A liquid flows within an annular region between the two cylinders under the action of a pressure gradient, \(d P / d z\) as shown in Figure P5.15. The inner cylinder is heated such that the viscosity of the liquid varies as a function of radial position according to \(\mu=\mu_{0} r\).

a. Derive the differential equation describing the motion of the fluid within the annulus.

b. What are the boundary conditions for the problem at \(r=r_{i}\) and \(r=r_{o}\) ?

c. Solve the equation for the velocity profile.

d. Where is the maximum velocity located? It is not at the midpoint of the annulus.

e. What is the volumetric flow rate within the annulus?

Transcribed Image Text:

FIGURE P5.15 Pressure Drop- dP/dz Z