Stress moments of nearly touching spheres in low Reynolds number flow

Abstract

If two spheres are nearly touching, and the flow around them is governed by the Stokes equations, the integral moments of the surface stress are singular functions of the gap width. The method used previously to calculate the singular terms in the zeroth moment (the force) and the antisymmetric first moment (the couple) is extended here to calculate the singular terms in the symmetric first moment (the stresslet) for motions perpendicular to the line of centres. It is shown that the reciprocal theorem requires unexpected relations between the newly found singularities and ones found previously. It is also shown that the singular terms can be used to improve the rate of convergence of series expressions for the stresslets. The series expressions then become valid for all separations of the spheres.