Transport of gas bubbles in capillaries

The pressure drop and wetting film thickness for isolated bubbles and bubble trains moving in circular and square capillaries are computed. An arclength-angle formulation of a composite lubrication equation allows for the numerical matching of the lubrication solution of the transition region to the static profiles away from the channel wall. This technique is shown to extend the classical matched asymptotic analysis of Bretherton for circular capillaries to higher capillary numbers Ca. More importantly, it allows the study of finite bubbles, which are shown to resemble infinitely long bubbles in film thickness and pressure drop if their lengths exceed the channel width. The numerical study of bubble trains, verified by a matched asymptotic analysis, shows a surprising result that the pressure drop across one member bubble is identical to that of an isolated bubble at low capillary numbers. This analysis of square capillaries neglects azimuthal flow and is only valid for Ca>3.0×10−3. Nevertheless the film radius and pressure drop of a bubble traveling in a square capillary above this capillary number are computed. These results are conveniently summarized in a correlation for the apparent viscosity of bubbles as a function of foam texture and capillary geometry and dimension.