Journal of Statistical Physics

, Volume 104, Issue 5–6, pp 1191–1251 | Cite as

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions

  • A. J. C. Ladd
  • R. Verberg


This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69, 70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.

Lattice-Boltzmann suspensions simulations of colloids hydrodynamic interactions 


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  1. 1.
    C. K. Aidun and Y. N. Lu, Lattice Boltzmann simulation of solid particles suspended in fluid, J. Stat. Phys. 81:49-61 (1995).Google Scholar
  2. 2.
    C. K. Aidun, Y. N. Lu, and E. Ding, Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation, J. Fluid Mech. 373:287-311 (1998).Google Scholar
  3. 3.
    R. C. Ball and J. R. Melrose, A simulation technique for many spheres in quasi-static motion under frame-invariant pair drag and Brownian forces, Physica A 247:444-472 (1997).Google Scholar
  4. 4.
    C. W. J. Beenakker, The effective viscosity of a concentrated suspension (and its relation to diffusion), Physica A 128:48-81 (1984).Google Scholar
  5. 5.
    O. P. Behrend, Solid-fluid boundaries in particle suspension simulations via the lattice-Boltzmann method, Phys. Rev. E 52:1164 (1995).Google Scholar
  6. 6.
    R. Benzi, S. Succi, and M. Vergassola, The lattice-Boltzmann equation-Theory and applications, Phys. Rep. 222:145 (1992).Google Scholar
  7. 7.
    H. Binous and R. J. Phillips, The effect of sphere-wall interactions on particle motion in a viscoelastic suspension of FENE dumbbells, J. Non-Newton. Fluid Mech. 85:63-92 (1999).Google Scholar
  8. 8.
    G. A. Bird, Molecular Gas Dynamics (University Press, London, Oxford, 1976).Google Scholar
  9. 9.
    L. Bocquet, J. Piasecki, and J.-P. Hansen, On the Brownian motion of a massive sphere suspended in a hard sphere fluid. 1. Multiple-time-scale analysis and microscopic expression for the friction coefficient, J. Stat. Phys. 76:505-526 (1994).Google Scholar
  10. 10.
    G. Bossis and J. F. Brady, Self-diffusion of Brownian particles in concentrated suspensions under shear, J. Chem. Phys. 87:5437 (1987).Google Scholar
  11. 11.
    J. F. Brady, Rheology of concentrated colloidal dispersions, J. Chem. Phys. 99:567-581 (1993).Google Scholar
  12. 12.
    J. F. Brady and G. Bossis, Stokesian dynamics, Ann. Rev. Fluid. Mech. 20:111 (1988).Google Scholar
  13. 13.
    J. F. Brady and J. F. Morris, Microstructure of strongly sheared suspensions and its impact on rheology and diffusion, J. Fluid Mech. 348:103-139 (1997).Google Scholar
  14. 14.
    H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface, Chem. Engng. Sci. 16:242-251 (1961).Google Scholar
  15. 15.
    M. P. Brenner, Screening mechanisms in sedimentation, Phys. Fluids 11:754-772 (1999).Google Scholar
  16. 16.
    R. E. Caflisch and J. H. C. Luke, Variance in the sedimentation speed of a suspension, Phys. Fluids 28:759 (1985).Google Scholar
  17. 17.
    A. A. Catherall, J. R. Melrose, and R. C. Ball, Shear thickening and order-disorder effects in concentrated colloids at high shear rates, J. Rheol. 44:1-25 (2000).Google Scholar
  18. 18.
    S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 1960).Google Scholar
  19. 19.
    H. Chen, Volumetric formulation of the lattice-Boltzmann method for fluid dynamics: Basic Concept, Phys. Rev. E 58:3955-3963 (1998).Google Scholar
  20. 20.
    H. Chen, S. Chen, and W. H. Matthaeus, Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method, Phys. Rev. A 45:R5339-5342 (1992a).Google Scholar
  21. 21.
    H. D. Chen, C. Teixeira, and K. Molvig, Realization of fluid boundary conditions via discrete Boltzmann dynamics, Int. J. Mod. Phys. C 9:1281-1292 (1998).Google Scholar
  22. 22.
