Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T08:36:11.508Z Has data issue: false hasContentIssue false
  • English
  • Français

The rheology and microstructure of concentrated non-colloidal suspensions of deformable capsules

Published online by Cambridge University Press:  23 September 2011

Jonathan R. Clausen ,
Daniel A. Reasor Jr  and
Cyrus K. Aidun
Jonathan R. Clausen
Affiliation:
G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
Daniel A. Reasor Jr
Affiliation:
G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
Cyrus K. Aidun*
Affiliation:
G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A detailed study into the rheology and microstructure of dense suspensions of initially spherical capsules is presented, where capsules are composed of a fluid-filled interior surrounded by an elastic membrane. This study couples a lattice-Boltzmann fluid solver to a finite-element membrane model creating a robust and scalable method for the simulation of these suspensions. A Lees–Edwards boundary condition is used to simulate periodic simple shear to obtain bulk rheological properties, and three-dimensional results are presented for capsules in the regime of negligible inertia, Brownian motion and colloidal interparticle forces. The simulation results focus on describing the suspension rheology as a function of the particle concentration and deformability, and relating these macroscopic rheological findings to changes at the particle level, i.e. the suspension microstructure. Several important findings are made: suspensions of deformable capsules are found to be shear thinning, and the initially compressive normal stresses associated with rigid spherical suspensions undergo rapid changes with moderate levels of particle deformation. These normal stress changes are particularly evident in the first normal stress difference, which undergoes a sign change at fairly minor levels of deformation, and the particle pressure, which decreases rapidly with increasing particle deformability. Changes in the microstructure as quantified by the single-body microstructure and the pair distribution function are reported. Also, results calculating particle self-diffusion are presented and related to changes in the normal stresses.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics Multiphase and Particle-laden Flows: Particle/fluid flow Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 685 , 25 October 2011 , pp. 202 - 234
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Present address: Sandia National Laboratories, Albuquerque, NM 87185, USA.

References

1. Acrivos, A., Mauri, R. & Fan, X. 1993 Shear-induced resuspension in a Couette device. Intl J. Multiphase Flow 19 (5), 797–802.CrossRefGoogle Scholar
2. Aidun, C. K. & Clausen, J. R. 2010 The lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42 (1), 439–472.CrossRefGoogle Scholar
3. Aidun, C. K. & Lu, Y. 1995 Lattice Boltzmann simulation of solid particles suspended in fluid. J. Stat. Phys. 81 (1), 49–61.CrossRefGoogle Scholar
4. Aidun, C. K., Lu, Y. & Ding, E. J. 1998 Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech. 373, 287–311.CrossRefGoogle Scholar
5. Bagchi, P. 2007 Mesoscale simulation of blood flow in small vessels. Biophys. J. 92 (6), 1858–1877.CrossRefGoogle ScholarPubMed
6. Bagchi, P. & Kalluri, R. M. 2010 Rheology of a dilute suspension of liquid-filled elastic capsules. Phys. Rev. E 81, 056320.CrossRefGoogle ScholarPubMed
7. Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100 (04), 831–853.CrossRefGoogle Scholar
8. Barthès-Biesel, D. 2009 Capsule motion in flow: deformation and membrane buckling. C. R. Physique 10 (8), 764–774.CrossRefGoogle Scholar
9. Barthès-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefGoogle Scholar
10. Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.CrossRefGoogle Scholar
11. Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order c 2 . J. Fluid Mech. 56 (03), 401–427.CrossRefGoogle Scholar
12. Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511–525.CrossRefGoogle Scholar
13. Biben, T. & Misbah, C. 2003 Tumbling of vesicles under shear flow within an advected-field approach. Phys. Rev. E 67 (3), 031908.CrossRefGoogle ScholarPubMed
