$[$Total $25$ points$]$ Consider flow in a circular tube, in which the heat flux varies axially according to the relation in the following figure.

where $L$ is the length of the tube, and a and $b$ are constants. The fluid enters the tube at temperature $T_{m,i}.$ Answer the following problems.

$n^{}$

$T_n$

$c_1$

$q_s^{\text{'}\text{'}}=a+$bsin$(\frac{x}{L})$

$($a$,10$ points$)$ Obtain an expression for the variation of the mean temperature $T_m(x)$ of the fluid as a function of $x$ along the tube. $($b$,10$ points$)$ Assuming a fully developed hydrodynamic and thermal conditions, obtain an expression for the tube surface temperature $T_s(x)$ as a function of $x$ along the tube. Develop an expression for the $x-$location at which this temperature is maximized. $($c$,5$ points$)$ Do you expect the outlet mean temperature of the fluid $T_{m,o}$ to increase or decrease if combined thermal and hydrodynamic entry conditions exist within the tube? Explain your answer.