    S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows, in Annual Review of Fluid Mechanics, J. L. Lumley, M. V. Dyke, and H. L. Reed, eds. (Palo Alto, California, 1998), pp. 329-364.Google Scholar
  23. 23.
    S. Chen, Z. Wang, X. Shan, and G. D. Doolen, Lattice Boltzmann computational fluid dynamics in three dimensions, J. Stat. Phys 68:379 (1992b).Google Scholar
  24. 24.
    S. Y. Chen, D. Martinez, and R. W. Mei, On boundary conditions in lattice Boltzmann methods, Phys. Fluids 8:2527-2536 (1996).Google Scholar
  25. 25.
    B. Cichocki and R. B. Jones, Image representation of a spherical particle near a hard wall, Physica A 258:273-302 (1998).Google Scholar
  26. 26.
    I. L. Claeys and J. F. Brady, Suspensions of prolate spheroids in Stokes flow. 1. Dynamics of a finite number of particles in an unbounded fluid, J. Fluid Mech. 251:411-442 (1993).Google Scholar
  27. 27.
    R. Cornubert, D. d'Humières, and C. D. Levermore, A Knudsen layer theory for lattice gases, Physica D 47:241 (1991).Google Scholar
  28. 28.
    R. G. Cox, The motion of suspended particles almost in contact, Int. J. Multiphase Flow 1:343-371 (1974).Google Scholar
  29. 29.
    R. I. Cukier, R. Kapral, and J. R. Mehaffey, Kinetic theory of the hydrodynamic interaction between 2 particles, J. Chem. Phys. 74:2494-2504 (1981).Google Scholar
  30. 30.
    B. Dubrulle, U. Frisch, M. Hénon, and J.-P. Rivet, Low-viscosity lattice gases, Physica D 47:27-29 (1991).Google Scholar
  31. 31.
    L. Durlofsky, J. F. Brady, and G. Bossis, Dynamic simulation of hydrodynamically interacting particles, J. Fluid Mech. 180:21 (1987).Google Scholar
  32. 32.
    D. A. Edwards, M. Shapiro, P. Bar-Yoseph, and M. Shapira, The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders, Phys. Fluids A 2:45 (1990).Google Scholar
  33. 33.
    D. L. Ermak and J. A. McCammon, Brownian dynamics with hydrodynamic interactions, J. Chem. Phys. 69:1352 (1978).Google Scholar
  34. 34.
    J. Feng, H. H. Hu, and D. D. Joseph, Direct simulation of initial-value problems for the motion of solid bodies in a Newtonian fluid. 1. Sedimentation, J. Fluid Mech. 261:95-134 (1994a).Google Scholar
  35. 35.
    J. Feng, H. H. Hu, and D. D. Joseph, Direct simulation of initial-value problems for the motion of solid bodies in a Newtonian fluid. 2. Couette and Poiseuille flows, J. Fluid Mech. 277:271-301 (1994b).Google Scholar
  36. 36.
    O. Filippova and D. Hänel, Grid-refinement for lattice-BGK models, J. Comput. Phys. 147:219 (1998).Google Scholar
  37. 37.
    A. L. Fogelson and C. S. Peskin, A fast numerical method for solving the three-dimensional Stokes equations in the presence of suspended particles, J. Comput. Phys. 79:50 (1988).Google Scholar
  38. 38.
    B. Fornberg, Steady incompressible flow past a row of circular cylinders, J. Fluid Mech. 225:625 (1991).Google Scholar
  39. 39.
    D. R. Foss and J. F. Brady, Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation, J. Fluid Mech. 407:167-200 (2000).Google Scholar
  40. 40.
    S. Fraden and G. Maret, Multiple light scattering from concentrated, interacting suspensions, Phys. Rev. Lett. 65:512 (1990).Google Scholar
  41. 41.
    U. Frisch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Lattice gas hydrodynamics in two and three dimensions, Complex Systems 1:649 (1987).Google Scholar
  42. 42.
    U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice gas automata for the Navier-Stokes equation, Phys. Rev. Lett. 56:1505 (1986).Google Scholar
  43. 43.
