An efficient scheme is presented for the numerical calculation of hydrodynamic interactions of many spheres in Stokes flow. The spheres may have various sizes, and are freely moving or arranged in rigid arrays. Both the friction and mobility matrix are found from the solution of a set of coupled equations. The Stokesian dynamics of many spheres and the friction and mobility tensors of polymers and proteins may be calculated accurately at a modest expense of computer memory and time. The transport coefficients of suspensions can be evaluated by use of periodic boundary conditions.

It is shown that the standard treatment of lubrication effects in many-particle hydrodynamic interactions leads to divergent three-particle contributions to the short-time translational self-diffusion coefficient. To resolve the problem the improved method to account for lubrication is proposed. The translational and rotational self-diffusion coefficients of the Brownian semidilute suspension are then evaluated up to terms of the second order in volume fraction.

We obtain the many-body hydrodynamic friction and mobility matrices describing the motion in a fluid of N hard-spheres with stick boundary conditions in the presence of a planar hard wall or free surface using ͑1͒ a multipole expansion of the hydrodynamic force densities induced on the spheres and ͑2͒ an image representation to account for the fluid boundary. The coupled multipole equations may be truncated at any order to give positive definite approximations to the exact friction and mobility matrices. An extension of the Bossis-Brady lubrication correction to the friction matrix is also discussed and included. The resulting method for computing the mobility matrix may be used for the Stokesian or Brownian dynamics simulation of N spheres subject to interparticle and external forces and imposed shear flow. We illustrate the method by performing Stokesian dynamics simulation of particles near a hard wall. The simulations exhibit the rapid convergence of the multipole truncation scheme including lubrication corrections.

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First-order virial expansion of short-time diffusion and sedimentation coefficients of permeable particles suspensions Phys. Fluids 23, 083303 (2011); 10.1063/1.3626196 Rotational and translational self-diffusion in concentrated suspensions of permeable particles We present a comprehensive computational study of the short-time transport properties of bidisperse hard-sphere colloidal suspensions and the corresponding porous media. Our study covers bidisperse particle size ratios up to 4 and total volume fractions up to and beyond the monodisperse hard-sphere close packing limit. The many-body hydrodynamic interactions are computed using conventional Stokesian Dynamics (SD) via a Monte-Carlo approach. We address suspension properties including the short-time translational and rotational self-diffusivities, the instantaneous sedimentation velocity, the wavenumber-dependent partial hydrodynamic functions, and the high-frequency shear and bulk viscosities and porous media properties including the permeability and the translational and rotational hindered diffusivities. We carefully compare the SD computations with existing theoretical and numerical results. For suspensions, we also explore the range of validity of various approximation schemes, notably the pairwise additive approximations with the Percus-Yevick structural input. We critically assess the strengths and weaknesses of the SD algorithm for various transport properties. For very dense systems, we discuss in detail the interplay between the hydrodynamic interactions and the structures due to the presence of a second species of a different size. C 2015 AIP Publishing LLC. [http://dx.

On the basis of a nonlinear kinetic equation for a moderately dense system of hard spheres and disks it is shown that shear and normal stresses in a steady-state, uniform shear flow contain singular contributions of the form t XI 3j2 for hard spheres, or IX I log[ XI for hard disks. Here Xis proportional to the velocity gradient in the shear flow. The origin of these terms is closely related to the hydrodynamic tails t -a~2 in the current-current correlation functions. These results also imply that a nonlinear shear viscosity exists in two-dimensional systems. An extensive discussion is given on the range of X values where the present theory can be applied, and numerical estimates of the effects are given for typical circumstances in laboratory and computer experiments.

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