Accuracy of the multipole expansion applied to a sphere in a creeping flow parallel to a wall

An example system is studied to discuss precision of the multipole expansion, applied to determine forces exerted on particles by a viscous low-Reynolds-number fluid flow. A single sphere in an ambient flow (pure shear, quadratic, and modulated shear) parallel to a close plane wall is considered. Forces and torques exerted by the ambient flow on a motionless sphere are evaluated. Their precision is determined and related to a multipole order of the truncation. Similar analysis is performed for a moving sphere with no ambient flow and for a freely moving sphere. Relative motion of the sphere with respect to the wall gives rise to strong lubrication interactions. It is analysed how these interactions affect accuracy of the pure multipole expansion, and what are the smallest distances where it becomes insufficient. An alternative precise method is applied, in which lubrication expressions are subtracted from the hydrodynamic forces and torques, and the residue is evaluated as a fast-convergent series of inverse powers of the distance between the sphere centre and the wall. The accuracy of this procedure is carefully analysed.