  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.
