  In clean Dirac electron systems such as graphene, electron-electron
interactions can dominate over other relaxation mechanisms such as phonon or
impurity scattering. It has been predicted that in this limit, collective
electron dynamics can be described by hydrodynamic equations. The prerequisites
for electron liquids are already fulfilled in current experiments and hints of
electron hydrodynamics have been identified in transport measurements. Here, we
show that a nozzle geometry, implemented for example in a graphene sample, can
cause a transition from subsonic to supersonic flow and provides an interesting
probe for the hydrodynamic regime of Dirac electrons. In particular, we predict
two distinct transport features that can be seen in the experimentally
measurable voltage characteristics on the exit side of the nozzle: a pronounced
negative local resistance, and an abrupt change of the electrostatic potential
induced by an electron shock wave. Our results pave the way for an experimental
identification of supersonic hydrodynamic electron flow and for the
experimental study of electron shock waves.
