A Spectral-Element Method for Particulate Stokes Flow

Abstract

A numerical procedure is presented for solving the equations of Stokes flow past a fixed bed of rigid particles and the equations describing the motion of a suspension of rigid particles upon which a specified force and torque is exerted, for general flow configurations and arbitrary particle shapes. The problem is formulated in terms of an integral equation of the first kind for the distribution of the boundary traction incorporating a Green function that observes the periodicity of the flow and the geometry of the boundaries of the flow, accompanied by appropriate boundary conditions and integral constraints. The integral equation is solved for two-dimensional flow by a spectral-element orthogonal-collocation method. Two important components of the numerical method are (a) preconditioning of the linear system that arises from the discretization of integral equation followed by reduction to remove the eigenfunctions corresponding to the null eigenvalue over each particle surface and (b) a physically motivated iterative solution of the master linear system based on particle clustering. It is found that, for the purpose of computing the force and torque exerted on fixed particles and the velocity of translation and angular velocity of rotation of freely suspended particles, the orthogonal collocation method has significant advantages over the trapezoidal discretization. The iterative solution of the integral equation converges even for closely spaced particles where each particle is treated as a cluster.