Consider

the steady$-$state fully$-$developed laminar flow of a viscous Newtonian fluid with constant physical properties between two infinitely wide permeable parallel plates separated by a distance H as shown in figure. There is a constant injection of the same fluid at the velocity V through the permeable plate at $\backslash (\backslash$mathrm$\{$y$\}=\backslash )$ H and constant withdrawal of the same incompressible fluid through the permeable plate at $\backslash ($ y$=0\backslash ).$ The continuity equation can be used to prove that the injection and withdrawal velocities at the upper and lower plates must be equal for the flow to be fully developed. The upstream $($entering$)$ temperature of the fluid is $\backslash (\backslash$mathrm$\{$T$\}\_\{1\}\backslash ).$ The temperature of the upper plate at $\backslash (\backslash$mathrm$\{$y$\}=\backslash$mathrm$\{$H$\}\backslash )$ is also $\backslash ($ T$\_\{1\}\backslash ),$ whereas the temperature of the lower plate at $\backslash (\backslash$quad y$=0\backslash )$ is $\backslash ($ T$\_\{0\}\backslash ),$ such that $\backslash ($ T$\_\{1\}>$T$\_\{0\}\backslash ).$ Since the flow is assumed to be laminar and fully developed, the velocity profile is given by $\backslash ($ u$\_\{$x$\}=$U$\_\{$m$\}\backslash$left$[1-\backslash$left$(1-\backslash$frac$\{2$ y$\}\{$H$\}\backslash$right$)^\{2\}\backslash$right$]\backslash )$ where $\backslash (\backslash$mathrm$\{$U$\}\_\{\backslash$mathrm$\{$m$\}\}\backslash )$ is the maximum fluid velocity.

a$)$ Simplify the general energy equation to obtain a differential equation for the temperature distribution in the fluid. Write all the relevant boundary conditions. State your assumptions clearly. Do not solve the differential equation, yet.

b$)$ Simplify the differential equation obtained in part $($a$)$ by neglecting viscous dissipation and energy transport in the $\backslash ($ x $\backslash )-$direction and solve the resulting equation.

c$)$ Scale the differential equation of part $($a$)$ to determine under which conditions it can be reduced to the form in part $($b$).$