An ideal liquid contained within a closed circular cylinder of radius $a$ rotates about the axis of the cylinder (assume this axis to be in the vertical $z$-direction).

(i) Prove that the equation of continuity and the boundary conditions are satisfied by the velocity $\mathbf{v}=\boldsymbol{\Omega} \times \mathbf{r}$, where $\boldsymbol{\Omega}=\Omega \hat{\mathbf{e}}_{z}$ is the angular velocity, with $\hat{\mathbf{e}}_{z}$ the unit vector in the $z$-direction, which depends only on time, and $\mathbf{r}$ is the position vector measured from a point on the axis of rotation.

(ii) Calculate the angular momentum $\mathbf{M}=\rho \int(\mathbf{r} \times \mathbf{v}) d V$ per unit length of the cylinder.

(iii) Suppose the the liquid starts from rest and flows under the action of an external force per unit mass $\mathbf{f}=(\alpha x+\beta y, \gamma x+\delta y, 0)$. By taking the curl of the Euler equation, prove that

$$\frac{d \Omega}{d t}=\frac{1}{2}(\gamma-\beta)$$

(iv) Find the pressure.