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On the base of the system of hydrodynamic equations we consider a model of formation and development of the hydrodynamic vortices in the nuclear matter during relativistic heavy-ion collisions, in astrophysical objects, and in powerful atmospheric phenomena such as typhoons and tornados. A new class of the analytic solutions of non-relativistic hydrodynamic equations for the incompressible liquid in the presence of a bulk sink are analyzed. The main feature of these solutions is that they describe non-stationary hydrodynamic vortices with the azimuth component of velocity exponentially or explosively growing with time. A necessary attribute of a system with such a behavior is a presence of a bulk sink, which provides the existence of the non-stationary vortex regime. These solutions are obtained by nullifying the terms in the Navier-Stokes equations, which describe viscous effects, exist and represent vortex structure with "rigid-body" rotation of the core and converging radial flows. With the help of our model we explain some typical features of the above physical systems from the unique point of view. [ABSTRACT FROM AUTHOR]