The methodology of "Stokesian dynamics," an efficient and accurate simulation technique for suspensions of spheres, is extended to non-spherical particles. The model system consists of rigid, non-Brownian prolate spheroids suspended in an incompressible Newtonian fluid at zero Reynolds number. The method is applied to calculate the hydrodynamic transport properties of unbounded dispersions of ellipsoids. Both "random" configurations and very orderly arrangements of particles are considered in order to probe the relation between the microstructure of the suspension and its macroscopically observable properties.
The simulation method is based on a microstructurally detailed description of the two-phase system and explicitly takes into account hydrodynamic interactions between the particles. Non-local singularity solutions for ellipsoids in Stokes flow are combined with Faxen laws using pair-wise additivity of velocities to construct a far-field approximation to the mobility tensor. The convergence problems associated with the long-ranged nature of viscous interactions at zero Reynolds number are handled using O'Brien's renormalization procedure. The Ewald summation technique is applied to accelerate the evaluation of the lattice sums generated by the periodic boundary conditions. Lubrication stresses between almost touching spheroids are added in a pair-wise manner to the mobility inverse. All the two-body resistance functions which diverge as the surface separation vanishes are computed to O(ε0), with ε the gap width, so that the singular behavior of the lubrication interactions is captured correctly for arbitrary relative orientations and relative motions of the particles.
The method is first illustrated for a finite number of particles in an unbounded fluid domain, and shown to be accurate and efficient. It is then applied to crystalline geometries of spheroids over the full concentration range from 0 to closest packing (74% by volume). The dependence of the hydrodynamic transport properties (sedimentation rate, diffusion coefficient, stress/rate-of-strain relation, permeability and hindered diffusivity) on the density of the dispersion, the aspect ratio of the particles and the lattice type is investigated. Equilibrium structures of hard ellipsoids generated by a Monte Carlo procedure are also considered. The high frequency limit of the hydrodynamic transport properties is computed and compared to the results for crystalline configurations, and to available experimental measurements. A discontinuous jump in some suspension properties is observed at the isotropic to nematic transition.
As a prelude to dynamic simulations, the compatibility of unit cells with pure straining flows is examined. It is demonstrated that no self-reproducing lattices exist in axisymmetric extensional flows, but a set of compatible basis vectors is derived. Planar straining fields on the other hand possess an infinite number of strain-periodic lattices.
