A Uniqueness Theorem for Incompressible Fluid Flows with Straight Streamlines

TL;DR

This paper proves that the only incompressible Euler flows with fixed straight streamlines are those generated by simple geometric configurations like spheres, cylinders, or planes, using differential geometry to explicitly integrate the equations.

Contribution

It provides a uniqueness theorem characterizing all such flows with straight streamlines using differential geometry and explicit integration of Euler equations.

Findings

Flows with fixed straight streamlines are limited to sphere, cylinder, or plane configurations.

Explicit solutions correspond to point, line, or plane sources at infinity.

The proof employs local differential geometry of line congruences.

Abstract

It is proven that the only incompressible Euler fluid flows with fixed straight streamlines are those generated by the normal lines to a round sphere, a circular cylinder or a flat plane, the fluid flow being that of a point source, a line source or a plane source at infinity, respectively. The proof uses the local differential geometry of oriented line congruences to integrate the Euler equations explicitly.