Write down the boundary conditions that are satisfied at the interface between two viscous fluids in motion. Briefly discuss the physical meaning of these boundary conditions.

A layer of incompressible fluid of density $\rho$ and viscosity $\mu$ flows steadily down a plane inclined at an angle $\theta$ to the horizontal. The layer is of uniform thickness $h$ measured perpendicular to the plane and the viscosity of the overlying air can be neglected. Using co-ordinates parallel and perpendicular to the plane, write down the equations of motion, and the boundary conditions on the plane and on the free top surface. Determine the pressure and velocity fields. Show that the volume flux down the plane is $\frac{1}{3} \rho g h^{3} \sin \theta / \mu$ per unit cross-slope width.

Consider now the case where a second layer of fluid, of uniform thickness $\alpha h$, viscosity $\beta \mu$, and density $\rho$ flows steadily on top of the first layer. Determine the pressure and velocity fields in each layer. Why does the velocity profile in the bottom layer depend on $\alpha$ but not on $\beta$ ?