$[4$ points total$]$ Two incompressible viscous fluids flow one on top of the other down an

inclined plate at an angle $.$ They both have the same density, $,$ but different viscosities ${}_1$

and ${}_2.$ The lower fluid has depth $h_1$ and the upper $h_2.$ Assuming that viscous forces from

the surrounding air is negligible and that the pressure on the free surface is atmospheric.

a$.[2$ points$]$ Starting from continuity and Navier$-$Stokes, show that the fluid pressure $P(y)=$

$P_a+gcos()(h_1+h_2-y)$

b$.[2$ points$]$ Show that the fluid velocity in the bottom layer is $v_x=[(h_1+h_2)y-\frac{1}{2}{y}^2]\frac{gsin}{{}_1}$