Geometrization of vortex and spiral flows in an ideal homogeneous fluid (Q1876444)

The authors present a theoretical study of steady vortex motion of homogeneous ideal heavy fluid with a free surface by methods of differential geometry. The main idea is based on suggestion that the velocity field is formed by geodesic flows on some surfaces. For steady flows, integral flow lines are geodesics on second-order surfaces being parameterized and located in the space occupied by the fluid. In this case both Euler and continuity equations are transformed into equations for inner geometry parameters. Conditions on external parameters are derived from boundary conditions of the problem. An investigation of properties of generalized Rankine vortex that is a vertical vortex flow contacting with the free surface is done. In addition to the classical Rankine vortex, these vortices are characterized by finite integral invariants. The constructed set of explicit solutions depends on a unique parameter, which can be determined experimentally through measurements of depth and shape of a near-surface hole produced by the vortex.