$5\mathrm{.}5.$ In cylindrical coordinates $(r,z,),$ the axial $(z)$ component of the Navier$-$Stokes equations is $($see Appendix III$)$

$(\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{t}}+u_r\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{r}}+\frac{u_{}}{r}\frac{del{u}_z}{\text{d}\text{e}\text{l}}+u_z\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{z}})$

$=-\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{z}}+g_z+[\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^2}\frac{de{l}^2{u}_z}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^2{u}_z}{del{z}^2}]$

$($a$)$ Show that the equation above reduces to

$\frac{dP}{dz}=\frac{}{r}\frac{d}{dr}(r\frac{d{u}_z}{dr})$

for steady uniform flow parallel to the $z$ axis $($with no free surface$).$ Explain specifically why each term either stays or vanishes.

$($b$)$ Solve the simplified equation for laminar flow in a capillary or narrow pipe. The solution should yield the famous Poiseuille equation of fluid mechanics,

$u_z=-\frac{1}{4}\frac{dP}{dz}(a^2-r^2)$

where $a$ is the pipe radius. Hint: The required boundary conditions for the two integrations are quite similar to those used in Section $5*7$

$($c$)$ What is the average velocity in the pipe?

$($d$)$ Express $u_z$ in terms of the average velocity.