Problem $3(30$ pts$)$: In Lecture $5,$ we covered the viscous heating problem for a concentric

cylinder separated with a thin fluid $($thermal conductivity $k,$ viscosity $)$ of thickness $b.$ The

outside cylinder $($ radiusR $)$ is moving at a constant angular velocity $$ and the inside cylinder

is stationary.

a$.$ Starting from shell balance, find the temperature profile given the boundary

conditions $T(x=0)=T_0$ and the conductive heat flux $(\frac{q_x}{A})$ at the outside surface

b$)$ is a constant $\frac{q_b}{A_b}*(20pts)$

b$.$ If $q_b=0,$ find the maximum temperature in the fluid. Sketch the temperature profile of

the fluid. $(10$ pts$)$