Active nematic fluids on Riemannian 2-manifolds

TL;DR

This paper develops a geometric, gauge-invariant computational framework to study active nematic fluids on arbitrary curved surfaces, revealing how geometry and topology influence their dynamics.

Contribution

It introduces a novel surface-based, gauge-invariant discretization method for active nematic fluids that accounts for curvature and topology effects.

Findings

Efficient simulation of active nematic flows on curved surfaces.

Stable numerical scheme preserving geometric properties.

Framework generalizable to active k-atic systems.

Abstract

Recent advances in cell biology and experimental techniques using reconstituted cell extracts have generated significant interest in understanding how geometry and topology influence active fluid dynamics. In this work, we present a comprehensive continuous theory and computational method to explore the dynamics of active nematic fluids on arbitrary surfaces without topological constraints. The fluid velocity and nematic order parameter are represented as the sections of the complex line bundle of a 2-manifold. We introduce the Levi-Civita connection and surface curvature form within the framework of complex line bundles. By adopting this geometric approach, we introduce a gauge-invariant discretization method that preserves the continuous local-to-global theorems in differential geometry. We establish a nematic Laplacian on complex functions that can accommodate fractional topological charges through the covariant derivative on the complex nematic representation. We formulate advection of the nematic field based on a unifying definition of the Lie derivative, resulting in a stable geometric semi-Lagrangian discretization scheme for transport by the flow. In general, the proposed surface-based method offers an efficient and stable means to investigate the influence of local curvature and global topology on the 2D hydrodynamics of active nematic systems. Moreover, the complex line representation of the nematic field and the unifying Lie advection present a systematic approach for generalizing our method to active $k$-atic systems.