https://hal-mines-paristech.archives-ouvertes.fr/hal-01519290Guémas, MarineMarineGuémasLadHyX - Laboratoire d'hydrodynamique - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueSVI - Surface du Verre et Interfaces - CNRS - Centre National de la Recherche ScientifiquePigeonneau, FranckFranckPigeonneauSVI - Surface du Verre et Interfaces - CNRS - Centre National de la Recherche ScientifiqueSellier, AntoineAntoineSellierLadHyX - Laboratoire d'hydrodynamique - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueRising bubble near a free surface: numerical and asymptotic studyHAL CCSD2013Bubblefree surfacesurface tensionStokes flowasymptotical expansionfilm drainage[PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]Pigeonneau, Franck2017-05-06 19:13:092022-06-26 10:00:382017-05-10 17:21:14enConference papersapplication/pdf1Phase separation is involved in many chemical processes and is generally limited by the collapse of inclusions at the free surface. For instance, the coalescence of bubbles in highly viscous Newtonian fluids is observed in various fields, such as geophysics or the glass industry. When a bubble rises through a liquid toward a free surface, we first observe the rising of the bubble driven by the buoyancy forces. In the second step corresponding to the drainage, a liquid film is created between the bubble interface and the free surface decreasing with the time. Both the bubble shape close to the free surface and the film thickness depend upon on the Bond number which is the ratio of gravity to surface tension forces. Under the assumption of the small Reynolds number, the inertial effects are neglected. Moreover, both the surface tensions of the free surface and the bubble are assumed uniform but theirs values can be different. We have already investigated the gravity-driven migration using a numerical method based on the boundary-integral method (Pigeonneau and Sellier (2011)). The aim of the current work is to develop an asymptotic solution when the Bond number is small. In the perturbation method, the interfaces and flow are developed following an asymptotic expansion for which the small parameter is the Bond number. The zeroth order corresponds to the case of undeformed interfaces (flat free surface and spherical bubble) which can be determined using bipolar coordinates. The hydrodynamic force at the zeroth order is obtained using the exact solution provided by Stimson and Jeffery (1926). The behavior of the film thickness is obtained from the momentum equation on the bubble. We compare the asymptotic predictions with the previous numerical results. Finally, the bubble and free surface shapes are investigated for different values of the surface tension ratio and small Bond number.