Streamfunction-vorticity formulation for incompressible viscid and inviscid flows on general surfaces
Abstract
This paper presents a streamfunction-vorticity formulation for the Navier--Stokes and Euler equations on general surfaces. Notably, this includes non-simply connected surfaces, on which the harmonic components of the velocity field play a fundamental role in the dynamics. By relying only on scalar and finite-dimensional quantities, our formulation ensures that the resulting methods give exactly tangential and incompressible velocity fields, while also being pressure robust. Compared to traditional methods based on velocity-pressure formulations, where one can only guarantee these structural properties by increasing the computational costs, this is a key advantage. We rigorously validate our formulation by proving its equivalence to the well understood velocity-pressure formulation under reasonable regularity assumptions. Furthermore, we demonstrate the applicability of the approach with numerical examples.
Cite
@article{arxiv.2512.20763,
title = {Streamfunction-vorticity formulation for incompressible viscid and inviscid flows on general surfaces},
author = {Tim Brüers and Christoph Lehrenfeld and Max Wardetzky},
journal= {arXiv preprint arXiv:2512.20763},
year = {2025}
}