Lorentz reciprocal theorem with body forces

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.