheral velocity components respectively . We compute the velocity components first using the solution given in equation (3.4) . Upon substituting the values obtained in equation (3.30) an easy computation leads to (z = i (En sin AM + Fn cos And) {4cn(l — 10..) 7"” + 4(1+pn)rp"} Figures (3.10—3.13) Show the vorticity for the previous regions . A similar problem , which is the flow in a rectangular cavity , has been studied by Joseph and Sturges (1978) [26] using biorthogonality series expansion method . They considered a two-dimensional rectangular cavity which is filled with liquid . The cavity is covered with a flat plate which is drawn across the top of the cavity with a constant velocity thus setting the fluid in motion . Their analytical solution was as accurate or more accurate than the finite difference solutions . The numerical method 53 n Complex roots An 1 2.240320591 + 3.798537445 i 2 3.004675284 + 9.735417974 i 3 3.410391280 + 15.530837167 i 4 3.689757283 + 21.290717883 i 5 3.903214445 + 27.035647620 i 6 4076030031 + 32.77275082‘3 i 7 4.221238428 + 38.505202650 i 8 4.346457904 + 44.234647703 i 9 4456531218 + 49.962028632 i 10 4.554729554 + 55.687926904 i 11 4.643366815 + 61.412721813 i 12 4724140232 + 67.136671915 i 13 4798331747 + 72.859959867 i 14 4.866933348 + 78582718565 i 15 4.930728420 + 84.305047137 i 16 4.990346418 + 90.027021102 i 17 5046300727 + 95748699057 i 18 5.099015556 + 101.470127200 i 19 5148845460 + 107.191342470 i 20 5196089803 + 112.912374774 i Table 3.1: Twenty first quadrant eigenvalues of (3.5) for a=3 . solution was studied by Burggraf (1966) [7] for flow in a square cavity and by Pan and Acrivas (1967) [37] for flow in deep and shallow cavities . Our analytical solution for the flow in a sectorial cavity is as accurate as that obtained by Joseph and Sturges (1978) [26] for flow in a rectangular cavity . However , our problem has not yet been studied using numerical methods that we can compare our results with . As in the case for a rectangular cavity we obtained here a steady flow involving a series of seperated closed streamlines and we showed the existence of corner eddies or (resistive eddies) of the type discussed by Moffatt (1964) [35] . 54 r @(r,§;10) \Il(r,—-};10) —%%%(r,§;10) Q%(r,—§;10) 1.0 —3.140 X 10’17 1.012 X 10"19 8.012 X 10"17 1.214 X 10'18 1.1 —9.299 X 10‘4 4.922 X 10‘9 —1.055 —1.999 X 10'6 1.2 —1.444 x 10‘3 7.022 x 10'9 —O.950 —4.197 x 10’7 1.3 1.483 X 10‘3 8.025 X 10"9 —1.056 7.270 X 10‘8 1.4 3.163 X 10"3 8.582 X 10‘10 —0.795 —4.044 X 10"7 1.5 —3.482 X 10"3 -—6.984 X 10‘9 —1.146 4.133 X 10‘7 1.6 7.454 X 10‘4 3.220 X 10"9 —l.005 —6.851 X 10"8 1.7 1.931 X 10’3 2.638 X 10’9 —0.883 —2.457 X 10‘7 1.8 -3.575 X 10"3 —6.497 X 10"9 —1.170 4.090 x 10’7 1.9 4.383 X 10‘3 8.354 X 10‘9 —0.824 —4.529 X 10’7 2.0 —4.643 X 10'3 —9.312 X 10‘9 —1.146 4.133 X 10’7 2.1 4.379 X 10'3 1.031 X 10’8 ~—0.919 —2.939 X 10‘7 2.2 —3.391 X 10'3 —1.191 X 10'8 —0.992 9.688 X 10"8 2.3 1.525 X 10‘3 1.412 X 10’8 —1.080 1.024 X 10"7 2.4 9.178 X 10‘4 —1.591 X 10’8 —0.935 -—8.346 X 10"8 2.5 —3.009 X 10’3 1.463 X 10’8 -—0.950 —4.l97 X 10‘7 2.6 3.838 X 10‘3 —7.304 X 10'9 —1.094 1.198 x 10‘6 2.7 --3.782 X 10’3 —-4.874 X 10’9 —1.233 —-8.590 X 10"? 2.8 4.272 X 10‘3 1.240 X 10‘8 —0.354 —2.165 X 10‘6 2.9 —4.384 X 10'3 —5.361 X 10'9 -—0.056 2.217 X 10'6 3.0 8.388 X 10‘14 —5.885 X 10‘17 4.120 X 10'13 ~2.430 X 10"16 Table 3.2: Error of the Papkovitch-Fadle series for N210 , a=3 and a = 7‘ Z. 55 r \l/(r,%; l5) @(r,—§; 15) -—%%‘g—(r,§;15) %%(r,—§;15) 1.0 —3.154 X 10‘17 1.004 X 10‘19 —-3.516 X 10"16 1.206 x 10‘18 1.1 —-1.226 X 10’4 —5.067 X 10‘10 —1.038 1.523 X 10—7 1.2 3.988 X 10'4 9.593 X 10"10 —0.973 1.499 X 10’8 1.3 -—4.049 X 10"4 —9.305 X 10"10 —-l.032 1.225 X 10—7 1.4 —8.287 X 10'4 —1.683 X 10“9 —l.066 1.257 X 10"7 1.5 —9.286 X 10"4 —l.847 X 10’9 —l.081 8.554 X 10’8 1.6 —7.013 X 10’4 —1.427 X 10‘9 —1.067 3.557 X 10‘8 1.7 1.470 X 10'4 2.403 X 10“10 -—0.993 ——1.574 X 10”8 1.8 1.206 X 10‘3 2.292 X 10’9 —0.991 —7.139 X 10"8 1.9 6.859 X 10'4 1.125 X 10’9 -0.972 —7.239 X 10"8 2.0 —1.238 X 10“3 —2.463 X 10‘9 —1.081 8.554 X 10'8 2.1 ——3.015 X 10‘4 -—2.252 X 10‘10 —0.982 6.696 X 10‘8 2.2 1.392 X 10"3 2.571 X 10‘9 —-0.963 —1.774 X 10‘7 2.3 “1.052 X 10‘3 —2.249 X 10"9 —1.036 1.633 X 10'7 2.4 1.169 X 10'5 1.484 X 10‘10 —0.997 —8.090 X 10’8 2.5 8.309 X 10‘4 1.999 X 10'9 —0.973 1.499 X 10‘8 2.6 —1.085 X 10'3 -3.051 X 10"9 -—1.048 —3.060 X 10"8 2.7 7.789 X 10'4 2.561 X 10‘9 —0.924 1.856 X 10'7 2.8 —1.336 X 10‘4 —4.483 X 10"10 —l.161 -5.414 X 10'7 2.9 —4.356 X 10‘4 —2.505 X 10'9 —0.623 8.700 X 10"7 3.0 7.327 X 10‘14 —5.841 X 10‘17 2.882 X 10‘13 —2.413 X 10"16 Table 3.3: Error of the Papkovitch-Fadle series for N215 ,a23 and 02 7r 4 . 56 y value 0 -1- _2» x value Figure 3.2: Contour lines of the stream function (3.4) of the flow in the sectorial r6g10nV={r,0:1SrS3, —§S€S§}. 57 Figure 3.3: Enlarged View of the corner at (r,6) 2 (1, —§) for the region in figure (3.2) showing the first corner eddy . y value FigUI‘e 3.4: Enlarged View of the corner at (r,0) 2 (3, —§) for the region in figure (3-2) Showing the first corner eddy . 58 y value 0 -1. -2» -3, o 075 i 115 2 2:5 3 x value Figure 3.5: Contour lines of the stream function (3.4) of the flow in the sectorial region V2{7‘,6:1§_r£3, —%S€S%}. Figure 3.6: Enlarged view of the bottom of the sectorial cavity in figure (3.5) . Figure 3.7: Enlarged view of the last edge eddy at the bottom of the region in figure (3.6) . y value 7 value -1.5 t- 59 Figure 3.8: Enlarged View of the corner at (r,0) 2 (3, —%) for the region in figure (3.5) showing the first corner eddy . O O A O O O Figure 3.9: Enlarged view of the corner at (736) 2 (1, —-§) for the region in figure (3.5) showing the first corner eddy . 61 Complex roots An :5 3.370703254 + 6.054102744 i 4.593078513 + 15444885446 i 5.238022153 + 24.625199349 i 5681498446 + 33.751926784 i 6.020153718 + 42.855983209 i 6.294247926 + 51.948130806 i 6.524514778 + 61.033183727 i 6.723061006 + 70.113642063 i 6.897577913 + 79.190941676 i 10 7.053258673 + 88.265969687 i 11 7.193775648 + 97.339305529 i 12 7.321821895 + 106.411344626 i 13 7.439431179 + 115482366591 i 14 7.548177057 + 124.552575035 i 15 7.649302051 + 133.622121941 i 16 7.743804453 + 142.691123171 i 17 7.832498415 + 151.759668679 i 18 7.916056628 + 160.827829424 i 19 7.995041303 + 169.895662168 i 20 8.069927115 + 178.963212888 i RIQUIuPCADlth—i {DOD Table 3.4: Twenty first quadrant eigenvalues of (3.5) for a22 . r \Il(r,—};10) \I/(r,———:-;10) —%%(r,§;10) 9%(n—g; 10) 1.00 —4.037 X 10‘17 -—1.958 X 10"20 2.214 X 10‘15 9.232 X 10‘20 1.05 3.846 X 10‘4 -7.926 X 10‘11 ——0.768 2.405 X 10"7 1.10 4.832 X 10‘4 7.204 x 10“11 —1.015 —7.784 x 10‘8 1.15 —3.950 X 10‘5 2.571 X 10‘10 —0.986 -2.195 X 10“9 1.20 —7.429 X 10‘4 —1.768 X 10"10 -1.053 1.478 X 10‘8 1.25 8.342 X 10‘4 -—3.507 X 10'12 —0.959 —1.049 X 10‘8 1.30 —3.847 X 10‘4 8.122 X 10‘11 —0.996 1.041 X 10'8 1.35 -—l.734 X 10‘4 —7.444 X 10’11 —1.049 —1.376 X 10‘8 1.40 6.067 X 10"4 4.457 X 10"11 —-0.922 1.641 X 10'8 1.45 —8.749 X 10'4 —-2.354 X 10‘11 —1.090 —1.620 X 10'8 1.50 1.017 X 10’3 1.535 x 10’11 —0.913 1.217 X 10‘8 1.55 —1.075 X 10"3 1.070 X 10"11 —1.071 —3.481 X 10’9 1.60 1.068 X 10‘3 —4.489 X 10‘12 —0.959 —1.049 X 10‘8 1.65 -9.942 X 10"1 4.507 X 10’11 —1.002 2.826 X 10‘8 1.70 8.469 X 10"4 ——1.212 X 10"10 —-1.030 —-4.236 X 10"8 1.75 —6.459 X 10‘4 2.271 X 10‘10 ——0.978 3.471 X 10"8 1.80 4.648 X 10‘4 —3.269 X 10“10 —0.938 2.193 X 10‘8 1.85 —4.139 X 10‘4 3.529 X 10‘10 —1.201 —1.378 X 10"7 1.90 5.234 x 10‘4 —2.449 x 10“10 —0.838 2.122 x 10‘7 1.95 —5.576 X 10‘4 3.177 X 10"11 —0.277 2.607 X 10‘7 2.00 1.349 X 10'11 4.393 X 10‘16 1.361 X 10‘10 8.109 X 10‘16 Table 3.5: Error of the Papkovitch-Fadle series for N210 , a22 and a 2 7’ Z. 63 r \Il(r,§;12) \Il(r,—§;12) —%—aa—gi(r,%;12) %(r,—§; 12) 1.00 ——7.694 X 10‘17 ——1.954 X 10’20 ~8.171 X 10'16 9.074 X 10"20 1.05 9.450 X 10‘5 -1.275 X 10"10 —l.065 8.537 X 10‘8 1.10 5.866 X 10'5 -—1.614 X 10’10 ——O.907 4.181 X 10‘8 1.15 2.894 X 10'4 -1.448 X 10’10 —1.010 -8.861 x 10‘9 1.20 4.592 X 10’4 —3.346 X 10’11 —0.969 7.153 X 10’9 1.25 —1.477 X 10'4 9.982 X 10“11 —O.973 1.994 X 10‘9 1.30 —5.621 X 10‘4 7.960 X 10“11 —1.082 —1.904 X 10’8 1.35 4.119 X 10’4 -9.714 X 10“11 —0.959 1.187 X 10"8 1.40 2.581 X 10"1 -4.196 X 10'11 —0.961 8.698 X 10‘9 1.45 —6.741 X 10‘4 1.282 X 10’10 —1.086 —2.058 X 10—8 1.50 5.964 X 10"4 —-8.581 X 10’11 —-0.917 1.859 X 10‘8 1.55 -—2.206 X 10"4 —2.631 X 10‘11 —1.056 —9.742 X 10‘9 1.60 —1.891 X 10‘4 1.278 X 10‘10 —0.973 1.994 X 10’9 1.65 4.785 X 10"4 —1.755 X 10‘10 —1.010 3.057 X 10‘10 1.70 ——6.066 X 10‘4 1.639 X 10"10 -0.991 4.303 X 10'9 1.75 5.912 X 10’4 —1.060 X 10‘10 —1.019 ——1.630 X 10’8 1.80 ——4.650 X 10‘4 1.865 X 10'11 ——0.971 3.573 X 10"8 1.85 2.581 X 10’4 8.332 X 10‘11 —1.010 —5.846 X 10’8 1.90 ——7.426 X 10"6 —l.808 X 10‘10 —1.111 5.496 X 10'8 1.95 -—-1.888 X 10’4 2.043 X 10‘10 —0.508 1.538 X 10"7 2.00 1.368 X 10‘11 4.386 X 10‘16 1.397 X 10‘10 8.130 X 10‘16 Table 3.6: Error of the Papkovitch-Fadle series for N212 , a22 and a 2 Ala 64 Figure 3.10: Vorticity in V2{7‘,9:1_<_7‘S3,— AH |/\ Figure 3.11: Vorticity in |/\ Ala V={nwigrg3,—ggag§y 65 Figure 3.12: Vorticity in V = {736;1 S r S 2 , _%r S 9 S? -2» Figure 3.13: Vorticity in V 2 {736: 1 g r S 2 , — NH MI: Chapter 4 Stokes Flow around a Bend 4. 1 Formulation A two—dimensional infinite channel Q with a. bend is filled with an incompressible fluid (see figure 4.1) . The fluid is set into motion by a fixed flow rate at infinity . In the absence of inertial terms we have Stoke’s flow ; thus we are considering solving the biharmonic problem : V411! 