A numerical method for the simulation of viscoelastic fluid surfaces

TL;DR

This paper introduces a novel numerical method for simulating deforming viscoelastic surfaces, capturing elastic to viscous behavior, and demonstrates its effectiveness through vesicle flow simulations and potential cytokinesis applications.

Contribution

We developed the first numerical approach for deforming viscoelastic surfaces based on the upper convected Maxwell model and integrated it with fluid dynamics using Finite Elements and ALE methods.

Findings

Accurate simulation of vesicle tumbling and tank-treading in shear flow.

Phase diagram illustrating the effect of viscoelastic parameters.

Potential application to simulate cytokinesis starting processes.

Abstract

Viscoelastic surface rheology plays an important role in multiphase systems. A typical example is the actin cortex which surrounds most animal cells. It shows elastic properties for short time scales and behaves viscous for longer time scales. Hence, realistic simulations of cell shape dynamics require a model capturing the entire elastic to viscous spectrum. However, currently there are no numerical methods to simulate deforming viscoelastic surfaces. Thus models for the cell cortex, or other viscoelastic surfaces, are usually based on assumptions or simplifications which limit their applicability. In this paper we develop a first numerical approach for simulation of deforming viscoelastic surfaces. To this end, we derive the surface equivalent of the upper convected Maxwell model using the GENERIC formulation of nonequilibrium thermodynamics. The model distinguishes between shear dynamics and dilatational surface dynamics. The viscoelastic surface is embedded in a viscous fluid modelled by the Navier-Stokes equation. Both systems are solved using Finite Elements. The fluid and surface are combined using an Arbitrary Lagrange-Eulerian (ALE) Method that conserves the surface grid spacing during rotations and translations of the surface. We verify this numerical implementation against analytic solutions and find good agreement. To demonstrate its potential we simulate the experimentally observed tumbling and tank-treading of vesicles in shear flow. We also supply a phase-diagram to demonstrate the influence of the viscoelastic parameters on the behaviour of a vesicle in shear flow. Finally, we explore cytokinesis as a future application of the numerical method by simulating the start of cytokinesis using a spatially dependent function for the surface tension.