Jet formation in bubbles bursting at a free surface

We study numerically bubbles bursting at a free surface and the subsequent jet formation. The Navier-Stokes equations with a free surface and surface tension are solved using a marker-chain approach. Differentiation and boundary conditions near the free surface are satisfied using least-squares methods. Initial conditions involve a bubble connected to the outside atmosphere by a preexisting opening in a thin liquid layer. The evolution of the bubble is studied as a function of bubble radius. A jet forms with or without the formation of a tiny air bubble at the base of the jet. The radius of the droplet formed at the tip of the jet is found to be about one tenth of the initial bubble radius. A series of critical radii exist, for which a transition from a dynamics with or without bubbles exist. For some parameter values, the jet formation is close to a singular flow, with a conical cavity shape and a large curvature or cusp at the bottom. This is compared to similar singularities investigated in other contexts such as Faraday waves.