Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 73 / 1 / b / ii

The minimum dissipation theorem fixes the boundary velocities and the far-field velocity: among admissible incompressible velocity fields, the Stokes flow minimizesFor a competitor $\mathbf{u}+\mathbf{v}$ with zero boundary data for $\mathbf{v}$, integration by parts eliminates the cross term and leaves $\mathcal{D}[\mathbf{u}+\mathbf{v}]-\mathcal{D}[\mathbf{u}]=2\mu \int \mathbf{e}(\mathbf{v}):\mathbf{e}(\mathbf{v})dV\ge 0$.

Extend the actual two-sphere velocity rigidly through sphere 2. It is an admissible field in the one-sphere exterior, with the same translation and rotation of sphere 1. It is continuous across the filled boundary, incompressible, and adds zero strain dissipation inside. Applying the theorem to the exact isolated-sphere flow therefore givesWith all applied couples and the second force zero, the boundary-work identity is ${\mathcal{D}}_{\mathrm{pair}}={\mathbf{F}}_1\cdot {\mathbf{U}}_1$. Thus $6\pi \mu a|{\mathbf{U}}_1{|}^2\le {\mathbf{F}}_1\cdot {\mathbf{U}}_1\le |{\mathbf{F}}_1||{\mathbf{U}}_1|$, which provesThis fixed-force comparison of minimum viscous dissipation uses a comparison at fixed actual velocity first; directly comparing different-force or different-velocity solutions would not justify the result.

If ${\mathbf{G}}_1\ne 0$, the power is instead ${\mathbf{F}}_1\cdot {\mathbf{U}}_1+{\mathbf{G}}_1\cdot {\mathbf{\Omega }}_1$, so the preceding estimate no longer bounds the force contribution alone. The two-sphere hydrodynamic mobility matrix generally has a nonzero self translation-rotation coupling: the freely moving second sphere reflects the first sphere's torque field. In the planar geometry this coupling produces a velocity perpendicular to $\mathbf{R}$; by choosing the sign and magnitude of the couple when the force has a component in that direction, its contribution to ${\mathbf{F}}_1\cdot {\mathbf{U}}_1$ can exceed the isolated force-only value. There is no universal force-only inequality with an additional applied couple. Special geometries can eliminate the coupling, but do not restore a general theorem.