An extended Bretherton model for long Taylor bubbles at moderate capillary numbers

When ( long) bubbles are transported in tubes containing a fluid, the presence of a thin film of fluid along the tube walls causes the velocity of the bubble to be different from the average fluid velocity. Bretherton [The motion of long bubbles in tubes, J. Fluid Mech. 10, 166 (1961)] derived a model to describe this phenomenon for pressure driven flows based on a lubrication approach coupled with surface deformation of the bubble. Bretherton found that the parameter governing the physics involved is the capillary number (Ca) which expresses the relationship between speed of the bubble, surface tension, and viscosity of the liquid. The results of Bretherton are here re-derived and analyzed in a slightly more perspicuous manner. Incorporating the condition that the bubble-film combination should fit inside the tube results in an expression very similar to the one found empirically by Aussillous and Quere [Quick deposition of a fluid on the wall of a tube, Phys. Fluids 12, 2367 (2000)] of the Taylor [Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech. 10, 161 (1961)] experimental data. Our expression is valid for Ca values up to Ca = 2.0, but approaches Bretherton's result for low values of Ca. The analysis is done in terms of the pressure buildup which originates from the interplay between surface tension and lubrication due to the thin layer of fluid near the tube wall. (C) 2014 AIP Publishing LLC.