Bubble length affects bubble speed in a rough microfluidic channel

We discuss the creeping motion of bubbles of different length in rough capillary tubes filled with carrier fluids. This extends the results of Bretherton\footnote{F.P.Bretherton, 1961, J. Fluid Mech., 10, 166.} for an infinite-length bubble at small capillary number $Ca$ in a circular tube. We first derive the asymptotic corrections to the speed owing to finite length. 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 bubble speed. Tube roughness leads to a carrier fluid flow in the azimuthal plane; this flow controls the relaxation of the bubble shape to its infinite length limit. For long-wavelength roughness, we find that the above decay length becomes much longer and even comparable to $R$. This implies a much-enhanced dependence of the bubble velocity on length. A shorter bubble should then catch up with a longer bubble ahead of it in the same channel. This mechanism may explain catch-up effects seen experimentally.\footnote{R.Ismagilov, private communication.}