119 806 THESIS - LIEEARY Michigan 8mm University : i! This is to certify that the thesis entitled AN APPROXIMATE PRESSURE DISTRIBUTION FOR A VISCOELASTIC FLUID IN CONICAL FLOW presented by Khuong Van Nguyen has been accepted towards fulfillment of the requirements for Master of Science Chemical Engineering degree in awn. Major professor Date November 1 , 1982 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES .ar— \— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. AN APPROXIMATE PRESSURE DISTRIBUTION FOR A VISCOELASTIC FLUID IN CONICAL FLOW By Khuong Van Nguyen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1982 ABSTRACT AN APPROXIMATE PRESSURE DISTRIBUTION FOR A VISCOELASTIC FLUID IN CONICAL FLOW By Khuong Van Nguyen A theory for estimating the isotropic pressure distribution for viscoelastic fluids in converging flow is developed. The theory assumes that to first order the kinematic structure of a pure radial converging flow is unaffected by the presence of a high molecular weight polymer. The normal stresses and shear stresses are estimated by using a retardation model for viscoelastic fluid. The isotropic pressure distribution is determined by integrating the radial com- ponent of the force balance. The results show that under certain conditions a positive minimum in the isotropic pressure occurs near the contraction and, thereby, provides a quantitative explanation of 'hazing.‘ Furthermore, a simple critefibn for flow instability is suggested which relates the applied pressure Pm, the fluid vis- cosity u,and the fluid retardation time X. Instability occurs when XPm/u _<_ 0.015. This thesis is dedicated to my parents and my wife, Linh Nguyen, whose love and concern have meant much to me. ii ACKNOWLEDGMENTS I would like to express my sincere appreciation to Dr. Charles A. Petty, without whose help this thesis would not have been possible. The partial financial support from National Science Foundation (Grant No. 71-1642) is also acknowledged. TABLE OF CONTENTS Page LIST OF FIGURES ........................ v NOTATION ........................... vii Chapter 1. INTRODUCTION ...................... 1 The Physical Problem ................. 1 Objectives ...................... 7 2. THEORY ......................... 9 Fundamental Equations ................ 9 Component Equations ................. 13 Radial Flow of a Newtonian Fluid ........... 14 3. RESULTS AND DISCUSSION ................. 25 4. CONCLUSIONS ...................... 39 LIST OF REFERENCES ...................... 40 iv LIST OF FIGURES Schematic of Flow Phenomenon ............. Schematic of Flow Pattern Predicted Theoretically . . . Definitions of Geometric Parameters .......... Stream Function for Axisymmetric Conical Flow of a Newtonian Fluid for d/D-+ O .............. Velocity Distribution for Axisymmetric Conical Flow of a Newtonian Fluid for d/D-+ O ......... Shear Stress Distribution for Axisymmetric Conical Flow of a Newtonian Fluid for d/D-+ O ........... Distribution of Compressive and Tensile Stresses for Axisymmetric Conical Flow of a Newtonian Fluid for d/D'+ O ........................ Pressure Distribution for Axisymmetric Conical Flow of a Newtonian Fluid for d/D'+ O ............. Locus of Zero Pressure for Axisymmetric Conical Flow of a Newtonian Fluid for d/D-+ 0 ............. Pressure Distribution for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant r, Case A) . Pressure Distribution for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D-+ O (constant w, Case A) . Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant w, Case A) . Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant Pm, Case A) Pressure Distribution at Center Line for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D +'O (Case A) ....................... Page 10 16 18 19 21 22 23 28 29 3O 33 Figure 3.4 3.5 3.6a 3.6b vi Distribution of Total Compressive and Tensile Stresses for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + 0 (Case A) ................. The Effect of Elasticity on the Growth of the 'Bulge' Region for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + 0 (Case A) .............. Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D-+ 0 (constant H, Case B) . Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D-+ D (constant P”, Case B) Page 34 36 37 38 “—1 llw lc ‘G-T: lid NOTATION Symbols Tube diameter Isotropic pressure Volumetric flow rate Radial coordinate Critical distance defined in Figure 3.2a Minimal distance defined in Figure 3.2a Strain rate tensor Total stress tensor Velocity vector Bulk average velocity Greek Symbols Retardation time defined by Equation (1.1) Viscosity coefficient appearing in Equation (1.1) Stream function Extra stress tensor vii CHAPTER 1 INTRODUCTION The Physical Problem Converging flows of Viscoelastic materials are encountered in numerous processes involving polymer melts and solutions. These flows have probably been studied as much for their unique and unusual behavior as for their pragmatic importance. For instance, an aqueous solution of polyacrylamide undergoing a 4:1 contraction at low Reynolds numbers can not only support a symmetric vortex flow off the axis of symmetry but also a flow 'bulge' simultaneously on the axis (see Figure 1.1a). Cable and Boger [19786.b] recently presented the results of a comprehensive experimental study of these structures and their stability. Just as curious is the phenomenon illustrated by Figure 1.1b, which shows a fairly large bubble retarded at the mouth of a contraction by an accelerating Viscoelastic fluid. Metzner [1967] attributed this phenomenon to the large normal stresses which usually accompany a converging Viscoelastic fluid. These normal stresses may also cause a minimum in the isotropic pressure just before a contraction and, thereby, explain the phenomenon known as 'hazing' illustrated in Figure 1.1c (Metzner et al., 1969). Fur- thermore, elastic forces also have important consequences in polymer extrusion as shown in Figure 1.1d and discussed by Denn [1980]. "Bubble" “BU'ge’, R°=TOO \\ 4:1 Contraction O “"“"<" ““““ "‘9' .TTT‘\-~L~__;:TEI a. b. (Cable & Boger,1978) (Uebler,1966) “Haze" / Clear Clear {1 1 \3§] (1.1) no will be used to estimate the distribution of forces in conical flow. In Equation (1.1) u is a shear viscosity and A is a retardation time. Although both of these material parameters should be con- sidered as scalar functions of the kinematic invariants of the strain rate 33 we follow the lead of many others (see, especially, the discussion in Black and Denn, 1975) and consider these as constants. Equation (1.1) contains the physical idea that the kinematic structure of the flow, as described by the strain rate :3 does not respond instantaneously to sudden change in the stress ;3 Unlike a Newtonian fluid, the flow is retarded by intrinsic elastic forces and thus a fluid element requires some finite amount of time to adjust to new surroundings. The temporal operator, which appears in Equation (1.1), is the familiar Oldroyd time derivative defined by CT‘ 0) m \2 a! ar CD + M: Q ‘0 a0) .+ f'\ '0 t: v “U1 + mfi ‘Q L; :3 £3 5t Even if the flow is steady in an Eulerian frame of reference, the remaining terms in Equation (1.2) still make important contributions for converging flows which are really unsteady in a Lagrangian sense. Objectives The purpose of this study is to develop some understanding of converging flows of viscoelastic fluids governed by Equation(1.1); Although the approach is elementary, it should nevertheless provide a basis for a more quantitative study. Our specific goal will be to estimate the distribution of forces (isotropic pressure, normal stresses, and shear stresses) induced by radial flow from a large reservoir to a small contraction. Because the underlying flow model includes an inward radial flow to a point (sink), the velocity and pressure necessarily become unbounded at the contraction. This is not a major concern for esti- mating the effects of elasticity far from this singularity but, unfortunately, most of the interesting results illustrated in Figure 1.1 occur near the contraction. Comparing the behavior of two models near their singular points may still provide some useful insights regarding the location and relative sizes of the flow domains which develop unusually large stresses. For example, radial flow of a Newtonian fluid toward a point sink generates a zero pressure distribution on a surface surrounding the contraction. The pressure is negative within this domain (see Figure 2.7). Obviously, this is unphysical and the fluid easily prefers a secondary motion rather than one which induces unbounded stresses. By comparing the patho- logical features of 'sink' flow for Newtonian and viscoelastic fluids some deductions regarding the onset of secondary motions may be inferred. 