Question $4.(20$ Points$)$ Flow through a

horizontal annulus $($shown$)$ can be

described through application of the

cylindrical Navier$-$Stokes equations:

$p(\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{t}}+v_r\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_{}}{r}\frac{del{v}_r}{\text{d}\text{e}\text{l}}-\frac{v_{}^2}{r}+v_z\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{z}})$

$=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{r}}+g_r+[\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{r}})-\frac{v_r}{r^2}+\frac{1}{r^2}\frac{de{l}^2{v}_r}{\text{d}\text{e}\text{l}{}^2}-\frac{2}{r^2}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+\frac{de{l}^2{v}_r}{del{z}^2}]$

$$:$(\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{t}}+v_r\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_{}}{r}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+\frac{v_r{v}_{}}{r}+v_z\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{z}})$

$=-\frac{1}{r}\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}}+g_{}+[\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{r}})-\frac{v_{}}{r^2}+\frac{1}{r^2}\frac{de{l}^2{v}_{}}{\text{d}\text{e}\text{l}{}^2}-\frac{2}{r^2}\frac{del{v}_r}{\text{d}\text{e}\text{l}}+\frac{de{l}^2{v}_{}}{del{z}^2}]$

$z$:$(\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{t}}+v_r\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_{}}{r}\frac{del{v}_z}{\text{d}\text{e}\text{l}}+v_z\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{z}})$

$=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}}+g_z+[\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^2}\frac{de\frac{l^2}{v_z}}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^2{y}_r}{del{z}^2}]$

Describe all assumptions $($and how those assumptions allow for cancellation of terms$)$

that allowed us to reduce the final form of the $z-$direction to the following solution $(5$

points$)$:

$0=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}}+[\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{r}})]$