A numerical study of the effect of surface tension and noise on an expanding Hele--Shaw bubble

In this paper, the dynamics of an interface under the influence of surface tension is studied numerically for flow in the Hele-Shaw cell, where the interface separates an expanding bubble of inviscid fluid from a displaced viscous fluid. Of special interest is the long-time behavior of the so-called q-pole initial data, whose motion is explicitly known and globally smooth for the zero surface tension flow. The numerical method is spectrally accurate and based upon a boundary integral formulation of the problem, together with a special choice for the frame of motion along the interface. In 64-bit arithmetic, a transition from the formation of side branches to tip splitting is observed as the surface tension is decreased. The tip splitting occurs on a time scale that decreases with the surface tension. This is consistent with some experimental observations. However, by increasing the arithmetic precision to 128 bits, it is found that this transition occurs at a yet smaller surface tension. The tip splitting is associated with the growth of noise in the calculation at unstable scales allowed by the surface tension, and a simple linear model of this growth seems to agree well with the observed behavior. The robustness of the various observed structures to varying amounts of noise is also investigated numerically. It is found that the appearance of side branches seems to be the intrinsic effect of surface tension, and the time scales for their appearance increases as the surface tension decreases. These results suggest, with some qualification, that surface tension acts as a regular perturbation to evolution from this initial data, even for long times.