89 research outputs found
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
Hyperbolic Models of Homogeneous Two-Fluid Mixtures
One derives the governing equations and the Rankine - Hugoniot conditions for
a mixture of two miscible fluids using an extended form of Hamilton's principle
of least action. The Lagrangian is constructed as the difference between the
kinetic energy and a potential depending on the relative velocity of
components. To obtain the governing equations and the jump conditions one uses
two reference frames related with the Lagrangian coordinates of each component.
Under some hypotheses on flow properties one proves the hyperbolicity of the
governing system for small relative velocity of phases.Comment: 14 page
Bubble effect on Kelvin-Helmholtz' instability
We derive boundary conditions at interfaces (contact discontinuities) for a
class of Lagrangian models describing, in particular, bubbly flows. We use
these conditions to study Kelvin-Helmholtz' instability which develops in the
flow of two superposed layers of a pure incompressible fluid and a fluid
containing gas bubbles, co-flowing with different velocities. We show that the
presence of bubbles in one layer stabilizes the flow in some intervals of wave
lengths.Comment: 19 page
Geometric evolution of the Reynolds stress tensor
14 pagesInternational audienceThe dynamics of the Reynolds stress tensor for turbulent flows is described with an evolution equation coupling both geometric effects and turbulent source terms. The effects of the mean flow geometry are shown up when the source terms are neglected: the Reynolds stress tensor is then expressed as the sum of three tensor products of vector fields which are governed by a distorted gyroscopic equation. Along the mean flow trajectories, the fluctuations of velocity are described by differential equations whose coefficients depend only on the mean flow deformation. If the mean flow vorticity is small enough, an approximate turbulence model is derived, and its application to shear shallow water flows is proposed. Moreover, the approximate turbulence model admits a variational formulation which is similar to the one of capillary fluids
Dynamics of shock waves in elastic-plastic solids
Submitted in ESAIM ProcedingsThe Maxwell type elastic-plastic solids are characterized by decaying the absolute values of the principal components of the deviatoric part of the stress tensor during the plastic relaxation step. We propose a mathematical formulation of such a model which is compatible with the von Mises criterion of plasticity. Numerical examples show the ability of the model to deal with complex physical phenomena
Dissipative Two-Fluid Models
dedicated to Guy Boillat : 13 pagesInternational audienceFrom Hamilton's principle of stationary action, we derive governing equations of two-fluid mixtures and extend the model to the dissipative case without chemical reactions. For both conservative and dissipative cases, an algebraic identity connecting equations of momentum, mass, energy and entropy is obtained by extending the Gibbs identity in dynamics. The obtained system is hyperbolic for small relative velocity of the phases
Ecoulements turbulents des eaux peu profondes
Les équations de Saint-Venant des eaux peu
profondes sans effet de tourbillon sont analogues aux équations de la dynamique des gaz
avec une loi d'état polytropique. Lorsque le tourbillon est pris en compte, on en déduit
des équations qui sont analogues aux écoulements turbulents des gaz avec une structure
simplifiée des équations du mouvement. Cette structure correspond aux équations avec un
tenseur des contraintes de Reynolds sans termes sources. Les équations d'évolution
prennent en compte seulement les effets géométriques. Une structure mathématique simple
de ces équations nous permet ainsi "l'intégration " des équations de Reynolds. Des lois
de conservation non-usuelles nécessaires pour l'étude des sauts hydrauliques turbulents
sont aussi déterminée
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