2 0 in 9 (4.1) \I! 2 0 on F1 111 2 l on F2 (4.2) velocity 2 0 on F1 UF2 (4.3) In (4.3) the no-slip conditions are imposed on the rigid boundaries of the region Q . From the symmetry of the region we are only concerned here with subregions I 85 II in figure (4.1) . The derivatives of KI}; give the velocity components in the X and Y directions respecively as 631 all}; . ___ ___. .4 BY ’ 1’ 0X (4) U: 66 67 P y 2 /Y r; I”’ x x I . x , x . , \er ‘\ . e1 1 ‘x 3 ~ x \\ . ‘\ ‘\ . ‘\ ‘ X Figure 4.1: Coordinate system of the channel with a bend . The derivatives of ‘11” give the velocity components _ at” ' new” 719 — 0r 1 LT— r 80 (4.5) where v. and 219 are the radial and peripheral velocity components respectively . Nondimensionalize all lengths by the minimum radius of curvature such that the constant gap width is a — 1 > 0 . In particular, the solutions l1}; and \Il]] of equations (4.1) - (4.3) in subregions (I) and (II) respectively , satisfy the following equations : 82 / / 02 2 f / val/[(11.3) = (W + W) QI,(.X.1) = 0 (4.6) in V12{X,Y:—oo(Y)) (4.10) satisfies equations (4.6) and (4.7) with the parabolic How 4100/) given by a) :3 .I' = _ - ."3 )fz KPOO) (a—1)3) + (a-—1)'2 (4.11) 69 satisfies (4.6) and the boundary conditions (4.7) . Also we have ~in)(Y) = Sn sin Sn COS[Sn(bY - 1)] — Sn (bY — 1) cos Sn sin[Sn(bY — 1)] WW) = Pn cosPn sin[Pn(bY — 1)] — 19,,(11/ — 1) sin P. cos[Pn(bY — 1)] (4.12) where 2 b 2 (4.13) a — 1 The eigenfunctions 3]") 85 691) satisfy the following boundary conditions : "'71 ~71. c1311") d1") 1)(0) : ¢i )(Cl—l) : 3}; (0) = :Y (CL—1) = 0 (4.14) An ’Tn d3") d3") ¢1’(0) = @1’(a-1) = {314(0) = % 0 , the first quadrant roots and are arranged in a sequence corresponding to increasing size of their real part and define S.n 2 Col 3 A :3 I l h—l [0 00 V where the overbar designates complex conjugate. Then WW) = 6%”) (n = 1,2, 3, ...> «ES—”’(Y) = W0”) (n = 1,2, 3, ...) are the eigenfunctions corresponding to the eigenvalues /\_n . Since the boundary data given in (4.7) and (4.9) are real then The following property of biorthogonality , see Joseph [23] , is satisfied by the eigen- functions and adjoint eigenfunctions dY = kn 6m (4.17) 0 —1 WW) /0 [2/7£""(Y),2Zém)(m( ] 1 '2 5ng and (1)” 2 kn 6m (4.18) The functions 1%”) & 123%") are related by M1") _ 2 2 ~00 CIT/.2 — —b Sn 21)? and the functions fl") & 175)”) are related by my” __2 2"(n) 372'- ”We and also —1— = ms: at") J23") A 1 - 6" P3 96%.") The adjoint eigenfunctions are given by ~ ¢]m)(l’) 2 51m)(Y) — 2 cosSm cos[Sm(bY — 1)] 255,“)(Y) = W0”) and wém’u') = cp'l‘m’o’) $1(Y) = 51"")(Y) + 2 sin Pm sin[Pm(bY — 1)] The constants kn $7, in in the biorthogonality conditions (4.17) - (4.18) by (4.19) (4.21) (4.23) are given 72 kn 2 —2(a — 1) cos4 Sn A (4.24) kn 2 —2(a — 1) sin4 Pn As for region (II) the series \I’n(r,6) 2 610(7) + Z Fn cos()\n9)q§1n)(r) (4.25) satisfies (4.8) and (4.9). We note that the solution \I’”(r,0) given in (4.25) actually is expressed as a linear combination of sin(/\n0) and cos()\n0) terms but due to the symmetry with respect to 0 in subregion (II) (see figure 4.1) , so the sin(/\,19) term is dropped out . In (4.25) \TIOU‘) is a steady circular flow ( i.e. flow between concentric circles) satisfying (4.8) and the boundary conditions given in (4.9) ; A 1 \I! r 2 . 2a210 a+a2—1r2—1 a) [(a2_1)2_4a210g2a]{( g )( > —4 a2 loga logr — 2 (a2 — 1) (1"2 log 1)} (4.26) Also, see Appendix A , , _ _.\ 2 A ¢]")(r) 2 an 7"" + bur 2" + an 7‘2 " + (Inf + " The coefficients an ,bn .cn ,(Zn and the corresponding eigenvalues An are glven In appendix A . In particular the eigenvalues /\n satisfy 73 sin 1,. = ifl X. (4.27) An = (ilogaHn ; fl = figs—5m — %) These eigenvalues are symmetrically located in the complex A plane ; we choose An , n > O , the first quadrant roots and are arranged in a sequence corresponding to increasing size of their real part and define A_n2Xn (n=1,2,3,...) where the overbar designates complex conjugate. Then 4"”(2) = 52%) (n = 1 , 2., 3. ...) is the eigenfunction corresponding to the eigenvalue A_n . Since the edge data given in (4.7) and (4.9) are real then F.n 2 Tn 4.2 Matching Eigenfunction Expansions We must now match the two solutions given in (4.10) and (4.25) . We shall require that the stream function and its first three partial derivatives with respect to X are continuous across the interface (i.e. when 6 2 61 and X 2 0) between region (I) and region (I1) . We impose C3 continuity of the stream function across subregion interfaces in order to determine the coefficients in the eigenfunction expansions . We require may) = tum. 61) (4.28) aw, . awn‘ 57mm — - 74 82W 82W WEI—(ow) = #0301) (4.30) 8341 834/ afi(0,}/) = 3%(T,61) (4.31) where OSY'ga—I and IS?” |/\ a: In equations (4.28) - (4.31) , across the interface , we have To prepare for the computation of the equations given in (4.28) - (4.31) we require first some partial derivatives across the interface . From figure (4.1) we have X 2 :c’cos(% — 6’1) —- y’sin(-"27- — 61) (4.33) Y 2 :v’sin( — 01) + y’cos(% — 01) to|=l where .7; 2 :c' + c0561 (4.34) y 2 y, + sinOl 75 Substituting (4.34) into equation (4.33) and making use of the following trigonometric identity cos(A+B) 2 cosA cosB —— sinAsinB we get X 2 :rcos(§ — ()1) — ysin<§ —61) Similarly, now using the trigonometric identity sin(.4+B) 2 sinAcosB + sinBcosA we get 73' Y 2 xsin<9 —61>+ ycos<§ —01) ——1 H H We have thus the following X 2 rcochos(-’21—91) — 7‘sin63in(%—-01) Y 2 rcosflsin(%—61) + rsinficos(%—01) — 1 where we replaced 2' and y by .r 2 rcosO r sinO a: II (4.35) (4.36) (4.37) 76 Using the identities (4.35) - (4.36) equations (4.37) can be simplified to X 2 r cos(0 — 01 + 321) (4.38) Y 2 7*sin(0 —61+%) _ 1 Differentiating both equations in (4.38) with respect to X we get 1: ngcos(0—01+;-‘)— r9375? sin(6—61+%) (4.39) 0 = 8T, sin(6—01+%) + Tag; COS(6—61+%) Across the interface of regions (I) & (II) (i.e. when 6261 , X 2 0) equations (4.39) reduce to E —— __.1. 3X .— r (4.40) — = 0 Differentiating the second equation in (4.38) twice with respect to X then upon using the chain rule we have 021' ‘ 86 2 °(9 0+.)+ ‘826 +29:fl cos(6—6+:) W" 8X 8‘” “‘ - 78X? “ax ax ‘ 2 (4.41) *4 0: [VI Similarly, differentiating the first equation in (4.38) twice with respect to X then using the chain rule we get 8% 80 2 7r 326 87‘ 86 . _ __ _ . __ __ _ _ . __ ~‘) __ __ __ 0 " ]aX2 ’ (8X) ] COS (0 91+ 2) [7 0):? + ‘ 6X 8X] 81“ (0 61+ (4. ~ lo|>l 4:. [\J V Making use of (4.40) equations (4.41) - (4.42) across the interface of regions (I) & (II) reduce to 027' _ _1_ 8X2 — r (4.43) (726’ _ 8X3 _ 0 In the same way if we differentiate both sides of equations (4.38) three times with respect to X and taking into account equations (4.40) and (4.43) then the equations across the interface of regions (I) 8: (Il) reduce to a3 _ ax: " 0 (4.44) 839 _ 2 9T — 73 From equations (4.40) , (4.43) 85 (4.44) we have the following values of partial derivatives across the interface(where 0 2 01 and X 2 0): it _ 329 _ F3T _ 0 3X — 8X2 — <9X3 _ fl — _1 3X — r (4.40) 321' __ 1 9X2 “ ave _ _ 8X3 - 7‘ Using (4.45) we can now compute the partial derivatives of ‘1! with respect to X , we have 78 8‘1! BKII 87- 8&1! ()6 W=EW+WW “W Across the interface this reduces to all! 1 8‘11 —- :2 ——— 4.47 8X 7‘ 09 ( ) For the second partial derivative 32g! _ 32W 87‘ + 841 027‘ + 32‘]? 80 + 8W 826 (448) 8X2 ‘ axar 8X 87‘ 8X? 3X80 8X 09 8X2 ' Using (4.45) and (4.47) , across the interface . the last equation reduces to 324 34 324 024 30 2 1an 1324 —.=.—.+_—-—. =-.—+-——— (4.49) 8X2 07“ 0X2 @92 3X 7" 07" r2 802 For the third partial derivative 83‘11 _ 82‘1’ 027° 33‘1’ 37‘ 8‘11 837“ 82W 827‘ 32W 820 em‘fififlfifimfi+EW+fiWW7fiWW 83W 80 3&1 836 824/ 826 ——— — —" — — — 4.50 33339 ax + 30 3X3 + 8X89 3X2 ( ) but we have the following partial derivatives 82w _ 8248; + (9sz 6X87' _ 81'? 8X 8r39 8X 82W _ 82Wfl + 824 8r 8X86 — 82 8 ' 898r 8A (4.51) 833» 82¢ 828 + (83489; + 834: _8_r_) 2.0; + am 821- 8x288 82 8X2 893 8 ' 8r862 4 ex 3031' 8X2 33W 31' 0‘41! <59 81' + (6987'? a/\ + aTC’62 dz\) 011‘ Across the interface they reduce to 02¢ _ _1 02¢ 0\8r _ r dear 82w _ __18’-’w 82k]! 81' 9 8A 89 — r 882 + agar 0A (4 5“) 83w _ law + 1 82¢ c7\236 — r2 063 T OTOG So across interface the third partial derivative in (4.50) reduces to 33W 9 02W 827“ + 83‘11 80 + 0‘]? 336 0X3 I 0X07" 8X2 8X2 89 8X 06 8X3 2 824/ 183W 1 8241 + 2 3‘11 (4 53) 1‘2 01‘39 7‘3 803 r2 87‘89 r3 30 ' Across interface we have thus the following partial derivatives 23: _ 422. 8\ — 1 36 32¢ _ 1% L3“? ' 6X7 _ r 'cr + r2 (962 (4 04) 83W ___1_'r)3\D __3_-3\D +13 0X3 _- r3 393 r2 0736 r3 0 In particular , across the interface , the partial derivatives of in; upon using (4.54) are given by 869 = 83$ = 0 3X 8A3 86 82‘ 83A 4 .___.0. — __Q — ___Q. — . 8x “ 8X2 “ 8X3 — 0 ( 55) 828 _ 1 86;, 8x2 — 7 87- Substituting the eigenfunction expansions given in (4.10) , (4.25) into the matching conditions (4.28) - (4.31) and making use of equations (4.54) and (4.55) we get the following matching equations : " , co ,7; 00 Cn “7n D71 “11 0 = ‘1’o(7‘)—‘I’0(})+ go Fncos<4461)4§’(r)— .2... (52—5, .3 ’(Y) + b21324) )(Y)} (4.56) 00 . 1 n 00 Cn ~n , Dn "n 0 = 2 mnsmonen (; 4: ’326) 1 2 4n) 0 _ ; (17“ + .20; Fn cos(/\n81) (7— (Ir — T? An 491 (7‘) — 2: [0,. 43’8") + D. 44%)] (458) 0° . 3 do”) 7‘ 1 n 2 n 0 = g Fn Sln()\n61) (E )‘n 1({7‘( ) — T—B)‘: ¢i )(T‘) _ fiAn 45] )(7‘)) — 2; [0.4548208 + Danna‘Ni/H (4.59) where r & Y in equations (4.56) - (4.59) are on the interface related by In the next section we describe how the coefficients Cn , Dn and Fn are determined using biorthogonality conditions satisfied by the Papkovich - Fadle eigenfunctions . 4.3 Biorthogonality Series Solution We can find the coefficients Cn , Dn and Fn by application of the biorthogo- nality conditions to the matching conditions given in equations (4.56) - (4.59) . To prepare for the application of the biorthogonality condition we need first to express the matching conditions given in (4.56) - (4.59) in vector form . Differentiating both sides of equation (4.56) twice and using that di’g") ’7' n (”'12 : b2 53; @(2 ) (4.61) d2"(n) A, n Ti)? 2 62 P3 @(2 ) then we can couple equations (4.56) and (4.58) as follows : 1 d3 ('1) ,. , n o : 7% .. ;——“’I.J’ — 4AM ’m = + Z Fn cos(/\n61) 0 [Si—2:9. ._ HQ] —00 d2¢(1")(1.) dr2 dY2 \:O drz .. Wm MY) _ Z on + 1),. (4 62) -oo 5%") $5.”)