Stokes flow of an evolving fluid film with arbitrary shape and topology

TL;DR

This paper introduces a novel coordinate-free, variational discretization method for modeling the Stokes flow of evolving viscous fluid films with arbitrary shapes and topologies, applicable to biological membranes.

Contribution

It develops a discrete, geometric variational framework that simplifies and stabilizes the simulation of complex fluid interfaces with arbitrary topology.

Findings

Provides a stable, structure-preserving variational integrator

Enables modeling of fluid films with complex shapes and topologies

Offers a fundamental geometric understanding of viscous film dynamics

Abstract

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 challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and directly translates to discretization. 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. 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 equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds.