Motion of a spherical particle in a viscous fluid along a slip wall

Abstract

Summary The motion of a solid spherical particle in the flow of a viscous fluid is calculated in the framework of Stokes equations, when the Navier condition of slippage applies on the wall and the no-slip condition applies on the particle. The problem for the flow around a sphere translating and rotating in a pure shear flow is solved, from the linearity of Stokes equations, as the sum of elementary problems: a sphere held fixed in a shear flow close to a wall, a sphere translating and rotating along a wall in a fluid at rest. The creeping flow is calculated analytically using the bispherical coordinates technique that provides the fluid velocity and pressure as series. The unknown coupled infinite series of coefficients are reduced by combinations from seven to four. This simplifies the formulation compared with earlier works on the translation and rotation problems. The infinite linear system for the coefficients is then solved using an extension of Thomas algorithm that allows to calculate O(10 5 ) terms in the series. Such a high number is necessary to obtain a good accuracy at small sphere to wall gaps down to 10 −5 sphere radius, in particular for a large slip on the wall. Accurate results for the force and torque on the particle are then obtained for each of the three elementary problems. On this basis, results are provided for the diffusion coefficient of a dilute suspension of freely rotating spheres parallel to a slipping wall and for the translational and rotational velocities of a freely moving sphere in a pure shear flow along such a wall. Simplified formulae are provided for application to various physical problems, like for instance the interpretation of velocity measurements at micro-scales by tracking of particles close to a hydrophobic wall.