Abstract. We undertake the mathematical analysis of a model describing equilibrium binary electrolytes surrounded by charged solid walls. The problem is formulated in terms of the electrostatic potential and the ionic concentrations which have prescribed spatial mean values. The free energy of the system is decomposed as the difference of the internal energy and entropy functionals. The entropy functional is the sum of an ideal entropy and an excess entropy, the latter taking into account nonideality due to electrostatic correlations at low ionic concentrations and steric exclusion effects at high ionic concentrations. We derive sufficient conditions to achieve convexity of the entropy functional, yielding a convex-concave free energy functional. Our main result is the existence and uniqueness of the saddle point of the free energy functional and its characterization as a solution of the original model problem. The proof hinges on positive uniform lower bounds for the ionic concentrations and uniform upper bounds for the ionic concentrations and the electrostatic potential. Some numerical experiments are presented in the case where the excess entropy is evaluated using the Mean Spherical Approximation.
Abstract. We study the mathematical properties of a nonequilibrium Langevin dynamics which can be used to estimate the shear viscosity of a system. More precisely, we prove a linear response result which allows to relate averages over the nonequilibrium stationary state of the system to equilibrium canonical expectations. We then write a local conservation law for the average longitudinal velocity of the fluid, and show how, under some closure approximation, the viscosity can be extracted from this profile. We finally characterize the asymptotic behavior of the velocity profile, in the limit where either the transverse or the longitudinal friction go to infinity. Some numerical illustrations of the theoretical results are also presented.
We present results on the ballistic and diffusive behavior of the Langevin dynamics in a periodic potential that is driven away from equilibrium by a space-time periodic driving force, extending some of the results obtained by Collet and Martinez in (J Math Biol, 56(6):765-792 2008). In the hyperbolic scaling, a nontrivial average velocity can be observed even if the external forcing vanishes in average. More surprisingly, an average velocity in the direction opposite to the forcing may develop at the linear response level-a phenomenon called negative mobility. The diffusive limit of the non-equilibrium Langevin dynamics is also studied using the general methodology of central limit theorems for additive functionals of Markov processes. To apply this methodology, which is based on the study of appropriate Poisson equations, we extend recent results on pointwise estimates of the resolvent of the generator associated with the Langevin dynamics. Our theoretical results are illustrated by numerical simulations of a two-dimensional system.
We consider a multiscale heat problem in civil aviation: determine the temperature field in a plane in flying conditions, with air conditioning. Ventilated electronic components in the bay bring a heat source, introducing a second scale in the problem. First, we present three levels of modelling for the physical phenomena, which are applied to the two sub-problems: the plane and the electronic component. Then, having reduced the complexity of the problem to a linear non-symmetric coercive PDE, we will use the reduced basis method for the electronic component problem. Résumé. Nous considérons un problème multi-échelle d'aérothermie en aviation civile. Nous souhaitons déterminer le champ de température dans un avion en conditions de vol, avec présence d'une climatisation. Des composantsélectroniques ventilés sont présents dans la soute, et constituent une source de chaleur, introduisant une deuxièmeéchelle dans notre problème. Dans un premier temps, nous présentons trois niveaux de modélisation pour le phénomène d'aérothermie, que nous appliquerons aux deux sous-problèmes: l'avion et le composantélectronique. Ensuite, nous appliquons la méthode des bases réduites au problème du composantélectronique, en considérant des simplifications de modélisation amenantà la résolution numérique d'une EDP elliptique linéaire coercive non-symétrique.
Abstract. This short note is an erratum to the article [R. Joubaud and G. Stoltz, Nonequilibrium shear viscosity computations with Langevin dynamics, Multiscale Model. Simul., 10 (2012), pp. 191-216]. We present required modifications in the proofs of Theorem 2 and 3.
To cite this version:Alexandre Ern, Rémi Joubaud, Tony Lelievre. Numerical study of a thin liquid film flowing down an inclined wavy plane. Physica D: Nonlinear Phenomena, Elsevier, 2011, 240 (21) AbstractWe investigate the stability of a thin liquid film flowing down an inclined wavy plane using a direct numerical solver based on a finite element/arbitrary Lagrangian Eulerian approximation of the free-surface Navier-Stokes equations. We study the dependence of the critical Reynolds number for the onset of surface wave instabilities on the inclination angle, the waviness parameter, and the wavelength parameter, focusing in particular on mild inclinations and relatively large waviness so that the bottom does not fall monotonously.In the present parameter range, shorter wavelengths and higher amplitude for the bottom undulation stabilize the flow. The dependence of the critical Reynolds number evaluated with the Nusselt flow rate on the inclination angle is more complex than the classical relation (5/6 times the cotangent of the inclination angle), but this dependence can be recovered if the actual flow rate at critical conditions is used instead.
We investigate numerically a density functional theory (DFT) for strongly confined ionic solutions in the canonical ensemble by comparing predictions of ionic concentration profiles and pressure for the double-layer configuration to those obtained with Monte Carlo (MC) simulations and the simpler Poisson-Boltzmann (PB) approach. The DFT consists of a bulk (ion-ion) and an ion-solid part. The bulk part includes nonideal terms accounting for long-range electrostatic and short-range steric correlations between ions and is evaluated with the mean spherical approximation and the local density approximation. The ion-solid part treats the ion-solid interactions at the mean-field level through the solution of a Poisson problem. The main findings are that ionic concentration profiles are generally better described by PB than by DFT, although DFT captures the nonmonotone co-ion profile missed by PB. Instead, DFT yields more accurate pressure predictions than PB, showing in particular that nonideal effects are important to describe highly confined ionic solutions. Finally, we present a numerical methodology capable of handling nonconvex minimization problems so as to explore DFT predictions when the reduced temperature falls below the critical temperature.
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