We study numerically bubbles bursting at a free surface. The Navier-Stokes equations with a free surface and surface tension are solved using a marker-chain approach. 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. The results show that the jet birth may occur with or without the formation of a tiny air bubble at the base of the jet. A series of critical radiuses exist, for which a transition from a dynamics with or without bubbles exist. This singularity is compared to the ones investigated in other contexts such Faraday waves.

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