I was invited by Mahadevan’s group to give a talk on Active Nematic Fluid on Curved Evolving Surfaces at Harvard Univerisity (Sep. 2024).

Abstract:

Membrane shape dynamics are crucial for understanding biological processes such as morphogenesis. We consider the models for evolving active nematic fluid films on complex geometries, a key feature of biological surfaces. The behavior of active matter on these surfaces is strongly influenced by their curvature and topology. While the governing differential equations are well-known in flat spaces, developing algorithms to solve these equations on surfaces with arbitrary geometry and topology remains challenging. In this talk, I will present a unifying variational and geometric framework that provides simple, robust, and structure-preserving discretization to solve these equations on general manifolds. We derive the discrete analogue of the evolving Stokes equations using a generalized Killing operator and the Onsager variational principle, uncovering the equations’ underlying structure and leading to a stable variational integrator. For nematodynamics, we introduce a nematic Laplacian on the complex line bundle to model nematic relaxation and use the Lie derivative to describe nematic advection. This geometric perspective leads to simple, unifying discretizations that are applicable to arbitrary curved, evolving surfaces.

References:

  1. Cuncheng Zhu, David Saintillan, & Albert Chern. (2024). Stokes flow of an evolving fluid film with arbitrary shape and topology. Arxiv.
  2. Cuncheng Zhu, David Saintillan, & Albert Chern. (2024). Active nematic fluids on Riemannian 2-manifolds. Arxiv.