(Y) (7.) N) Now applying the operator a_1 0 —1 ~(m) /' 7(m) 1' / [2). (my). 0)) W 0 1 2 to both sides of equation (4.62) and letting dd>§"’(Y+1) .4 ,, ~ 0 -1 v}: —-—d.—— U... = / MW”). gnu/)1 0 1 '2 m air2 ' in) /' WA: 0.51 (3 +1) — dY (4.63) 0 ._1 ~ 0 —1 Vii?” Q... = / WW) «LEW/)1 A W (4.64) 0 1 22 (in ._ ‘9‘”) d7? dY2 :0 0-1 0 “-1 ainuy) f0 [7594” $§m’(3")l ( ) W = 0 (4.65) Expanding the integrand in (4.65) gives _.zgn>(y).zim)(Y) + (53%”) + 259430) WW) 83 Making the change of variables Z = 1))” — 1 , this integrand becomes an odd function of Z integrated on [—1 1] . This can be easily seen by substituting the definitions of the eigenfunctions and adjoint eigenfunctions given in equations (4.12) , (4-21) and (4.22) into this integrand and simplify it through multiplication . Hence by making use of the Biorthogonality Condition we get 0 = Qm — km C,n + ZR. Umn cos(/\n61) (4.66) NoW applying the operator 0-1 A A 0 —1 . / MWY) . 2.4%”) cl)" 0 1 2 . to both sides of equation (4.62) and letting 0 _1 __1_d¢§"’(r'+1) A a—l A A Y+1 dr Um... = / MW”). WW) 0 1 2 d2ag")(Y+1) dr2 mi.)— )3; ¢§”’(Y + 1) —— dY (4.67) O 1 d$otrl Also the following integral vanishes since , A A 0 —1 WY) / [45mm . 44%)] W = 0 (469) ° 1 ‘3 4W) EXpanding the integrand in (4.69) gives —4’£~")(Y)$£m’) 434mm Making the change of variables Z = b)” — 1, this integrand becomes an odd function of Z integrated on {—1 1] . This can be easily seen by substituting the definitions of the eigenfunctions and adjoint eigenfunctions given in equations (4.12) , (4.21) and (4- 23) into this integrand and then simplify it through multiplication . Hence upon using the Biorthogonality Condition we get 0 = Q... — 2.. D... + ZR. (7...... cos()\n61) (4.70) Next; we need to couple equations (4'7) & (4.59) and apply the biorthogonality con C15 tion for a sectorial cavity (See Section 2.2) . The biorthogonality condition is given by /1 [dilmkfl . ¢>§m)(r)l BU) dr = P; 6.7... (4.71) 85 where .497.) = 44:77.) — — W) (4.72) 44%): gm.) and 4.“’(r)= 39%) here )7. = 1 + 2A3. (4.73) and 212 0 13(7): _3- L 2T3 272 By (4-72) 85 (4.73) we have dizéinm‘) (n) #n ,(n) ~ 4.2 — o. (r) + .2—4 (7) then 1 (n) Cl2¢in)(7‘l . ,(n) 2 2 (n) - c6] ()— 2r — 209 (7‘) — —)\n (7)1 (7“) (4(4) 7‘ d7? " 7‘ MultiIDIying both sides of equation (4.59) by 7'3 and using (4.74) then equations (4-57) and (4,59) can be coupled as . dd>(n) In R 0 so 37‘ .73- (7‘) - AZ 0i )(7‘) 73(1)”) = :FnAn sin()\n01) -— 2 O —°° 42dr (4'77) 1 a 31313,. W..- = / [747(7) . wém)(7)]B(7) . came—1N7 (478) an then making use of the Biorthogonality Condition we get 0 a -~2AmFmsin(/\m01)P;. + ZinFannsinonel) — :[C V...+D..14mnl (4.79) 87 From equations (4.66) , (4.70) & (4.79) we have the following infinite system of equations to be solved for the coefficients Cn , Dn and Fn 0 = Qm — km Cm + 23°00 Fn Umn cos()\n61) 0 = Qm — I‘m Dm + :‘f’w Fn Umn cos()\n61) 0 :1 —2AmFm Sln(Am61)P;1 + 2:000 An1:i'z.}:gmn Sin(An61) — Ziooo[Cann + Dnl/V'mn] (4.80) To rewrite the infinite system of equations (4.80) in matrix form let K : 9diagm (Am F; sin /\m¢91) [W = diagm(cos /\m(}1) H = diagm(/\m sin Amdl) (4.81) E = diagm(fi) _ ° _1__ G — dzagm (Em) then first equation in (4.80) in matrix form becomes EQ + (EUAI)F : C (4.82) Similarly the second equation in (4.80) becomes GQ + (00111)]? = D (4.83) CD a) and the third equation in (4.80) becomes (RH—[OF — VC — WD :: O (4.84) Substituting equations (4.82) — (4.83) into (4.84) we get 0 = (RH — K) F _ V(E Q + EUM F) _ ill/(0Q + CUM F) This gi ves VE Q + WGQ = (RH — K — VEUM _ WGUM) F Therefore we have F = (RH — K — VEUM — WGUM)-1(VEQ + WGQ) C = EQ + EUMF (4.85) D = 0(1) + GUMF provided that the inverse of S exists . Here 5 = RH —— K —_ VEUM — WGUM We note here that the Ci's and D273 are given explicitly in terms of the Fi's . System (4.85) represents an infinite system of equations for the components of the Vec ‘ . . . tol F ; this system is to be solved by truncation . g 8.9 4.4 Numerical Results and Discussion _7T The first example we consider is flow in the region where a = 3 , b = 1 and 01 _. 2 (See Figure 4.1) . The two subregions (I) & (II) shown in figure 4.1 are thus given respectively by V1={X,Y:—oo 0 , the first quadrant roots and are arranged in a sequence corresponding to increasing size of their real part and define )1_n=Tn (7121,2,3,...) where the overbar designates complex conjugate. Then 111—W) = 265%) (n = 1.2. 3. ~-> is the eigenfunction corresponding to the eigenvalue A_.n . Since the edge data given in (5.2) and (5.3) are real then 5.2 Matching Eigenfunction Expansions We must now match the two solutions given in (5.8) - (5.9) . We shall require that the stream function and its first three partial derivatives with respect to 6 are continuous across the interface at 9 : 01 and 6' = —()1 between region I and region II . We impose C 3 continuity of the stream function across subregion interfaces in order to determine the coefficients in the eigenfunction expansions . We require 111,030.) = 411102—01) (5.13) %%(1‘.91) = a§911(1".—91) (5.14) 382%50301) = 8;:2”(r’,_91) (515) 9%4. 1) = 95% 2—61) (5.16) where 7‘1 S 71,7‘ <7‘2 To prepare for the computation of the equations given in (5.13) - (5.16) we require first some partial derivatives across the interface . From figure (5.1) we have y = 2'; + (T1+T'2)Sln61 (5.17) IL' :- —X 'l- (7’1+7‘2)C0861 or in polar coordinates we have 'rsin9 = 1", sin 9' + (r1+ r2)sin 91 (5.18) 7‘cos9 = —'r' cos 9' + (r1+ 7‘2)cos 91 Differentiating both equations in (5.18) with respect to 9 we get _ - ' 6:77" ,' ' ae’ rcos9 — 51119 ‘57 + 7 cos9 1.5-9— .. (5.19) - __ 2:1 ,- ra_6' —rsm9 _ cos9 06 + rsm9 39 Across the interface of regions (I) & (II) (i.e. equations (5.19) reduce to 7‘ cos 91 —r sin 91 Solving the two equations in (5.20) We note that (5.21) is still valid when 9 118 —sin91—— — cos 91 — for 9 = 91 i I 33; + r c0591 31" ’ ' 39 — r sm91 r99 31" 797 and '55 7' —T T 0 E 2 . (5.20) we get Differentiating both sides of equations (5.19) with respect to 9 and taking into consideration equation (5.21) then the equations across the interface of regions (I) & (II) reduce to —7'sin91 — —sin91C’6"2 Q.) —r cos 91 — cos 91 362 Solving the two equations in (5.22 get I 321' + r cos 91 I . '. — 7‘ 81116] J'— ) for 3.2? 529’ and W , 2 + 1" sin91 (2%) 2 I I + r cos 91 (—%%> and considering (5.21) we (5.23) Differentiating both sides of equations (5.19) with respect to 9 twice and taking into account equations (5.21) and (5.23) then the equations across the interface of 11.9 regions (I) & (II) reduce to 3 __ - a3r’ . 30’ 321" I 339’ . 30' -7'C0301 — —Sln61 W + 3C0$61 51—9 87 + T C0361 W — 7‘ C0501 —9 . a3 ’ . 39’ 82 ’ I . 339' I . 39' 3 rs1n91 = ——cos91 ears — 351119137 8972 — r s1n91 50—3 + r sm91 —9 (5.24) Solving the two equations in (5.24) for 836 and 331' and making use of equations 863 393 (5.21) & (5.23) we get (5.25) From equations (5.21) , (5.23) & (5.25) we have the following values of partial derivatives across the interface where 9 = 91 and 9' = —91: 2:. _ L9, i; 0 <96 — 392 393 59—9, _ L F19 — 7.’ (5.26) 321" _ , 1 592 — 7 (1 + 7) 3 I 2 3 i; = 45) — 3(5) — 2(5) Consider the function F : cos()\n9') (15(7") (5.27) We compute the first three partial derivatives of F by using the chain rule . Making 120 use of equation (5.26) these partial derivatives of F across the interface reduce to 3;; = 1,. Sim/v.6.) 09' 4(1“) W 3275‘ = —X:’. cosunei) (954') 50') + cos(1.0.)%.— 56') 31933 = -/\.. sin(/\n91) 20") (43. (-‘?.—‘2.—)3 —— 33963,) + 3A.. Sin()\n91) 3599'- g"; ’(r') (5.28) Similarly if we have the function G = 5(r’) (5.29) then by setting An = 0 in F we get using equation (5.28) the following partial derivatives on the interface between regions (I) & (II) 6%? = 0 3,39% = 2.”... 4'0“) (530) 7939? : 0 Using equation (5.17) we have 732 = X2 + y2 = [~75 _ (71+7‘2) 008191].2 + [y — (7“; +73) sin91]2 (5.31) = r2 — 27‘(r1+r2) cos(9—91) + (r1+r2)2 In particular 011 the interface 121 Substituting the eigenfunction expansions given in (5.8) - (5.9) into the matching conditions (5.13) - (5.16) and making use of equations (5.28) , (5.30) and (5.32) we get the following matching equations : I o = (ma—00(6))! + f: Encos()\n91) ¢§"I(r) _ i Fncos()\n91) 51100) 9:91 (5.33) 0 °° . (n) °° - 39' (n) I = —: EnAn S1H(An01) 451 (7) — Z: FnAn51n(An01) W ¢1 (T) (5.34.) 327" A, I 00 2 I(n) 0 : -- W $00") —' Z EnAnCOS()‘n01)(.D1 (T) '— 9=91 ‘m 00 a2 ' -’n I 80' 2 n , Z acosim.) 59: 21‘ In) — 213(55) <11 m) (5.35) 0 = Z ETA?l sin(/\n91) ¢§nl(7‘) + I 3 I I I 00 I 30 836 86 827‘ I I - (n) . 2 _ __ _ . _ __ (72) 20:0 E. An 51n(/\n91) [C51 (7 ){An (89) 393} 3 80 392 (151 (1')] (5.36) In the next section we describe how the coefficients En and E. are determined using biorthogonality conditions satisfied by the Papkovich - Fadle eigenfunctions . 5.3 Biorthogonality Series Solution We can find the coefficients En and E. by application of the biorthogonality condition for a sectorial cavity (See Section 2.2) to the matching conditions given in (5.33 - 5.36) The biorthogonality condition is given by 7214120). 74%)] 8(7) ' dr = P; 5,... (5.37) where here and 51%) = .2125) — — 5.00) (5.38) 214712): £71m and ¢£""(r)=¢im)(r) '11,. 2 1+ 2): 5713 0 30") = _3- L 2T3 2T2 To prepare for the application of the biorthogonality condition we need first to express the matching conditions given in (5.33) - (5.36) in vector form . From (5.38) we have 1 n n 1” n 1 n 75:51 )0") = 4’0) — 7.531(7) + 55‘. ’(r) so if we substitute it into equation (5.35) then we can couple equations (5.33) & (5.35) as I—fi 16 O A ‘3 v | 164 O A ‘3 Wt... .. ( 0 ) + 2 En cos()\n91) .. 15%) - r5325) .27 . + Z Fn cos()\n91) lu—A y Ix) 0,201)’ 152' '(n1I n('CCI—9) 9109—;ng 1 (7“) + E E. cos()\n91) (5.39) --00 -(n)(7~) Now applying the operator /WMPWL 26525 7'1 :lI to both sides of equation (5.39) and letting 0 w.=/7WPWLwWWHm0( )dr ’1 45W0—r4NW> (5.40) .. 0170’) W... — / [vim’(r1.¢4’”’(r)18(v) dr . 