4.7 The Stokeslet

The Stokes solution for the flow past a sphere owes its existence to the linearity of the equations and to the simplicity of the geometry of the problem. But, for more irregular geometries, recourse must be made to numerical techniques. One such technique is the boundary integral method. Here, the surfaces of the particles and the walls around them are simulated in terms of distributions of the fundamental solutions of the Stokes equations. Such methods are well known in fluid mechanics, where it is shown that mass point sources, dipoles, and quadrupoles can be used to simulate a variety of flows.

In rigid-particle fluid mechanics, the mass point source has zero strength, because the particles have constant volume, so that the lowest order singularity-type solution that is relevant is the dipole. This is made of a source-sink pair and represents a force acting at a point. In the case of incompressible fluid motions at low Reynolds numbers, this singularity solution is known as the stokeslet.

To introduce the stokeslet, we consider a rigid particle moving slowly in a viscous, incompressible fluid. The particle induces a fluid velocity whose magnitude decreases slowly with distance. Far from the particle, the particle appears as a point so that we may regard that velocity as being produced by a point force. But, as we get closer to the particle, its finite size becomes apparent, and it is evident that each element of area on its surface exerts a force on the...