14. Brady, J. F. 1993 Brownian motion, hydrodynamics, and the osmotic pressure. J. Chem. Phys. 98 (4), 3335–3341.CrossRefGoogle Scholar
15. Brady, J. F. & Bossis, G. 1988 Stokesian Dynamics. Annu. Rev. Fluid Mech. 20 (1), 111–157.CrossRefGoogle Scholar
16. Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.CrossRefGoogle Scholar
17. Breedveld, V., van den Ende, D., Bosscher, M., Jongschaap, R. J. J. & Mellema, J. 2001b Measuring shear-induced self-diffusion in a counterrotating geometry. Phys. Rev. E 63 (2), 21403.CrossRefGoogle Scholar
18. Breedveld, V., van den Ende, D., Bosscher, M., Jongschaap, R. J. J. & Mellema, J. 2002 Measurement of the full shear-induced self-diffusion tensor of noncolloidal suspensions. J. Chem. Phys. 116 (23), 10529–10535.CrossRefGoogle Scholar
19. Breedveld, V., van den Ende, D., Jongschaap, R. J. J. & Mellema, J. 2001b Shear-induced diffusion and rheology of noncolloidal suspensions: time scales and particle displacements. J. Chem. Phys. 114 (13), 5923.CrossRefGoogle Scholar
20. Chen, L. B., Ackerson, B. J. & Zukoski, C. F. 1994 Rheological consequences of microstructural transitions in colloidal crystals. J. Rheol. 38, 193–216.CrossRefGoogle Scholar
21. Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30 (1), 329–364.CrossRefGoogle Scholar
22. Clausen, J. R. 2010 The effect of particle deformation on the rheology and microstructure of noncolloidal suspensions. PhD thesis, Georgia Institute of Technology.Google Scholar
23. Clausen, J. R. & Aidun, C. K. 2009 Galilean invariance in the lattice-Boltzmann method and its effect on the calculation of rheological properties in suspensions. Intl J. Multiphase Flow 35, 307–311.CrossRefGoogle Scholar
24. Clausen, J. R. & Aidun, C. K. 2010 Capsule dynamics and rheology in shear flow: particle pressure and normal stress. Phys. Fluids 22, 123302.CrossRefGoogle Scholar
25. Clausen, J. R., Reasor, D. A. & Aidun, C. K. 2010 Parallel performance of a lattice-Boltzmann/finite element cellular blood flow solver on the IBM Blue Gene/P architecture. Comput. Phys. Commun. 181 (6), 1013–1020.CrossRefGoogle Scholar
26. Coupier, G., Kaoui, B., Podgorski, T. & Misbah, C. 2008 Noninertial lateral migration of vesicles in bounded Poiseuille flow. Phys. Fluids 20, 111702.CrossRefGoogle Scholar
27. Danker, G. & Misbah, C. 2007 Rheology of a dilute suspension of vesicles. Phys. Rev. Lett. 98 (8), 088104.CrossRefGoogle ScholarPubMed
28. Deboeuf, A., Gauthier, G., Martin, J., Yurkovetsky, Y. & Morris, J. F. 2009 Particle pressure in a sheared suspension: a bridge from osmosis to granular dilatancy. Phys. Rev. Lett. 102 (10), 108301.CrossRefGoogle Scholar
29. Ding, E. J. & Aidun, C. K. 2003 Extension of the lattice-Boltzmann method for direct simulation of suspended particles near contact. J. Stat. Phys. 112 (3), 685–708.CrossRefGoogle Scholar
30. Dupin, M. M., Halliday, I., Care, C. M., Alboul, L. & Munn, L. L. 2007 Modeling the flow of dense suspensions of deformable particles in three dimensions. Phys. Rev. E 75 (6), 66707.CrossRefGoogle ScholarPubMed
31. Eckstein, E. C., Bailey, D. G. & Shapiro, A. H. 2006 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79 (01), 191–208.CrossRefGoogle Scholar
32. Eilers, H. 1941 Die viskosität von emulsionen hochviskoser stoffe als funktion der konzentration. Colloid Polym. Sci. 97 (3), 313–321.Google Scholar
33. Einstein, A. 1906 Zur Theorie der Brownschen Bewegung. Ann. Phys. (Leipzig) 19, 371–381.CrossRefGoogle Scholar
34. Einstein, A. 1911 Berichtigung zu meiner Arbeit: eine neue Bestimmung der Moleküldimensionen. Ann. Phys. (Leipzig) 34 (3), 591–592.CrossRefGoogle Scholar
35. Foss, D. R. & Brady, J. F. 1999 Self-diffusion in sheared suspensions by dynamic simulation. J. Fluid Mech. 401, 243–274.CrossRefGoogle Scholar
36. Frisch, U., d’Humières, D., Hasslacher, B., Lallemand, P., Pomeau, Y. & Rivet, J.-P. 1987 Lattice gas hydrodynamics in two and three dimensions. Complex Syst. 1 (4), 649–707.Google Scholar
37. Gadala-Maria, F. 1979 The rheology of concentrated suspensions. PhD thesis, Standford University.Google Scholar
38. Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.CrossRefGoogle Scholar
39. Ginzbourg, I. & Adler, P. M. 1994 Boundary flow condition analysis for the 3-dimensional lattice Boltzmann model. J. Phys. II 4 (2), 191–214.Google Scholar
