A new and very general technique for simulating solid-uid 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 uid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-ow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic uctuations in the uid 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 the companion paper, extensive numerical tests of the method, for stationary ows, time-dependent ows, and nite Reynolds number ows, are reported.

We present a new method to compute the absolute free energy of arbitrary solid phases by Monte Carlo simulation. The method is based on the construction of a reversible path from the solid phase under consideration to an Einstein crystal with the same crystallographic structure. As an application of the method we have recomputed the free energy of the fcc hard-sphere solid at melting. Our results agree well with the single occupancy cell results of Hoover and Ree. The major source of error is the nature of the extrapolation procedure to the thermodynamic limit. We have also computed the free energy difference between hcp and fcc hard-sphere solids at densities close to melting. We find that this free energy difference is not significantly different from zero: −0.001<ΔF<0.002.

A new and very general technique for simulating solid-uid suspensions has been described in a previous paper (Part I); the most important feature of the new method is that the computational cost scales linearly with the number of particles. In this paper (Part II), extensive numerical tests of the method are described; for creeping ows, both with and without Brownian motion, and at nite Reynolds numbers. Hydrodynamic interactions, transport coe cients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated.

The lattice-Boltzmann method has been refined to take account of near-contact interactions between spherical particles. First, we describe a comprehensive solution to the technical problems that arise when two discretized surfaces come into contact. Second, we describe how to incorporate lubrication forces and torques into lattice-Boltzmann simulations, and test our method by calculating the forces and torques between a spherical particle and a plane wall. Third, we describe an efficient update of the particle velocities, taking into account the possibility that some of the differential equations are stiff.

Accurate values for the hydrodynamic transport properties of random dispersions of hard spheres have been determined by numerical simulation. The many-body hydrodynamic interactions are calculated from a multipole-moment expansion of the force density on the surface of the solid particles; the singular lubrication forces are included exactly for pairs of particles near contact. It has been possible to calculate the transport properties of small periodic systems, at all packing fractions, with uncertainties of less than 1%; but for larger systems we are limited computationally to lower order, and therefore less accurate, moment approximations to the induced force density. Nevertheless, since the higher-order moment contributions are short range they are essentially independent of system size and we can use small system data to correct our results for larger systems. Numerical calculations show that this is a reliable and accurate procedure. The ensemble-averaged mobility tensors are strongly dependent on system size, with deviations from the thermodynamic limit varying as N−1/3. We show that these deviations can be accounted for by a straightforward calculation based on the length of the unit cell and the suspension viscosity and mobility. The remaining number dependencies are small. Problems involved in implementing this method for larger numbers of particles are considered, and alternative methods are discussed.

Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at
moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random
arrays of spheres. The drag force on the spheres, and hence the permeability of the
arrays, is calculated as a function of the Reynolds number at solid volume fractions
up to the close-packed limits of the arrays. At Reynolds numbers up to O(102), the
non-dimensional drag force has a more complex dependence on the Reynolds number
and the solid volume fraction than suggested by the well-known Ergun correlation,
particularly at solid volume fractions smaller than those that can be achieved in
physical experiments. However, good agreement is found between the simulations
and Ergun's correlation at solid volume fractions approaching the close-packed limit.
For ordered arrays, the drag force is further complicated by its dependence on the
direction of the flow relative to the axes of the arrays, even though in the absence
of fluid inertia the permeability is isotropic. Visualizations of the flows are used to
help interpret the numerical results. For random arrays, the transition to unsteady
flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are
discussed.

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