Question $4$

Consider the combined gravity$-$Conette driven flow between two inclined, parblel plates, us depicted in the figure. The lower plate is stationary, while the upper plate moves at constant upward velocity $V_{\text{t}\text{o}\text{p}\text{.}}.$ The plates are very long and wide and they are separated by a small gap of length $h.$ The fluid has viscosity $$ and density $.$

Figure $3$: Inclined gravity and Consette driven flow.

$($a$)$ What assumptions can be made about this flow scenario?

$($b$)$ Write the complete Navier$-$Stokes equations governing this flow, including known Boundary Conditions.

$($c$)$ Simplify and reduce the governing equations to a second order ordinury differential equation $($ODE$).$ Justify your assumptions.

$($d$)$ Find the solution to this ODE using appropriate Boundary Conditions.