# Dynamic boundary conditions for membranes whose surface energy depends on the mean and Gaussian curvatures

## TL;DR

This paper derives dynamic equations and boundary conditions for fluid membranes with surface energy depending on both mean and Gaussian curvatures, extending static models to account for membrane motion.

## Contribution

It introduces a dynamic framework for membrane shape equations and boundary conditions considering curvature-dependent surface energy.

## Key findings

- Derived dynamic shape equation for membranes
- Generalized boundary conditions on contact lines
- Extended classical Young-Dupré condition to dynamic case

## Abstract

Membranes are an important subject of study in physical chemistry and biology. They can be considered as material surfaces with a surface energy depending on the curvature tensor. Usually, mathematical models developed in the literature consider the dependence of surface energy only on mean curvature with an added linear term for Gauss curvature. Therefore, for closed surfaces the Gauss curvature term can be eliminated because of the Gauss-Bonnet theorem. In [18], the dependence on the mean and Gaussian curvatures was considered in statics. The authors derived the shape equation as well as two scalar boundary conditions on the contact line. In this paper-thanks to the principle of virtual working-the equations of motion and boundary conditions governing the fluid membranes subject to general dynamical bending are derived. We obtain the dynamic 'shape equa-tion' (equation for the membrane surface) and the dynamic conditions on the contact line generalizing the classical Young-Dupr{\'e} condition.