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

Abstract

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 ratioγ 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 ,γ). The long-time shapes are found to deeply depend upon both Bo 1 andγ 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 ,γ). The long-time film thickness exponentially decays in time and a so-called thinning rate α for which the numerical behaviors and a simple model reveal two basic behaviors: (i) at small Bond number, α behaves as 1/Bo 1 and (ii) at large Bond number, α is nearly constant. In addition, it is found that in the entire range of the quantity χ = (1 +γ)Bo 1 /(2γ), the thinning rate α is well approximated by the function 1/(18 χ) + α ∞ with α ∞ ≈ 0.158. Such a result also permits one to estimate the typical drainage time versus the initial bubble radius a, the liquid density ρ and viscosity µ, the gravity and the free surface, and bubble surface tensions.

Bag film breakup of droplets in uniform airflows

“…Developing new theoretical models in place of (3.3) whose predictions match better with the late-time neck drainage behaviour observed within the We and Oh range of interest is out of the scope of the current work. We note briefly that the drainage behaviour of bag films under the influence of aerodynamic pressure difference observed here bears resemblance to the drainage of liquid films between a free air-water surface and a buoyancy-driven air bubble (Pigeonneau & Sellier 2011;Kočárková, Rouyer & Pigeonneau 2013;Guémas, Sellier & Pigeonneau 2015), where film drainage models are developed based on lubrication assumptions (see § 4.2 of Magnaudet & Mercier (2020) and references therein for more detailed discussions). More specifically, Pigeonneau & Sellier (2011) also showed a deviation from exponential decay of bubble film thickness under asymptotically large surface tension, which is ascribed to a finite-time singularity and contrasts with the thin-film RT instability mechanism proposed by Villermaux & Bossa (2009).…”

The Life of a Surface Bubble

“…However they find an algebric decay at low Bo and large surfactants concentration which reduces the drainage rate (long living bubbles). The presence and dynamics of surfactants may change the equilibrium shape of the bubble, as previously observed for a bubble and a free surface with unequal surface tensions [26]. Additionally, the presence of surface tension gradients not only rigidifies the interfaces but also affects the usual monotonous thickness decay from top to bottom.…”

The Life of a Surface Bubble

“…However they find an algebric decay at low and large surfactants concentration which reduces the drainage rate (long living bubbles). The presence and dynamics of surfactants may change the equilibrium shape of the bubble, as previously observed for a bubble and a free surface with unequal surface tensions [ 35 ]. Additionally, the presence of surface tension gradients not only rigidifies the interfaces but also affects the usual monotonous thickness decay from top to bottom.…”