The force on an axisymmetric body in linearized, time-dependent motion: a new memory term

Abstract

In contrast to the steady Stokes equations for creeping motion, the time-dependent linearized Navier–Stokes equations have only been solved for very restricted geometries, the solution for the sphere being the sole solution for an isolated finite body. In the present paper, the linearized Navier–Stokes equations are further explored and a simple expression is derived which relates the force on an arbitrary axisymmetric body in oscillatory motion to the solution for the stream function in the far field. This result is applied to the case of a slightly eccentric spheroid and it is shown that the total hydrodynamic force contains four terms, three of which correspond to the classical solutions for the Stokes drag, added mass and Basset force on the perturbed sphere; the fourth term is only present when the body is non-spherical. In contrast to the three classical forces, the new term is not a simple power of the dimensionless frequency parameter iL2ω/ν, in which L is a length-scale, ω is the frequency of oscillation and ν is the kinematic viscosity of the fluid. A Laplace superposition is then used to find the force on the spheroid in an arbitrary axisymmetric motion with velocity U(t). The new memory term decays faster than the Basset force at large times and is bounded at short times.