14(3) ¢1n> (5.41) w [1110(7) — 410(I’)]0=61 am: WWWLITWHMM dr -lfiumo (5.42) then by making use of the Biorthogonality Condition we get 0 = Qm + Em cos(/\m91) P; + :En an cos(/\n91) + E: E. Van cos()\n91) —00 (5.43) Next we need to couple equations (5.34) & (5.36) . By (5.26) 85 (5.32) we have 69! 3 030! 89’ 3 69! 2 601 2 _ _ __ = 2 t) __ __ __ A”(86) 093 (An +“)(ae) + 3(66) + (80) so we can rewrite equation (5.36) as follows 86’ 327" '(n) 0 = ZAiEnsinOngi) in)(7‘) — 3 Z)“ F"Sin()‘"61)w D9? 1 ._00 —’.\'.> (r') + T —00 0° , n I 80’ 3 86, 2 7'1 7’ ZAnFn51n()\n91)¢g )(r){[()\i+ 2) (8—9) + 3 (59) + (—_+':—2—) (5.44) , 2 If we divide equation (5.34) by (3%) then we can couple (5.34) & (5.36) or equivalently ( 5.44) as 0 o. ( ) = 219.1,. Sln()\n91) 0 + E F”)... sin(/\n91)< . 5119’ 52% =’(n) .I ‘3‘55 592 .1 (I) + (5.45) 0 Now applying the operator /. [4&m’(r’) , WM) 842*) :l dr' to both sides of equation (5.45) and letting Rm” = /. [44“)(4’) , 4.5.7444) Bu“) 4%")(4) dr’ (5.46) ) dr' (5.47) then making use of the Biorthogonality Condition we get 0 = —/\mFmsin(Am61)P,; + Z AnEann sin()\n01) + Z AnFnSmnsin(An91) (5.48) -00 —00 From equations (5.43) & (5.48) we have the following infinite system of equations to be solved for the coefficients En and Fn 125 Now applying the operator / MW) . 4§m’(r’>1 BM) H dr’ to both sides of equation (5.45) and letting Rm = / {/(m)(r , 7~')(7~’]B )5§”M() dr’ (5.46) ) dr’ (5.47) then making use of the Biorthogonality Condition we get 0: —Am Fm sin(Am 01) P" + :An En Rmn sin(An 91) + :An Fn Smn sin(An 01) (5.48) —00 —O<} From equations (5.43) & (5.48) we have the following infinite system of equations to be solved for the coefficients En and Fn 1‘26 0 = Qm + Em cos(A,,101)P,’,'1 + 23°00 En an cos(An61) + 23°00 FnWmn cos(An61) 0 = —AmFm sin(Am01)P,; + :3; 1.19.3..." sin(An61) + 2:000 AnFnSmn Sin(/\n61) (5.49) To rewrite the infinite system of equations (5.49) in matrix form let D = diagm(P;l) § L = diagm(Am) H = diagm(sin Am61) and [1’ = diagm(cos Am91) then first equation in (5.49) in matrix form becomes (DK + LHV)E + (Kl/V)F = —Q (5.50) and similarly the second equation in (5.51) becomes (LHR)E + LH(S — D)F = O (5.51) We can eliminate E from (5.50) and (5.51) to obtain [(DK + LHV)G’1R + Ix'l/V]F = -—Q (5.52) provided that the inverse of G exists . Here G : LHR R: LH(D—H) Equation (5.52) represents an infinite system of equations for the components of the vector F. 5.4 Numerical Results and Discussion The first example we consider is flow in the channel where r1 = 1 . r2 = 2 and 191 = % (See Figure 5.1) . The two subregions (I) 8: (11) shown in figure 5.1 are thus given respectively by V1={r,6:ISrSQ,—§S¢9£%} (553) and V11={T’,9’ 137" S? , —g$9, 3%} (5-54) The complex eigenvalues given in (5.12) were found numerically accurate to eight decimal places using a subroutine from the IMSL Library based on Muller’s method with deflation (See Table 5.1) . A discussion of these eigenvalues is given in Appendix A . Equations (5.49) form an infinite system of equations to be solved for the coefficients En and Fn , n = i1, i2, . To solve this system of equations we must truncate the infinite sums appearing in (5.49) to finite ones . We do this by replacing the lower and upper limits of summations by —N and N . respectively. The truncated systems are then solved using a subroutine from the LINPACK Library based on Gaussian elimination with partial pivoting to compute the LU factorization of the complex matrix and then solves it . The coefficients of the truncated system that are given as integrals are integrated numerically using a subroutine from the IMSL Library based on Gaussian Method. The convergence of the solution of the truncated equations was then checked nu- merically at twenty mesh points on the interface between subregions (I) & (II) . Convergence is reached when N is ‘20 where the two solutions 0?] , \P 11 together with their first three partial derivatives with respect to 9 given in equations (5.13) - (5.16) match very well along the interface . The matching slightly improves if N is increased to ‘26 (See tables 5.2 and 5.3) . The solution remains unchanged if N is larger than 26. Figure (5.2) shows the peripheral velocity profile for a circular flow (i.e. flow between two circles) which is contributed by the 410(7") term of the solution \P1(r,0) in subregion (1) given in equation (5.8) or in case of subregion (II) is contributed by the {13003) term of the solution 911(r'fl') given in equation (5.9) . Figures (5.3) - (5.5) show the peripheral ( azimuthal ) velocity profiles v9 , 120' at different angles in subregions (I) & (II) that are contributed by the infinite sum term in equations (5.8)-(5.9) ; these infinite sum terms add a small perturbation to the the velocity profile of the circular flow . Figure (5.3) shows that close to 0 = 0 we have negligible contribution from the infinite sum term so we almost have a circular flow ; as the angle increases we have a slight increase in the tilt of the profile towards the shorter circular edge r = 1 compared with that of the circular flow profile , the tilt reaches it’s maximum when 0 = 700 ; this tilt is because the corrugation of the two edges of the channel are horizontally out—of—phase (i.e. a horizontal shift to the left of the edge F1 in figure (5.1) does not yield F2) . As 6 = 700 is slightly increased an abrupt and significant change in direction of the tilt towards the larger circular edge r = 2 occurs since the flow gets close to the interface where we have a change in the sign of the curvature of the channel (See figure 5.4) . At the interface the tilt reaches it’s maximum then as the flow moves to subregion (II) the tilt decreases then when 9' = -—750 starts tilting slightly towards the shorter circular edge 7" = 1 as is clear in figure (5.5) ; close to (9' = 0 i.e. away from the interface we have no contribution from the infinite sum term so the flow is almost circular . The velocity components for subregion (I) are given by 041 ._ 1% ”UT 37‘1 _7‘30 v9: and for subregion (II) the velocity components are given by v , = (5.56) The pressure P and the velocity components are related by the following equations 3P 1 2 avg _ 2 .-. ‘57 - V ”r — 5'14 — $555 (55‘) 89 : ’"(V ”9 + $589 _ F5) (5'58) Plugging equation (5.8) into (5.55) we can compute the velocity components then using (5.57)-(5.58) we get for subregion (I) 284:4 = :3; AnEn Sin(An9) [4(1—Anyway: + 4(1+An)rxn-1] aP _ 24 (5-59) 391 — 1610g22—9 + 23°00 En WSW") [4 An (An —1)Cn 7"“ + 4A,. (An +1) Mn] Similarly plugging equation (5.9) into (5.56) we compute the velocity components then using (5.57)-(5.58) we get for subregion (II) 2554 = -2310 1,. F... sin(An6') [4(1—A,.)c..(r’)-An-1 + 4(1+An) (r')*"“] 8P 35y- _ 1610g2242—9 — 233» Fn COSWO') [4 An (1,. -1>cn(r')-"n + 4A,. (4.. +1) (74M (5.60) 1330 Integrating equations (5.59) and (5.60) and taking into consideration that the pressure is continuous along the interface between regions (I) & (II) (i.e. P1 2 P11 along 9 = —-9' = %) then up to a constant the pressure is given by P1 = 1610757242—9w _ g) + @6n 2‘11. E. sinmne) [40. — mar—An + 4(1+ A.) 7.44] P” : 16103242—9(6I + 5) “ (5.62) 23°00 Fn “WW (4% -1)cn(r')‘*" + 4(1+ A.) (7’)“) In figure (5.6) we have a graph of the pressure contributed by the linear terms in equations (5.61)-(5.6‘2) which actually arise if we only have a circular flow . The contributions to the pressure at the center-line of the channel (i.e. where r = 1.5 or r' = 1.5) by the infinite sum terms appearing in (5.61)-(5.6‘2) are shown in figures (5.7)-(5.8) where we note that the graphs of P1 . P11 are the same thus the pressure oscillates along the center line of the channel . The second example we consider is flow in the channel where r1 = 1 , r2 = ‘2 and 91 = 343 (See Figure 5.1) . The two subregions (I) & (11) shown in figure 5.1 are thus given respectively by V1={r,6zlgrg’2,—-Eg€g§-} (5.63) and V” ={r’.0’ :1 gr’ _<_ 2 , —g g 6’ g E} (5.64) 131 so we decreased the angle in example one from to g . The graph of the 5 pressure along the center-line of the channel is almost the same as of the previous example thus the pressure oscillates . Figures (5.9) - (5.10) show the contributions of the infinite sum terms appearing in equations (5.8) - (5.9) on the azimuthal velocity profile at different angles in subregions (I) & (II) . Close to 9 = 0 we have negligible contribution so flow is almost circular ; as the angle increases we have a slight increase in the tilt of the profile of the circular flow towards the shorter circular edge r = 1 and this increase reaches it’s maximum when 9 = 250 then the effect of the interface - where we have a change in the sign of the curvature - starts to be felt . As 9 = 250 is slightly increased a sudden and significant change in the direction of the tilt towards the larger circular edge r = 2 . The tilt reaches it’s maximum at the interface where 9 = g . As flow moves to subregion (II) we have a decrease in tilt of the circular flow profile away from the shorter edge 1" = 1 then when 9' = —300 the profile starts slightly to tilt towards the circular shorter edge ; close to 9' = O i.e. away from the interface contribution is negligible so we almost have a circular flow . The third example we consider is How in the channel where r; = 1 , r2 = 2 and _ 37f ' . . . - - 91 — 7 (See Figuie 5.1) . The two subieg1ons (I) & (II) shown in figure 5.1 are thus given respectively by V1={r,9:1§r§2.—'T§9ST} (5.65) and I I I 377 I 37l- V11={T,0.1£7‘S.2.