A viscous fluid is confined between an inner, impermeable cylinder of radius $a$ with centre at $(x, y)=(0, a)$ and another outer, impermeable cylinder of radius $2 a$ with centre at $(0,2 a)$ (so they touch at the origin and both have their axes in the $z$ direction). The inner cylinder rotates about its axis with angular velocity $\Omega$ and the outer cylinder rotates about its axis with angular velocity $-\Omega / 4$. The fluid motion is two-dimensional and slow enough that the Stokes approximation is appropriate.

(i) Show that the boundary of the inner cylinder is described by the relationship

$$r=2 a \sin \theta,$$

where $(r, \theta)$ are the usual polar coordinates centred on $(x, y)=(0,0)$. Show also that on this cylinder the boundary condition on the tangential velocity can be written as

$$u_{r} \cos \theta+u_{\theta} \sin \theta=a \Omega,$$

where $u_{r}$ and $u_{\theta}$ are the components of the velocity in the $r$ and $\theta$ directions respectively. Explain why the boundary condition $\psi=0$ (where $\psi$ is the streamfunction such that $u_{r}=\frac{1}{r} \frac{\partial \psi}{\partial \theta}$ and $\left.u_{\theta}=-\frac{\partial \psi}{\partial r}\right)$ can be imposed.

(ii) Write down the boundary conditions to be satisfied on the outer cylinder $r=4 a \sin \theta$, explaining carefully why $\psi=0$ can also be imposed on this cylinder as well.

(iii) It is given that the streamfunction is of the form

$$\psi=A \sin ^{2} \theta+B r^{2}+C r \sin \theta+D \sin ^{3} \theta / r$$

where $A, B, C$ and $D$ are constants, which satisfies $\nabla^{4} \psi=0$. Using the fact that $B=0$ due to the symmetry of the problem, show that the streamfunction is

$$\psi=\frac{\alpha \sin \theta}{r}(r-2 a \sin \theta)(r-4 a \sin \theta)$$

where the constant $\alpha$ is to be found.

(iv) Sketch the streamline pattern between the cylinders and determine the $(x, y)$ coordinates of the stagnation point in the flow.