A fluid flows through a circular pipe of radius, \(r_{o}\). The flow is fully developed. The pipe is oriented vertically with respect to gravity (z-direction) and also has a constant pressure gradient imposed upon it in the opposite flow direction.

a. Pare down the Navier-Stokes equations and justify why you are making the assumptions you make.

b. What are the boundary conditions for this problem?

c. The solution is of the form:

\[v_{z}=\frac{1}{4 \mu}\left[\frac{d P}{d z}-ho g_{z}\right]\left[r^{2}-r_{o}^{2}\right]\]

What is the volumetric flow rate and under what conditions is it 0 ?

d. Is the flow irrotational?

e. What are the contributions to viscous dissipation from this velocity profile?