 Force-free inclusions increase rotational resistance

At fixed sphere rotation in an annular Stokes flow, add rigid particles with zero net force and torque. Extend their actual rigid velocities through their interiors. The extension is an admissible incompressible competitor in the particle-free annulus, with zero interior rate-of-strain tensor. The Minimum-dissipation theorem for Stokes flow gives $D\ge D_0$. The boundary-work identity gives $D=\mathbf{\Omega }\cdot \mathbf{G}$, because the stationary outer wall and the freely moving particles contribute no power. Equality would require the extension to equal the unique particle-free flow, whose strain cannot vanish on a finite particle volume when $\Omega \ne 0$. Thus the resistance component along the imposed rotation strictly increases.