= Force-free inclusions increase rotational resistance
{title2=$\boldsymbol\Omega\cdot\mathbf G>\boldsymbol\Omega\cdot\mathbf G_0$}
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 $D=\boldsymbol\Omega\cdot\mathbf G$, 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 $\Omega\ne0$. Thus the resistance component along the imposed rotation strictly increases.
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