The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers

Abstract

Assuming that the Stokes flow past an arbitrary particle in a uniform stream is known for any three non-coplanar directions of flow, then the force on the body to O(R), for any direction of flow, is given explicitly in terms of these Stokes velocity fields. The Reynolds number (R) based on the maximum particle dimension is assumed small. For bodies with certain types of symmetry it suffices merely to know the Stokes resistance tensor for the body in order to calculate this force. In this case the resulting formula is identical to that of Brenner (1961) and Chester (1962). However, for bodies devoid of such symmetry, their formula is incomplete—there being an additional force at right angles to the uniform stream which remains invariant under a reversal of the flow at infinity. As this additional force is a lift force, it follows that the Brenner-Chester formula furnishes the correct drag on a body of arbitrary shape; moreover, this drag is always reversed to at least O(R) by a reversal of the uniform flow at infinity.Exactly analogous formulae are derived using the classical Oseen equations, and it is shown that although this gives both the correct vector force on bodies with the above types of symmetry and the correct drag on bodies of arbitrary shape, it gives in general an incorrect lift component for completely arbitrary particles.Finally, the singular perturbation result for the force on an arbitrary body is extended to terms of O(R2log R). This higher-order contribution to the force is given explicitly in terms of the Stokes resistance tensor, and has the property of being reversed by a reversal of the flow at infinity, regardless of the geometry of the body.These results are collected in the Summary at the end of the paper.