Direct Simulation of the Motion of Solid Particles in Viscoelastic Fluids

Direct Simulation of the Motion of Solid Particles in Viscoelastic Fluids A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY PETER YIJIAN HUANG IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY January 1997

Acknowledgment I want to express my sincere gratitude and appreciation to my advisor, Professor Daniel D. Joseph, for his help, advice and encouragement throughout the course of my study at the University of Minnesota. I feel so lucky to have him as my advisor. It is he who introduced me to the fantastic world of viscoelastic fluid dynamics. The past five years at the University of Minnesota has been a very significant, unforgettable and enjoyable time for me. I wish to thank Professors Ahmed Sameh, Gordon Beavers, Thomas Lundgren, Mitchell Luskin, John Lowengrub and Suhav Patankar for agreeing to be members of my preliminary and final oral examination committee and for their guidance and advice. Many thanks to Professor Marchel J. Crochet of the Université Catholique de Louvain who hosted my visit in Belgium in 1993 and allowed me access to the EVSS version of POLYFLOW code. And thanks to Professor Howard Hu of the University of Pennsylvania for his updating particle-mover code and his help with my particle simulation research. I also wish to acknowledge all the faculty of the Department of Aerospace Engineering and Mechanics, especially Professors Theodore Wilson, Amy Alving for their encouragement and help. I am very thankful to Kathy Kosiak, especially for her kindly help for my Belgium trip, in addition to all AEM staff members for their generous help and support throughout the years. Many of my fellow colleagues from Joseph s lab have made my life, study and work much easier. I want to thank them all, especially Runyuan Bai, Jimmy Feng, Timothy Hall, Adam Huang, Todd Hesla, Joe Liu, Clara Mata and Walter Wang for their kindly collaboration and support in many ways. Also many kindly friends in Minnesota, Dan & Barbara Richter, Glen and Bridget Kenadjian, Weld Ransom and Susan Schultz, their families and many others have made me feel at home here in Minnesota, and I greatly appreciate their encouragement and support as well. ii

I would like to acknowledge the National Science Foundation, (HPCC Grand Challenge grant; Fluid, Particulate and Hydraulic Systems), the DOE (Department of Basic Energy Sciences), the US Army (Mathematics and the Army High Performance Computing Research Center), the Schlumberger Foundation and the Minnesota Supercomputer Institute for supporting the research I have conducted in the past five years at the University of Minnesota. My family has been an integral part of my accomplishments, and I thank them very much for their continuous love, understanding and support. The special love and support of my wife, Qiwen Lin, has made a substantial difference in my life in the past ten years. She has helped make my dreams come true, and our lovely son, Kenny (Kaichong), little twin daughters, Laura & Claire, bring lots of hope and joy to our lives. Also, I am very grateful to my mother Xiu-Mei and my father Shui-Long for their dedication as parents, their help in taking care of our children and more specially for preparing me to face the challenges throughout my life. iii

Abstract This thesis presents numerical studies on the motion of solid particles in a twodimensional channel in Newtonian and viscoelastic fluids. The results agree well with experimental observations. Chapter 2 studies the stable orientation of a sedimenting long particle in Newtonian fluids. Results show that the main restoring mechanism which turns the major axis of the ellipse perpendicular to the stream is the high pressure at the "stagnation point" on the front surface where the shear stress vanishes. Chapter 3 uses the static calculation technique to investigate the drag on a circular cylinder and the velocity profile in its wake as functions of the wall blockage and properties of the fluid. Wall proximity shortens the wake and increases the drag, this effect is reduced by fluid elasticity. Shear-thinning decreases the drag and shortens the wake. A negative wake appears for the strongest wall blockage. Chapters 4 and 5 do direct simulation of the motion of solid particles in viscoelastic fluids. The mechanisms of viscoelastic effects are identified with normal stresses on the particle's surface by modifying the pressure distribution; the normal component of the extra stresses on the particles vanishes. Chapter 4 gives the results of the motion of a circular cylinder in Couette and Poiseuille flows of Oldroyd-B fluids. Both neutrally and non-neutrally buoyant cylinders are considered. The stable equilibrium position of neutrally buoyant particles is determined by a competition among inertia, elasticity, shear-thinning and the channel blockage ratio in both shear flows, and influenced by the curvature of the inflow velocity profile in Poiseuille flow. Chapter 5 studies the migrations and stable orientations of elliptic particles settling in Oldroyd-B fluids. Critical elasticity and Mach numbers are borders between vertical and horizontal turning of particles. Critical elasticity number is a function of channel blockage ratio, particle aspect ratio and retardation/relaxation time ratio. Tilted equilibria can be generated by shear thinning fluids. Inertia turns the particle perpendicular to the stream, while the normal stresses turn it along the stream. Two ellipses settling in a viscoelastic fluid attract, chain and rotate toward vertical. iv

