This problem is about viscous heating. A Newtonian fluid of constant physical properties $($ and $k)$ flows down a circular pipe. Taking the $z-$axis downwardly along the center of the pipe $($and parallel to the gravity $g)$ and assuming that the flow is incompressible and laminar, we have

$v_z(r)=v_{\text{m}\text{a}\text{x}}[1-(\frac{r}{R}{)}^2],$

where $R$ is the inner radius of the pipe and

$v_{\text{m}\text{a}\text{x}}=\frac{R^2}{4},$

in which

$=g-\frac{dp}{(d)z}$

is a constant. The other components of the velocity are zero.

Suppose that the pipe is insulated and let $T_0$ denote the temperature of the fluid for $z0.$ You are asked to find the temperature profile in the flind for $z>0.$

$($a$)$ List assumptions you would impose on the thermal energy balance equation.

$($b$)$ Set up the partial differential equation for the temperature field.

$($c$)$ Assume that the conductive heat transfer in the $z-$direction is negligible compared to the convective heat transfer in the same direction and let $T(r,z)=f(r)+g(z).$ Find two ordinary differential equations, one for $f(r)$ and the other for $g(z).$

$($d$)$ Find the temperature field involving arbitrary constants.

$($e$)$ Impose boundary conditions at $r=0$ and $R$ and determine some of the constants in $T(r,z).$

$($f$)$ Show that the other boundary condition at $z=0$ cannot be satisfied.

$($g$)$ The macroscopic energy balance written for a control volume $V_f($i$.$e$.,$ volume fixed in space$)$ takes the form

$\frac{d}{dt}{\displaystyle \underset{V_f}{\overset{\phantom{}}{}}}(u+\frac{1}{2}{v}^2+)dV+{\displaystyle \underset{S_f}{\overset{\phantom{}}{}}}(h+\frac{1}{2}{v}^2+)v_{*}{n}_{d}S$

$=-{\displaystyle \underset{S_f}{\overset{\phantom{}}{}}}{n}_{*}{q}_{d}S+{\displaystyle \underset{V_f}{\overset{\phantom{}}{}}}RdV$