Consider steady, laminar, vertical flow under the influence of gravity in a long, thin channel between two parallel plates. The plate at $x=0$ is fixed in space, while the plate at $x=H$ is moving upward at speed $u,$ as sketched in Figure $1.$ The plates are of length $L($aligned with hat$(y))$ and width $W($aligned with hat$(z))$ and are separated by a height $H($aligned with hat$(x)$;$HWL).$ The left wall is at $x=0,$ the right wall is at $x=H,$ and the space between them is filled with fluid of viscosity $$ and density $.$ The magnitude of gravitational acceleration is $g.$ You may assume the flow is laminar and fully$-$developed and that the pressure is the same everywhere.

$($a$)$ Starting with the Navier$-$Stokes momentum equation in the $y$ direction, show which terms are negligible and simplify the equation to an ODE. For each term you neglect, clearly state why.

$($b$)$ Find the general solution to the ODE developed above, which may contain yetunknown constants of integration.

Figure $1$: Channel setup for solo question.

$($c$)$ Define two boundary conditions that you can use to solve for the constants of integration.

$($d$)$ Using the boundary conditions you identified, determine the velocity profile $v_y(x)$ in terms of any of the following: $,,W,L,H,g,u.($If it's helpful in checking your work, you may choose to make up values and plot the velocity profile you derived.$)$

$($e$)$ Find the net upward volumetric flow rate in terms of any of the following: $,,W,L,H,g,u.$