    M. A. Gallivan, D. R. Noble, J. G. Georgiadis, and R. O. Buckius, An evaluation of the bounce-back boundary condition for lattice Boltzmann simulations, Int J. Numer. Meth. Fluids 25:249-263 (1997).Google Scholar
  44. 44.
    C. K. Ghadder, On the permeability of unidirectional fibrous media: A parallel computational approach, Phys. Fluids 7:2563 (1995).Google Scholar
  45. 45.
    I. Ginzbourg and P. M. Adler, Boundary condition analysis for the three-dimensional lattice-Boltzmann model, J. Phys. II France 4:191 (1994).Google Scholar
  46. 46.
    I. Ginzbourg and D. d'Humières, Local second-order boundary methods for lattice-Boltzmann models, J. Stat. Phys. 84:927 (1996).Google Scholar
  47. 47.
    R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph, and J. Periaux, A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies, Comput. Method Appl Math Engng 184:241-267 (2000).Google Scholar
  48. 48.
    A. Greenbaum, Iterative methods for solving linear systems (Society for Industrial and Applied Mathematics, Philadelphia, 1997).Google Scholar
  49. 49.
    R. D. Groot and P. B. Warren, Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation, J. Chem. Phys. 107:4423-4435 (1997).Google Scholar
  50. 50.
    J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986).Google Scholar
  51. 51.
    J. Happel and H. Brenner, Low-Reynolds Number Hydrodynamics (Martinus Nijhoff, Dordrecht, 1986).Google Scholar
  52. 52.
    E. H. Hauge and A. Martin-Löf, Fluctuating hydrodynamics and Brownian motion, J. Stat. Phys. 7:259 (1973).Google Scholar
  53. 53.
    X. He and L.-S. Luo, Lattice-Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys. 88:927 (1997).Google Scholar
  54. 54.
    X. He, Q. Zou, L.-S. Luo, and M. Dembo, Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model, J. Stat. Phys. 87:115-136 (1997).Google Scholar
  55. 55.
    M. W. Heemels, M. H. J. Hagen, and C. P. Lowe, Simulating solid colloidal particles using the lattice-Boltzmann equation, J. Comput. Phys. 164:48-61 (2000).Google Scholar
  56. 56.
    F. Higuera, S. Succi, and R. Benzi, Lattice gas dynamics with enhanced collisions, Europhys. Lett. 9:345 (1989).Google Scholar
  57. 57.
    R. J. Hill, D. L. Koch, and A. J. C. Ladd, Inertial flows in ordered and random arrays of spheres, J. Fluid Mech, Submitted (1999).Google Scholar
  58. 58.
    P. J. Hoogerbrugge and J. M. V. A. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhys. Lett. 19:155 (1992).Google Scholar
  59. 59.
    W. G. Hoover, T. G. Pierce, C. G. Hoover, J. O. Shugart, C. M. Stein, and A. L. Edwards, Molecular-dynamics, smoothed-particle applied mechanics, and irreversibility, Comput. Math. Appl. 28:155-174 (1994).Google Scholar
  60. 60.
    A. Jasberg, A. Koponen, M. Kataja, and J. Timonen, Hydrodynamical forces acting on particles in a two-dimensional flow near a solid wall, Comput. Phys. Comm. 129:196-206 (2000).Google Scholar
  61. 61.
    D. J. Jeffrey and Y. Onishi, Calculation of the resistance and mobility functions of two unequal rigid spheres in low-Reynolds-number flow, J. Fluid Mech. 139:261 (1984).Google Scholar
  62. 62.
    D. D. Joseph, Y. J. Liu, M. Poletto, and J. Feng, Aggregation and dispersion of spheres falling in viscoelastic liquids, J. Non-Newton Fluid Mech. 54:45-86 (1994).Google Scholar
  63. 63.
    D. L. Koch and A. J. C. Ladd, Moderate Reynolds number flows through periodic and random arrays of aligned cylinders, J. Fluid Mech. 349:31 (1997).Google Scholar
  64. 64.
    D. L. Koch and E. S. G. Shaqfeh, Screening in sedimenting suspensions, J. Fluid Mech. 224:275 (1991).Google Scholar
  65. 65.
