Two large flat plates are inclined at an angle B with

respect to the horizontal as shown below. The lower

plate is fixed while the top plate moves to the left at the

speed U$.$ Two parallel fluid layers fill the space between

the plates. The lower fluid has a density p$1$ and viscosity

m$1$ and the upper fluid has a density p$2$ and viscosity m$2.$

The flow is incompressible, steady, two$-$dimensional,

and parallel. It is known that the pressure gradient in

the x$$ direction is zero.

a$)$ write out the appropriate continuity and NS equations, list your approximations and cross out terms using the following code: S$-$steady, C$-$Continuity, Hx$,$y$,$z $-$ Homogeneity with direction, $2$D$-$two dimensional

b$)$ write the approximate boundary condintions at the bottom wall, the interface and the top wall, use pref $=0$ for refrence pressure

c$)$ solve the equations for velocity and pressure in the two fluids

d$)$ find the expression for the upper plate velocity at which the velocity at the interface between the two fluids is $0.$

Hint: consider the flow of the two liquids as two functions $,$ u$1$ and u$2,$ in respective domains $(0$