Bubble interaction model for hydrodynamic unstable mixing.

The analytic model for the evolution of single and multiple bubbles in Rayleigh-Taylor mixing is presented for the system of arbitrary density ratio. The model is the extension of Zufiria's potential theory, which is based on the velocity potential with point sources. We present solutions for a single bubble, at various stages, from the model and show that the solutions for the bubble velocity and curvature are in good agreement with numerical results. We demonstrate the evolution of multiple bubbles for finite density contrast and investigate dynamics of bubble competition, whereby leading bubbles grow in size at the expense of neighbors. The model shows that the growth coefficient alpha for the scaling law of the bubble front depends on the Atwood number and increases logarithmically with the initial perturbation amplitude. It is also found that the aspect ratio of the bubble size to the bubble height exhibits a self-similar behavior in the bubble competition process, and its values are insensitive to the Atwood number. The predictions of the model for the similarity parameters are in accordance with experimental and numerical results.