Generating, from scratch, the near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres in low Reynolds number flow

Abstract

The motion of rigid spherical particles suspended in a low Reynolds number fluid can be related to the forces, torques and stresslets acting upon them by 22 scalar resistance functions, commonly notated $X^A_{11}$, $X^A_{12}$, $Y^A_{11}$, etc. Near-field asymptotic forms of these resistance functions were derived in Jeffrey and Onishi (J. Fluid Mech., 1984) and Jeffrey (Phys. Fluids A, 1992); these forms are now used in several numerical methods for suspension mechanics. However, the first of these important papers contains a number of small errors which make it difficult for the reader to correctly evaluate the functions for parameters not explicitly tabulated. This short article comprehensively corrects these errors, and adds formulae that were originally omitted from both papers, so that the reader can verify and implement the equations independently. The corrected expressions, rationalised and using contemporary nondimensionalisation, are shown to match mid-field values of these scalars which are calculated through an alternative method. A Python script to generate and evaluate these functions is provided.