The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid

Abstract

In a recent paper by Lawrence & Weinbaum (1986) an unexpected new behaviour was discovered for a nearly spherical body executing harmonic oscillations in unsteady Stokes flow. The force was not a simple quadratic function in half-integer powers of the frequency parameter λ2 = −ia2ω/ν, as in the classical solution of Stokes (1851) for a sphere, and the force for an arbitrary velocity U(t) contained a new memory integral whose kernel differed from the classical t−½ behaviour derived by Basset (1888) for a sphere. A more general analysis of the unsteady Stokes equations is presented herein for the axisymmetric flow past a spheroidal body to elucidate the behaviour of the force at arbitrary aspect ratio. Perturbation solutions in the frequency parameter λ are first obtained for a spheroid in the limit of low- and high-frequency oscillations. These solutions show that in contrast to a sphere the first order corrections for the component of the drag force that is proportional to the first power of λ exhibit a different behaviour in the extreme cases of the steady Stokes flow and inviscid limits. Exact solutions are presented for the middle frequency range in terms of spheroidal wave functions and these results are interpreted in terms of the analytic solutions for the asymptotic behaviour. It is shown that the force on a body can be represented in terms of four contributions; the classical Stokes and virtual mass forces; a newly defined generalized Basset force proportional to λ whose coefficient is a function of body geometry derived from the perturbation solution for high frequency; and a fourth term which combines frequency and geometry in a more general way. In view of the complexity of this fourth term, a relatively simple correlation is proposed which provides good accuracy for all aspect ratios in the range 0.1 < b/a < 10 where exact solutions were calculated and for all values of λ. Furthermore, the correlation has a simple inverse Laplace transform, so that the force may be found for an arbitrary velocity U(t) of the spheroid. The new fourth term transforms to a memory integral whose kernel is either bounded or has a weaker singularity than the t−½ behaviour of the Basset memory integral. These results are used to propose an approximate functional form for the force on an arbitrary body in unsteady motion at low Reynolds number.