A fluid with constant properties $($viscosity $$

and density $)$ is kept between two

concentric cylinders of height $L,$ along

which the pressure is the same. The inner

cylinder has a radius of $r_l,$ and the outer one

has a radius of $r_2.$ The cylinders can rotate

around an axe in axial direction passing

through their center.

In Case $1,$ the outer cylinder rotates with

angular velocity $$ while the inner cylinder

is kept fixed. In Case $2,$ the inner cylinder

rotates with angular velocity $$ while the

outer cylinder is kept fixed. With the

rotational motion of the cylinders, the fluid

is set in motion in tangential direction with

Here, the tangential velocity sto only changes with the radial direction $r.$ The velocity profile for both

cases are shown in the figure. Note that tangential velocity is the product of angular velocity and radius:

$w=I.$

a$)$ Draw the forces acting on the control volume dr $d$ d $x$ dz given here.

Please do not make shell balance.

b$)$ Using the Navier$-$Stokes equations given below, find the tangential velocity profile va$($r$)$ for Case $1$

and Case $2.$

r$-$direction:

$(\frac{v_r}{t}+v_r\frac{v_r}{r}+\frac{v_{}}{r}\frac{v_r}{}-\frac{v_{}^2}{r}+v_z\frac{v_r}{z})=-\frac{P}{r}+g_r+[\frac{}{r}(\frac{1}{r}\frac{}{r}(r{v}_r))+\frac{1}{r^2}\frac{{}^2{v}_r}{{}^2}-\frac{2}{r^2}\frac{v_{}}{}+\frac{{}^2{v}_r}{z^2}]$

$-$direction:

z$-$direction:

$(\frac{(hat(v_z))}{t}+v_r\frac{v_z}{r}+\frac{v_{}}{r}\frac{v_z}{}+v_z\frac{v_z}{z})=-\frac{P}{z}+g_z+[\frac{1}{r}\frac{}{r}(r\frac{(hat(del))v_z}{r})+\frac{1}{r^2}\frac{{}^2{v}_z}{{}^2}+\frac{{}^2{v}_z}{z^2}]$