Flow past a sphere undergoing unsteady rectilinear motion and unsteady drag at small Reynolds number

The flow induced by a sphere which undergoes unsteady motion in a Newtonian fluid at small Reynolds number is considered at distances large compared w ith sphere radius a. Previous solutions of the unsteady Oseen equations (Oc kendon 1968; Lovalenti & Brady 1993b) for rectilinear motion are refined. T hree-dimensional Fourier transforms of the disturbance field are integrated over Fourier space to derive new concise equations for the velocity field and history force in terms of single history integrals. Various slip-velocity profiles are classified by the ratio A of the particl e relative displacement, z(p)'(t') - z(p)'(tau'), to the diffusion length, l(D)' == 2[v(t' (_) tau')](1/2), where v, is the kinematic viscosity of the fluid. Most previous studies are concerned with largedisplacement motions for which the ratio is large in the long-time limit. It is shown using asym ptotic calculations that the flow at any point at large distance z past a s phere for arbitrary large-displacement and non-reversing motion is the same as the steady-state laminar wake if z is expressed in terms of the time el apsed since the particle was at that point in the laboratory frame. The poi nt source solution for the remainder of the far flow is also valid for the unsteady case. A start-up motion with slip velocity V-p' = gamma'(t')(-1/2), t' > 0, is in vestigated for which A is finite. A self-similar solution for the flow fiel d is obtained in terms of space coordinates scaled by the diffusion length, u' = au(ss)(eta)/t' where eta = r'/2(vt')(1/2). The unsteady Oseen correct ion to the drag is inversely proportional to time. When A is small in the long-time limit (a small-displacement motion) the fl ow field also depends on the space coordinates in terms of eta. The distrib ution of' the streamwise velocity u(z) is symmetrical in z.