Local analysis and topological bifurcations of free surfaces

The recently developed method of the local analysis\footnote{J. Eggers and M. A. Fontelos, Panoramas et Synth\`{e}ses \textbf{38}, 69 (2013).} is applied in order to determine generic singularities and topological bifurcations of free surfaces and interfaces in liquids and gases. In the presented physical examples the surface geometry is described by a family of smooth maps with the state variables and a set of control parameters. We determine the order of Taylor expansion about a point of interest defining the multiplicity of a singular germ, codimension, unfolding of singularity and bifurcation diagram.