The steady motion of a particle of arbitrary shape at small Reynolds numbers

Abstract

The results given by Brenner & Cox (1963) for the resistance of a particle of arbitrary shape in translation at small Reynolds numbers are generalized. Thus we consider here a single particle of arbitrary shape moving with both translation and rotation in an infinite fluid, the Reynolds number R of the fluid motion being assumed small. With the additional assumption that the motion is steady with respect to some inertial frame of reference, we calculate both the force and couple on the body as an expansion in the Reynolds number to O(R2 In R). This force and couple are expressed entirely in terms of various Stokes flows for the given body in rotation or translation.A discussion is given of the form taken by the formulae for the force and couple for cases in which the body possesses symmetry properties. Quantitative results are obtained for both a spheroid and a dumb-bell-shaped body in pure translation and also for a translating rotating sphere and for a dumb-bell-shaped body in pure rotation.The application of the general results to ‘quasi-steady’ problems is considered, with particular reference to a freely falling spheroid (of small eccentricity) which is shown to orientate itself so that it is broad-side on to its direction of motion.Finally the general results are compared with those that would be obtained by the use of the Oseen equations. By consideration of a particular example it is shown that the Oseen equations do not in general give the correct value of the force on the body to O(R).