Effective Rheology of Bubbles Moving in a Capillary Tube
  We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train from considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles. 
Santanu Sinha, Alex Hansen, Dick Bedeaux and Signe Kjelstrup
physics.flu-dyn