'Hna attractiveness of this suggestion lies in the sim- plicity of the analysis which follows. CHAPTER 2 THEORY Fundamental Equations Figure 2.1 shows the geometry of the flow. Spherical coordi- nates are used with F = 0 located at the contraction; the parameter d represents the diameter of the small capillary. The flow is entirely radial and axisymmetric, so the velocity vector can be written as (2.1) §:> [I II) 1 r\ S) (D V VB 3 V where Gr is the radial component. The continuity equation (see page 83 of Bird et al., 1960) requires Azh Ur :- con/S'l‘an't ) (2.2) which shows that Gr is unbounded for ;.+ 0 if we insist on (2.1) thoughout the flow domain. ///// \E ///// o4- Figure 2.1. Definition of Geometric Parameters \ \\\\\\\\\\. \\\\\\\\\\ 11 The bulk average velocity Mb 3 ‘l'Q/TL’. a; (2.3) J and the diameter of the small capillary d are used as characteristic velocity and length scales. The stresses are made dimensionless by using the volumetric flow rate Q, the viscosity coefficient u. and the diameter d. Thus, a dimensionless applied pressure at r = w is defined as Pa, _-—__. “A5 Ace (2.4) /“ <1 It will be useful to remember that increases in 0 correspond to decreases in P0° for fixed Pm. The governing dimensionless equation of motion for steady state creeping flow is Va:- 1‘. O (2.5) where "—1 II I "U H + 01 (2.6) 12 The constitutive model proposed for ;:is given by Equation (1.1). In dimensionless form this becomes ’2: = 2(5 + W83) (2'7) .- g at where the Weissenberg number, defined by w E A uh (2'8) :1 compares an intrinsic time for retardation with a characteristic time for the flow, i.e., fi%u This parameter and Pm determine the distribution of forces in the infinite reservoir. Boundary conditions for the problem studied include no slip at the wall, A A UV.(P,TE/Z) : O 3 (2-9) symmetry about the axis, 3.3: .. o . (2.10) 29 v.0 and, P- ‘a, . (2.11) 13> u a" v 13 Component Equations For axisymmetric flows in spherical coordinates the two rele- vant component equations (made dimensionless) associated with Equation (2.5) are _:_P_ - I?“ t...) (2.12) r r" ' “(zr. Sme') YSMG )0 (_2590 -+ 25¢4S) r 9 J 1- 75' = i3 3‘: (" z") + rs' 0 5.9 _(2’,, sine) (2.13) m The nine component equations for (2.7) in spherical coordinates are tedius, but not difficult to derive. For axisymmetric radial flow, the result is (see Bhatnagar and Rajagopalan, 1967) Z" = z 23.: .. 2w {11min} 28%)} (2.14) 14 tfl =39: +2w51.u,>ur r r )r' 2 Z = Z = .3— 23! W { u 3 ur re or r 9 'i 3,1 )rJG } Zr¢ at,“ = 20‘ 1 iff 1: 0 Radial Flow of a Newtonian Fluid (2.15) (2.16) (2.17) (2.18) Setting w = 0 in Equations (2.14)-(2.17) gives the components of the stress for Newtonian fluid. Inserting the results into Equations (2.12) and (2.13) gives two equations for the unknown functions p(r,e) and ur(r,e). Actually, continuity has already imposed itself on the radial dependence of ur(r,e), so an ordinary differential equation for the stream function 0(9), defined by 15 ' A“? (2.19) results from the requirement that ‘ 1 DP -_-_ 3.13._ (2.20) )0. Jr )r )9 The stream function is easily found to be A A 2 Y(0') :: ‘l’.(o) €040- ) (2.21) where ‘i’(0) = -9. (2.22) and Q is the volumetric flow rate. Figure 2.2 shows how the stream 11' function changes with 6 for 0 3.6 g The dimensionless velocity (Gr/uh) can be determined by insert- ing (2.21) into (2.19) with the result that «safe . (2.23) r!» U :-..3... ' a Note that the no slip condition is satisfied at the surface 6 = %-. Also, Ur +0 as r +-m and, unfortunately, ”r +.w as r + 0. This 16 O 2 90 4‘5 0 700 9:0 0 Me) Me) Figure 2.2 Stream Function for Axisymmetric Conical Flow Of a Newtonian Fluid for d/D + 0 17 velocity field has been used previously by several authors (see Introduction) as a zero order approximation for converging flows of polymer solutions. This is the analog for conical flow of Stokes' classical result for creeping flow past a sphere. Figure 2.3 shows the behavior of ”r with e for several spherical surfaces. Note that the flow has decayed significantly within a few capillary diameters from the contraction. Figure 2.4 shows the distribution of shear stresses made dimen- sionless with liii-and given by TI 23,, -.