$2$B$.6$ Flow of a film on the outside of a circular tube $($see Fig. $2$B$,6).$ In a gas absorption experiment a viscous fluid flows upward through a small circular tube and then downward in laminar flow on the outside. Set up a momentum balance over a shell of thickness $r$ in the film,

Fig. $2$B$\mathrm{.}6$ Velocity distribution and $z-$momentum balance for the flow of a falling film on the outside of a circular tube.

as shown in Fig. $2$B$\mathrm{.}6.$ Note that the "momentum in$"$ and "momentum out" arrows are always taken in the positive coordinate direction, even though in this problem the momentum is flowing through the cylindrical surfaces in the negative $r$ direction.

$($a$)$ Show that the velocity distribution in the falling film $($neglecting end effects$)$ is

$v_z=\frac{g{R}^2}{4}[1-(\frac{r}{R}{)}^2+2a^{2}ln(\frac{r}{R})]$

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

$($c$)$ Show that the result in $($b$)$ simplifies to Eq$.2,2-21$ if the film thickness is very small.