Stability of bubbles in a Hele–Shaw cell

Abstract

The linear stability of steadily moving bubbles in a Hele–Shaw cell is investigated. It is shown analytically that without the effect of surface tension, the bubbles are linearly unstable with the stability operator having a continuous spectrum. For small bubbles that are circular, analytical calculations also show that any amount of surface tension stabilizes a bubble. Numerical calculations suggest that the branch of bubble solutions that, in the limit of large area, corresponds to the McLean–Saffman finger is stable for any nonzero surface tension. However, the decay rate of disturbances on the McLean–Saffman branch depends appreciably on the bubble size even for large bubbles. This suggests that the stability results on this branch cannot be immediately extrapolated to the McLean–Saffman fingers. For another branch of bubble solution, which in the limit or large area corresponds to the first of the Romero–Vanden-Broeck finger solutions, numerical evidence suggests that it is unstable to one symmetric and one antisymmetric mode for any surface tension. The symmetric unstable mode tends to break the tip of the bubble and the growth rate of this mode is unaffected by further increase in bubble size, once the bubble is large enough. This suggests that there is an analogous instability for the finger, and this agrees with the numerical findings of Kessler and Levine [Phys. Rev. A 33, 2632 (1986)]. Agreement is noted in the quantitative comparison of the growth rate with the predictions [Tanveer, Phys. Fluids 30, 2318 (1987)] on the limiting growth rate for the symmetric unstable mode for the first Romero–Vanden-Broeck branch of finger solution.

Files

Additional details