-. - a case sine (2.24) ,~s Tre E 1 corresponds to a pressure drop needed for a flow rate 0 through a capillary tube having an L/d of only 1/128! Thus the magnitude of the shear stress is extremely small even within one diameter of the singular point. On the other hand, the normal stresses are much larger in magnitude. However, the shear stresses still play a key role for the Newtonian case since the normal stress terms in Equation (2.12) balance each other identically. One curi- osity about Newtonian sink flows follows from Equation (2.24) which shows that the stress is zero at the rigid interface. Thus, the velocity and the shear stress are both zero on the same surface, a feature not normally associated with real viscous flows. The compressive and tensile stresses are given by 18 lJr I 0 20" 45° 70° 90° 2.0 -1 F 1.0 -2 _- -3 ’— r=CL5 -14 )- ¢ D ._._.., Q \ -5 _. I § \ a \ ~ \ \ . \ \ -6 —’ I ‘—d l Ub Figure 2.3. Velocity Distribution for Axisymmetric Conical Flow of a Newtonian Fluid for d/D + 0 19 Figure 2.4. Shear Stress Distribution for Axisymmetric Conical Flow of a Newtonian Fluid for d/D + O 20 trr = ‘6 5.2123- (2'25) 9" z¢¢ : tea :1. 3 00529 (2.26) Y3 Note that t" + a... +2.27 ,0 (2.27) Figure 2.5 shows that these normal stresses are an order of magnitude larger than the shear stresses for this flow. Thus, 'sink' flow should be an excellent kinematical model to develop insight regarding viscoelastic flows (Chapter 3). The pressure distribution for w = O is given by p ____ pa _+ l—3coo‘6» (2.23) r3 Figure 2.6 shows that P = P0° for e = 54.75° and all r > 0. For a < 54.75°, P < Pm; and, for e > 54.75°, P > Pm. This qualitative feature of 'sink' flow has been verified experimentally by Bond [1925]. Figure 2.7 shows the locus of zero pressure for two different values of Pm. Clearly no real flow will show this type of behavior and the 'bulge' where negative pressures occur will be replaced by another flow pattern having more than one velocity component. In 20,. _L O I Normal Stress. Zii '40; Figure 2.5 1.0 2.0 21 2.0 1.0 45° F ' 90° T= 0.5 Distribution of Compressive and Tensile Stresses for Axisymmetric Conical Flow of a Newtonian Fluid for d/D+O 22 16 14_ 12- 10- P "P f=0.5 1.0 2.0 Figure 2.6 Pressure Distribution for Axisymmetric Conical Flow of a Newtonian Fluid for d/D + 0 90° 23 8:0 A O 1.0" N=.60 PP°° ’ - >a=90° .5 Figure 2.7 Locus of Zero Pressure for Axisymmetric Conical Flow of a Newtonian Fluid for d/D + O 24 the next chapter we examine the effect of elasticity on the distri- bution of stresses and, in particular, show how the negative bulge region changes with the Neissenberg number. An interesting result which follows is that for large enough w the region can be eliminated. CHAPTER 3 RESULTS AND DISCUSSION Two approximations to the pressure field were examined. Both approaches assume that the radial flow given by Equation (2.23) can be used as a first approximation. Thus, the components of}; can easily be computed for this flow and from Equations (2.14)-(2.17) we Obta I n O a _L l+cosflfi ~23}. c0510 |+C0$99 (3.2) 2'“ 2r3( ) ‘1 r‘( X ) '5 =__'_5_ 14-01529 -229. l'i'COSQ‘O’z 3.3 w “A ) 8 r,,( ) ( ) Cy, : {370+ c.0529) .1 9%(1‘103526) sin 24} (3.4) 25 26 The isotropic pressure distribution to first order was cal- culated by simply integrating Equation (2.12). If "r were an actual solution to Equations (2.12) and (2.13), then this would give the same result as integrating Equation (2.13). However, because w > 0, these two procedures yield different results because Equation (2.20) is not necessarily satisfied for ”r defined by Equation (2.23). Therefore, in what follows, Case A denotes the procedure which deter- mines P so the radial component of the balance equation is satisfied everywhere. For Case B, Equation (2.13) is integrated from 0 = 0 to e. The pressure distribution for e = O is determined by inte- grating (2.12) along the axis. Thus, this procedure yields a pres- sure field which satisfies the e-component of the force balance everywhere and the radial component balance only on the axis. The two results are Case A: - - ' 3c. 29+) 3.5 10,, P .. 