—TSQST} (5.66) to 1’1 . The graph of the 101:! .b. 132 pressure along the center-line of the channel is almost the same as of the previous example thus the pressure oscillates . Figures (5.11) - (5.12) show the contributions of the infinite sum terms appearing in equations (5.8) - (5.9) on the azimuthal velocity profile at different angles in subregions (I) & (II) . Close to 9 = 0 we have negligible contribution so flow is almost circular ; as the angle increases we have a slight increase in the tilt of the profile of the circular flow towards the shorter circular edge r = 1 and this increase reaches it’s maximum when 9 = 1200 then the effect of the interface — where we have a change in the sign of the curvature - starts to be felt . As 9 : 1200 is slightly increased a sudden and significant change in the direction of the tilt towards the larger circular edge r = 2 . The tilt reaches it’s maximum at the interface where 9 : T’ . As flow moves to subregion (II) we have a decrease in tilt of the circular flow profile away from the shorter edge r, = 1 then when 9' = —1200 the profile starts slightly to tilt towards the circular shorter edge ; close to 9' = 0 i.e. away from the interface contribution is negligible so we have almost a circular flow . In conclusion , for Stoke’s flow in the periodically curved channel shown in figure (5.1) the pressure oscillates along the center-line . Away from the interface the azimuthal velocity profile is almost that of a circular flow (i.e. flow between two circles) . As the flow gets close to the interface - where there is a change in the sign of the curvature of the edges of the channel - an abrupt and significant change in the direction of the tilt of the profile occurs . 133 Complex roots An :23 3.370703254 + 6.054102744 i 4.112343180 + 10.802931474 i 4.593078513 + 15.444885446 i 4.951563568 + 20.045543966 i 5.238022153 + 24.625199349 i 5.476764440 + 29.192525396 i 5.681498446 + 33.751926784 i 8 5.860740743 + 38.305901960 i 9 6.020153718 + 42855983209 i 10 6.163696600 + 47.403166826 i 11 6.294247926 + 51.948130806 i 12 6.413966841 + 56.491353726 i 13 6.524514778 + 61.033183727 i 14 6627197730 + 65.573880523 i 15 6723061006 + 70.113642063 i 16 6.812954353 + 74.652622031 i 17 6897577913 + 79.190941676 i 18 6.977515421 + 83.728698023 i 19 7.053258673 + 88.265969687 i 20 7.125225878 + 92.802821085 i 21 7.193775648 + 97.339305529 i 22 7.259217815 + 101.875467545 i ‘73 7.321821895 + 106.411344626 i ~103Cfihi>00l0|~4 24 7.381823795 + 110.946968584 i 25 7.439431179 + 115.482366591 i 26 7.494827805 + 120.017562008 i Table 5.1: Twenty six first quadrant eigenvalues of (5.12) for r1 = 1 , r2 = 2 . 134 r \I11(r,%;26) \I111(7",—%;26) a'7‘,12{-(7',12226) 9%611(r',—%;26) 1.00 -—1.02 X 10'15 4.25 X 10‘15 —1.05 X 10'16 -—3.83 X 10‘14 1.05 4.6729 X 10‘3 4.6709 X 10‘3 —8.9683 X 10’2 5.3292 X 10‘2 1.10 2.5230 X 10'2 2.2229 X 10‘2 —0.145851 0.036050 1.15 6.2300 X 10“2 6.2300 X 10'2 ~0.120010 -0.079385 1.20 0.108765 0.108765 -—0.140809 -0.154503 1.25 0.162127 0.162127 -0.195566 —-0.195451 1.30 0.222234 0.222234 —0.241518 -—0.240113 1.35 0.287813 0.287815 —0.281368 —0.281603 1.40 0.357404 0.357404 —0.315298 —0.315829 1.45 0.429648 0.429646 -—0.341935 —0.341364 1.50 0.503211 0.503214 -—0.358559 —0.358712 1.55 0.576708 0.576707 —0.364515 —0.365027 1.60 0.648766 0.648763 —0.360302 —0.359003 1.65 0.718035 0.718044 —0.341848 —0.343913 1.70 0.783053 0.783041 -0.314057 —0.311460 1.75 0.842449 0.842459 -0.270792 —0.273487 1.80 0.894645 0.894641 -0.220592 —-—0.218062 1.85 0.938187 0.938185 —0.156742 —0.160129 1.90 0.971200 0.971202 —-0.098868 —0.088786 1.95 0.992247 0.992251 —-0.050556 —0.017843 2.00 0.999999 1.000000 —2.05 X 10'13 4.79 X 10‘17 Table 5.2: Convergence of the Papkovitch-Fadle series for N226 , r1 = 1 , 7‘2 2 and 61 2 135 r 87329221- r,%;26) %1(r',—%;26) %p51(r,%;26) 8373311( ', %;26) 1.00 —3.5 x 10‘15 3.7 X 10’13 —1.8 X 10‘13 —4.0 X 10’12 1.05 —1.377885 —1.386545 —2.515556 —2.068390 1.10 —1.695860 —1.703743 —1.622384 —1.768306 1.15 -0.622091 -—0.630688 —0.116687 —2.054909 1.20 —0.703067 —0.711739 -0.431854 —1.986255 1.25 ——1.221263 —1.226234- ——2.417017 —2.288905 1.30 —1.432121 —1.428847 —3.727090 —3.301293 1.35 —1.654442 —1.648300 -—3.807945 —3.729874 1.40 —1.860136 —1.863625 -—4.799951 —4.801738 1.45 —2.099256 —2.102848 -6.856380 —6.922190 1.50 —-2.294335 —2.286830 —7.099842 —7.149344 1.55 —2.448117 —2.455342 —10.192682 —10.148095 1.60 —2.649893 —2.649045 —13.068916 —13.030107 1.65 —2.651345 —2.638694 —l2.1062l2 —12.130149 1.70 ——2.805537 —2.830969 —-21.236649 —21.275853 1.75 ——2.615561 —2.594777 —16680361 —16694264 1.80 —2.605717 —2.596951 —27.287956 —27.278368 1.85 —2.123435 ——2.155192 —26.3081157 ——26.292548 1.90 —1.764723 —1.777291 —26402980 ~26405178 1.95 —1.714488 —1.563494 —43.511228 —43.547924 2.00 —-5.7 X 10‘12 -—6.4 X 10‘15 -—1.4 X 10‘10 8.2 X 10-13 Table 5.3: Convergence of the Papkovitch-Fadle series for N=26 and 61 : T121,T2:2 136 figure 5.2: Azimuthal velocity profile for a circular flow contributed by 1110(7') or ‘IIO (7") term given in (5.8) - (5.9) . V 15 Daqrvvl V o Doqz... 0.00002 0.00001 Figure 5.3: Contribution of the infinite series term in (5.8) on the azimuthal velocity Profile of the flow at different values of the angle 9 in subregion (I) given in (5.53). 137 V on n-q:..- 7 5 no.2... o o o o V l5 D-qzoo- Figure 5.4: Contribution of the infinite series term in (5.8) on the azimuthal velocity profile of the flow at different angles 9 in subregion (I) given in (5.53) close to the interface (i.e. when 9 = g) . V -90 D-qz... V -l5 D-vrool 0.2 0.15 0.1 0.05 I. 1.6 1.0 V I l o o - o w v. V 40 0.9".- 0.92 c.01 ‘ 1.6 1.: z ‘4 -0.°) -o.02 V -:a o.qg... 0.0092 0.00015 c.0001 0.00005 , r' - ,. -°.oooos 1'2 1' —o.oool -c.ooois o Dace..- V -60 D-qrool Figure 5.5: Contribution of the infinite series term in (5.9) on the azimuthal velocity Profile of the flow at different angles 9' in subregion (II) given in (5.54). 138 1’ £11 Subz-q1en 1 -o.5 P1 -o.25 PA ' 0.25 Pt 0.: 171: P in Subroqion xx I Figure 5.6: Pressure contributed by the circular flows \Ilo(r) & (1700") terms given in (5.8) 85 (5.9) respectively. 9 1n Subraulan x 0.0075 o.cos 0.0015 -1 s —1. —o.: W 1 x 5 ‘ -D.°Oa5 —a.ocs P in Subrovion r 0.25 o.2 0.15 0.1 0.05 c I o as o s 0.75 1 A as 1 : Figure 5.7: Pressure at the center-line of the channel (i.e. where r = 1.5) contributed by the infinite series term in (5.8) of subregion (I) given in (5.53) . ! Ln Subrovlon Ix 0.008 9.096 0.00. 9.002 —o.ooz 7 Sn Subtoqton II Figure 5.8: Pressure at the center—line of the channel (i.e. where 7" = 1.5) contributed by the infinite series term in (5.9) of subregion (If) given in (5.54) . 0 Door..- 5 D-qzool Figure 5.9: Contribution of the infinite series term in (5.8) on the azimuthal velocity PIOfile of the flow at different angles 9 in subregion (I) given in (5.63) . 140 —45 DOVE..- —40 Door-.- V —35 D-qr-o. V -39 00.1..- 0.02 0.005 O. QO‘ °-°‘ o. 002 . . . . ‘ .‘ug ' ‘0-01 I 2 1 1 ‘ l I 2 0°: X.1.1 -0. 00¢ ‘°'°’ —0. 00‘ —25 Boat... —zo Douro-- 0.004v 0.004 0.002 0.002 1. z 1. 4 1.1.0 1.: 1.4 .0 1.0 -0.002 '°-°°’ —0.ooq —15 Door... V —10 D-qzo-n 0.004 0 004 c. 002 °~°°3 1. z 1 4 1. s 1. u 1.2 1.4 1.0 1.0 " -°.°°2 'O.°°2 -°.0°‘ v —s 0.9:...) ‘ 0 Doug..- Figure 5.10: Contribution of the infinite series term in (5.9) on the azimuthal velocity profile of the flow at different angles 9' in subregion (II) given in (5.64) . -7 2. 10. i -1.15 10 0 Door..- 0 n 0 IE 15 Door-.- 1: 0.q:... 00 0.0:..- 0. 0.0001 0 0.00005 0. . 5 I B l 2 I. ‘ 1.6 1.5 -°.O°OO§ ‘0. 002 -0.0001 '° °°‘ -° 00‘ 120 D-qroo- V 130 0.3:... V 135 DOGZOQI 0.00 o_: 0.02 0.05 1.2 1. 1.5 1.0 2 ' 1.2 1 1.0 1.0 2 ‘ -0.02 —0.05 —0.04 ~°~X Figure 5.11: Contribution of the infinite series term in (5 .8) on the azimuthal velocity Profile of the flow at different angles 9 in subregion (1) given in (5.65) . 141 -135 00020-- V -130 D.¢t... 0': 0.06 0.10 °_°‘ 0'1 0.02 0.03 2 t. . 1.2 1. 1.0 1.0 1.2 1. 1.‘ 1.0 11 ' -o.oz -0.os _°_°‘ 70’: —o.os -125 D.q:... v -120 D.qr..o O 02 0.005 0.00. 0 OX 0.002 r' r’ 1.2 1. 1.6 1.0 2 1.2 1. 1.6 1.. “i _°.°1 —o.ooz -0.00‘ -O.°2 ~0.00‘ V -105 0.9:... V -75 0.0:... 0.0002 °'°°‘ 0.00015 0.002 °-°°°1 0.00005 1.2 1.4 1.6 1.0 ‘ -0 00005 1.2 1. 1.6 1.0 ‘ -°-°°2 -o.ooo1 _°_°°. -o.ooo15 V -‘O Doqzool -;V ‘30 906‘... 0.000015 2_ 1° 0.0000% 1 1 -‘ 5. 10 a 6 1 U 2 -‘ ~. _ " -c 1.2 1 1.6 1.0 2" ‘3' 1° -1 10 - .00001 - _‘ -0.000015 -2. 1o -15 0.9:... Figu re 5.12: Contribution of the infinite series term in (5.9) on the azimuthal velocity profile of the flow at different angles 9' in subregion (II) given in (5.66) . Chapter 6 Discussion The first purpose of this thesis is to develop an eigenfunction expansion method for solving problems of Stokes flow (creeping motion) mainly in sectorial cavities and some other flow geometries which have never been done before . The theory leads to a set of eigenfunctions , self—adjoint eigenfunctions , biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion . The resulting infinite system of linear equations are then solved by truncation . This algorithm is illustrated by solving the problem of slow steady flow induced in a sectorial cavity by shearing the incompressible fluid on the top with a moving P1 ate or belt . This problem has not been solved before either analytically or numer- ical 1y . The biorthogonality condition we derived for a sectorial cavity enabled us to Use the resulting algorithm to solve Stokes flow problems in geometries having a seCtOrial subregion . The Solution we obtained for a sectorial cavity was as accurate as that obtained by Joseph and Sturges [26] for a rectangular cavity . However , our problem has not yet, been studied using numerical methods that we can compare our results with . AS in the case for a rectangular cavity we obtained a. steady flow involving a series 0f Separated closed streamlines and we showed the existence of corner eddies ( or resi'stive eddies ) of the type discussed by Moflatt (1964) [35] . From comparing sev- 14‘2 143 eral sectorial cavities we noticed an increase in the number of edge eddies when the length of the cavity increases and a. decrease in the rate of convergence as the width of the cavity increases . Convergence of the biorthogonality method is very fast . Only a small number of terms in this expansion is needed to produce fairly accurate approximations . We required at most 15 terms from the truncated infinite system of equations to acheive convergence and the error was less than six percent . Convergence and completeness of our method for Stokes flow in sectorial cavities has not been proved . As for edge problems which arise in elasticity and Stokes flow in cavities , Joseph [‘23] and Joseph and Sturges ['26] established sufficient conditions to guarantee convergence of the biorthogonal series solution for all types of edge data which might be expected in applications . Joseph. Sturges & Warner (1982) [28] added new theorems to the theory of convergence of the biorthogonal series and also proved completeness of the biorthogonal series of biharmonic eigenfunctions using the method of Titchmarsh . There are ways other than the application of the biorthogonality condition for the computation of the coefficients of the eigenfunction expansion , for example , inver— sion methods and boundary-collocation techniques . 