Minimum-dissipation theorem for Stokes flow

Among all incompressible velocity fields with the same prescribed boundary velocity and far-field behavior, the Stokes flow minimizes viscous dissipation. If $\mathbf{v}=\mathbf{u}+\mathbf{w}$, where $\mathbf{u}$ is the Stokes solution and $\mathbf{w}$ has homogeneous boundary data, integration by parts and the Stokes equation eliminate the cross term, leaving

The Minimum-dissipation theorem for Stokes flow compares fields with fixed boundary velocities. Introducing a freely moving rigid inclusion can reduce velocity and total power at fixed applied force while increasing the resistance at fixed velocity. Extending the inclusion's rigid velocity through its interior adds zero viscous dissipation and gives a trial field in the domain without that inclusion, proving the fixed-velocity comparison.