Active Flows on Curved Surfaces

TL;DR

This paper develops a numerical method to study how Gaussian curvature affects active turbulence and vortex formations on curved surfaces, revealing curvature-dependent flow behaviors and alignment patterns.

Contribution

It introduces a covariant numerical approach for the generalized Navier-Stokes equation on surfaces, exploring the impact of curvature on active turbulence phenomena.

Findings

Vortex chains align with minimal curvature lines.

Turbulence depends on the sign and gradient of Gaussian curvature.

Qualitative agreement with experiments on active nematic liquid crystals.

Abstract

We consider a numerical approach for a covariant generalised Navier-Stokes equation on general surfaces and study the influence of varying Gaussian curvature on anomalous vortex-network active turbulence. This regime is characterised by self-assembly of finite-size vortices into linked chains of anti-ferromagnet order, which percolate through the entire surface. The simulation results reveal an alignment of these chains with minimal curvature lines of the surface and indicate a dependency of this turbulence regime on the sign and the gradient in local Gaussian curvature. While these results remain qualitative and their explanations are still incomplete, several of the observed phenomena are in qualitative agreement with experiments on active nematic liquid crystals on toroidal surfaces and contribute to an understanding of the delicate interplay between geometrical properties of the surface and characteristics of the flow field, which has the potential to control active flows on surfaces via gradients in the spatial curvature of the surface.