I will be presenting a talk in the geometric mechanics section at the 16th World Congress on Computational Mechanics (WCCM)/4th Pan American Congress on Computational Mechanics (PANACM) in summer 2024.

This is a work with Prof. Albert Chern and Prof. David Saintillan.

For a sneak peak into the work, here is the abstract of the talk:

Viscous flow of evolving film with arbitrary shape and topology

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modeling of lipid bilayers in cells. While the governing equations were formulated by Scriven in 1960, solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenging task. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. The contribution is two-fold. First, we replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. This allows us to gain a fundamental understanding of the underlying mechanics. For a general embedded surface, the velocity field is a section of the pullback bundle from the tangent bundle of the ambient space via the embedding. Through the pullback connection, the strain rate tensor is derived from the derivative of the velocity field. This coodinate-free abstraction directly leads to a discretization for the strain rate tensor on discrete meshes. Second, we construct a discrete analogue of the system using the Onsager variational principle, which, in a smooth context, governs the flow of a viscous medium. According to this principle, the velocity of the viscous film minimizes a dissipation measure known as the Rayleighian, which is a function of the strain rate tensor. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equation via the variational principle. This approach results in a standard linear saddle-point problem that can be efficiently solved using the augmented Lagrangian method.