Fluid Dynamics in Curvilinear Coordinates without Fictitious Forces

TL;DR

This paper presents a geometric approach to fluid dynamics in curvilinear coordinates that eliminates fictitious forces by solving for Cartesian momentum components, unifying relativistic and nonrelativistic frameworks.

Contribution

It introduces a method to remove fictitious forces in curvilinear coordinates by working in three dimensions and employing a geometric spacetime approach.

Findings

Fictitious forces can be eliminated in curvilinear coordinates using Cartesian momentum components.

The geometric approach provides insights into the unity of relativistic and nonrelativistic fluid dynamics.

The method simplifies the formulation of fluid equations on curved meshes.

Abstract

Use of curvilinear coordinates is sometimes indicated by the inherent geometry of a fluid dynamics problem, but this introduces fictitious forces into the momentum equations that spoil strict conservative form. If one is willing to work in three dimensions, these fictitious forces can be eliminated by solving for rectangular (Cartesian) momentum components on a curvilinear mesh. A thoroughly geometric approach to fluid dynamics on spacetime demonstrates this transparently, while also giving insight into a greater unity of the relativistic and nonrelativistic cases than is usually appreciated.