7d .27.)- ( ) V__‘_/_ (8.25 cosae .. 0.18:.0546 +8.53) . r6 Case B: p __ p :__1__ 3co52er+1 __ (3.6) )1". (10.68 (.0520 -4.5cos4e .1 22,31) . re 27 The effect of the Weissenberg number on the pressure distribu- tion for a fixed distance from the apex is illustrated in Figure 3.1a. Note the decrease in the critical angle where P = P°° as N increases. Thus, as the elastic effect increases the amount of fluid driven to the contraction under a favorable pressure drop decreases. Fur- thermore, the magnitude of IAPI seems to be affected more on the axis (a = 0) than elsewhere. At 0 90°, w does not diminish |AP| at all. Figure 3.1b shows that for N 0.1 a fluid particle on the axis experiences more than a four-fold increase in AP over a distance of less than 2d. Thus, the fluid experiences a significant accelera- tion along this path. A comparable retarding effect (i.e., AP < 0) occurs near the surface at 6 = 90°. Figure 3.1b seems to convey the idea that one mechanism near the axis causes the fluid to move toward the apex while another is responsible for flow toward the apex near the wall. Figures 3.2a and b attempt to show most of the important qualitative effects of the Neissenberg number on the pressure dis- tribution. First note the dramatic shift away from the Newtonian result (cf. Figure 2.7). For P0° > 0.606 and w = 0.1, a region of negative pressures no longer exists. Moreover, for fixed Pm, increasing N also eliminates this phenomenon. It is tempting to conjecture that the 'bulge' of negative pressures would be replaced in a real fluid by a secondary flow. Thus, a criteria for removing the existence of negative pressures 28 VV=2 05° Figure 3.1a. Pressure Distribution for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant r, Case A) Figure 3.1b. 29 r /l':1.5 r=2.5 r=311 A—JD—e 4'50 NOD Pressure Distribution for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant w, Case A) P:>lfi, 0 ‘ f $83900 .5 1.0 Figure 3.2a. Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant N, Case A) Figure 3.2b. Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + 0 (constant R”, Case A) 32 near the contraction becomes a stability criterion. Therefore, we tentatively offer this observation as a hypothesis for further study: Inatnbittty Hypothesis. A sufficient condition for the occurrence of a secondary motion on the axis of flow is WP“, <0.06 (3.7) Equation (3.7) follows directly from Equation (3.5) by setting P = O and determining the limits of the negative pressure 'bulge' on the axis a = 0. If Inequality (3.7) is satisfied, then two real roots exist. Also shown in Figures 3.2a and b are lines of minimum pressure. The physical significance of this is made clearer by examining Figure 3.3. Note once again the dramatic effect the Weissenberg number has on the isotropic pressure distribution. The result por- trayed here shows that a positive minimum in P occurs about one and a half diameters from the contraction. If the polymer were saturated with air at Pm, then 'hazing' would occur (see Figure 1.1c). Figure 3.4 shows the variation of the total normal stresses on the axis. On the axis, r and rm denote, respectively, the point where o [AP| = O and the position of minimum pressure. From Equation (3.5) it follows that ”’1' C) VV Figure 3.3. Pressure Distribution at Center Line for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + 0 (Case A) 34 + W=.15 3.0 _ 2..- / Trr (Compressive Stress) -5-1 Total Normal Sress, T“ \ T99 ( Tensile Stress ) -3.S_) -01 Distribution of Total Compressive and Tensile Stresses for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D‘+ 0 (Case A) Figure 3.4. 