Motion of gas bubbles, considered as massless bodies, affording deformations within a prescribed family of shapes, in an incompressible fluid under the action of gravitation and surface tension

A model allowing to describe motion and coalescence of gas bubbles in a liquid under the action of gravitation and surface tension is proposed. The shape of the bubbles is described by a pre-defined family of mappings, for example ellipsoids with a fixed volume and the effects of the gas motions inside the bubbles are neglected. The motion of a bubble is obtained in a Lagrangian form using the D'Alembert principle of virtual works. The set of equations is numerically solved with the help of the fictitious domain technique in which the Navier-Stokes equations in the domain formed by both fluid and gas are considered. The equations governing the bubbles motion are imposed by introducing Lagrange multipliers on the bubbles boundaries. Numerical results in 2D and 3D are presented