On Euler equation for incoherent fluid in curved spaces

TL;DR

This paper explores hodograph equations for the Euler fluid equations in curved spaces, deriving solutions including stationary states for specific geometries like cones and spheres in various dimensions.

Contribution

It introduces methods to construct hodograph equations in curved spaces using geodesic integrals, expanding solution classes for Euler equations in non-Euclidean geometries.

Findings

Derived hodograph equations for curved spaces with constant pressure

Constructed solutions including stationary states

Analyzed specific cases like cones and spheres in detail

Abstract

Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph equations. These hodograph equations provide us with various classes of solutions of the Euler equation, including stationary solutions. Particular cases of cone and sphere in the 3-dimensional Eulidean space are analysed in detail. Euler equation on the sphere in the 4-dimensional Euclidean space is considered too.