The diagram below shows two co$-$rotating screws that pull$-$up a Newtonian fluid out of a tank. Assuming $HL,HR,$ and $HW,$ the zone in the dashed rectangle can be modeled as two parallel plates moving in opposite directions.

$J_2$

a$.$ Write the differential momentum balance for a Newtonian fluid.

b$.$ Write $($in mathematical form$)$ the boundary conditions.

c$.$ Determine the velocity profile.

d$.$ Determine the stress profile.

e$.$ Determine the force exerted by the fluid on each wall?

f$.$ Can you obtain an expression for the stress from the Newton's first law of viscosity method $($i$.$e$.$ Chapter $1)?$ Explain.