On equilibrium states of fluid membranes
arXiv:2303.16882 · doi:10.1063/5.0152423
Abstract
The paper studies the equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Using a continuum description of the membrane's motions based on the surface Navier--Stokes equations with bending forces, the paper derives differential equations governing the mechanical equilibrium. The equilibrium conditions are found to be independent of lateral viscosity and relate tension, pressure, and tangential velocity of the fluid. These conditions suggest that either the lateral fluid motion ceases or non-decaying stationary flow of mass can only be supported by surfaces with Killing vector fields, such as axisymmetric shapes. A shape equation is derived that extends the classical Helfrich model with an area constraint to membranes of non-negligible mass. Furthermore, the paper suggests a simple numerical method to compute solutions of the shape equation. Numerical experiments conducted reveal a diverse family of equilibrium configurations. The stability of equilibrium states involving lateral flow of mass remains an unresolved question.
References in corpus (8)
- Modelling fluid deformable surfaces with an emphasis on biological interfaces
- Incompressible fluid problems on embedded surfaces: Modeling and variational formulations
- Solving the incompressible surface Navier-Stokes equation by surface finite elements
- Arbitrary Lagrangian--Eulerian finite element method for curved and deforming surfaces. I. General theory and application to fluid interfaces
- Numerical computations of the dynamics of fluidic membranes and vesicles
- The formulation of the Navier-Stokes equations on Riemannian manifolds
- A numerical approach for fluid deformable surfaces with conserved enclosed volume
- A penalty finite element method for a fluid system posed on embedded surface