Particle motion in Stokes flow near a plane fluid-fluid interface. Part 2. Linear shear and axisymmetric straining flows

Abstract

We consider the motion of a sphere or a slender body in the presence of a plane fluid–fluid interface with an arbitrary viscosity ratio, when the fluids undergo a linear undisturbed flow. First, the hydrodynamic relationships for the force and torque on the particle at rest in the undisturbed flow field are determined, using the method of reflections, from the spatial distribution of Stokeslets, rotlets and higher-order singularities in Stokes flow. These fundamental relationships are then applied, in combination with the corresponding solutions obtained in earlier publications for the translation and rotation through a quiescent fluid, to determine the motion of a neutrally buoyant particle freely suspended in the flow. The theory yields general trajectory equations for an arbitrary viscosity ratio which are in good agreement with both exact-solution results and experimental data for sphere motions near a rigid plane wall. Among the most interesting results for motion of slender bodies is the generalization of the Jeffrey orbit equations for linear simple shear flow.