A shallow layer of fluid of viscosity $\mu$, density $\rho$ and depth $h(x, t)$ lies on a rigid horizontal plane $y=0$ and is bounded by impermeable barriers at $x=-L$ and $x=L$ $(L \gg h)$. Gravity acts vertically and a wind above the layer causes a shear stress $\tau(x)$ to be exerted on the upper surface in the $+x$ direction. Surface tension is negligible compared to gravity.

(a) Assuming that the steady flow in the layer can be analysed using lubrication theory, show that the horizontal pressure gradient $p_{x}$ is given by $p_{x}=\rho g h_{x}$ and hence that

$$h h_{x}=\frac{3}{2} \frac{\tau}{\rho g} .$$

Show also that the fluid velocity at the surface $y=h$ is equal to $\tau h / 4 \mu$, and sketch the velocity profile for $0 \leqslant y \leqslant h$.

(b) In the case in which $\tau$ is a constant, $\tau_{0}$, and assuming that the difference between $h$ and its average value $h_{0}$ remains small compared with $h_{0}$, show that

$$h \approx h_{0}\left(1+\frac{3 \tau_{0} x}{2 \rho g h_{0}^{2}}\right)$$

provided that

$$\frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$$

(c) Surfactant at surface concentration $\Gamma(x)$ is added to the surface, so that now

$$\tau=\tau_{0}-A \Gamma_{x},$$

where $A$ is a positive constant. The surfactant is advected by the surface fluid velocity and also experiences a surface diffusion with diffusivity $D$. Write down the equation for conservation of surfactant, and hence show that

$$\left(\tau_{0}-A \Gamma_{x}\right) h \Gamma=4 \mu D \Gamma_{x}$$

From equations (1), (2) and (3) deduce that

$$\frac{\Gamma}{\Gamma_{0}}=\exp \left[\frac{\rho g}{18 \mu D}\left(h^{3}-h_{0}^{3}\right)\right]$$

where $\Gamma_{0}$ is a constant. Assuming once more that $h_{1} \equiv h-h_{0} \ll h_{0}$, and that $h=h_{0}$ at $x=0$, show further that

$$h_{1} \approx \frac{3 \tau_{0} x}{2 \rho g h_{0}}\left[1+\frac{A \Gamma_{0} h_{0}}{4 \mu D}\right]^{-1}$$

provided that

$$\frac{\tau_{0} h_{0} L}{\mu D} \ll 1 \quad \text { as well as } \quad \frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$$