Viscous fluid with dynamic viscosity $\mu$ flows with velocity $\left(u_{x}, u_{y}, u_{z}\right) \equiv\left(\mathbf{u}_{H}, u_{z}\right)$ (in cartesian coordinates $x, y, z)$ in a shallow container with a free surface at $z=0$. The base of the container is rigid, and is at $z=-h(x, y)$. A horizontal stress $\mathbf{S}(x, y)$ is applied at the free surface. Gravity may be neglected.

Using lubrication theory (conditions for the validity of which should be clearly stated), show that the horizontal volume flux $\mathbf{q}(x, y) \equiv \int_{-h}^{0} \mathbf{u}_{H} d z$ satisfies the equations

$$\nabla \cdot \mathbf{q}=0, \quad \mu \mathbf{q}=-\frac{1}{3} h^{3} \nabla p+\frac{1}{2} h^{2} \mathbf{S}$$

where $p(x, y)$ is the pressure. Find also an expression for the surface velocity $\mathbf{u}_{0}(x, y) \equiv$ $\mathbf{u}_{H}(x, y, 0)$ in terms of $\mathbf{S}, \mathbf{q}$ and $h$.

Now suppose that the container is cylindrical with boundary at $x^{2}+y^{2}=a^{2}$, where $a \gg h$, and that the surface stress is uniform and in the $x$-direction, so $\mathbf{S}=\left(S_{0}, 0\right)$ with $S_{0}$ constant. It can be assumed that the correct boundary condition to apply at $x^{2}+y^{2}=a^{2}$ is $\mathbf{q} \cdot \mathbf{n}=0$, where $\mathbf{n}$ is the unit normal.

Write $\mathbf{q}=\nabla \psi(x, y) \times \hat{\mathbf{z}}$, and show that $\psi$ satisfies the equation

$$\nabla \cdot\left(\frac{1}{h^{3}} \nabla \psi\right)=-\frac{S_{0}}{2 \mu h^{2}} \frac{\partial h}{\partial y}$$

Deduce that if $h=h_{0}$ (constant) then $\mathbf{q}=\mathbf{0}$. Find $\mathbf{u}_{\mathbf{0}}$ in this case.

Now suppose that $h=h_{0}(1+\epsilon y / a)$, where $\epsilon \ll 1$. Verify that to leading order in $\epsilon, \psi=\epsilon C\left(x^{2}+y^{2}-a^{2}\right)$ for some constant $C$ to be determined. Hence determine $\mathbf{u}_{0}$ up to and including terms of order $\epsilon$.

[Hint: $\nabla \times(\mathbf{A} \times \hat{\mathbf{z}})=\hat{\mathbf{z}} \cdot \nabla \mathbf{A}-\hat{\mathbf{z}} \nabla \cdot \mathbf{A}$ for any vector field $\mathbf{A} .]$