Unsteady dynamics of long gas bubbles travelling in a liquid-filled capillary tube*

The steady propagation of a long gas bubble through a viscous liquid within a circular tube is a classical problem in fluid mechanics since the seminal studies of Bretherton and Taylor. This problem has been extensively studied for small capillary and Reynolds numbers, where a steady flow is achieved. However, recent experimental and computational studies revealed that if the Weber number of the flow is sufficiently larger than unity, the rear meniscus of the bubble becomes unstable and time-dependent patterns arise. In this work, we employ linear global stability analysis and direct numerical simulations to study the occurrence and origin of this instability. For capillary numbers in the range Ca=0.005-0.04, we show that the flow becomes unstable when the Weber number grows above values in the range We=9-16. The instability is due to a non-monotonic pressure profile established along the rear meniscus of the bubble, which surface tension is unable to oppose beyond a certain threshold in the Weber number. The linear global stability analysis and DNS predict bubble shapes, stability boundaries and time-dependent patterns in good agreement with each other. The stability boundary is identified well by a modified Weber number which describes the competition of pressure and capillary forces acting on the rear meniscus of the bubble.