Abstract

We consider a numerical approach for a covariant generalized Navier–Stokes equation on general surfaces and study the influence of varying Gaussian curvature on anomalous vortex-network active turbulence. This regime is characterized 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.

Highlights

  • To model fluids on curved surfaces is a problem which dates back to Scriven,[1] who derived a covariant Navier–Stokes (NS) equation and established the coupling between spatial curvature and fluid flow

  • As these vector fields cannot be described by the vorticity-stream function formulation, the approach is only applicable for surfaces, where harmonic vector fields are trivial, which are only connected surfaces.[18]

  • In addition to a few microscopic models for active nematics, which do not account for hydrodynamic effects,[41–43] continuum models, with numerics based on the vorticity-stream function formulation[13,14] and the generally applicable approach for polar liquid crystals[25] as well as first experimental work on active nematic liquid crystals comprised of microtubules and kinesin, which are constrained to lie on a toroidal surface,[43] no results are available

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Summary

INTRODUCTION

To model fluids on curved surfaces is a problem which dates back to Scriven,[1] who derived a covariant Navier–Stokes (NS) equation and established the coupling between spatial curvature and fluid flow. We will here consider a modeling approach for active flows and extend studies for a generalized Navier–Stokes (GNS) equation on a sphere[30] to toroidal surfaces. This GNS equation describes internally driven flows through higher-order hyperviscosity-like terms in the stress tensor. In addition to a few microscopic models for active nematics, which do not account for hydrodynamic effects,[41–43] continuum models, with numerics based on the vorticity-stream function formulation[13,14] and the generally applicable approach for polar liquid crystals[25] as well as first experimental work on active nematic liquid crystals comprised of microtubules and kinesin, which are constrained to lie on a toroidal surface,[43] no results are available.

MATHEMATICAL MODEL
NUMERICAL APPROACH
Reformulation
Time discretization
Space discretization
Passive flows on torus
Active flows on sphere
RESULTS
Active flows on toroidal surface
CONCLUSIONS
Methods
Full Text
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