$($U$)$

${}_c>P_0$

A rod of length $L$ and radius $R_1$ moves downward at a constant velocity of $U_x.$ The rod is concentrically covered by another cylinder with radius of $R_2.$ A Newtonian fluid flows between the gap $($annular region$)$ of these concentrically aligned cylindrical containers. The pressure $P_L$ is greater than $P_o.$

a$)$ Obtain a steady$-$state model equation for the velocity profile of fluid in an annular region of the illustrated piston, far from the end disturbance $($edge effects are negligible$).$

b$)$ Write appropriate boundary conditions and solve the model equation obtained in part a$).$

c$)$ Sketch the velocity profile you obtained in part b$).$

Hint: Use modified pressures $P=p+gh,\text{'}h\text{'}$ is the distance 'upward' that is$,$ in the direction opposed to gravity from same preselected reference plane. Hence, in this problem, $h=-L.$