A layer of thickness $h$ of fluid of density $\rho$ and dynamic viscosity $\mu$ flows steadily down and parallel to a rigid plane inclined at angle $\alpha$ to the horizontal. Wind blows over the surface of the fluid and exerts a stress $S$ on the surface of the fluid in the upslope direction.

(a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.

(b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields.

(c) Determine the volume flux and show that there is no net flux if

$$S=\frac{2}{3} \rho g h \sin \alpha$$

Draw a sketch of the corresponding velocity profile.

(d) Determine the value of $S$ for which the shear stress on the rigid plane is zero and draw a sketch of the corresponding velocity profile.