New solutions for steady bubbles in a Hele–Shaw cell

Abstract

Exact solutions are presented for steadily moving bubbles in a Hele–Shaw cell when the effect of surface tension is neglected. These solutions form a three-parameter family. For specified area, both the speed of the bubble and the distance of its centroid from the channel centerline remain arbitrary when surface tension is ignored. However, numerical evidence suggests that this twofold arbitrariness is removed by the effect of surface tension, i.e., for given bubble area and surface tension, solutions exist only when the bubble velocity and the centroid distance from the channel centerline attain one or more isolated values. From a limited numerical search, no nonsymmetric solutions could be found; however, a branch of symmetric bubble solutions that was not found in earlier work was found. This branch corresponds to one of the Romero–Vanden-Broeck branch of finger solutions when the bubble size is large. A new procedure for numerical calculations of bubble solutions in the presence of surface tension is presented and is found to work very well for reasonably large bubbles, unlike the previous method of Tanveer [Phys. Fluids 29, 3537 (1986)]. The precise power law dependence of bubble velocity on surface tension for small surface tension is explored for bubbles of different area. Agreement is noted with recent analytical results for a finger.