Question $2[20$ marks$]$

Consider a viscous film of liquid draining uniformly down the side surface of a vertical static solid rod of radius $\backslash ($ a $\backslash )$ as shown in Figure Q$2.$ Assume the rod is infinitely long and the film already approaches a terminal or fully developed draining flow of constant outer radius $\backslash ($ b $\backslash ).$ Use a three$-$dimensional cylindrical coordinate system in the analysis, in which $\backslash (\backslash$theta $\backslash )$ is the tangential direction. The tangential and radial components of the fluid velocity are both equal to zero, i$.$e$.\backslash ($ u$\_\{\backslash$theta$\}=$u$\_\{$r$\}=0\backslash ).$ The $\backslash ($ z $\backslash )$ component of fluid velocity $\backslash ($ u$\_\{$z$\}\backslash )$ is only a function of $\backslash ($ r $\backslash )$ and independent of $\backslash (\backslash$theta $\backslash )$ and $\backslash ($ z $\backslash ).$ The flow is in the steady state. The gravitational acceleration $\backslash ($ g $\backslash )$ is in the positive $\backslash ($ z $\backslash )$ direction. Assume there is zero pressure gradient of fluid in the $\backslash ($ r $\backslash )$ and $\backslash (\backslash$theta $\backslash )$ direction. Assume that the fluid shear stress is zero on the free surface of the fluid film $($where $\backslash ($ r$=$b$)\backslash ).$ Apply the non$-$slip boundary condition on the rod surface $($where $\backslash ($ r$=$a $\backslash )).$

$($a$)$ Derive the mathematical expression for the fluid velocity profile $\backslash ($ u$\_\{$z$\}\backslash )$ as a function of $\backslash ($ r $\backslash ),$ by using fluid density $\backslash (\backslash$rho $\backslash ),$ dynamic viscosity $\backslash (\backslash$mu $\backslash ),$ gravitational acceleration $\backslash ($ g $\backslash ),$ rod radius $\backslash ($ a $\backslash )$ and film outer radius $\backslash ($ b $\backslash )$ as inputs.

$($b$)$ Derive the mathematical expression for the fluid shear stress profile $\backslash (\backslash$tau$\_\{$r z$\}\backslash )$ as a function of $\backslash ($ r $\backslash ),$ by using fluid density $\backslash (\backslash$rho $\backslash ),$ dynamic viscosity $\backslash (\backslash$mu $\backslash ),$ gravitational acceleration $\backslash ($ g $\backslash ),$ rod radius $\backslash ($ a $\backslash )$ and film outer radius $\backslash ($ b $\backslash )$ as inputs.

$($c$)$ Derive the mathematical expression for the fluid wall shear stress $\backslash (\backslash$tau$\_\{$r z$\}\backslash )$ on the rod surface $($where $\backslash ($ r$=$a $\backslash )),$ by using fluid density $\backslash (\backslash$rho $\backslash ),$ dynamic viscosity $\backslash (\backslash$mu $\backslash ),$ gravitational acceleration $\backslash ($ g $\backslash ),$ rod radius $\backslash ($ a $\backslash )$ and film outer radius $\backslash ($ b $\backslash )$ as inputs.

Figure Q$2$