35 2.02 wv’ (3-3) 55 II and l 2.54 WA :7 II (3.9) The hazing phenomenon discussed by Metzner et al. [1969] could pos- sibly be used to test these predictions. Finally, Figure 3.5 shows how the size of the negative pressure 'bulge' varies with N for two values (Hi Pm. The parameter ArB simply denotes the width of this region on the axis. For a fixed Neissenberg number, say w = 0.25, the 'bulge' obviously does not exist for Pm,< 0.3. However, by decreasing Pco or by increasing the flow rate 0 (see Equation (2.4)) a 'bulge' will appear and may trigger an instability. Figures 3.6a and b summarize the results obtained by calcu- lating the pressure distribution using the e-component of the force balance. Differences between this case and the previous one should be apparent. Ar Figure 3.5. The Effect of Elasticity on the Growth of the 'Bulge' Region for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + 0 (Case A) 37 VV=431 8=C> A r 0Pm P430, ‘VV=O Figure 3.60. F f -—)-9=900 Locus of Zero Pressure for Axisymmetric Conical Flow of a Viscoelastic Fluid for d/D + O (constant P”, Case B) CHAPTER 4 CONCLUSIONS First-order approximate pressure distributions for viscoelastic fluids in conical flow were examined in this study. The results show that under certain conditions a positive minimum in the isotropic pressure occurs near the contraction. If the polymer were saturated with air at Pm, 'hazing' would occur at a location proportional to one-third of the volumetric flow rate (rman/B). A simple experi- ment on the 'hazing' phenomenon is recommended to verify the predic- tion. In addition, a negative pressure region was observed in this study along the centerline near the apex. It was postulated that this 'bulge' of negative pressure region would be replaced in a real flow by second motions. A condition for removing these negative pressures from the flow domain was derived. The result, which relates the applied pressure Pm to the fluid viscosity and retardation time, is Apm >/ 0.015 [A This interesting expression deserves further study. 39 LIST OF REFERENCES LIST OF REFERENCES Ackerberg, R.C., 1965. "The Viscous Incompressible Flow Inside a Cone," J. Fluid Mech. 21:1, 47. Bhatnager, R.K., and Rajagopalan, R., 1967. "Secondary Flow of Rivlin-Ericken Fluids Between Two Concentric Spheres," Rheo Acta 6, 15. Bird, R.B., Stewart, N.E., and Lightfoot, E.N., 1960. Transport Phenomena, Wiley, New York. Black, J.R., and Denn, M.M., 1976. "Converging Flow of a Visco- elastic Liquid," J. of Non-Newtonian Fluid Mechanics 1, 83. Bond, w.N., 1925. "Viscous Flow Through Wide-Angled Cones," Phil. Mag. 6, 1058. Cable, P.J., and Boger, D.V., 1978a. "A Comprehensive Experimental Investigation of Tubular Entry Flow of Viscoelastic Fluids: Part I. Vortex Characteristics in Stable Flow," A.I.Ch.E.I. 24:5, 869. , 1978b. "A Comprehensive Experimental Investigation of Tubular Entry Flow of Viscoelastic Fluids: Part II. The Velocity in Stable Flow," A.I.Ch.E.J. 2436, 992. Denn, M.M., 1980. Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J. Duda, J.L., and Vrentas, J.S., 1973. “Entrance Flow of Non-Newtonian Fluids," Trans. of the Society of Rheology 1251, 89. Harrison, N.T., 1920. "The Pressure in a Viscous Liquid Moving Through a Channel with Diverging Boundaries," Proc. Camb. Phil. Soc. 19, 307. Kaloni, P.N., 1965a. '"On Creeping Flow of a Viscoelastic Liquid in Converging Channel," J. of Phy. Society of Japan 3951, 132. , 1965b. "0n the Flow of an Elastico-Viscous Fluid in a Conical Duct," J. of Phy. Society of Japan 29:4, 610. 40 41 Leslie, F.M., 1960. "The Slow Flow of a Viscoelastic Liquid Past a Sphere,“ Quant. J. Mech. and Applied Math. 11, 36. Metzner, A.B., 1967. "Behavior of Suspended Matter in Rapidly Accelerating Viscoelastic Fluids: The Uebler Effect," A.I.Ch.E.J. 13:2, 316. Metzner, A.B., Uebler, E.A., and Chan Man Fong, C.F., 1969. "Con- verging Flow of Viscoelastic Materials," A.I.Ch.E.J. 15, 750. Oka, 5., and Takami, A., 1967. "The Steady Slow Motion of a Non- Newtonian Liquid Through a Tapered Tube," Japanese J. of Applied Physics 634, 1967. Perera, M.G.N., and Walters, K., 1976. "Long-Range Memory Effects in Flows Involving Abrupt Changes in Geometry," J. of Non- Newtonian Fluid Mechanics 2, 49. Shfimmer, P., 1967. "Zum Fliererhalten Nicht-Newtonscher FlUssig- keiten in Konischen Dusen," Rheo. Acta 6, 192. Strauss, K., 1975. "Theoretical Rheology," Appl. Sci. Publ., Barking, p. 56. Tanner, R.I., 1966. "Plane Creeping Flows of Incompressible Second- Order Fluids," The Physics of Fluids 936, 1246. Uebler, R.B., 1966. "Pipe Entrance Flow of Elastic Liquids," Ph.D. Thesis, Univ. Delaware, Newark, Delaware. Wagner, M.G., and Slattery, J.C., 1971. I'Slow Flow of a Non-Newtonian Fluid Past a Droplet," A.I.Ch.E.J. 11:5, 1198. ER HR “‘"lllliilliilllill) 1111111111)”