Vol. 7, No. 2, 2019

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

Sergey Gavrilyuk and Henri Gouin

Vol. 7 (2019), No. 2, 131–157
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. Rosso and Virga (Proc. Roy. Soc. Lond. A 455:1992 (1999), 4145–4168) considered the dependence on the mean and Gaussian curvatures in statics and under a restrictive assumption of the membrane inextensibility. 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 without the membrane inextensibility assumption. We obtain the dynamic “shape equation” (equation for the membrane surface) and the dynamic conditions on the contact line generalizing the classical Young–Dupré condition.

Keywords
surface energy, curvature tensor, dynamic boundary conditions
Physics and Astronomy Classification Scheme 2010
Primary: 45.20.dg, 68.03.Cd, 68.35.Gy, 02.30.Xx
Mathematical Subject Classification 2010
Primary: 74K15, 76Z99
Secondary: 92C37
Milestones
Received: 12 October 2018
Revised: 20 February 2019
Accepted: 2 April 2019
Published: 27 May 2019

Communicated by Pierre Seppecher
Authors
Sergey Gavrilyuk
Aix Marseille Univ
CNRS, IUSTI, UMR 7343
Marseille
France
Henri Gouin
Aix Marseille Univ
CNRS, IUSTI, UMR 7343
Marseille
France