Table of Contents Acknowledgments...ii Abstract... iv Table of Contents... v List of Figures...vii Chapter 1. Introduction... 1 Chapter 2. The turning couples on an elliptic particle settling in a Viscous fluid...8 2.1. Introduction...8 2.2. Basic equations...10 2.3. Numerical analysis...13 2.3.1. Interpolation of velocity...13 2.3.2. Stress tensor on a surface node... 14 2.3.3. Pressure and stress torques on a particle surface... 16 2.4. Results and discussions...19 2.4.1. Shear stress and position of stagnation and separation points... 20 2.4.2. Pressure... 24 2.4.3. Torques due to viscous tractions...30 2.5. Conclusions...31 Chapter 3. Wall effects on the flow of viscoelastic fluid around a circular cylinder... 32 3.1. Introduction...32 3.2. Formulation of the problem... 36 3.3. Results and discussions...39 3.3.1. Wall effects for a Newtonian fluid...40 3.3.2. Effects of elasticity in an infinite domain... 42 3.3.3. Effects of shear thinning in an infinite domain...46 3.3.4. Wall effects for a viscoelastic fluid...47 3.4. Concluding remarks... 53 Chapter 4. Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids... 55 4.1. Introduction...55 4.2. Numerical method...59 4.3. Particle migration in a Couette flow... 61 v

4.3.1. Neutrally buoyant particles... 62 4.3.2. Non-neutrally buoyant particles...67 4.4. Particle migration in a Poiseuille flow...70 4.4.1. Neutrally buoyant particles... 70 4.4.2. Non-neutrally buoyant particles...80 4.5. The effects of shear thinning in sedimentation... 84 4.6. Conclusions...85 Chapter 5. Direct simulation of the simulation of elliptic particles in Oldroyd-B fluids...88 5.1. Introduction...88 5.2. Governing equations... 99 5.3. Normal stress at the boundary of a rigid body... 102 5.4. Numerical method...103 5.5. Critical Reynolds number for horizontal turning of an ellipse falling in a Newtonian fluid between close walls...106 5.6. Universal features of equilibrium orientation and position of an elliptical particle... 108 5.7. Critical values of elasticity number for Maxwell models ( λ 2 = 0)... 109 5.8. Critical values of Mach number for Maxwell models ( λ 2 = 0)... 120 5.9. Unstable tilted equilibria...123 5.10. Effects of the retardation and relaxation time ratio...127 5.11. Sedimentation of ellipses when λ 2 λ 1 = 1 8... 130 5.11.1. Critical elasticity number E...130 5.11.2. Streamlines and dynamics of turning ellipses...131 5.11.3. Effects of the particle aspect ratio a/b...134 5.11.4. Interaction of two ellipses...134 5.12. Shear thinning... 136 5.13. Conclusions...138 Bibliography... 141 vi

