Question $3.$

Consider a liquid film with a thickness of $H$ flowing on a wall as given in the figure below. The flow of the incompressible liquid is laminar and steady. The solid wall moves upward at a speed of $U_B.$ The length and width of the wall could be assumed $L$ and $W,$ respectively.

a$)$ Derive a differential equation defining the velocity profile of the liquid film.

b$)$ Find the particular solution for the model equation derived in part $($a$)$ using the proper boundary conditions.

c$)$ Sketch the velocity and shear stress profiles for the liquid film.

d$)$ What should be the minimum value of $U_B$ to prevent the downward flow of the liquid film?