Exact solutions of the hydrodynamic equations derived from partially invariant solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a “reduced” system of $n$ equations and an additional equation for an “extra” function $w$. In this case, the “reduced” system, in which $w=0$, admits a Lie group $G$. Taking a certain partially invariant solution of the “reduced” system with respect to this group as a “seed” solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the “seed” solution on the invariants of the group $G$ has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.