Past exam of the advanced fluid dynamics course of the University of Oxford / 2018 / 2 / a

Use the same constant dynamic viscosity $\mu$ for both incompressible Stokes flows, and let ${\mathbf{f}}^{(a)}$ be body force per unit volume. Their equations and Newtonian fluid stress tensor arewhere $e_{ij}^{(a)}=({\partial }_i{u}_j^{(a)}+{\partial }_j{u}_i^{(a)})/2$ is the strain-rate tensor. Consider the cross-work fluxBy the product rule,The pressure terms vanish by incompressibility. Symmetry of the stress tensor removes antisymmetric velocity gradient contributions, leaving $2\mu e^{(1)}:e^{(2)}$ in both cross terms; they cancel. Apply the divergence theorem, with $\mathbf{n}$ outward from $V$, and rearrange:This is the Lorentz reciprocal theorem with body forces. It depends on reciprocal linear viscous response in the same fluid; arbitrary unrelated constitutive laws would not supply the cross-term cancellation.