Thermodynamically consistent picture of the phase-field model of vesicles: Elimination of the surface tension

In two recent papers [D. Jamet and C. Misbah, Phys. Rev. E 76, 051907 (2007); 78, 031902 (2008)], we considered a thermodynamically consistent model for vesicles and membranes, where we dealt, in the first paper, with the membrane local incompressibility condition, while in the second one with the bending energy and the derivation of a constitutive law of the composite fluid: ambient fluid+membrane. This is the last paper of this series and focuses on the elimination of surface tension (inherent in phase-field models), retaining the thermodynamically consistent model. We write down the complete set of equations and the full constitutive law for membranes embedded in a Newtonian fluid.