Consider the pressure gradient driven, steady, incompressible, fully developed flow of two

fluids of viscosities $1$ and $2$ between two parallel, infinite plates as shown in figure. The

flow is at small velocities and is driven by a pressure difference per unit length equal to $P$

The top fluid is lighter than the bottom one. Answer the following:

$($a$)$ Write the simplified continuity and Navier$-$Stokes equations to model this flow.

$($b$)$ Write four boundary conditions which govern the flow.

$($c$)$ Derive the expressions for velocity distribution in the fluids in terms of pressure gradient

and viscosities ${}_1$ and ${}_2$