I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks.
An incompressible thermal conducting fluid is contained between two infinite horizontal plates separated by a distance $H$. Initially the plates and fluid are at rest at temperature $T_0$. At time $t = 0$ the upper plate is raised to temperature $T_1 > T_0$ and moved horizontally at speed $U$.
(a) Assuming the fluid flow is lamina, show that the velocity and temperature field equations reduce to $v=p=0$ , $ \rho u_t(y,t)=\mu u_{yy}(y,t)$ and $\rho cT_t(y,t)=kT_{yy}(y,t) + \mu (u_y(y,t))^2$
(b) Scale the equations. What dimensionless groups govern the temperature and velocity fields?
(c) Qualitatively discuss the processes set into play and indicate the time scales involved.
(d) Roughly how long does it take before steady conditions are reached.
(e) Determine the steady state temperature and velocity fields.