Active nematic fluids on Riemannian 2-manifolds

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.

细胞生物学的最新进展以及使用重组细胞提取物的实验技术，使得人们对理解几何形状和拓扑结构如何影响活性流体动力学产生了极大的兴趣。在这项工作中，我们提出了一种全面的连续理论和计算方法，用于探索无拓扑约束的任意表面上的活性向列流体的动力学。流体速度和向列序参量被表示为二维流形的复线丛的截面。我们在复线丛的框架内引入了列维 - 奇维塔联络和曲面曲率形式。通过采用这种几何方法，我们引入了一种规范不变的离散化方法，该方法保留了微分几何中的连续局部 - 全局定理。我们在复函数上建立了一个向列拉普拉斯算子，它可以通过复向列表示上的协变导数容纳分数拓扑荷。我们基于李导数的统一定义制定了向列场的平流，从而得到了一种用于流输运的稳定的几何半拉格朗日离散化方案。一般来说，所提出的基于表面的方法为研究局部曲率和全局拓扑对活性向列系统的二维流体动力学的影响提供了一种高效且稳定的手段。此外，向列场的复线表示和统一的李平流为将我们的方法推广到活性$k$-atic系统提供了一种系统的方法。