Creeping Flows

 The creeping flow of an incompressible Newtonian fluid satisfies the Stokes equation

(1)

and the equation of continuity given as

(2)

Taking the divergence of (1) and using (2), in the absence of body force one finds

(3)

Reciprocity Theorem

Suppose  are two independent solutions (velocity and stress) to (1) and (2). Then

(4)

where S is a closed surface bounding any fluid volume. (See Happel and Brenner, page 85 for the proof).

Minimum Energy Dissipation Theorem

 The dissipation rate in creeping flow is less than any other incompressible, continuous motion consistent with the same boundary condition. (See Happel and Brenner for the proof.)