List of Figures Figure 2.1. Triangle element e with three vertex nodes (i, j, k) and three midpoints. Figure 2.2. Particle surface node k and the N k elements surrounding it. Figure 2.3. Coordinates of the surface of the ellipse. The x-axis is along the wall of the channel. P (X i, Y i ) is the center of the ellipse and k is the node on the surface. α p is the turning angle of the ellipse, r = ξ k 2 + ηk 2 forces F n and F t. is the distance between node k and the center P. a 1 and b 1 are the arms of the Figure 2.4. (a) Snapshots of the motion of the ellipse from rest to periodic oscillation. (b) Snapshots of the motion of the ellipse in one whole cycle. This cycle covers time steps from itime = 232 to 264. Figure 2.5. Real time at different time steps covered in one cycle of oscillation. Figure 2.6(a). Pressure contour for a sedimenting ellipse at three time steps where (x, y, α p ) indicates the position of the ellipse; (b). Streamline contour plots for the sedimenting ellipse at three time steps where (x, y, α p ) indicates the position of the ellipse; (c). Vorticity contour plots for the sedimenting ellipse in three different stages where (x, y, α p ) indicates the position of the ellipse. Figure 2.7. The shear stress vanishes at the stagnation points corresponding to dividing streamlines. (a) α p =0: the ellipse is horizontal; (b) α p >0: the ellipse tilts up on the right; (c) α p <0: the ellipse tilts down on the right. Figure 2.8. Shear stress distribution on the ellipse for a motion cycle with time steps from itime = 232 to 264. The unit of the shear stress is dyne/cm 2. Figure 2.9. Torque distribution due to shearing on the ellipse for a motion cycle with time steps from itime = 232 to 264. The unit of the torque is dyne. Figure 2.10. Pressure distribution on the ellipse for a motion cycle with time steps from itime = 232 to 264. The unit of the pressure is dyne/cm 2. Figure 2.11. Pressure torque distribution on the ellipse surface for a motion cycle with time steps from itime = 232 to 264. The unit of the torque is dyne. Figure 2.12. Total torque distribution due to pressure on the ellipse for a motion cycle with time steps from itime = 232 to 264. The unit of the torque is dyne. Figure 2.13. Total torque distribution due to viscosity on the ellipse for a motion cycle with time steps from itime = 232 to 264. The unit of the torque is dyne. Figure 3.1. Geometry of the problem. Figure 3.2. A typical mesh used for β=0.025. It has 2420 nodes and 1184 elements. Figure 3.3. Wall effects on the drag coefficient of a cylinder in a Newtonian flow. C d is the standard drag coefficient for an infinite domain (Sucker & Brauer 1975). vii

Figure 3.4. Velocity distribution on the centerline of an unbounded flow field around a cylinder. Oseen's approximate solution at Re=0.1 is also shown for comparison. Figure 3.5. Wall effects on the velocity distribution along the centerline of the flow field. Re=1. Figure 3.6. The effect of Deborah number on the drag of a cylinder in an unbounded flow of an Oldroyd-B fluid. C d0 is the drag coefficient computed for a Newtonian fluid (De=0). A data point of Hu & Joseph (1990) for Re=10 is also shown for comparison. Figure 3.7. The velocity distribution along the centerline of the wake of an unbounded viscoelastic flow field around a cylinder. (a) The wake is lengthened by elasticity. Also note the recirculation zone behind the cylinder (-0.7<x/d<-0.5) at Re=10. (b) The elastic effect increases with Re. For Re =0.1, the Newtonian and viscoelastic profiles are indistinguishable. Figure 3.8. The effect of shear thinning on the drag coefficient of a circular cylinder in an unbounded flow. Re =1, De =1.0. Figure 3.9. Effect of shear-thinning on the velocity distribution along the centerline of an unbounded flow field around a cylinder. Re=1. Figure 3.10. Wall effects on the drag coefficient for a Newtonian fluid and an Oldroyd-B fluids at two Deborah numbers. Re=1. The curves are plotted in two parts (a and b) to show details at low blockage. Figure 3.11. Wall effects on the drag coefficient for viscoelastic fluids with or without shear thinning. Re =1. Figure 3.12. Effects of blockage on the velocity distribution along the centerline of the channel. The fluid is viscoelastic without shear-thinning. Re =1, De =1. Figure 3.13. Comparison of velocity distributions along the centerline of the wake at different blockage ratios. Solid lines represent Newtonian profiles at Re=1; dashed lines represent profiles of an Oldroyd-B fluid at Re=1 and De=1. Figure 3.14. Effects of shear-thinning on the velocity distribution along the centerline of the channel. Re=1, β=0.33. Figure 4.1. Lateral migration of a solid particle in Couette flow between two walls moving in opposite directions. W is the width of the channel. Y c is the distance of the cylinder center from the wall. We call the final equilibrium value of Y c /W the standoff distance. Figure 4.2. Lateral migration of a neutrally buoyant particle released from different initial positions in Couette flow of an Oldroyd-B fluid (β=0.25, Re =5, De=1.0). Y 0 indicates the initial position of the particle. V init is the initial velocity of the particle. Figure 4.3. The effect of the Reynolds number on the migration of a neutrally buoyant particle in Couette flow of an Oldroyd-B fluid (β=0.25, De=1.0). The centerline of the channel is no longer a stable equilibrium when the Reynolds number is small. viii

