Use of the Basset coefficient in the calculation of the velocity of a spheroid slowing down in a viscous incompressible fluid after a sudden impulse

Abstract: The motion of a spheroid in a viscous incompressible fluid after a sudden small impulse in the direction of the symmetry axis is studied on the basis of the linearized Navier-Stokes equations. The time-dependence of the spheroid velocity follows by a Fourier transform from the frequency- dependence of the impedance involving friction coefficient, body mass, and added fluid mass. A term proportional to the square root of frequency in the asymptotic high frequency expansion of the impedance, with a Basset coefficient, describes the initial decay in time from the initial value determined by the effective mass. It is shown from numerical evidence based on the exact multipole expansion of the solution for the flow field in terms of spheroidal wavefunctions that the knowledge of the Basset coefficient is insufficient for a reliable estimate of the deviations from a simple two-pole approximation to the complete behavior in time. The two-pole approximation can be calculated from the effective mass and the steady state friction coefficient.