$(10$ points$)$

From the Navier$-$Stokes equation in the $-$direction for incompressible flow in cylindrical coordinates as

$-$component:

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

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

Find the most general case of purely circulating motion

$V_{}(r),V_r=V_z=0$

for a flow with no slip between two fixed concentric cylinders of which radii are $"$a$"$ and $"b"$