Figure 4.4. The effect of Deborah number on the migration of a neutrally buoyant particle in Couette flow of an Oldroyd-B fluid (β=0.25, Re=1.0). Figure 4.5. The effect of the blockage ratio β=a/r on the migration of a neutrally buoyant particle in Couette flow of an Oldroyd-B fluid (De=1.0, Re =0). The cylinder will touch the wall when the center of the cylinder is at Y c /W=0.5β in the channel. Figure 4.6. Streamlines for the Couette flow around a cylinder under conditions specified in figure 5, β=0.25 and t * =689. At this t * the particle has drifted to its equilibrium position. Figure 4.7. Streamlines for the Couette flow around a cylinder under conditions specified in figure 5, β=0.50 and t * =196.5. The particle is still migrating to the wall. The computation fails for larger t *. Figure 4.8. The effect of shear-thinning on a neutrally buoyant particle in Couette flow of an Oldroyd-B fluid (β=0.25, Re=0, De=2.8). The cylinder will touch the wall when the center of the cylinder is at Y c /W=0.125 in the β=0.25 channel. Figure 4.9. Trajectories of non-neutrally buoyant particles in Couette flow of an Oldroyd-B fluid (β=0.25, Re =5, De=1.0). (a) Particles with density ρ s smaller than the fluid ρ f ; (b) particles with density ρ s larger than the fluid ρ f. Figure 4.10. Equilibrium position of a non-neutrally buoyant particle in Couette flow of an Oldroyd-B fluid with De=1.0 and a Newtonian fluid (β=0.25, Re=5). Figure 4.11. Lateral migration of a solid particle in a plane Poiseuille flow. U 0 is the maximum velocity at the centerline. (X c, Y c ) is the center of the cylinder. Figure 4.12. Trajectories of a neutrally buoyant particle released at different initial positions in Poiseuille flow of an Oldroyd-B fluid (β=0.25, Re=5, De =0.2). Figure 4.13. The effect of the Reynolds number on a neutrally buoyant particle released in Poiseuille flow of an Oldroyd-B fluid (β=0.25, De=0.2). Figure 4.14. The effect of the blockage ratio β =a/r on the motion of a neutrally buoyant particle in Poiseuille flow of an Oldroyd-B fluid (Re=5, De =0.2). The cylinder will touch the wall when the center of the cylinder is at Y c /d=0.5 where d is the diameter of the particle. Figure 4.15. Streamlines for the Poiseuille flow around a cylinder when Re=5, De=0.2 and β=0.25. The particle has reached its equilibrium position which is determined by a balance of compressive normal stresses on the open side near the centerline where the streamlines are crowded and lubrication forces on the other side. Figure 4.16. Streamlines for the Poiseuille flow around a cylinder when Re=5, De=0.2 and β=0.50. The particle has drifted to an equilibrium position very close to the wall. Streamlines on the open side near the centerline are crowded. ix

Figure 4.17. The effect of normal stresses on a neutrally buoyant particle released in Poiseuille flow of an Oldroyd-B fluid without wall effects (β=0.025, Re=0). The cylinder drifts toward the center of the channel. Figure 4.18. The effect of normal stresses on a neutrally buoyant particle in Poiseuille flow of an Oldroyd- B fluid in the limiting case of no inertia (β=0.25, Re=0). Figure 4.19. The effect of normal stresses on a neutrally buoyant particle released in Poiseuille flow of an Oldroyd-B fluid (β =0.25, Re=5). The cylinder will touch the wall when the center of the cylinder is at Y c /W=0.125 in this β=0.25 channel. (a) The particle trajectories; (b) the standoff distances. Figure 4.20. The effect of shear-thinning on the migration of a neutrally buoyant particle in Poiseuille flow of an Oldroyd-B fluid (β=0.25, Re=5, De=0.2, λ 3 /λ 1 =100). The cylinder will touch the wall when the center of the cylinder is at Y c /W=0.125 in the β=0.25 channel. Figure 4.21. Trajectories of non-neutrally buoyant particles in Poiseuille flow of an Oldroyd-B fluid (β=0.25, Re=5, De=0.2) when (a) the density of the particle ρ s is smaller than the density of the fluid ρ f and (b) the density of the particle ρ s is larger than the density of the fluid ρ f. Figure 4.22. Trajectories of non-neutrally buoyant particles in Poiseuille flow of an Oldroyd-B fluid (β=0.10, Re=5, De=0.2) when (a) the density of the particle ρ s is smaller than the density of the fluid ρ f and (b) the density of the particle ρ s is larger than the density of the fluid ρ f. Figure 4.23. The stable position of a non-neutrally buoyant particle in Poiseuille flow of an Oldroyd-B fluid. Re=5. A Newtonian curve for β=0.25 is also shown for comparison. Figure 4.24. The effects of shear thinning on the sedimentation of particle in an Oldroyd-B fluid. β=0.25. Figure 5.1. Slow flow around an elliptic particle. θ is the tilt angle of the major axis and the horizontal. α is the angle of a typical surface node of the particle. There are two stagnation points s where the shear stress vanishes and two other points c where the streamlines are most crowded. The shear stresses and normal stresses are strongest at c. The nonlinear contribution of the normal stresses in slow flow at each point on the ellipse is given by Ψ 1 (0) γ 2 where Ψ 1 (0) > 0 is the coefficient of the first normal stress difference and γ is the shear rate. These normal stresses are compressive and give rise to high pressure at points near where the flow is fast, opposing the high pressure at points s which turn the ellipse broadside-on (θ=180 o ) (Joseph & Feng, 1996). If the normal stresses at points c dominate at all θ, the major axis of the ellipse will align with gravity (θ=90 o ). Figure 5.2. The orientation and migrations of particles in slow flow of a fluid dominated by viscoelasticity can be determined from Stokes flow around that body, with large normal stresses where the shear rates on the body are large, near the places where the streamlines are crowded (see figure 5.1) (a) There are two symmetric orientations with respect to gravity for a falling square (cube); x

