A rod of radius KR moves upward with a constant speed vo through a cylindrical container of inner radius R containing a Newtonian fluid. The liquid circulates in the cylinder, moving upward along the moving central rod and moving downward near the fixed container wall. Flows similar to this occur in the seals of some reciprocating machinery, for example. in the annular space between piston rings.

a. Simplify the equation of continuity and the relevant component of the equation of motion. List and number all assumptions you made. Label each cancelled term in the equations of change with the number corresponding to the assumption used

b. Consider the problem where the annular region is quite narrow - that is, where ( is just slightly less than unity. In that case, the annulus may be approximated by a thin plane slit and the curvature can be neglected. The neglect of curvature leads to the approximation (due to r = R)

Use the dimensionless variables ( = vz/vo and ( = r/R. Instead of 4, it is more convenient, to use ( = (( -() / (1 -() in this part of the problem. Show that in this limit, the velocity distribution in the annular region, far from the end disturbances, is given by

In evaluating one of the three constants, make use of the fact the net flow rate in the annular region is zero, due to the circulation liquid in the enclosed cylinder.

Transcribed Image Text:

Rod of radius KR moves upward with velocity Cylinder of length L and inner radius R (with L>> R) dv, d²v 1 d dr r dr dr2 22