Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biological membranes

In this paper we derive thermodynamically consistent higher order phase �eld models for the dynamics of vesicle membranes in incompressible viscous uids. We start with basic conservation laws and an appropriate version of the second law of thermodynamics and obtain generalizations of models introduced by Du, Li and Liu [5] and Jamet and Misbah [13]. In particular we derive a stress tensor involving higher order derivatives of the phase �eld and generalize the classical Korteweg capillarity tensor