State the minimum dissipation theorem for Stokes flow in a bounded domain.

Fluid of density $\rho$ and viscosity $\mu$ fills an infinite cylindrical annulus $a \leq r \leq b$ between a fixed cylinder $r=a$ and a cylinder $r=b$ which rotates about its axis with constant angular velocity $\Omega$. In cylindrical polar coordinates $(r, \theta, z)$, the fluid velocity is $\mathbf{u}=(0, v(r), 0)$. The Reynolds number $\rho \Omega b^{2} / \mu$ is not necessarily small. Show that $v(r)=A r+B / r$, where $A$ and $B$ are constants to be determined.

[You may assume that $\nabla^{2} \mathbf{u}=\left(0, \nabla^{2} v-v / r^{2}, 0\right)$ and $(\mathbf{u} \cdot \nabla) \mathbf{u}=\left(-v^{2} / r, 0,0\right) .$ ]

Show that the outer cylinder exerts a couple $G_{0}$ per unit length on the fluid, where

$$G_{0}=\frac{4 \pi \mu \Omega a^{2} b^{2}}{b^{2}-a^{2}} .$$

[You may assume that, in standard notation, $e_{r \theta}=\frac{r}{2} \frac{d}{d r}\left(\frac{v}{r}\right)$.]

Suppose now that $b \geq \sqrt{2} a$ and that the cylinder $r=a$ is replaced by a fixed cylinder whose cross-section is a square of side $2 a$ centred on $r=0$, all other conditions being unchanged. The flow may still be assumed steady. Explaining your argument carefully, show that the couple $G$ now required to maintain the motion of the outer cylinder is greater than $G_{0}$.