40. Goddard, J. D. & Miller, C. 1967 Nonlinear effects in the rheology of dilute suspensions. J. Fluid Mech. 28 (04), 657–673.CrossRefGoogle Scholar
41. Higuera, F. J. & Jimenez, J. 1989 Boltzmann approach to lattice gas simulations. Europhys. Lett. 9 (7), 663–668.CrossRefGoogle Scholar
42. Hinch, E. J. & Leal, L. G. 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52 (04), 683–712.CrossRefGoogle Scholar
43. Hinch, E. J. & Leal, L. G. 1973 Time-dependent shear flows of a suspension of particles with weak Brownian rotations. J. Fluid Mech. 57 (04), 753–767.CrossRefGoogle Scholar
44. d’Humières, D., Ginzburg, I., Krafczyk, M., Lallemand, P. & Luo, L.-S. 2002 Multiple-relaxation-time lattice Boltzmann models in three dimensions. Phil. Trans. R. Soc. Lond. A 360 (1792), 437–451.CrossRefGoogle ScholarPubMed
45. Jeffrey, D. J., Morris, J. F. & Brady, J. F. 1993 The pressure moments for two rigid spheres in low-Reynolds-number flow. Phys. Fluids A 5 (10), 2317–2325.CrossRefGoogle Scholar
46. Junk, M. & Yong, W. A. 2003 Rigorous Navier–Stokes limit of the lattice Boltzmann equation. Asymptotic Anal. 35 (2), 165–185.Google Scholar
47. Kaoui, B., Ristow, G. H., Cantat, I., Misbah, C. & Zimmermann, W. 2008 Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow. Phys. Rev. E 77 (2), 21903.CrossRefGoogle ScholarPubMed
48. Karnis, A. & Mason, S. G. 1967 Particle motions in sheared suspensions XXIII. Wall migration of fluid drops. J. Colloid Interface Sci. 24 (2), 164–169.CrossRefGoogle Scholar
49. Keller, S. R. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47.CrossRefGoogle Scholar
50. Kennedy, M. R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput. Fluids 23 (2), 251–278.CrossRefGoogle Scholar
51. Krieger, I. M. & Dougherty, T. J. 1959 A mechanism for non-Newtonian flow in suspensions of rigid spheres. J. Rheol. 3 (1), 137–152.Google Scholar
52. Kulkarni, P. M. & Morris, J. F. 2008 Suspension properties at finite Reynolds number from simulated shear flow. Phys. Fluids 20 (4), 040602.CrossRefGoogle Scholar
53. Lac, E., Barthès-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
54. Ladd, A. J. C. 1994a Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 271, 285–309.CrossRefGoogle Scholar
55. Ladd, A. J. C. 1994b Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311–339.CrossRefGoogle Scholar
56. Ladd, A. J. C. & Verberg, R. 2001 Lattice-Boltzmann simulations of particle–fluid suspensions. J. Stat. Phys. 104 (5), 1191–1251.CrossRefGoogle Scholar
57. Leighton, D. T. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
58. Lin, C. J., Peery, J. H. & Schowalter, W. R. 1970 Simple shear flow round a rigid sphere: inertial effects and suspension rheology. J. Fluid Mech. 44 (01), 1–17.CrossRefGoogle Scholar
59. Loewenberg, M. & Hinch, E. J. 1996 Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech. 321, 395–419.CrossRefGoogle Scholar
60. Lorenz, E., Caiazzo, A. & Hoekstra, A. G. 2009 Corrected momentum exchange method for lattice Boltzmann simulations of suspension flow. Phys. Rev. E 79 (3), 036705.CrossRefGoogle ScholarPubMed
61. MacMeccan, R. M. 2007 Mechanistic effects of erythrocytes on platelet deposition in coronary thrombosis. PhD thesis, Georgia Institute of Technology.Google Scholar
62. MacMeccan, R. M., Clausen, J. R., Neitzel, G. P. & Aidun, C. K. 2009 Simulating deformable particle suspensions using a coupled lattice-Boltzmann and finite-element method. J. Fluid Mech. 618, 13–39.CrossRefGoogle Scholar
63. Marchioro, M. & Acrivos, A. 2001 Shear-induced particle diffusivities from numerical simulations. J. Fluid Mech. 443, 101–128.CrossRefGoogle Scholar
64. McNamara, G. R. & Zanetti, G. 1988 Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61 (20), 2332–2335.CrossRefGoogle ScholarPubMed
65. Mewis, J., Frith, W. J., Strivens, T. A. & Russel, W. B. 1989 The rheology of suspensions containing polymerically stabilized particles. AIChE J. 35 (3), 415–422.CrossRefGoogle Scholar
66. Misbah, C. 2006 Vacillating breathing and tumbling of vesicles under shear flow. Phys. Rev. Lett. 96 (2), 028104.CrossRefGoogle ScholarPubMed
67. Morris, J. F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J. Rheol. 43 (5), 1213–1237.CrossRefGoogle Scholar
68. Morris, J. F. & Brady, J. F. 1998 Pressure-driven flow of a suspension: Buoyancy effects. Intl J. Multiphase Flow 24 (1), 105–130.CrossRefGoogle Scholar
