= Lorentz reciprocal theorem with body forces
{c}
For two incompressible <Stokes flows> with the same <dynamic viscosity>, $\nabla\cdot\sigma^{(a)}+\mathbf f^{(a)}=0$. The divergence of $u_i^{(2)}\sigma_{ij}^{(1)}-u_i^{(1)}\sigma_{ij}^{(2)}$ equals $-\mathbf u^{(2)}\cdot\mathbf f^{(1)}+\mathbf u^{(1)}\cdot\mathbf f^{(2)}$. The cross-<strain-rate tensor> products cancel by reciprocity. Integrating gives equality of the two surface cross-work terms after adding the corresponding volume body-force work. The outward domain normal must be used consistently, especially at an exterior domain's inner surface.
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