    A. Koponen, Simulations of Fluid Flow in Porous Media by Lattice-Gas and Lattice-Boltzmann Methods, Ph.D. thesis, University of Jyväkylä, Finland (1998).Google Scholar
  66. 66.
    A. J. C. Ladd, Hydrodynamic interactions in a suspension of spherical particles, J. Chem. Phys. 88:5051 (1988).Google Scholar
  67. 67.
    A. J. C. Ladd, Hydrodynamic transport coefficients of random dispersions of hard spheres, J. Chem. Phys. 93:3484 (1990).Google Scholar
  68. 68.
    A. J. C. Ladd, Short-time motion of colloidal particles: Numerical simulation via a fluctuating lattice-Boltzmann equation, Phys. Rev. Lett. 70:1339 (1993).Google Scholar
  69. 69.
    A. J. C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part I. Theoretical foundation, J. Fluid Mech. 271:285 (1994a).Google Scholar
  70. 70.
    A. J. C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part II. Numerical results, J. Fluid Mech. 271:311 (1994b).Google Scholar
  71. 71.
    A. J. C. Ladd, Hydrodynamic screening in sedimenting suspensions of non-Brownian spheres, Phys. Rev. Lett. 76:1392 (1996).Google Scholar
  72. 72.
    A. J. C. Ladd, Sedimentation of homogeneous suspensions of non-Brownian spheres, Phys. Fluids 9:491-499 (1997).Google Scholar
  73. 73.
    A. J. C. Ladd, M. E. Colvin, and D. Frenkel, Application of lattice-gas cellular automata to the Brownian motion of solids in suspension, Phys. Rev. Lett. 60:975 (1988).Google Scholar
  74. 74.
    A. J. C. Ladd and D. Frenkel, Dynamics of colloidal dispersions via lattice-gas models of an incompressible fluid, in Cellular Automata and Modeling of Complex Physical Systems, P. Manneville, N. Boccara, G. Y. Vichniac, and R. Bidaux, eds. (Berlin-Heidelberg, 1989), pp. 242-245.Google Scholar
  75. 75.
    A. J. C. Ladd and D. Frenkel, Dissipative hydrodynamic interactions via lattice-gas cellular automata, Physics of Fluids A 2:1921 (1990).Google Scholar
  76. 76.
    A. J. C. Ladd, Effects of container walls on the velocity fluctuations of sedimenting spheres, Unpublished work (2000).Google Scholar
  77. 77.
    L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, London, 1959).Google Scholar
  78. 78.
    L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Massachusetts, 1969).Google Scholar
  79. 79.
    C. E. Leith, Stochastic backscatter in a subgrid-scale model-Plane shear mixing layer, Phys. Fluids A 2:297-299 (1990).Google Scholar
  80. 80.
    A. Levine, S. Ramaswamy, E. Frey, and R. Bruinsma, Screened and unscreened phases in sedimenting suspensions, Phys. Rev. Lett. 81:5944 (1998).Google Scholar
  81. 81.
    M. Loewenberg and E. J. Hinch, Numerical simulation of a concentrated emulsion in shear flow, J. Fluid Mech. 321:395-419 (1996).Google Scholar
  82. 82.
    C. P. Lowe and D. Frenkel, Short-time dynamics of colloidal suspensions, Phys. Rev. E 54:2704-2713 (1996).Google Scholar
  83. 83.
    C. P. Lowe, D. Frenkel, and A. J. Masters, Long-time tails in angular momentum correlations, J. Chem. Phys. 103:1582-1587 (1995).Google Scholar
  84. 84.
    J. H. C. Luke, Decay of velocity fluctuations in a stably stratified suspension, Phys. Fluids. 12:1619-1621 (2000).Google Scholar
  85. 85.
    L.-S. Luo, Unified theory of lattice Boltzmann models for nonideal gases, Phys. Rev. Lett. 81:1618-1621 (1998).Google Scholar
  86. 86.
    A. Madja, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions (Springer-Verlag, New York, 1984).Google Scholar
  87. 87.
    R. S. Maier, R. S. Bernard, and D. W. Grunau, Boundary conditions for the lattice Boltzmann method, Phys. Fluids 8:1788-1801 (1996).Google Scholar
  88. 88.
