### A model for dip-coating of a two liquid mixture.

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We consider a free boundary problem of a two-dimensional Navier-Stokes shear flow. There exist a unique global in time solution of the considered problem as well as the global attractor for the associated semigroup. As in [1] and [2], we estimate from above the dimension of the attractor in terms of given data and the geometry of the domain of the flow. This research is motivated by a free boundary problem from lubrication theory where the domain of the flow is usually very thin and the roughness...

Dans ce papier, nous étudions un problème de contrôle par les coefficients issu de la lubrification élastohydrodynamique. La variable de contrôle est l’épaisseur du fluide. Le phénomène de cavitation est pris en compte par le modèle Elrod-Adams, connu pour ses performances dans la conservation des débits d’entrée et de sortie. L’idée est de régulariser dans l’équation d’état le graphe d’Heaviside, en l’approchant par une suite de fonctions monotones et régulières. Nous dérivons les conditions d’optimalité...

The purpose of this paper is to study a control by coefficients problem issued from the elastohydrodynamic lubrication. The control variable is the film thickness.The cavitation phenomenon takes place and described by the Elrod-Adams model, suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The idea is to use the penalization in the state equation by approximating the Heaviside graph whith a sequence of monotone and regular functions....

The impacts of the two-beam interference heating on the number of core-shell and embedded nanoparticles and on nanostructure coarsening are studied numerically based on the non-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20 nm) metallic bilayers. The model incorporates thermocapillary forces and disjoining pressures, and assumes dewetting from the optically transparent substrate atop of the reflective support layer,...

In this paper, using the asymptotic expansion, we prove that the Reynolds lubrication equation is an approximation of the full Navier–Stokes equations in thin gap between two coaxial cylinders in relative motion. Boundary layer correctors are computed. The error estimate in terms of domain thickness for the asymptotic expansion is given. The corrector for classical Reynolds approximation is computed.

A small vicinity of a contact line, with well-defined (micro)scales (henceforth the “microstructure”), is studied theoretically for a system of a perfectly wetting liquid, its pure vapor and a superheated flat substrate. At one end, the microstructure terminates in a non-evaporating microfilm owing to the disjoining-pressure-induced Kelvin effect. At the other end, for motionless contact lines, it terminates in a constant film slope (apparent contact...

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy...

We consider the identification of a distributed parameter in an elliptic variational inequality. On the basis of an optimal control problem formulation, the application of a primal-dual penalization technique enables us to prove the existence of multipliers giving a first order characterization of the optimal solution. Concerning the parameter we consider different regularity requirements. For the numerical realization we utilize a complementarity function, which allows us to rewrite the optimality...

The aim of this work is to deduce the existence of solution of a coupled problem arising in elastohydrodynamic lubrication. The lubricant pressure and concentration are modelled by Reynolds equation, jointly with the free-boundary Elrod-Adams model in order to take into account cavitation phenomena. The bearing deformation is solution of Koiter model for thin shells. The existence of solution to the variational problem presents some difficulties: the coupled character of the equations, the nonlinear...