Consider the Couette flow depicted in Fig. P13.15. The bottom plate is stationary and adiabatic, and the top plate is moving at the velocity U 1 and is cooled by an ambient fluid that is at...

Consider the Couette flow depicted in Fig. P13.15. The bottom plate is stationary and adiabatic, and the top plate is moving at the velocity U₁
and is cooled by an ambient fluid that is at temperature T_∞.

(a) Using first-order slip and temperature-jump conditions, and assuming that α = α_T
= 1, prove that the temperature profile will be

Where

(b) Prove that

where T_m
is the mean (mixed-cup) temperature and