0111‘ algorithm can also be used for solving problems other than those of Stokes flow . It may be applied to elasticity which is governed by the biharmonic equation . The biOI‘thogonal series expansion method may also be used for solving flow problems Where the resulting governing equation is not the biharmonic equation . For example , JOSQIDh [2'2] used biorthogonal series expansion method in his study of the free surface on t-1'1e round edge of a flowing liquid filling a torsion flow viscometer . The biorthogonality condition is derived for a general class of fourth-order boundary Value problems with variable coefficients for a. wide class of Stokes flow domains ; in- Chllcling the sectorial cavity . These biorthogonality conditions are properties satisfied 144 by the eigenfunctions and self~adjoint eigenfunctions . They enable us to compute the coefficients of the eigenfunction expansion solution for the stream function for several Stokes flow problems . The deficiences in the biorthogonality conditions de— rived by Joseph and Sturges [‘26] and others is that they are derived for some particular fourth-order boundary value problem corresponding to the special region they consid- ered . However , the biorthogonality conditions we derived are not restricted to one particular Stokes flow problem , in addition , the coefficients of the resulting fourth- order boundary value problems we considered need not be constant as in previous literature . We can generalize our method to other linear partial differential equation boundary- value problems . If the partial differential equation is separable , and the resulting fourth-order boundary value problem is in the form of the generalized equation given in this thesis , then our biorthogonality condition can be used to compute the co~ efficients of the eigenfunction expansion solution for such boundary-value problems as well . However , the method of separation of variables is useful only to domains bounded by coordinate surfaces , thus the applicability of our algorithm is limited to domains of simple shape . Another purpose of this research is to describe a matched eigenfunction expansion method for solving Stokes flow whose geometry is composed of contiguous simple Sha'IDes . For the flow around a bend , the flow region is decomposed into a semi- i“fir-lite rectangular subregion and into a sectorial one ; this enables the stream func— tion to be represented by means of an expansion of Papkovich-Fadle eigenfunctions in eEach of these two subregions . The coefficients in these expansions are obtained by mat Ching them and their first three partial derivatives across the common interface in a weak sense and then making use of the biorthogonality conditions we derived for seCtorial and other flow regions . 145 The same matching technique was used to solve Stokes flow in curved channels . This channel has sectorial subregions and thus the biorthogonality conditions we derived for sectorial geometries can be applied to find the coefficients in the eigenfunction expansions . The matching method can only be applied in principle to domains that are decom- posable into rectangular or sectorial subregions or both . The advantage here is that the geometries do not need to be described in the same coordinate system . Although the matching technique is done in a weak sense for solving Stokes flow in curved channels and flow around a bend , however , we were able to couple the equa- tions in such a way such that the resulting stream function we obtained is actually C 3 continuous across the subregion interfaces . The matching across the interface was very good . The stream function and its first three partial derivatives matched across the interface with an error less than one per- cent . Convergence was relatively fast . An advantage of the method is that the region can be treated without truncation of the domain as is the case when using numerical methods based on finite differences or finite elements . Instead , a global approximation to the solution is found . In conclusion , we have developed an eigenfunction expansion method to treat the biharmonic equation arising in Stokes flow . The method seems to be powerful in treating problems whose boundaries are composed of contiguous curvilinear regions especially for the sectorial region . The biorthogonality conditions can also be applied to other geometries which can be described by separable coordinate system . Appendix A Eigenfunctions and Eigenvalues for Stokes Flow in sectorial Regions A. 1 Eigenfunctions We seek the solution $03 0) of the biharmonic problem a? 10 ' 1 a? 2 4 . ._ __ __ : . v w(7,0) .. (W + N?" + r2392) \I/(r,0) 0 (A1) \Il(a,9) = wag) = 22“) = gage) = 0 (A2) ()7‘ 07' Using separation of variables we assume a solution of the form @(r, 6) = To) 1/(0) (A3) Substituting (A3) into (A.1) then separable solutions of (A.1) are governed by the reduced equation 146 <>—-< >21” -o (A.4) We seek eigenfunctions in the r direction requiring W) = _12 (A5) where A is a complex constant . From (A.5) Y is given by l"(0) = linear combination {sin A0 , cos A6} (A6) Substituting equation (A6) into (A.4) then T(r) satisfies the following equation r4 T“) + ~27~3 Tm — (1+-212)7~2 Tm + (1+2A2)1~T(1) + A2 (AZ—4) T = 0 (A.7) The latter is Euler’s equation whose characteristic equation after simplifying it reduces to a (cw—‘2)? — 2A2([a—1]2+1>+ A4 = 0 01‘ (,\2—a2)(A2—[a—2]2) = 0 so the characteristic. roots are a=iA,2iA 148 Thus T is given by T(r) = linear combination {TA , 'r‘A , r2_’\ , 7‘2“} (A.8) The solution of (A.1) is therefore given by \IJ(7‘, 0) = 2 [EA sin A0 + F; cos A0] T(r;A) (A.9) ,\ where T(7‘;A) = a,\ 7"\ + In 7"'\ + c,\ 7‘2—'\ + (11 7‘2“ (A.10) A.2 Eigenvalues The linear combinations (A.10) may be formed into a countably infinite set of eigen- functions (1530(7) = T(r; An) associated with the eigenvalues An . We need the solution (A.9) to satisfy the boundary conditions (A?) . The eigenfunctions 06$”)(r) are thus the combinations (A.10) which satisfy the homogeneous side-wall conditions T(a) = T(b) = T(a) = T(b) = 0 (A11) Substituting equation (A.10) into the conditions (A.11) we get four linear homoge- neous equations whose coefficient matrix is designated as bA b—A bz—A b2+A M = (A1?) Aa’\‘1 -—Aa"\”l (’2 — A) al‘A (2 + A) a1“ Ab*—‘ —A b"‘-1 (2 — A)b1'A (2 + A) 19+A The linear equations are solvable if and only if detM = 0 (A.13) Expanding the determinant given in (A.13) then after an easy long computation it simplifies to 0 = 8(1—A2)ab — 46+“ 111-2" + 4A2a'1b3 — 4a1’2’\ 121+“ + 4A2 a3b-1 (A.14) but 1+2,\ 1-2A 1—2,\ 1+2,\ , a 2A a -2A _4a b — 4(1 1) : —4ab E + _g (‘A15) and also we have 2 8(1-—A2)ab + 41122-le + 4A2a3b‘1 = Sab + 4abA2 E — 9] (A.16) a Using (A.15) - (A.16) in (A.14) then after dividing both sides by —4ab equation (A.14) could be rewritten as 150 [6)" — (a)? - leF =0 M or a a b 9 i ‘— : - — -— . . ..smh[(log b) A] :l: [b a] A (A18) or equivalently 'n[z'(1 3)A] — ii 3 9 A (A19) SI 0g 1) — 2 b a ' In particular the eigenvalues An satisfy sin An = i0 An A (A20) A,,,::(ilog%)An ; ,02—1—(g—%) 2log-alZ There are countably infinite number of eigenvalues An of (A.‘20) which are sym- metrically located in the complex A plane . we make use of the eigenvalues with positive real parts . The eigenvalues An (72 : 1, ‘2, 3, ...) are the first quadrant roots of (A20) ordered according to the size of their real parts . The eigenvalues with a negative index are defined by Then , using (A.10) , we have gal—"M = 31"”(2) Using the boundary conditions (A.11) then the coefficients an , bn , c.n & dn of the eigenfunctions T(r;An) : WW) satisfy the following equations 151 0 = a’\" (1n + a"\" bn + (124" on + a2+’\" (In 0 = An<1’\”‘1 an — Ana—A"-1 bn + (2 — An)a1"\" cn + (An + 2)al""\n dn (A21) 0 = b": an + b-An bn + 52"“! on + 122“" dn 0 = Anbknrl an — Ankh-1b” + (2 — An)bl-"n on + (An + 2)b*n+1 dn Solving equations (A21) then the coefficients may be determined to within an arbi- trary multiplicative constant ; so if we normalize dn then we get ,\ (1 — Mm“ an = 1—rin a2 _JTTTF) 1‘ (2) bn = An 11”" (44—) 1‘ (3) (A22) n 12 ,, 1 - (3)9 C71 — Anl—l a2\ l—An b2\ (1_ (3)2An) dn 2 1 A.3 Discussion of Eigenvalues The eigenvalues given in (A20) were analyzed in several papers . These roots can be computed by iteration but analytical solutions were found . A method was suggested by Burniston and Siewart [8] for obtaining exact analytical closed-form formulae for the zeros of the following more general form of (A20) a sinC = C (A23) where a is an arbitrary assigned complex constant. The advantage of their method is that they give the zeros through formulae containing only real integrals and not complex contour integrals . They make use of the theory of the Riemann-Hilbert 152 boundary—value problem in the theory of analytic functions . A less complicated method was proposed by Anastasselou and Ioakimidis [1] where they solve a simple discontinuity problem instead of the more complicated Riemann-Hilbert problem , moreover , their approach requires less computations for the