The Resistance of an Arbitrary Body in Confined Unsteady Stokes Flow

Abstract

In this article, we address resistance forces and torques acting onto a body with arbitrary shape moving in an unsteady Stokes flow. We start analyzing the functional form of the expressions for forces and torques, which depend on the frequency parameter and on the position of the body in the domain of the fluid, and determining the asymptotic limits for high and low frequencies. In this regard, we show that, for high frequencies (hence short times), forces and torques are obtained by the associated hydrodynamic problems considering ideal potential flows, independently of the geometry of the problem. Afterwards, with the aim of obtaining expressions for forces and torques valid in the entire range of frequencies, we extend to the unsteady case the reflection method, largely employed in the theory of the steady Stokes flows. In this way, general expressions are provided in terms of the Faxén operators of the body and the Green function associated with the geometry of the confinement, that are valid, to the leading order, at any frequency, independently of the geometry of the problem. Finally, as the application of the general expressions, explicit relations for the resistance forces acting onto a spherical body with no-slip boundary conditions near a plane wall with full-slip boundary conditions are obtained, valid over the entire frequency range, provided that the distance between the plane and the sphere is larger than one sphere radius.