Fundamental solution of unsteady Stokes equations and force on an oscillating sphere near a wall

Abstract

We derive the Green's function of unsteady Stokes equations near a plane boundary with no-slip boundary conditions. This provides flow due to an oscillating point force acting on fluid bounded by a wall. Our derivation is different from previous theories and resolves the apparent discrepancies of the reported results. Two-dimensional Fourier transform of the solution with respect to horizontal coordinates is given via elementary functions in a more compact form than by the previous theories. The tensorial Green's function in real space is reduced to two Hankel transforms of order zero. We derive a simple form for the real-space solution in the two limiting cases of a distance to the wall much larger and much smaller than the viscous penetration depth. We demonstrate the applicability of this form by obtaining results for the force exerted on a sphere oscillating near the wall. Using the integral equation on surface traction whose kernel is the fundamental solution, we derive the force in the limits of a distant wall and low frequency. The wall correction to the force decays as the inverse third power of the source to the wall separation distance, much faster than the inverse first power of the classical Lorentz solution for the time-independent problem. Our results significantly extend the range of parameters for which the force admits a simple closed-form solution. Small biological swimmers propelled by inherently unsteady swimming gait generate flows driven by derivatives of the point source and we provide an example of a wall-bounded solution of this type. We demonstrate that frequency expansion is an efficient way of studying the Green's functions in confined geometry that gives the complete series solution for channel geometry.