Problem $1$: Momentum Transfer between Rotating Walls

The space between two coaxial cylinders is filled with an oil at constant temperature $($density: $1000k\frac{g}{m^3}$ and dynamic viscosity $0\mathrm{.}1k\frac{g}{m}-s).$

The radius of the inner cylinder is $1$ inch.

The radius of the outer cylinder is $2$ inches.

The inner cylinder is stationary. The outer cylinder is rotating at $3$ radians$/$second$.$

Your goal is to determine the velocity distribution in the fluid, the shear stresses on the walls of the two cylinders!

A$.$ First, draw an illustrative figure that represents the above scenario with a clear depiction of angular velocities that you specified at both cylindrical surfaces

L$-$y$.$ I. Ientity the co$-$ordinate system, the velocity component of interest and the direction along which it is going to exhibit the strongest variation. Pick the appropriate governing equation from the list below and eliminate all the zero terms

Cylindrical coordinates $(r,,z)$:

$(\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{t}}+v_r\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_{}}{r}\frac{del{v}_r}{\text{d}\text{e}\text{l}}+v_z\frac{del{v}_r}{\text{d}\text{e}\text{l}\text{z}}-\frac{v_{}^2}{r})=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{r}}+[\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r{v}_r))+\frac{1}{r^2}\frac{de{l}^2{v}_r}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^2{v}_r}{del{z}^2}-\frac{2}{r^2}\frac{del{v}_0}{\text{d}\text{e}\text{l}}]+g_{,}$

$(\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{t}}+v_r\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{r}}+\frac{v_0}{r}\frac{del{v}_{}}{\text{d}\text{e}\text{l}}+v_z\frac{del{v}_{}}{\text{d}\text{e}\text{l}\text{z}}+\frac{v_r{v}_{}}{r})=-\frac{1}{r}\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}}+[\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(v_{}))+\frac{1}{r^2}\frac{de{l}^2{v}_0}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^2{v}_{}}{del{z}^2}+\frac{2}{r^2}\frac{del{v}_r}{\text{d}\text{e}\text{l}}]+g_{}$

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

C$.$ Integrate your equation twice and obtain a general solution for the fluid velocity distribution in the annular region

D$.$ Using the two boundary conditions, solve for the two integration constants $C_1$ and $C_2$ and report the final expression for the velocity distribution.

E$.$ You look up a book on Transport Phenomena and the equation for calculating shear stresses in cylindrical co$-$ordinate systems is given as:

${}_r={}_r=-[r\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(\frac{v_{}}{r})+\frac{1}{r}\frac{del{v}_r}{\text{d}\text{e}\text{l}}]$