We examined a simple wire coating device in class earlier in the semester. However, a colleague has expressed concern that viscous heating may be causing damage to the coating fluid. To determine whether this is the case, we need to determine the temperature profile in the fluid.

The idealized system geometry is shown in the figure below. The coating fluid is Newtonian with constant viscosity and density. We assumed that the device is very long $($so end effects can be neglected$),$ that the flow is at steady state, and that we may neglect any pressure gradient in the fluid. Under these conditions, the velocity profile in the fluid is:

$v_r=0$;$,v_{}=0$;$,v_z=V\frac{ln(\frac{r}{R})}{ln()}$

We may assume that the wire $($inner cylinder$)$ and outer cylinder are both at constant temperature $T_0,$ and we may assume that the temperature in the fluid is a function of only the radial coordinate $(r).$

$($a$)[40$ points$]$ Write the appropriately simplified energy equation describing the temperature in the fluid.

$($b$)[30$ points$]$ Specify the boundary conditions that should be used in solving this equation.

$($c$)[30$ points