Question One $(12$ Marks$)$

In the diagram below, oil of viscosity $$ fills the gap between two very long, parallel plates, both of which are moving with the velocities as indicated. The plates are a total distance of $h$ apart and inclined at an angle of $$ degrees. Assume that the flow between the plates is fully$-$developed, steady and laminar and that there is no pressure gradient in the $x-$or $z-$directions. Edge effects may be neglected.

$1\mathrm{.}1$ Write down an appropriate full set of Navier$-$Stokes equations $($continuity and momentum$)$ for the above case without simplifying. List all assumptions that apply to this problem in mathematical terms.

$(6)$

$1\mathrm{.}2$ Simplify the above equations to derive a differential equation for $v_4$ as a function of $y,$ but do not attempt to solve the differential equation