  We discuss the creeping motion of plugs of negligible viscosity in rough
capillary tubes filled with carrier fluids. This extends Bretherton's research
work on the infinite-length bubble motion in a cylindrical or smooth tube for
small capillary numbers Ca. We first derive the asymptotic dependence of the
plug speed on the finite length in the smooth tube case. This dependence on
length is exponentially small, with a decay length much shorter than the tube
radius R. Then we discuss the effect of azimuthal roughness of the tube on the
plug speed. The tube roughness leads to an unbalanced capillary pressure and a
carrier fluid flux in the azimuthal plane. This flux controls the relaxation of
the plug shape to its infinite-length limit. For long-wavelength roughness, we
find that the above decay length is much longer in the rough tube, and even
becomes comparable to the tube radius R in some cases. This implies a
much-enhanced dependence of the plug speed on the plug length. This mechanism
may explain the catch-up effect seen experimentally.
