Steady flows of ideal incompressible fluid

TL;DR

This paper explores a novel connection between fluid mechanics and differential geometry by analyzing special steady solutions to the Euler equations, called Gavrilov flows, revealing their geometric structure and providing numerical examples.

Contribution

It introduces Gavrilov flows as a new class of steady Euler solutions and describes their geometric structure and conditions for axisymmetric cases.

Findings

Derived PDE system for axisymmetric Gavrilov flows

Identified conditions for the consistency of these PDEs

Presented numerical examples with specific pressure and isobaric surface geometries

Abstract

A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. Local structure of a Gavrilov flow is described in terms of geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for an axisymmetric Gavrilov flow and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus.