TABLE OF CONTENTS Let us consider two generally unrelated Stokes flows with velocities u and u ? and associated modified stress tensors ? and ? ?. Using Eq. (3.3.22) in conjunction with Eq. (6.1.1c), we obtain the Lorentz reciprocal identity
which is the counterpart of Green's second identity for harmonic functions (Lorentz, 1907). For brevity, in the ensuing discussion we shall omit the qualifier "modified" when we refer to the modified pressure and stress.
Assuming that the two flows are free of singular points within a certain control volume, we integrate Eq. (6.8.1) over the control volume and use the divergence theorem to obtain
where D is the boundary of the control volume, and f = ? n is the boundary traction; the unit normal vector n may point either into or out from the control volume.
The major strength of the reciprocal identities (6.8.1) and (6.8.2) is that they allow us to obtain information about a certain flow without having to solve the equations of Stokes flow explicitly, but merely by using information about another flow. This will be demonstrated now by discussing certain characteristic examples.
Consider the flow around a rigid particle that is held stationary in an incident ambient flow with velocity u ?. The particle causes a disturbance flow with velocity u D, which is added to the ambient flow to give the total...
