Stokes problems of multiparticle systems: A numerical method for arbitrary flows

Abstract

A boundary element method (BEM) is developed and used to compute the arbitrary Stokes flow of a system of N particles of arbitrary shape and size embedded in a Newtonian fluid. The particles can be assumed to have a specified deformation (this feature is included here in anticipation of some applications in biophysics). Hydrodynamic and nonhydrodynamic forces are assumed to act in general; Brownian forces are neglected; and the Stokes flow field is assumed to be arbitrary. The method is benchmarked against the exact solution of two equal spheres in free settling motion by Goldman, Cox, and Brenner [Chem. Eng. Sci. 2 1, 1151 (1966)]. The handling of a finite bounding surface (i.e., arbitrary flow field, container walls) is tested with the calculation of the effective shear viscosity of a dilute suspension of monosized rigid spheres. The benchmarks show that the method performs satisfactorily. Also reported are the results for three spheres, arranged either in equispaced linear or triangular configuration (fixed or free settling under gravity). The effect of the hydrodynamic interaction with a rigid plane boundary (half‐space problem) is automatically treated by the use of the special image of the Kelvin state [N. Phan‐Thien, J. Elasticity 1 3, 231 (1983)]. This is illustrated by considering the problem of free settling of particles near a rigid no‐slip planar boundary; specifically, some results for two equal spheres are reported. It is shown that the particle motion near the solid plane is fundamentally different from its motion in the far field, and thus the relative divergence of particles and the reversal of their angular velocity is observed.