Annular falling film. Falling film reactors are commonly used in continuous photochemical processes. The example given in the figure below $^5$ is for an advanced oxidation proce a$.$ Develop a shell balance for the laminar falling film on the inner wall and obtain the velocity profile. Neglect any gas drag on the interface. The inner radius is R$.$

v$\_$z$=\backslash$rho g$/2\backslash$mu $($R$+\backslash$delta $)^2$ln$($r$/$R$)-\backslash$rho g R$^2/4\backslash$mu $[($r$/$R$)^2-1]$

b$.$ Obtain the volumetric flow rate.

V$=\backslash$pi $\backslash$rho g$/\backslash$mu $\{($R$+\backslash$delta $)^4/4[2$ ln$(1+\backslash$delta $/$R$)-3/2]+$R$^2($R$+\backslash$delta $)^2/2-$R$^4/8\}$

c$.$ Can you derive an explicit expression for the film thickness, $\backslash$delta $?$