Elongation of material lines and vortices by Euler flows on two-dimensional Riemannian manifolds

TL;DR

This paper investigates how the curvature of two-dimensional Riemannian manifolds affects fluid flow, specifically the elongation of material lines and vortices, extending existing concepts to curved surfaces and illustrating curvature's role in vortex filamentation.

Contribution

It derives a formula for the second derivative of particle distance squared, incorporating curvature effects, and extends hyperbolic domain concepts to curved surfaces.

Findings

Negative curvature accelerates material line elongation.

Curvature influences vortex filamentation on curved surfaces.

Examples demonstrate curvature's impact on flow dynamics.

Abstract

We study the influence of the differential geometry of the flow domain on the motion of fluids on two-dimensional Riemannian manifolds, particularly on the elongation of material lines and vortices. We derive a formula for the second order time derivative of the square of the distance between close fluid particles and show that a curvature term appears. The elongation of a material line is accelerated by negative curvature. The use of this expression extends Haller's definition of hyperbolic domains to flows on curved surfaces. The need to consider curvature effects is illustrated by three examples. The example of a curved two-dimensional torus implies that the filamentation of vortices can be triggered by negative curvature.