Consider a fluid flowing in the annular region between two concentric cylindrical

pipes $($azimuthal flow$).$ The inner cylinder has a radius $\text{'}{R}_i\text{'},$ and the outer cylinder

has a radius $\text{'}{R}_0\text{'}.$ The inner cylinder is stationary, while the outer cylinder rotates

with a constant angular velocity $\text{'}\text{'}.$ The flow is incompressible, steady, and

laminar. The viscosity of the fluid is $\text{'}\text{'}.$

$($a$)$ Derive the velocity profile for azimuthal flow $v_{}(r)$ for the fluid between the

two cylinders, if there are no radial or axial velocity components show the

Navier Stoke's equation simplifies to the following equations:

$\frac{1}{r}*\frac{d}{dr}(r*{}_r)=0,$&$,{}_r(r)=*\frac{d{v}_{}}{dr}$

$($b$)$ Determine the shear stress distribution, ${}_{re}(r),$ across the annular gap.