Low-Reynolds-number rising of a bubble near a free surface at vanishing Bond number

Abstract

This work considers a nearly spherical bubble and a nearly flat free surface interacting under buoyancy at vanishing Bond number Bo. For each perturbed surface, the deviation from the unperturbed shape is asymptotically obtained at leading order on Bo. The task appeals to the normal traction exerted on the unperturbed surface by the Stokes flow due to a spherical bubble translating toward a flat free surface. The free surface problem is then found to be well-posed and to admit a solution in closed form when gravity is still present in the linear differential equation governing the perturbed profile through a term proportional to Bo. In contrast, the bubble problem amazingly turns out to be over-determined. It however becomes well-posed if the requirement of horizontal tangent planes at the perturbed bubble north and south poles is discarded or if the term proportional to Bo is omitted. Both previous approaches turn out to predict for a small Bond number, quite close solutions except in the very vicinity of the bubble poles. The numerical solution of the proposed asymptotic analysis shows in the overlapping range Bo=O(0.1) and for both the bubble and the free surface perturbed shapes, a good agreement with a quite different boundary element approach developed in Pigeonneau and Sellier [“Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface,” Phys. Fluids 23, 092102 (2011)]. It also provides approximated bubble and free surface shapes whose sensitivity to the bubble location is examined.