$3.$ Consider the fully$-$developed steady$-$state laminar flow of a Newtonian fluid in a vertical pipe of radius R which is rotated at the constant angular velocity $\backslash (\backslash$omega $\backslash ),$ and also is subject to an axial pressure gradient and gravity forces in the negative $\backslash ($ z $\backslash )-$direction.

a$)$ Determine the relative forms of the continuity and momentum equations.

b$)$ Write the boundary conditions on the equations in part $($a$).$

c$)$ Solve the above to obtain the velocity profile for the circumferential velocity.

d$)$ Determine the pressure profile $\backslash (\backslash$mathrm$\{$P$\}(\backslash$mathrm$\{$r$\},\backslash$mathrm$\{$z$\})\backslash ).$

e$)$ Determine the axial velocity profile.

f$)$ Scale these differential equations to determine under what conditions the pressure forces can be neglected relative to the gravity forces. Express your answer in terms of dimensionless group which must be very large or very small in order to neglect the pressure forces.