A sphere in a second degree polynomial creeping flow parallel to a plane, impermeable and slipping wall

Abstract

The motion of a solid and no-slip spherical body immersed in a Newtonian liquid near a motionless, plane and impermeable slip wall is investigated, in the creeping flow approximation, using on the wall the Navier slip boundary condition. The considered cases are as follows (i) a sphere either translating or rotating parallel to the wall in a quiescent liquid; (ii) a sphere either held fixed or freely-suspended in a modulated, linear or quadratic ambient shear flow. For each case, the velocity and pressure fields about the sphere together with the associated physical quantities whenever relevant (the force, torque, non-zero stresslet component on the sphere and its translational and angular velocities) are expressed in bipolar coordinates as infinite series, the coefficients of which are governed by an infinite linear system. This system is solved numerically by first truncating at an order depending on the relevant quantity and on the sphere location and wall slip length and then applying either a Gaussian elimination or a Thomas’ algorithm for inverting a large tridiagonal matrix. Handy formulae for all key quantities are also derived as asymptotic expansions for a sphere-wall gap that is large compared with the sphere radius. The sensitivity of the computed associated normalized friction factors (force, torque, stresslet) and mobilities (translational and angular velocities) to both the sphere location and the wall slip length are then discussed.