$3(25pt)$ Consider steady, fully developed flow of water between two vertical parallel plates a distance $h$ apart, where the left plate is fixed and the right plate moves upwards with a velocity U as shown below. For fully developed flow, the $y-$component of velocity, $v,$ is zero and $x-$component, $u,$ is a function of $y$ only. There is no imposed pressure difference between top and bottom.

Work in the coordinate system shown.

$($a$)(10$ pt$)$ Simplify the $x-$momentum equation below to a form specific to this problem. Clearly state the reason for dropping each term.

$($b$)(5$pt$)$ State the mathematical boundary conditions.

$($c$)(5pt)$ Derive an expression for the velocity profile $u(y)$

$($d$)(5pt)$ Derive an expression for the shear stress at the left wall.

The general form of the $x-$momentum equation for $2-$D incompressible flow is written as

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+u\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}=-\frac{1}{}\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{x}}+\frac{}{}(\frac{de{l}^{2}u}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2})+g_x$