the one with flat faces perpendicular is unstable; the stable orientation is when the line between vertices is parallel to gravity; (b) a rectangle or cylinder with flat end (Joseph and Liu 1993) tends to line up with the longest line in the body parallel to gravity. Falling bodies tilt when the longest line in the body is not an axis of symmetry; this is called shape tilting; (c ) Falling ellipses tend to straighten out, line up and chain; (d) Perturbed chains straighten out. Figure 5.3. Sedimentation of an elliptic particle in a 2-D channel. W is the width of the channel. (X, Y) is the position of the center of the elliptic particle. θ is the orientation of its major axis. Figure 5.4. Orientation θ and trajectory Y/W of an ellipse (a/b=1.5) settling in a Newtonian fluid (De =0) between close walls (R/a=5). At Re=0.31, the ellipse turns vertical and executes a damped oscillation as it drifts to the channel center; at Re=0.82, the ellipse turns horizontal as it migrates to the channel center. Figure 5.5. (a) A plastic long particle with round ends used in the experiment; (b) the long particle turns its longside parallel to the stream under the effects of lubrication in a Newtonian fluid when the channel is very narrow and the settling is very slow. Figure 5.6(A). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R /a=4 at elasticity number below critical: E=0.05, Re=0.89, De =0.044, M=0.20. The ellipse migrates to the center and turns horizontal. Figure 5.6(B). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R/a=4 at elasticity number above critical: E=0.2, Re =1.15, De =0.23, M=0.51. The ellipse migrates to the center and turns vertical. Figure 5.7(A). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R/a=5 at elasticity number below critical: E=0.2, Re =1.00, De =0.50, M=0.71. The ellipse migrates to the center and turns horizontal. Figure 5.7(B). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R/a=5 at elasticity number above critical: E=0.5, Re =0.82, De =0.16, M=0.37. The ellipse migrates to the center and turns vertical. Figure 5.8(A). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R /a=10 at elasticity number below critical: E=1.0, Re =De =M=0.44. The ellipse migrates to the center and turns horizontal. Figure 5.8(B). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R/a=10 at elasticity number above critical: E=2.0, Re=0.49, De =0.97, M=0.69. The ellipse migrates to the center and turns vertical. Figure 5.9(A). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R /a=20 at elasticity number below critical: E=2.0, Re=0.56, De =1.11, M=0.79. The ellipse migrates to the center and turns horizontal. xi

Figure 5.9(B). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R/a=20 at elasticity number above critical: E=3.0, Re=0.58, De =1.73, M=1.0. The ellipse migrates to the center and turns vertical. Figure 5.10(A). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R /a=30 at elasticity number below critical: E=2.0, Re=0.59, De =1.18, M=0.84. The ellipse migrates to the center and turns horizontal. Figure 5.10(B). The orientation θ and trajectory Y/W of an ellipse of ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage ratio R/a=30 at elasticity number above critical: E=4.0, Re=0.59, De =2.36, M=1.18. The ellipse migrates to the center and turns vertical. Figure 5.11. Critical elasticity number E c vs. blockage ratio R/a for ellipses of aspect ratio a/b=1.5 falling in Maxwell fluids. When E is less than critical, inertia dominates as in a Newtonian fluid and the ellipse ultimately falls with its long axis horizontal for all values of the velocity however lager; when E is greater than critical the ellipse ultimately falls with its long axis vertical for all Mach numbers less than the critical one. Figure 5.12. The orientation θ and trajectory Y/W of an ellipse of aspect ratio a/b=1.5 falling in a Maxwell fluid in a channel with blockage R/a=10 for E = De / Re = 1.6 and different M. M=0.62 (Re =0.49, De=0.78) is subcritical; M=1.34 (Re=1.06, De=1.70) is supercritical; M=1.03 (Re =0.81, De=1.30) is marginally supercritical (see section 9). Figure 5.13. Torque on a fixed ellipse of aspect ratio a/b=1.5 in a uniform stream of Maxwell fluid as a function of the orientation of the ellipse for an off-center (Y/W=0.58) and on center position (Y/W=0.50) in a channel with blockage R/a=10 and E=1.6, M=1.03, Re=0.81, De=1.30. There are two zeros for tilted angles in the off-center case, but the only zero torque positions at the channel center are for 90 o (vertical) and 180 o (horizontal). Figure 5.14. The orientation θ and trajectory Y/W of an ellipse of aspect ratio a/b=1.5 falling from two different initial conditions in a Maxwell fluid. R/a=10, E=1.6, Re =0.81, De=1.30 and M=1.03. The ellipse is initially at rest. Figure 5.15. Perturbed orientation θ and trajectory Y/W of the ellipse in figure 5.14 starting from rest near the proposed equilibrium at θ= 148 o and Y/W=0.58. The starting conditions are θ=148 o and Y/W=0.58. The particle migrates to the center and turns broadside on. Figure 5.16. Comparison of the orientation of falling ellipses in a Maxwell fluid λ 2 =0 to an Oldroyd-B fluid with λ 2 /λ 1 =1/8. (R/a=10, a/b=1.5). Here M is much less than one and vertical and horizontal orientations depend on the critical elasticity number which is higher when λ 2 /λ 1 =0. (a) The particle settles with vertical orientation when E is above the critical ; (b) the same particle rotates to a horizontal orientation when E is below the critical but to a vertical orientation when inertia is turned off (E= ). xii

Figure 5.17. Orientation angle and trajectory of an ellipse as a function of fall distance for different elasticity number corresponding to a fixed Reynolds number (Re 0.47) in an Oldroyd-B fluid with λ 2 /λ 1 =1/8 when R/a=10 and a/b=1.5. Figure 5.18. Streamlines for the flow around an elliptic particle in an Oldroyd-B fluid under conditions specified in figure 5.17, R/a=10, a/b=1.5, Re=0.46, De=1.85, E=4.0, M=0.92. (a) The particle is turning clockwise; (b) the particle has reached an equilibrium in which the major axis is along the stream. Figure 5.19. Streamlines for the flow around an elliptic particle in an Oldroyd-B fluid under conditions specified in figure 5.17, R/a=10, a/b=1.5, Re=0.44, De =0.18, E=0.40, M=0.08. (a) The particle is turning counter-clockwise; (b) the particle has reached an equilibrium in which the major axis is across the stream. Figure 5.20. Orientation θ and trajectory Y/W of an ellipse of different aspect ratio falling in an Oldroyd-B fluid with λ 2 /λ 1 =1/8 (R/a=10, Re=0.61, De=0.97, E=1.58, M=0.769). Figure 5.21. The motion of two elliptic particles in an Oldroyd-B fluid (R/a=10, a/b=1.5, θ ο =90 o ). The number on the particles shows different time step. (a) Particles are released one on top of the other; (b) particles are released side by side. Figure 5.22. The effect of shear thinning on the turning of an elliptic particle in an Oldroyd-B fluid (R/a=10, a/b=1.5, λ 2 /λ 1 =1/8, E=1.56). Figure 5.23. Orientation θ and trajectory Y/W of an ellipse released with different initial positions and tilt angles in an Oldroyd-B fluid with shear thinning. ( λ 2 λ 1 = 1 8, R /a =10, a/b=1.5, Re=0.69, De =1.07, E=1.56, M=0.86). The ellipse migrates to a stable equilibrium off-center with a certain tilt angle. xiii