Energy Variational Modeling and Numerical Simulation of Open Membranes in Stokes Flow

Lipid bilayer membranes are fundamental biological structures that serve as cellular boundaries, mediating transport, signaling, and maintaining structural integrity. This study introduces a novel mathematical model for open membranes immersed in Stokes flows, accounting for membrane elasticity, line tension at the open edge, and fluid-membrane interactions. The model is derived from an energy functional that incorporates Helfrich bending energy and a line energy associated with the open edge. By balancing dissipation in both the bulk fluid and the membrane surface, following the maximal dissipation principle, we derive the governing equations within an energy variational framework. Assuming axisymmetry and employing a boundary integral reduction, we transform the 3D problem into an effectively 1D problem, for which we develop a finite element-based numerical method to solve the resulting moving boundary problem. Several numerical examples are provided to validate the model and compare the results with existing studies.