I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated.
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 $U$. Assume the fluid flow is laminar.
I now need to scale the following two equations that I have derived: $ \rho u_t(y,t)=\mu u_{yy}(y,t)$ and $\rho cT_t(y,t)=kT_{yy}(y,t) + \mu (u_y(y,t))^2$.
I decide to let $y=Ly'$, $u=Uu'$, $t=t_{0}t'$ and $T=T_{0}+(T_{1}-T_{0})T'$ but I am having trouble with the scaling.
I ended up with $u'_{t'}=\frac{\mu t_{0}}{L^2 \rho}u'_{y'y'}$ and $T'_{t'}=\frac{kt_{0}}{\rho c L^2}T'_{y'y'} + \frac{\mu U^2 t_{0}}{L^2 \rho c (T_{1}-T_{0})}u'_{y'}$, but I'm not convinced this is correct. Apologies for the long question.