$(20$ pts$)2)$ A cylindrical rod of diameter $D$ and length $L$ is being pulled horizontally at a constant velocity $U_0$ through a Newtonian liquid with dynamic viscosity $.$ The rod moves between two fixed plates, each separated from the rod by a small distance $t.$ The velocity distribution of the liquid around the rod is given by: $u=U_o[1-(\frac{y}{t}{)}^2].$

where $y=0$ is at the surface of the rod, and $y=t$ is at the fixed plate. End effects of the cylindrical rod can be neglected.

a$)$ Derive an expression for the shear stress $$ developed in the liquid.

b$)$ Derive an expression the force $F$ required to pull the rod at the constant velocity Uo$.$

c$)$ Draw the velocity distribution $u$ and shear stress distribution $$ in the liquid around the rod.

Fixed plate