The singularity method in unsteady Stokes flow: hydrodynamic force and torque around a sphere in time-dependent flows

Abstract

The equations for the hydrodynamic force and torque acting on a sphere in unsteady Stokes equations under different flow conditions are solved analytically by means of the singularity method. This analytical technique is based on the combination of suitable singularity solutions (also called fundamental solutions) such as primary Stokeslets, potential dipoles, or higher-order singularities, to construct the flow field. The different flows considered here include four examples: (1) a rotating sphere in a viscous flow, (2) a stationary sphere in a time-dependent shear flow, (3) a sphere with free rotation in a simple shear flow, as well as (4) a stationary sphere in a time-dependent axisymmetric parabolic flow. Our paradigm is to derive the fundamental solutions in unsteady Stokes flows and to express the solutions as a convolution integral in time using the time–space fundamental solutions. Next the Laplace transform is used to determine the strength of the distributed singularities that induce the velocity field around a stationary or rotating sphere. Then we use the computed strength of the singularities to derive hydrodynamic force and torque. In particular, for the problem of a stationary sphere in unsteady axisymmetric parabolic flow, our solution for the time-dependent force acting on the sphere consists of five force components – the well-known quasi-steady Stokes drag, the added mass term, the Basset historic (memory) force, and two additional memory forces. The first additional memory force due to the rate change of velocity, we find, is similar to the result obtained by Lawrence & Weinbaum (J. Fluid Mech., vol. 171, 1986, pp. 209–218) for the ostensibly unrelated setting of a slightly deformed translating spheroid. The second additional memory force comes from the effect of the rate change of acceleration and is found for the first time in this study to the best of our knowledge.