3 years ago

Hydrodynamic Vortex on Surfaces

Clodoaldo Grotta Ragazzo, Humberto Henrique de Barros Viglioni

Abstract

The equations of motion for a system of point vortices on an oriented Riemannian surface of finite topological type are presented. The equations are obtained from a Green’s function on the surface. The uniqueness of the Green’s function is established under hydrodynamic conditions at the surface’s boundaries and ends. The hydrodynamic force on a point vortex is computed using a new weak formulation of Euler’s equation adapted to the point vortex context. An analogy between the hydrodynamic force on a massive point vortex and the electromagnetic force on a massive electric charge is presented as well as the equations of motion for massive vortices. Any noncompact Riemann surface admits a unique Riemannian metric such that a single vortex in the surface does not move (“Steady Vortex Metric”). Some examples of surfaces with steady vortex metric isometrically embedded in \(\mathbb {R}^3\) are presented.

Publisher URL: https://link.springer.com/article/10.1007/s00332-017-9380-7

DOI: 10.1007/s00332-017-9380-7

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