Spherical particle in Poiseuille flow between planar walls

We study a spherical mesoparticle suspended in Newtonian fluid between plane-parallel walls with incident Poiseuille flow. Using a two-dimensional Fourier transform technique we obtain a symmetric analytic expression for the Green tensor for the Stokes equations describing the creeping flow in this geometry. From the matrix elements of the Green tensor with respect to a complete vector harmonic basis, we obtain the friction matrix for the sphere. The calculation of matrix elements of the Green tensor is done in large part analytically, reducing the evaluation of these elements to a one-dimensional numerical integration. The grand resistance and mobility matrices in Cartesian form are given in terms of 13 scalar friction and mobility functions which are expressed in terms of certain matrix elements calculated in the spherical basis. Numerical calculation of these functions is shown to converge well and to agree with earlier numerical calculations based on boundary collocation. For a channel width broad with respect to the particle radius, we show that an approximation defined by a superposition of single-wall functions is reasonably accurate, but that it has large errors for a narrow channel. In the two-wall geometry the friction and mobility functions describing translation-rotation coupling change sign as a function of position between the two walls. By Stokesian dynamics calculations for a polar particle subject to a torque arising from an external field, we show that the translation-rotation coupling induces sideways migration at right angles to the direction of fluid flow.