Shape of long gas bubbles propagating in square capillaries

We present the results of a systematic analysis of the shape of the thin lubrication film surrounding a long gas bubble transported by a liquid flow in a square capillary. Direct numerical simulations are performed for a range of capillary and Reynolds numbers Ca=0.002-0.5 and Re=1-2000, and very long bubbles, up to 20 times the channel hydraulic diameter. In agreement with previous studies, when Ca>0.05 the bubble exhibits an axisymmetric shape, whereas for Ca<0.05 the bubble flattens at the centre of the channel wall and thick liquid lobes are left at the corners. When Ca<0.01, the thin film at the centre of the wall assumes a saddle-like shape, which leads to the formation of two constrictions at the sides of the liquid film profile. The resulting cross-stream capillary pressure gradients drain liquid out of the thin-film, whose thickness decreases indefinitely as a power-law of the distance from the bubble nose. We report detailed values of the centreline and minimum film thickness along the bubble, bubble speed, and cross-sectional gas area fraction, at varying Ca and Re. Inertial effects retard the formation of the saddle-shaped film at the channel centre, which may never form if the bubble is not sufficiently long. However, the film thins at a faster rate as Re increases