Figure Q$2$ ilustrates the fully developed Couette flow of a Newtonian fluid having

viscosity $,$ density $$ and thermal conductivity $k,$ between two infinite paralel plates

separated by a distance $L.$ The upper plate at $y=L$ moves with a constant positive

velocity vr while the lower plate remains stationary. The lower and upper plates are

kept at uniform temperatures, $T_0$ and $T_1$ respectively. The physical properties of the

fluid can be assumed to be independent of temperature.

Figure Q$2.$ Flow of Newtonian fluid between two infinite paralel plates.

a$)$ First, identify the non$-$vanishing velocity components and their derivatives. Then,

if the lemperature varies linearly across the plates as per Figure Q$2,$ determine

if the resulting velocity profile is similar to the profile shown in Figure Q$2\mathrm{.}1.$

Figure Q$2\mathrm{.}1.$ Fuly developed Newtonian fluid flow.

b$)$ Provide a postulation on the temperature profile across the two plate slabs.

c$)$ The temperature rises and the amount of heat transfer through the walls are of

interest in engineering applications. Use the velocity profile derived in part $($a$)$

and the postulation in part $($b$),$ prove that the lemperature proflie of the fluid is

given by:

$\frac{T-T_0}{T_1-T_0}=\frac{y}{L}[1+\frac{1}{2}P{r}_{-E}(1-\frac{y}{L})]$

Where the Prandtl number is defined as $\frac{cy}{k}$ and the Eckert number, E is defined as

$\frac{v_1^2}{c_4(T_1-r_6)}.$ Include in all reasonable assumptions made throughout the derivation.

d$)$ Show that the heat flux at the upper wall $)=(L$ is given by:

$q=-k\frac{T_1-T_b}{L}(1-\frac{1}{2}Pr*E)$