With simple scale analysis, equations in cylindrical coordinate system

$($figure below$)$ can be converted to cartesian coordinate system.

Consider a flow is described by following mass conservations equations

and momentum equation in cylindrical coordinates.

Mass conservation equation:

$\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_r}{r}+\frac{1}{r}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{z}}=0$

Momentum equation in r$-$direction:

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

$,=-\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{r}}+(\frac{de{l}^2{v}_r}{del{r}^2}+\frac{1}{r}\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{r}}-\frac{v_r}{r^2}+\frac{1}{r^2}\frac{de{l}^2{v}_r}{\text{d}\text{e}\text{l}{}^2}-\frac{2}{r^2}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+\frac{de{l}^2{v}_r}{del{z}^2})+F_r$

If the flow is infinitely far from the $r=0$ origin of the coordinate system, the local

three$-$directional increments $r,r,z$ become analogous to three Cartesian

increments $x,y,z$ measured away from the local point $(r,,z)$ in the flow

field. Show that in the limit $r,$ the transformation $rx,ry,z$

$z$ leads to the collapse of above equations into corresponding mass and

momentum conservation equations in Cartesian $(x,y,z)$ coordinate system given

below.

Mass conservation:

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}+\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{z}}=0$

$x-$momentum equation

p$....($equation and exact question is in the image below$)$