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

At fixed outer boundary velocity, extending the inclusion velocity rigidly through its interior makes the particle flow an admissible competitor for the particle-free Minimum-dissipation theorem for Stokes flow. Its interior rate-of-strain tensor vanishes. For a linear background, the increase in viscous dissipation is force times relative translation, torque times relative rotation, plus the contraction of the particle stresslet tensor with background strain.

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.

A dilute suspension of identical rigid spheres raises the dynamic viscosity by a fraction $5\varphi /2$, where $\varphi$ is the particle volume fraction. At imposed trace-free rate-of-strain tensor $E$, a sphere of radius $a$ adds viscous dissipation $20\pi \mu a^3(E:E)/3$. Multiplication by the number density $3\varphi /(4\pi a^3)$ gives $5\mu \varphi E:E$, which added to $2\mu E:E$ proves the formula. This is the leading noninteracting-sphere term; hydrodynamic interactions enter at higher concentration.

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.

Take the original sphere on the tube axis, with diagonal positive hydrodynamic resistance matrix $R_0$ and axial force $W>0$. Add a freely moving neutral sphere and extend its rigid velocity through its interior. The resulting field is admissible in the original domain, so its dissipation obeys $D>q^T{R}_{0}q\ge R_{zz}{V}_z^2$ for any finite nonoverlapping placement of the extra sphere. Since only the first sphere does external work, $D=W{V}_z$, giving the displayed bound without assuming zero transverse velocity or rotation. Strictness follows because equality would require the original nontrivial analytic flow to have zero strain throughout the added ball.