derivation of numerical results by using an n-point numerical integration rule for the evaluation of integrals. The only significant analytical information that was available before a closed-form solution was found were asymptotic expressions given by Hardy [20] , [21] ; for example , (n ~ (472-51); + i{—lna + ln(4n+1)7r} , a > 0 (A24) _I The eigenvalues (A23) arise in Stoke’s flow problems as we have seen in previous chapters, for example , flow in sectorial regions , flow in a rectangular strip and flow between concentric cylinders . They are particularly important in studies of plane biharmonic functions in infinite or semi-infinite strips, as , for example , in the determination of the stress field in a thin plate in either plane strain or flexure [5] , [15] , [36] . Equation (A23) , see Fettis [16] , and for real 0 have a finite number of real roots and an infinite number of pairs of conjugate complex ones . Since equation (A23) is unchanged if C is replaced by —C , the complex roots of (A23) are symmetrically situated in each of the four quadrants of the complex C-plane . For this last reason and without loss of generality we will be restricted to roots in the first quadrant . For our case and from equation (A20) we have b 0:. lb(-—g) , 01 Let g(:r)=a:2—‘2:rlna: $21 Since g(1) = 1 , then upon computing the first derivative it easily follows that y(r) Z 1 for :1: Z 1 so we conclude that 0 > 1 Thus for our case the equation sinA = ,0A has A = 0 as its only simple real root ( since ,0 > 1 ) , see Fettis [16], all other zeros occur in complex conjugate pairs , lying symmetrically in the four quadrants of the A-plane . However the equation sinA : —,0A has two real roots , namely , A = 0, iln(a/b) as is clear from (A1?) and (A.19) , the rest occur in complex conjugate pairs , lying symmetrically in the four quadrants of the A-plane . Also from the asymptotic formulae (A241) we note the real part increases with n so that enabled us to arrange them in the previous section in a sequence of increasing 154 real part . In conclusion , from the previous two sections the series \Il(r,0) = Z [EnsinAn0 + FncosAn0] (550(7) (A25) --00 satisfies equations (Al) and (A2) where ¢gn)(r) = an TA” + bn T—An + Cn 7,2—An + dn 7.2+)”, The coefficients (1,, ,b,, ,0" ,dn and the corresponding eigenvalues An are given in equations (A22) and (A20) respectively . These eigenvalues are symmetrically located in the complex A plane ; we choose An , n. > 0 , the first quadrant roots and are arranged in a sequence corresponding to increasing size of their real part and define [\9 00 v ¢§_n)(r) = We) (n = 1, is the eigenfunction corresponding to the eigenvalue A_n . If the edge data are real then E—nZE—n i F—n:?n Appendix B Axisymmetric Flow in a Torus B.1 Biharmonic equation in Toroidal Coordinates We consider next toroidal coordinates a , fl , (35 related to the rectangular coordinates :1: , y , z , see [19] and [32] , by the formulas c sinha cos <15 c sinha sin 0) c sinfl (I: z t : 7 Z : cosha — cosfl ’ J cosh a — cosfl cosha — cosfl where OSa 0 is a scale factor . This coordinate system is useful for solving boundary value problems involving the domain bounded by a torus , or the domain bounded by two intersecting spheres . If a point has cylindrical coordinates r , qb and z , then c sinh a cosh a -— cos B csinfl (‘2 | cosh a — cos B The corresponding triply orthogonal system of surfaces consists of the toroidal sur- faces (Anchor rings) a = const. , described by the equation 155 156 a = CODSI [3 = const Figure B.1: Toroidal coordinates in a meridian plane 2 2 C 2 (r — ccotho') + 2 = . smh oi the spheres (Spherical caps or lenses) I 0 = const. , described by the equation 2 (z — ccotfl)2 + 73.: (sififl) and the planes 05 = const. , ( see figure 1 ) . It should be noted that all spheres intersect in the circle r = c , z = 0 . We seek a solution u(a, [3, o’) of the biharmonic equation in toroidal coordinates , namely , V4U(as'3, ¢) = 0 03-1) 157 We use separation of variables . To begin with consider Laplace’s operator in toroidal coordinates 8 0a 0 8a 1 02a szh—h— h—h— —— B2 “ 180(20a)+100(200)+h§0¢2 ( ) Here It _ (cosha — cosfl)3 1 — c35inha h __ csinha 2 — cosha — cosfl Clearly from (B2) we can separate the variable o in Laplace’s equation and ul— timately in the biharmonic equation . So seeking an eigenfunction solution in a direction it is necessary to require that 0) satisfies the following equation 0% a—w- 2 —/.£2 u (B.3) where ,u is a constant . After substituting (B.3) into (B2) we note that we cannot separate the other variables in this new equation . However , ifwe introduce a new unknown function 22 by making the substitution u = k(a,fl) v (BA) where k = \/2 sinha (cosh a —— cos mi 158 then the Laplace operator after a long computation and making use of (B3) becomes V211 = 32—\/2cosho' — 2cosfl Rv (8.5) c where Rv = sinha(cosha — cos.0)(vo.o. + 2235) + [2 + cosha(cosha — 3cosfl)] v0 — 2 sin/3 sinha '05 + Ix’v (B.6) with K given by K : 33:11—01 (4 coshsa — 3 cosha sinhzoz — cosfl sinh2a — 4 cosfl cosh2a) #2 cosh gngaCOSH (B.7) Making use of the Laplace’s operator written in the form (8.5) then the biharmonic equation (8.1) in terms of '0 reduces to (120)“)! + (R7066 + cotha (Rv)o_ + 132) + , 2 (Ho) 4 smh a Long simplification of (BS) and finally dividing both sides of the equation by the common factor (cosh a — cos 0) gives rise to the following equation in v , 159 0 = sinha (vaaaa + 2vaafifi + vflfififi) + 6 COSha (vaaa + ”0133) g ' I) coshza _ 9 11.2 (lg ' coshza _ #2 ) + (2 smha + "' sinha “sinha ’Ufifi + 2 smha + 5 sinha 2 sinha vac 1_7_' _ cosh3a __ c) 2 cosha) {(Q ' _ _3_ cosh201 cosh‘ a) + (2 0051101 m ~11 557: ”a + 16 Sinha 2 The. + T»: — 2 42-“? —1 l} ’1 (sinh a + 25inha v (13.9) From equation (B.9) we can now separate the variable 0 also . Thus seeking an eigenfunction solution in a direction it is necessary to require also that 0 satisfies the following equation v = F(O)B(fl)¢(¢) 03-10) such that B satisfies the equation B"(fl) + 123(5) = 0 (B.11) Where V is a constant . Also by (B3) (1) satisfies 51¢) + u2<1>(c,'>) = 0 (3.1?) Then after a long expansion the reduced equation in terms of a for the biharmonic equation is given by 160 0 = sinho'(F(“) — 21/217”) + W‘F) + 6cosha 17(3) —- 6V2coshaF(1)-— ”2(25111110 + 2% — 2“? )F + 1 (1—23 sinhza+5’cosh2a—2p2) 17(2) sinh a sinha sinha H 3 . 2 4 + (77 COSha - 93112-3 — 2/‘2 50—5123) F11) + (33 Slnlla __ §cosh a + cosh a) F sinh a sinh a 1—6' 2 sinha sinhza _ 2 ( 2—u2 1 ) F 4" sinhg a 2sinh a (13.13) or it could be simplified to 0 : sinh 0' FM) + 6 cosha [7(3) + —i—- {(5 — 2n?) + (£3 — 21/2) sinh2 a} Fm sinh a 2 sh . 2 . c h4 - 2 2- 2) + fi{(l§l — 61/2)smh o' — (1+3/l2lj [7(1) + sin1hcv{0S asinhgod # + (1/4 — 12—31/2 +%)sinhga + (211,2 —-2)1/'2 — “—22 — g} F (13.14) Let F(a) : A(s) , s = cosha Substituting the last equation into (B.14) and finally multiplying by sinh a then 24(3) will satisfy the following equation (52—1)3 A“) + 12s(s2 —1)2 14(3) + {(371—2112)(32—1)2 + 2182(82—1) + (5—2;12)(32—1)} 241(2) + {(26—81/2)s(32—l) + 4(1—p2)s} A“) + {(u4—%V2+1%)(52 — 1) + W1 + '2(;i"?—1)z/2 — %(p2+3)}A = 0 (13.15) 161 Thus A satisfies the following equation written in self-adjoint form (pA‘2))H + (cl/1(1)), + rA = 0 (B.16) Lv : (1 —32)vss — 232), + [uQ— % —(1—52)- #2] v Qv = (1 — 32)2vss — 2s(1-—.52)vs + (l — s2) [12 — — — (1 —sz)_1 #2] 1) (Bl?) then equation (B15) can thus be rewritten as 0 = L(Q(A)) — 2(1—82m”) + 43.4“) — 2112(1—32)_1A — 2(1/him + A 4 1—52 (B18) or 0 = L(Q(A)) — 2 <1—s2) A”) >+{(zfl—3)+ (1 ...2) 2}] A+ A 4 1—52 (3.19) the last equation reduces to The quadratic operator has the following roots {2 i \/4 — 4(1—4,1t2)}=1:l: 2n wit—4 so we can rewrite equation (B20) as [Q — (1+?u)1l [<2 — (1410110) = 0 (B21) the two operators in the last equation are commutative so A satisfies also 1Q — (1:210!)ch — (1+2/tlll(A) = 0 (1322) Solutions of the equation [Q — (1+211)I](A) = 0 or equivalently (1—82)2 Am — 28(1—52) A“) + ((1/2——)(1—s2) — (p2+2;i+1)} A :2 0 (B23) are given by 163 A(3) 2 linear combination {P:j:(s) , Qfi::(s)} (B24) 2 2 where P and Q are the associated Legendre functions of the first and second kinds, respectively . As for the solutions of [Q — (1_2#)Il(A) = 0 (3.25) or equivalently (1 —82)2 Am — 23(1—32) All) + {(1/2 —§)(1—52) — (,u2 —2;L+ 1)} A = 0 (B26) they are given by A(s) = linear combination ( P::1§1(s) , Q“:l(s)} (B27) By superposition of the solutions given in equations (B4) and (B.10) then the solution of the biharmonic equation (BI) is given by u I Z {6* }{e*“°}(P,/_11(8la Q)_:(s),P,,_+:(s). 