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Dynamic boundary conditions for membranes whose surface energy depends on the mean and Gaussian curvatures
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.Comment: Mathematics and Mechanics of Complex Systems, mdp, In pres
Hamilton's Principle and Rankine-Hugoniot Conditions for General Motions of Mixtures
In previous papers, we have presented hyperbolic governing equations and jump
conditions for barotropic fluid mixtures. Now we extend our results to the most
general case of two-fluid conservative mixtures taking into account the
entropies of components. We obtain governing equations for each component of
the medium. This is not a system of conservation laws. Nevertheless, using
Hamilton's principle we are able to obtain a complete set of Rankine-Hugoniot
conditions. In particular, for the gas dynamics they coincide with classical
jump conditions of conservation of momentum and energy. For the two-fluid case,
the jump relations do not involve the conservation of the total momentum and
the total energy.Comment: Extended version of meccanica 34: 39-47, 199
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