Interception of two spheres with slip surfaces in linear Stokes flow

Abstract

The interception of two spherical particles with arbitrary size in an infinite linear ambient Stokes flow is considered. The particle surfaces allow for slip according to the Navier–Maxwell–Basset law relating the shear stress to the tangential velocity. At any instant, the flow is computed in a frame of reference with origin at the centre of one particle using a cylindrical polar coordinate system whose axis of revolution passes through the centre of the second particle. Taking advantage of the axial symmetry of the boundaries of the flow in the particle coordinates, the problem is formulated as a system of integral equations for the zeroth, first, and second Fourier coefficients of the boundary traction with respect to the meridional angle. The force and torque exerted on each particle are determined by the zeroth and first Fourier coefficients, while the stresslet is determined by the zeroth, first, and second Fourier coefficients. The derived integral equations are solved with high accuracy using a boundary element method featuring adaptive element distribution and automatic time step adjustment according to the inter-particle gap. The results strongly suggest the existence of a critical value for the slip coefficient below which the surfaces of two particle collide after a finite interception time. The critical value depends on the relative initial particle positions. The particle stress tensor and coefficients of the linear and quadratic terms in the expansion of the effective viscosity of a dilute suspension in terms of the concentration in simple shear flow are discussed and evaluated. Surface slip significantly reduces the values of both coefficients and the longitudinal particle self-diffusivity.