II$.(20\backslash \%)$ A liquid with density $\backslash (\backslash$rho $\backslash )$ and viscosity $\backslash (\backslash$mu $\backslash )$ flows inside an infinite, rotating, circular cylinder. The axial pressure drops from $\backslash ($ p$\_\{1\}\backslash )$ to $\backslash ($ p$\_\{2\}\backslash ),$ over a length of $\backslash ($ L $\backslash ).$ The cylinder has a radius, $\backslash ($ R $\backslash ),$ and angular velocity, $\backslash (\backslash$omega $\backslash ).$ You may assume steady, fully$-$developed, incompressible flow, and neglect the effect of gravity.

a$)$ Simplify the governing equations. State assumptions clearly.

b$)$ List the appropriate boundary conditions.

c$)$ Solve for the velocity distribution in the cylinder $($both the axial and circumferential velocity components are non$-$zero, i$.$e$.$ functions of $\backslash ($ r $\backslash )).$

d$)$ Solve for the pressure $($it is expected to be a function of both $\backslash ($ r $\backslash )$ and $\backslash ($ z $\backslash )).$

e$)$ Plot the axial and circumferential velocity components and pressure profiles as a function of $\backslash ($ r $\backslash )$ using the computer $($choose reasonable values for quantities that may be needed; use either air or water$)$