A Newtonian fluid of viscosity, \(\mu\), and density, \(ho\), is contained in between two vertical pipes of diameter, \(d_{o}\) and \(d_{i}\). The situation is shown in Figure P10.22. If left to its own devices, the fluid would begin to drain from between the pipes under the influence of gravity. To stop the fluid from draining out, the inner pipe is lifted at some velocity \(v_{i}\).

a. Derive the differential equation governing the fluid flow between the two pipes.

b. Specify the boundary conditions.

c. Solve the equation to get the velocity profile and the volumetric flow rate of fluid.
(this integral may help)
\[\int\left[a\left(r_{o}^{2}-r^{2}\right)+b \ln \left(\frac{r}{r_{o}}\right)\right] r d r=\frac{1}{2} a r_{o}^{2} r^{2}-\frac{1}{4} r^{4}+\frac{1}{2} b r^{2} \ln \left(\frac{r}{r_{o}}\right)-\frac{1}{4} b r^{2}\]

d. What inner pipe velocity is required to keep the fluid from draining out?

Transcribed Image Text:

FIGURE P10.22 g d Z