Problem $2(20$ points$)$ Drawn from A$.$ Gerhart et$.$ Al$,9-$th Ed$,$ Chapter $6$

Concepts: Navier$-$Stokes equation in rectangular coordinates

Two immiscible, incompressible, viscous fluids having the same densities, but different viscosities are contained between two infinite, horizontal, parallel plates. The bottom plate is fixed, and the upper plate moves with a constant velocity U $.$ Determine the velocity at the interface. Express your answer in terms of $\backslash ($ U$,\backslash$mu$\_\{1\}\backslash ),$ and $\backslash (\backslash$mu$\_\{2\}\backslash ).$ The motion of the fluid is caused entirely by the movement of the upper plate, that is$,$ there is no pressure gradient in the $\backslash ($ x $\backslash )-$direction. The fluid velocity and shearing stress are continuous across the interface between the two fluids. Assume laminar flow.