Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T07:51:41.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation

Published online by Cambridge University Press:  26 April 2006

Anthony J. C. Ladd
Anthony J. C. Ladd
Affiliation:
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 271 , 25 July 1994 , pp. 285 - 309
Copyright
© 1994 Cambridge University Press

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.)

References

Allen, M. P. & Tildesley, D. J. 1987 Computer Simulation of Liquids. Clarendon.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bird, G. A. 1976 Molecular Gas Dynamics. Oxford University Press.
Bird, G. A. 1990 The direct simulation Monte Carlo method: Current status and perspectives. In Microscopic Simulations of Complex Flows (ed. M. Marechal). Plenum.
Bossis, G. & Brady, J. F. 1987 Self-diffusion of Brownian particles in concentrated suspensions under shear. J. Chem. Phys. 87, 5437.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Ann. Rev. Fluid. Mech. 20, 111.Google Scholar
Chapman, S. & Cowling, T. G. 1960 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.
Chen, S., Wang, Z., Shan, X. & Doolen, G. D. 1992 Lattice Boltzmann computational fluid dynamics in three dimension. J. Statist Phys 68, 379.Google Scholar
Cornubert, R., D'Humières, D. & Levermore, C. D. 1991 A Knudsen layer theory for lattice gases. Physica D 47, 241.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21.Google Scholar
Ermak, D. L. & McCammon, J. A. 1978 Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69, 1352.Google Scholar
Fogelson, A. L. & Peskin, C. S. 1988 A fast numerical method for solving the three-dimensional Stokes equations in the presence of suspended particles. J. Comput. Phys. 79, 50.Google Scholar
Frisch, U., Hasslacher, B. & Pomeau, Y. 1986 Lattice gas automata for the Navier–Stokes equation. Phys. Rev. Lett. 56, 1505.Google Scholar
Frisch, U., D'Humières, D., Hasslacher, B., Lallemand, P., Pomeau, Y. & Rivet, J.-P. 1987 Lattice gas hydrodynamics in two and three dimension. Complex Systems 1, 649.Google Scholar
Hansen, J. P. & McDonald, I. R. 1986 Theory of Simple Liquids. Academic.
Happel, J. & Brenner, H. 1986 Low-Reynolds Number Hydrodynamics. Martinus Nijhoff.
Higuera, F., Succi, S. & Benzi, R. 1989 Lattice gas dynamics with enhanced collisions. Europhys. Lett. 9, 345.Google Scholar
Hoef, M. A. van der 1992 Simulation study of diffusion in lattice-gas fluids and colloids. PhD thesis, University of Utrecht, Utrecht, Netherlands.
Hoef, M. A. van der, Frenkel, D. & Ladd, A. J. C. 1991 Self-diffusion of colloidal particles in a two-dimensional suspension: are deviations from Fick's law experimentally observable? Phys. Rev. Lett. 67, 3459.Google Scholar
Hoogerbrugge, P. J. & Koelman, J. M. V. A. 1992 Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys. Lett. 19, 155.Google Scholar
Kao, M. H., Yodh, A. G. & Pine, D. J. 1993 Observation of Brownian motion on the time scale of the hydrodynamic interaction. Phys. Rev. Lett. 70, 242.Google Scholar
Karrila, S. J., Fuentes, Y. O. & Kim, S. 1989 Parallel computational strategies for hydrodynamic interactions between rigid particles of arbitrary shape in a viscous fluid. J. Rheol. 33, 913.Google Scholar
Koelman, J. M. V. A. & Hoogerbrugge, P. J. 1993 Dynamic simulations of hard-sphere suspensions under steady shear. Europhys. Lett. 21, 363.Google Scholar
Ladd, A. J. C. 1988 Hydrodynamic interactions in a suspension of spherical particles. J. Chem. Phys. 88, 5051.Google Scholar
Ladd, A. J. C. 1991 Dissipative and fluctuating hydrodynamic interactions between suspended solid particles via lattice-gas cellular automata. In Computer Simulation in Materials Science (ed. M. Meyer & V. Pontikis). Kluwer.
Ladd, A. J. C. 1993 Short-time motion of colloidal particles: Numerical simulation via a fluctuating lattice-Boltzmann equation. Phys. Rev. Lett. 70, 1339.Google Scholar
Ladd, A. J. C. 1994 Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311.Google Scholar
Ladd, A. J. C., Colvin, M. E. & Frenkel, D. 1988 Application of lattice-gas cellular automata to the Brownian motion of solids in suspension. Phys. Rev. Lett. 60, 975.Google Scholar
Ladd, A. J. C. & Frenkel, D. 1989 Dynamics of colloidal dispersions via lattice-gas models of an incompressible fluid. In Cellular Automata and Modeling of Complex Physical Systems (ed. P. Manneville, N. Boccara, G. Y. Vichniac & R. Bidaux. Springer.
Ladd, A. J. C. & Frenkel, D. 1990 Dissipative hydrodynamic interactions via lattice-gas cellular automata. Phys. Fluids A 2, 1921.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Addison-Wesley.
McNamara, G. R. & Alder, B. J. 1992 Lattice Boltzmann simulation of high Reynolds number fluid flow in two dimensions. In Microscopic Simulations of Complex Hydrodynamic Phenomena (ed. M. Mareschal & B. L. Holian). Plenum.
McNamara, G. R. & Alder, B. J. 1993 Analysis of the lattice Boltzmann treatment of hydrodynamics. Physica A 194, 218.Google Scholar
McNamara, G. R. & Zanetti, G. 1988 Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61, 2332.Google Scholar
Sulsky, D. & Brackbill, J. U. 1991 A numerical method for suspension flow. J. Comput. Phys. 96, 339.Google Scholar
Tran-Cong, T. & Phan-Thien, N. 1989 Stokes problems of multiparticle systems: A numerical method for arbitrary flows. Phys. Fluids A 1, 453.Google Scholar
Yen, S. M. 1984 Numerical solution of the nonlinear Boltzmann equation for nonequilibrium gas flow problems. Ann. Rev. Fluid Mech. 16, 67.Google Scholar
Zanetti, G. 1989 The hydrodynamics of lattice gas automata. Phys. Rev. A 40, 1539.Google Scholar
Zhu, J. X., Durian, D. J., Müller, J., Weitz, D. A. & Pine, D. J. 1992 Scaling of transient hydrodynamic interactions in concentrated suspensions. Phys. Rev. Lett. 68, 2559.Google Scholar