$3-$ In a gas absorption experiment a viscous Newtonian fluid flows upward through a small circular tube and then downward in laminar flow on the outside $(30$ points$).$ Answer this problem using two approaches: i$)$ using Newton's second law of motion and ii$)$ using Navier$-$Stokes equation.

a$)-$ Obtain the velocity profile in the falling film.

b$)$ obtain an expression for the mass rate of flow in the film.