Pouisselle Flow $($Pressure Driven Flow$)$

$2.$ Assume that a flow of SAE$30$ oil is driven steadily, within a flat, planar channel $($infinitely long through the page$)$ using pressure. Two pressure sensors are located at $10$ m intervals along the pipe. The outlet pressure sensors reads different outlet pressures on two different days. We will assume that the flow remains laminar and fully$-$developed within the channel. Determine the following,

a$.$ Using differential continuity and the Navier Stokes equations, derive an expression for the velocity profile of the fluid within the channel. Record each step of the derivation.

b$.$ Identify the number and locations of boundary conditions required to solve this equation.

c$.$ Identify the boundary conditions in the flow scenario above that can be used to solve for the velocity profile.

d$.$ Calculate and plot $($ y $-$location $V.S.V(y))$ the velocity profile through the channel for the two different days. Compare and describe the differences between the two flows.

Please i need typed and $100\%$ correct solution. Plz its really important. its fluid mechanics course