    D. O. Martinez, W. H. Matthaes, S. Chen, and D. C. Montgomery, On boundary conditions in lattice Boltzmann methods, Phys. Fluids 6:1285-1298 (1994).Google Scholar
  89. 89.
    G. R. McNamara and B. J. Alder, Analysis of the lattice Boltzmann treatment of hydrodynamics, Physica A 194:218 (1993).Google Scholar
  90. 90.
    G. R. McNamara and G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett. 61:2332 (1988).Google Scholar
  91. 91.
    R. W. Mei, L. S. Luo, and W. Shyy, An accurate curved boundary treatment in the lattice Boltzmann method, J. Comput. Phys. 155:307-330 (1999).Google Scholar
  92. 92.
    J. R. Melrose and R. C. Ball, The pathological behavior of sheared hard-spheres with hydrodynamic interactions, Europhys. Lett. 32:535-540 (1995).Google Scholar
  93. 93.
    J. J. Monaghan, Smoothed particle hydrodynamics, Annu. Rev. Astron. Astr. 30:543-574 (1992).Google Scholar
  94. 94.
    J. P. Morris, P. J. Fox, and Y. Zhu, Modeling Low Reynolds Number Incompressible Flow Using SPH, J. Comput. Phys. 136:214-226 (1997).Google Scholar
  95. 95.
    G. P. Muldowney and J. J. L. Higdon, A spectral boundary-element approach to 3-dimensional Stokes flow, J. Fluid Mech. 298:167-192 (1995).Google Scholar
  96. 96.
    N.-Q. Nguyen and A. J. C. Ladd, Lubrication forces in lattice-Boltzmann simulations, Unpublished work (2000).Google Scholar
  97. 97.
    H. Nicolai and E. Guazzelli, Effect of the vessel size on the hydrodynamic diffusion of sedimenting spheres, Phys. Fluids 7:3 (1995).Google Scholar
  98. 98.
    D. R. Noble, S. Y. Chen, J. G. Georgiadis, and R. O. Buckius, A consistent hydrodynamic boundary-condition for the lattice Boltzmann method, Phys. Fluids 7:203-209 (1995).Google Scholar
  99. 99.
    S. A. Orszag and V. Yakhot, Reynolds-number scaling of cellular-automaton hydrodynamics, Phys. Rev. Lett. 56:1691-1693 (1986).Google Scholar
  100. 100.
    H. C. Öttinger, Stochastic Processes in Polymeric Fluids (Springer-Verlag, Berlin, 1996).Google Scholar
  101. 101.
    T. N. Phung, J. F. Brady, and G. Bossis, Stokesian dynamics simulation of Brownian suspensions, J. Fluid Mech. 313:181-207 (1996).Google Scholar
  102. 102.
    C. Pozrikidis, On the transient motion of ordered suspensions of liquid drops, J. Fluid Mech. 246:301-320 (1993).Google Scholar
  103. 103.
    D. W. Qi, Lattice Boltzmann simulations of particles in nonzero Reynolds number flows, J. Fluid Mech. 385:41-62 (1999).Google Scholar
  104. 104.
    Y. H. Qian, D. d'Humières, and P. Lallemand, Lattice BGK models for the Navier- Stokes equation, Europhys. Lett. 17:479-484 (1992).Google Scholar
  105. 105.
    S. R. Rastogi, N. J. Wagner, and S. R. Lustig, Rheology, self-diffusion, and microstructure of charged colloids under simple shear by massively parallel nonequlibrium Brownian dynamics, J. Chem. Phys. 104:9234-9248 (1996).Google Scholar
  106. 106.
    D. H. Rothman, Cellular-automaton fluids: a model for flow in porous media, Geophys. 53:509-518 (1988).Google Scholar
  107. 107.
    A. S. Sangani and A. Acrivos, Slow flow past periodic arrays of cylinders with application to heat transfer, Int. J. Multiphase Flow 8:193 (1982).Google Scholar
  108. 108.
    A. S. Sangani and G. B. Mo, An O(N) algorithm for Stokes and Laplace interactions of particles, Phys. Fluids 8:1990-2010 (1996).Google Scholar
  109. 109.
