A sphere in a second degree polynomial creeping flow parallel to a wall

Comprehensive results are provided for the creeping flow around a spherical particle in a viscous fluid close to a plane wall, when the external velocity is parallel to the wall and varies as a second degree polynomial in the coordinates. By linearity of Stokes equations, the solution is a sum of flows for typical unperturbed flows: a pure shear flow, a 'modulated shear flow', for which the rate of shear varies linearly in the direction normal to the wall, and a quadratic flow. Solutions considered here use the bipolar coordinates technique. They complement the accurate results of Chaoui and Feuillebois (2003) for the pure shear flow. The solution of Goren and O'Neill (1971) for the quadratic flow is reconsidered and a new analytical solution is derived for the ambient modulated shear flow. The perturbed flow fields for these two cases are presented in detail and discussed. Results for the force and torque friction factors are provided with a 5 × 10 ^-17 accuracy as a reference. For the quadratic flow, there is a force and a torque on a fixed sphere. A minimum value of the torque is found for a gap of about 0.18 a, where a is the sphere radius. This minimum is interpreted in term of the corresponding flow structure. For the modulated shear flow, there is only a torque. The free motion of a sphere in an ambient quadratic flow is also determined.