Slow viscous gravity-driven interaction between a bubble and a free surface with unequal surface tensions

The axisymmetric gravity-driven dynamics of a bubble rising toward a free surface is addressed for gas-liquid interfaces having unequal surface tensions. The liquid flow is governed by the Stokes equations which are here solved using a boundary element method in axisymmetric configuration. Within this framework, two dimensionless numbers arise: the Bond number Bo(1) based on the surface tension of the bubble interface and the surface tension ratio (gamma) over cap. comparing the free surface and bubble surface tensions. Under a careful and discussed selection of the code key settings (number of boundary elements, initial bubble location, and distance beyond which the free surface is truncated), it has been possible to numerically and accurately track in time the bubble and free surface shapes for several values of (Bo(1), (gamma) over cap). The long-time shapes are found to deeply depend upon both Bo(1) and (gamma) over cap and also to compare well with the shapes predicted in Princen and Mason ["Shape of a fluid drop at a fluid-liquid interface. II. Theory for three-phase systems," J. Colloid. Sci. 20, 246-266 (1965)] using a hydrostatic model in which both surfaces are touching. Similarly, the drainage dynamics of the liquid film thickness between the bubble and the free surface depends on (Bo(1), (gamma) over cap). The long-time film thickness exponentially decays in time and a so-called thinning rate alpha for which the numerical behaviors and a simple model reveal two basic behaviors: (i) at small Bond number, alpha behaves as 1/Bo(1) and (ii) at large Bond number, alpha is nearly constant. In addition, it is found that in the entire range of the quantity chi = (1 + (gamma) over cap) Bo(1)/(2 (gamma) over cap), the thinning rate alpha is well approximated by the function 1/(18 chi) + alpha(infinity) with alpha(infinity) approximate to 0.158. Such a result also permits one to estimate the typical drainage time versus the initial bubble radius a, the liquid density. and viscosity mu, the gravity and the free surface, and bubble surface tensions. (C) 2015 AIP Publishing LLC.