Viscous dissipation

For an incompressible Newtonian fluid, the nonnegative rate at which viscosity converts mechanical energy into heat is

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.

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.