    P. N. Segré, O. P. Behrend, and P. N. Pusey, Short-time Brownian motion in colloidal suspensions-Experiment and simulation, Phys. Rev. E 52:5070-5083 (1995).Google Scholar
  110. 110.
    P. N. Segré, E. Herbolzheimer, and P. M. Chaikin, Long-range correlations in sedimentation, Phys. Rev. Lett. 79:2574 (1997).Google Scholar
  111. 111.
    A. Sierou and J. F. Brady, Accelerated Stokesian dynamics simulations, J. Fluid Mech. (2001)Google Scholar
  112. 112.
    P. A. Skordos, Initial and boundary conditions for the lattice Boltzmann method, Phys. Rev. E 48:4823-4842 (1993).Google Scholar
  113. 113.
    J. A. Somers and P. C. Rem, in Shell Conference on Parallel Computing, G. A. van der Zee, ed. (1988).Google Scholar
  114. 114.
    P. Tong and B. J. Ackerson, Analogies between colloidal sedimentation and turbulent convection at high Prandtl numbers, Phys. Rev. E 58:R6931-R6934 (1998).Google Scholar
  115. 115.
    S. O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys. 100:25-37 (1992).Google Scholar
  116. 116.
    M. A. van der Hoef, D. Frenkel, and A. J. C. Ladd, Self-diffusion of colloidal particles in a two-dimensional suspension: are deviations from Fick's law experimentally observable?, Phys. Rev. Lett. 67:3459 (1991).Google Scholar
  117. 117.
    J. C. van der Werff and C. G. de Kruiff, Hard-sphere colloidal dispensions: the scaling of rheological properties with particle size, volume fraction, and shear rate, J. Rheol. 33:421 (1989).Google Scholar
  118. 118.
    J. C. van der Werff, C. G. de Kruiff, C. Blom, and J. Mellema, Linear viscoelastic behavior of dense hard-sphere dispersions, Phys. Rev. A 39:795-807 (1989).Google Scholar
  119. 119.
    R. Verberg, I. M. de Schepper, and E. G. D. Cohen, Viscosity of colloidal suspensions, Phys. Rev. E 55:3143-3158 (1997).Google Scholar
  120. 120.
    R. Verberg and A. J. C. Ladd, Simulation of low-Reynolds-number flow via a time-independent lattice-Boltzmann method, Phys. Rev. E 60:3366-3373 (1999).Google Scholar
  121. 121.
    R. Verberg and A. J. C. Ladd, Lattice-Boltzmann model with sub-grid scale boundary conditions, Phys. Rev. Lett 84:2148-2151 (2000a).Google Scholar
  122. 122.
    R. Verberg and A. J. C. Ladd, Simulations of erosion in narrow fractures, Water Resources Res., Submitted: Preprint at wrr00.pdf (2000b).Google Scholar
  123. 123.
    D. A. Weitz, D. J. Pine, P. N. Pusey, and R. J. A. Tough, Nondiffusive Brownian motion studied by Diffusing-Wave Spectroscopy, Phys. Rev. Lett. 63:1747 (1989).Google Scholar
  124. 124.
    Y. Zhu, P. J. Fox, and J. P. Morris, A Pore-Scale Numerical Model for Flow through Porous Media, Int. J. Numer. Anal. Methods Geomech. 23:881-904 (1999).Google Scholar
  125. 125.
    J. X. Zhu, D. J. Durian, J. Müller, D. A. Weitz, and D. J. Pine, Scaling of transient hydrodynamic interactions in concentrated suspensions, Phys. Rev. Lett. 68:2559 (1992).Google Scholar
  126. 126.
    D. P. Ziegler, Boundary conditions for lattice-Boltzmann simulations, J. Stat. Phys. 71:1171-1177 (1993).Google Scholar
  127. 127.
    A. Z. Zinchenko and R. H. Davis, An efficient algorithm for hydrodynamical interaction of many deformable drops, J. Comput. Phys. 157:539-587 (2000).Google Scholar

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© Plenum Publishing Corporation 2001

Authors and Affiliations

  • A. J. C. Ladd
    • 1
  • R. Verberg
    • 1
  1. 1.Chemical Engineering DepartmentUniversity of FloridaGainesville

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