Modeling and Simulation of Open Membranes in Stokes Flow with Mixed-Dimensional Coupling

TL;DR

The paper addresses the coupled dynamics of open lipid membranes in Stokes flow by formulating a mixed-dimensional PDE system that couples 3D bulk fluid, 2D membrane elasticity, and 1D edge line tension. It develops an axisymmetric boundary-integral reduction and a hybrid boundary-element/finite-element method (BEM-FEM) to solve the effectively 1D problem, augmented by a local mesh-refinement strategy to resolve edge singularities. The membrane energy combines Helfrich curvature elasticity with edge line tension, while inextensibility and incompressibility are enforced via Lagrange multipliers, and the dynamics are governed by a balance of bulk/membrane dissipation and interfacial forces. Numerical experiments demonstrate accurate edge dynamics, boundary-layer formation near the open edge, and multiscale fluid–membrane coupling, including comparisons with reduced spherical-cap models. This framework provides a first fully coupled, multiscale tool for open membranes in viscous flows, with potential extensions to 3D, area compressibility, and applications to vesicle healing and synthetic membrane systems.

Abstract

In this work, we present a mathematical and computational framework to model the dynamics of open lipid bilayer membranes interacting with ambient Stokes flow. The model explicitly couples the three-dimensional viscous fluid, the two-dimensional membrane surface, and its one-dimensional free edge. We develop an axisymmetric hybrid BEM-FEM method that solves the problem with an effective one-dimensional formulation. A key component is a local mesh refinement strategy designed to accurately resolve singularities and boundary layers originating at the membrane edge. Several numerical examples are provided to showcase its ability to capture intricate edge dynamics and multiscale fluid-membrane coupling.

Figures (14)

• Figure 1: Illustration for an open membrane surface $\Gamma$ with boundary $\partial\Gamma$. $\bm{n}$ is the unit normal of $\Gamma$, $\bm{e}$ is the tangent and $\bm{\nu}$ is the co-normal along the edge $\partial\Gamma$.
• Figure 1: Left: numerical error of the radial velocity for different mesh size using both uniform mesh and locally refined mesh; Right: radial single-layer density $\xi^r$ computed using different regularizing parameters $\varepsilon = 1, 0.5, 0.001$ with $N=32$.
• Figure 2: Numerical solution of $F$, normalized by $\gamma/\mu$, for different membrane widths $R_{o,0}-R_{i,0}=16,\,32,\,64,\,128$, compared with the analytical solution for $R_{o,0}=\infty$.
• Figure 3: Comparison of numerical results (solid) and analytical solutions (dashed) for the value $F$, normalized by $\gamma/\mu$, and the dimensionless hole area $A/(\pi R_{i,0}^2)$ over time, for viscosity ratios $\mu_{\Gamma}/(\mu R_{i,0}) = 0,\,0.1,\,0.3,\,0.5$.
• Figure 4: Time evolution of the total energy of the membrane for the three cases.