69. Morris, J. F. & Katyal, B. 2002 Microstructure from simulated Brownian suspension flows at large shear rate. Phys. Fluids 14 (6), 1920–1937.CrossRefGoogle Scholar
70. Noble, D. R., Chen, S., Georgiadis, J. G. & Buckius, R. O. 1995 A consistent hydrodynamic boundary condition for the lattice Boltzmann method. Phys. Fluids 7 (1), 203–209.CrossRefGoogle Scholar
71. Nott, P. R. & Brady, J. F. 1994 Pressure-driven suspension flow: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
72. Nourgaliev, R. R., Dinh, T. N., Theofanous, T. G. & Joseph, D. 2003 The lattice Boltzmann equation method: theoretical interpretation, numerics and implications. Intl J. Multiphase Flow 29 (1), 117–169.CrossRefGoogle Scholar
73. Papir, Y. S. & Krieger, I. M. 1970 Rheological studies on dispersions of uniform colloidal spheres II. Dispersions in nonaqueous media. J. Colloid. Interface Sci. 34 (1), 126–130.CrossRefGoogle Scholar
74. Parsi, F. & Gadala-Maria, F. 1987 Fore-and-aft asymmetry in a concentrated suspension of solid spheres. J. Rheol. 31 (8), 725–732.CrossRefGoogle Scholar
75. Phung, T. N., Brady, J. F. & Bossis, G. 1996 Stokesian dynamics simulation of Brownian suspensions. J. Fluid Mech. 313, 181–207.CrossRefGoogle Scholar
76. Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities. J. Fluid Mech. 361, 117–143.CrossRefGoogle Scholar
77. Reasor, D. A., Clausen, J. R. & Aidun, C. K. 2011 Coupling the lattice-Boltzmann and spectrin-link methods for the direct numerical simulation of cellular blood flow. Intl J. Numer. Meth. Fluids, doi:10.1002/fld.2534.CrossRefGoogle Scholar
78. Roscoe, R. 1967 On the rheology of a suspension of viscoelastic spheres in a viscous liquid. J. Fluid Mech. 28 (02), 273–293.CrossRefGoogle Scholar
79. Russel, W. B., Saville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.CrossRefGoogle Scholar
80. Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–146.CrossRefGoogle Scholar
81. Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol. 46, 1031–1056.CrossRefGoogle Scholar
82. Sierou, A. & Brady, J. F. 2004 Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506, 285–314.CrossRefGoogle Scholar
83. Singh, A. & Nott, P. R. 2003 Experimental measurements of the normal stresses in sheared Stokesian suspensions. J. Fluid Mech. 490, 293–320.CrossRefGoogle Scholar
84. Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37 (01), 129–149.CrossRefGoogle Scholar
85. Sukumaran, S. & Seifert, U. 2001 Influence of shear flow on vesicles near a wall: a numerical study. Phys. Rev. E 64 (1), 11916.CrossRefGoogle Scholar
86. Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138 (834), 41–48.Google Scholar
87. Vahala, G., Keating, B., Soe, M., Yepez, J., Vahala, L. & Ziegeler, S. 2009 Entropic, LES and boundary conditions in lattice Boltzmann simulations of turbulence. Eur. Phys. J. Special Topics 171 (1), 167–171.CrossRefGoogle Scholar
88. Vlahovska, P. M. & Gracia, R. S. 2007 Dynamics of a viscous vesicle in linear flows. Phys. Rev. E 75 (1), 016313.CrossRefGoogle ScholarPubMed
89. Wagner, A. J. & Pagonabarraga, I. 2002 Lees–Edwards boundary conditions for lattice Boltzmann. J. Stat. Phys. 107 (1), 521–537.CrossRefGoogle Scholar
90. Wu, J. & Aidun, C. K. 2009 Simulating 3D deformable particle suspensions using lattice Boltzmann method with discrete external boundary force. Intl J. Numer. Meth. Fluids 62 (7), 765–783.CrossRefGoogle Scholar
91. Wu, J. & Aidun, C. K. 2010 A method for direct simulation of flexible fiber suspensions using lattice-Boltzmann equation with external boundary force. Intl J. Multiphase Flow 36, 202–209.CrossRefGoogle Scholar
92. Yurkovetsky, Y. & Morris, J. F. 2008 Particle pressure in sheared Brownian suspensions. J. Rheol. 52 (1), 141–164.CrossRefGoogle Scholar
93. Zarraga, I. E., Hill, D. A. & Leighton, D. T. 2000 The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44 (2), 185–220.CrossRefGoogle Scholar
94. Zhang, J., Johnson, P. C. & Popel, A. S. 2007 An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows. Phys. Biol. 4, 285.CrossRefGoogle ScholarPubMed
95. Zinchenko, A. Z. & Davis, R. H. 2002 Shear flow of highly concentrated emulsions of deformable drops by numerical simulations. J. Fluid Mech. 455, 21–62.CrossRefGoogle Scholar