How Does Viscosity Affect the Velocity of Coaxial Tubes in Fluid Dynamics?

[Mythbusters]

We have a combination of two very long coaxial tubes with radii R and 2R. The tubes are placed vertically, the space between the tubes is filled with a heavy fluid of viscosity. The outer tube glides stationary down under the action of gravity, the inner tube is at rest. Both tube ends are open to the amosphere. Mass of the outer tube per unit length is m=M/L. Find the tube velocity U.

In the present geometry stress is calculated as stress=viscosity*du/dr (z-direction)

Need help to figure out how I will attack this problem, thanks!

Mythbusters - "Well, here's your problem"

 

[diazona]

Stress in this case would be F/A I think, i.e. force on the outer tube divided by area. You'd probably have to use the average area of the inner and outer tubes. Then just integrate the velocity gradient from R to 2R to find the velocity of the outer tube.

Disclaimer: I am far from an expert on fluid dynamics...