6253(3)) (cosh a — cos 1'3) Luv 2 2 2 2 (8.28) NIH where s : cosh a 164 B.2 Axisymmetric Stokes flow in a torus We consider axisymmetric Stoke’s flow in a. torus . We thus seek a solution u(a , fl , ¢l of the following equation given in toroidal coordinates , see [19] , E4240, ,3, a) = 0 (13.29) We use separation of variables . To begin with consider the operator E2 in toroidal coordinates a 1 an a l au 2 _ ,2 _ __ __ _____ , Eu _ {a (w a) + W WW) (3.30) Here to _ sinha cosh o- — cos 5 l h = Z (cosho — c033) we note from (8.30) that separating variables is not possible in the equation E2 u = 0 . However , if we introduce a new unknown function v by making the substitution u 2 Ma, ,8) 1) (13.31) where k = c smha 3 (cosh a - c033)? then the E2 operator after a long computation and making use of (8.30) becomes 165 E221 = lsinha (cosho — cosfl)'% RU (8.32) C where Bi) 2 (cosha — cos B)('v(m + 2233) + (cscha — cotha cosfl — sinha) vo, — ‘2 sin6 v3 + Av (3.33) with A given by {(1 — Alcsch‘2 o')(cosha — cos B) + 4 cosfl} (8.34) ut-IH Making use of the E2 operator written in the form (8.32) then the Stoke’s equation (8.29) ,namely E2(E2u) = 0 in terms of v reduces to (BMW + (RUM); + cotha (Rv)0 + [i (sin25 + sinh2 a)(cosha — cos m—z — % cosfl (cosh a — cosfl)‘1 — csch2 a] Rv :: 0 (8.35) but since §(sin2)3 + sinh2 a) (cosh o- -— cos-(3)4 — % cosfi(cosha — cos ,3)‘1 : %(cosha — cos l3)‘2 {sin23 + sinh2cr — '2 cos ,3 cosha + '2 cos2 B} : %(cosho' — cos ,B)‘2 {cosh2 o' — '2 cosfl cosha + cos2 [3} = %(cosha — cos ,3)"2 (cosh o — cos ,3).2 = i 166 then equation (8.35) can be simplified to 1 (RD) + (BMW + cotha (Rv)o, + (Z — CSCh20z> Rv : O (8.36) GO A tedious simplification of equation (8.36) gives after dividing both sides of the equation by the common factor (cosh a — cos 3) 0 = ”053,63 + 'ZvOOaB + vaaaa + ‘2 cot-ha vac-0' + ‘3 cot-ha vow _ G + 3csc112a) vac, + (E — 2cschza) v55 + 3 cotha (cschza -— %) va 2 + (1% — §csch2 or — 3 csch4 a) v (8.37) Now equation (8.37) is separable . If we write 0 = F(CY) G03) (8.38) then the reduced equation of (8.37) is given by O = FG”) + 2Fl'zlGl2) + FMG + ‘2cothaFl3lG + 2cothaF(l)G(2) — (l + 3csch20') FmG + (3 — 2csch2a) F0”) 2 +3cotha(csch2cr — %) FmG' + (3 — %csch?oz — 3csch4a) F0 6 p—l (B39) 167 Dividing the latter equation by G then equation (8.39) is seperable if G satisfies OI‘ o”(3) + flow) = 0 (3.40) where l/ is a constant . After substituting (8.40) into equation (8.39) the reduced equation in terms of a for the Stoke’s equation is given by 0 = F”) — 21/2Fl2) + V4F + 2cothaFl3) — 21/2 cothaFm -— G + 3CSCl120) 13(2) — z/2 (-5— - 2cscl12a) F + 3cotlia(csc112a —- 1) 17(1) 2 2 + (9— —- §csch2a — 3csch4a) F 16 (8.41) or it could be simplified to 0 = F“) + 2cotha 17(3) — G + 21/2 + 3csc112cr) F0) + {3 cotha(csch2a — %) — 21/2 cotha} F“) (8.42) + (V’ ~31} + 3 + (21/2 — %) cschzo' — 3csch4a) F 16 Let F(a) = I\'(s) . s = cosha 168 Substituting the last equation into (8.42) then K(s) will satisfy the following equa- tion 0 = (32—1)2 K“) + 83(52—1)K(3) + {(2—25—‘21/2)(32—1) + 6} Km + (1—41/2)5K(1)+{V“ — 31/2 + 3% + (21/2 — i) (s2 —1)—1 — 3 (s2 —1)_2}K Then K satisfies the following equation written in self-adjoint form I! (plx’m) + (qu) + 7'11, 2 0 (8.44) Define the operator L in terms of the new variable 3 as follows _ (1— 32)—1] v (8.45) VPIF‘ Lv = (1—32)v$3 — 2.935 —)—[1/2 — then equation (8.43) can thus be rewritten as 0 = L2(K) + 2 L(K) + (1—41/2) K (3.46) 169 The quadratic operator has the following roots (2 i \/4 — 4(1—4%)) = 1 :l: 21/ ml H so we can rewrite equation (8.46) as [L + (141/)1] [L + (1+2u)1)(1<) = 0 (3.47) The two operators in the last equation are commutative so K satisfies also II o [L + (1+21/)I] [L + (1 _21/)1](1<) (3.48) Solutions of the equation [L + (1—2u)1)(1<) = 0 or equivalently (1—52)1{(2l — 23 K“) + {(V_§)(V_l) _ (1—32)-1} K = 0 (3.49) are given by K(s) .—.. linear combination {133-1(3) , Qi_%(s)} (8.50) 170 where P and Q are the associated Legendre functions of the first and second kinds, respectively . As for the solutions of [L + (1+2u)1](1\.’) = o (3.51) or equivalently (1—32) K”) — 2.5 K“) + {(u+7)(u+§-) — (1—52)-1} K = 0 (3.52) they are given by K(s) 2 linear combination {P:+1§(s) , 1 2(5)} (8.53) By superposition of the solutions given in equations (8.31) and (8.38) then the solu- tion of the Stoke’s equation (8.29) is given by sinh a u = 234%. com} {1014343333 103.1(5), 344(8)} ~ (3.54) (cosh oz — cos 6)? where Bibliography [1] [21 [3] [4] l5] [6] [7] [8l [9] [10] BC. Anastasselou and N .I. Ioakimidis , A new method for obtaining exact analytical formulae for the roots of transcendental functions , Letters in Mathematical Physics 8 (1984) 135-143 . .1. Bear , Dynamics of Fluids in Porous Media , Elsevier , New York , 1972 . R. Berker , Integration des équations du mouvement d’un fluide visqueux incompressible , Handbuch der physik , Ed. 5. Flugge , Vol. VIII/2 , Springer , Berlin , 1-384 . G.A. Bliss , A Boundary Value Problem For A System Of Ordinary Linear Differential Equations Of The First Order , American Mathe- matical Society , Transactions 28 (1926) 561-584 . V.T. Buchwald , Eigenfunctions of plane elastostatics , I. The strip , Proc. Roy. Soc. London, Ser.A. 277 (1964) 385-400 . H. Bouwer, Groundwater Hydrology, McGraw—Hill , New York , 1978 . O.R. Burggraf , Analytical and numerical studies of the structure of steady seperated flows , J. Fluid Mech. 24 (1966) 113-151 . 8.8. Burniston and CE. Siewert , Exact analytical solutions of the transcendental equation a sin6 = 6 , SIAM J. Appl. Math. 24 (1973) 460-466 . A.J. Chorin and J.E. Marsden , A Mathematical Introduction to Fluid Mechanics , Springer-Verlag , 1990 . Z. Dagan, S. Weinbaum and R. Pfeffer , An infinite-series solution for the creeping motion through an orifice of finite length , J. Fluid Mech. 115 (1982) 505-523 . Z. Dagan, S. Weinbaum and R. Pfeffer , General theory for the creeping motion of a finite sphere along the axis of a circular orifice , J. Fluid Mech. 117(1982) 143-170 . Z. Dagan, S. Weinbaum and R. Pfeffer , Axisymmetric stagnation flow of a spherical particle near a finite planar surface at zero Reynolds number , J. Fluid Mech. 122(1982) 273-294 . 171 172 [13] F. A. L. Dullien , Porous Media-Fluid Transport and Pore Structure , Academic , New York , 1979 . [14] S. Eskinazi , Principles OfFluid Mechanics , Boston : Allyn and Bacon, Inc. , 1968 . [15] FA. Gaydon and W.M. Shepherd , Generalized plane stress in a semi- infinite strip under arbitrary end-load , Ibid., 281 ( 1964) 184-206 . [16] H.E. Fettis , Complex roots of sinz = a2. cosz = az.andcosh z = a2 , Mathematics Of Computation 30 (1976) 541-545 . [17] Y.C. F ung and SS. Sobin , Pulmonary Alveolar Blood Flow , Bioengi- neering Aspects of the Lung , ed. West , J.B. , Dekker , 1977 , Chapter 4 . [18] RD. Gregory , The semi-infinite strip at _>_ 0 , -1 S y S 1 ; complete- ness of the Papkovich-Fadle eigenfunctions when ¢$r(0,y),¢yy(0,y) are prescribed , J. Elasticity 10 (1980) 57-80 . [19] J. Happel and H. Brenner , Low Reynolds Number Hydrodynamics , N.J. : Prentice-Hall , 1965 . [20] G.H. Hardy , On the zeroes of the integral function :r — sinx = 2n+1 Zfo(—1)"‘1;n+l, , Messenger of Mathematics 31 (1902) 161-165 . [21] G.H. Hardy , On the zeroes of certain integral functions , Messenger of Mathematics 32 (1903) 36-45 . [22] DD. Joseph , Slow motion and viscometric motion ; Stability and bifurication of the rest state of a simple fluid , Arch. Rational Mech. Anal. , 56(1974) 99-156 . [23] DD. Joseph , The convergence of biorthogonal series for biharmonic and Stokes flow edge problems , Part I , SIAM J. Appl. Math. 33(1977) 337-347 . [24] DD. Joseph , A new separation of variables theory for problems of Stokes flow and elasticity , In : Trends in Applications of Pure Mathe- matics to Mechanics, Vol. II (H. Zorski,Ed.) London : Pitman , 1979. [25] DD. Joseph and R.L. Fosdick , The free surface on a liquid between cylinders rotating at different speeds , Part I , Arch. Rational Mech. Anal. , 49 (1973) 321-380 . [26] DD. Joseph and L. Sturges , The convergence of biorthogonal series for biharmonic and Stokes flow edge problems , Part II , SIAM J. Appl. Math. 34 (1978) 7-26 . [27] DD. Joseph and L. Sturges , The free surface on a liquid filling a trench heated from its side , J. Fluid [Mech. , 69 (1975) 565-589 . 173 [28] DD. Joseph , L. Sturges and W.H. Warner , Convergence of biorthogo- [‘29] [30] [31] [321 [33] [431 nal series of biharmonic eigenfunctions by the method of Titchmarsh , Archive for Rational Mechanics and Analysis 78 (1982) 223-274. Y.Y. Jung and DD. Joseph , Stoke’s flow in a trench between concen- tric cylinders , SIAM J. Appl. Math. 34 (1978) 247-285 . W.E. Langlois , Slow Viscous Flow , N.Y. : Macmillan Company , 1964. M.J. Lighthill , The position of the shock-wave in certain aerodynamic problems , Quart. J. Mech. Appl. Math. 1 (1948) 309-318 . N.N. Lebedev , Special Functions And Their Applications , Prentice- Hall , 1965 . C.H. Liu and DD. Joseph , Stokes flow in wedge-shaped trenches , J. Fluid Mech. 42 (1977) 443-463 . G.H. Liu and DD. Joseph , Stokes flow in conical trenches , SIAM J. Appl. Math. 34 (1978) 286-296 . H.K. Moffatt , Viscous and resistive eddies near a sharp corner , J. Fluid Mech. 18 (1964) p. 1. G.A. Nariboli , Eigen-functions for the strip problem , Mathematika 12 (1965) 58-64 . F. Pan and A. Acrivos , Steady flows in rectangular cavities , J. Fluid Mech. 28 (1967) 643-655 . T.N. Phillips , Singular matched eigenfunction expansions for Stokes flow around a corner , IMA Journal of Applied Mathematics 42 (1989) 13-26 . T.N. Phillips and AR. Davies , On semi-infinite spectral elements of Poisson problems with reentrant boundary singularities , J. Computa- tional Appl. Math. 21 (1988) 173-188 . A. Karageorghis and T.N. Phillips , Spectral collocation methods for Stokes flow in contraction geometries and unbounded domains , J. Comput. Phys. 80 (1989) 314-330 . T.N. Phillips and A. Karageorghis , Efficient direct methods for solv- ing the spectral collocation equations for Stokes flow in rectangularly decomposible domains , SIAM J. Sci. Stat. Comput. 10 (1989) 89-103 . T.N. Phillips and A. Karageorghis , A conforming spectral colloca- tion strategy for Stokes flow through a channel contraction , Applied Numerical Mathematics 7 (1991) 329-345 . L.C. Poteete , A matching technique for solving Stokes flow problems , Ph.D. Dissertation, Dept. Math. , Michigan State Univ. , 1993 . 174 [44] R.C.T. Smith , The bending of a semi-infinite strip , Austral. J. Sci. Res. 5 (1952) p. 227. [45] DA. Spence , A note on the eigenfunction expansion for the elastic strip , SIAM J. Appl. Math. 42 (1982) 155-173 . [46] SA. Trogdon and D.D. Joseph , Matched eigenfunction expansions for slow flow over a slot, J. Non-Newtonian Fluid Mech. 10(1982) 185-213. [47] C.Y. Wang , Exact solutions of the unsteady Navier-Stokes equations , Appl. Mech. Rev. 42 (1989) 8269-8282 . [48] C.Y. Wang , Flow in Narrow Curved Channels , Journal of Applied Mechanics 47 (1980) 7—10 . [49] C.Y. Wang , Perturbation Methods (Lecture Notes) , National Taiwan University , 1972 . [50] J .Y. Y00 and D.D. Joseph , Stokes flow in a trench between concentric cylinders , SIAM J. Appl. Math. 34 (1978) 247-285 . MICHIGRNSTQTE UNIV. LIBRARIES [I][I[lll][I][llllllllllllll[lllllllllllll 3129301.298986