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International audienceThe axisymmetric gravity-driven dynamics of a bubble rising toward a free surface isaddressed for gas-liquid interfaces having unequal surface tensions. The liquid flowis governed by the Stokes equations which are here solved using a boundary elementmethod in axisymmetric configuration. Within this framework, two dimensionlessnumbers arise: the Bond number Bo1 based on the surface tension of the bubbleinterface and the surface tension ratio ˆ γ comparing the free surface and bubble surfacetensions. Under a careful and discussed selection of the code key settings (numberof boundary elements, initial bubble location, and distance beyond which the freesurface is truncated), it has been possible to numerically and accurately track in timethe bubble and free surface shapes for several values of (Bo1, ˆ γ). The long-time shapesare found to deeply depend upon both Bo1 and ˆ γ and also to compare well withthe shapes predicted in Princen and Mason [“Shape of a fluid drop at a fluid-liquidinterface. 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 drainagedynamics of the liquid film thickness between the bubble and the free surface dependson (Bo1, ˆ γ). The long-time film thickness exponentially decays in time and a so-calledthinning rate α for which the numerical behaviors and a simple model reveal twobasic behaviors: (i) at small Bond number, α behaves as 1/Bo1 and (ii) at largeBond number, α is nearly constant. In addition, it is found that in the entire rangeof the quantity χ = (1 + ˆ γ)Bo1/(2 ˆ γ), the thinning rate α is well approximated by thefunction 1/(18 χ) + α∞ with α∞ ≈ 0.158. Such a result also permits one to estimatethe typical drainage time versus the initial bubble radius a, the liquid density ρ andviscosity μ, the gravity and the free surface, and bubble surface tensions

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