A layer of oil of thickness a and viscosity o floats on top of a layer of water of thickness b and viscosity w. Both layers are contained between two large plates. The lower plate is stationary and the upper plate moves with velocity U.
a) Derive an expression for the speed of the water-oil interface,
b) Derive an expression for the volumetric flow rates of water and oil per unit depth,
as functions of a, b, o, w, and U.