Guy Bayada

The transition between the Stokes equations and the Reynolds equation: A mathematical proof

The Reynolds equation is used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. Using the asymptotic expansion in the Stokes equations, we show the existence of singular perturbation phenomena whenever the two surfaces are in relative motion. We prove that the Reynolds equation is an approximation of the Stokes equations and that the kind of convergence is strongly related with the boundary conditions on the velocity field.

An Average Flow Model of the Reynolds Roughness Including a Mass-Flow Preserving Cavitation Model

An average Reynolds equation for predicting the effects of deterministic periodic roughness, taking Jakobsson, Floberg, and Olsson mass flow preserving cavitation model into account, is introduced based upon the double scale analysis approach. This average Reynolds equation can be used both for a microscopic interasperity cavitation and a macroscopic one. The validity of such a model is verified by numerical experiments both for one-dimensional and two-dimensional roughness patterns.

On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions

The asymptotic behaviour of a Stokes flow with Tresca free boundary friction conditions when one dimension of the fluid domain tends to zero is studied. A specific Reynolds equation associated with variational inequalities is obtained and uniqueness is proved.

Inertial effects in the asymptotic behavior of a thin film flow

Assemien, A., Bayada, G. and M. Chambat, Inertial effects in the asymptotic behavior of a thin film flow, Asymptotic Analysis 9 (1994) 177-20S. We study the dependence with respect to the domain of the constants appearing in some Sobolev embeddings. It shows that the inertial effects are negligible at the first order for the asymptotic behavior of a thin film flow. The influence of the inertial effects is obtained at the second order in the asymptotic expansions of the pressure and velocity, with full convergence argument for chosen boundary conditions.

A Neumann-Neumanndomain decomposition algorithm for the Signorini problem

Abstract In this paper, we propose a Neumann-Neumann algorithm to approximate a frictionless static Signorini contact problem between two elastic bodies and we prove its convergence. The Neumann-Neumann algorithm is a parallel one, in which we have to solve a Dirichlet problem and then a Neumann one, simultaneously on each domain. The primary feature of this new algorithm is the retention the natural interface between the two bodies as a numerical interface for the domain decomposition.

Viscoelastic fluids in a thin domain

The present paper deals with viscoelastic flows in a thin domain. In particular, we derive and analyse the asymptotic equations of the Stokes-Oldroyd system in thin films (including shear effects). We present a numerical method which solves the corresponding problem and present some related numerical tests which evidence the effects of the elastic contribution on the flow.

HOMOGENIZATION OF A NONLOCAL ELASTOHYDRODYNAMIC LUBRICATION PROBLEM: A NEW FREE BOUNDARY MODEL

This paper deals with the homogenization of a lubrication problem, using two-scale convergence techniques. We study in particular the Reynolds–Hertz model, which takes into account elastohydrodynamic deformations of the upper surface, when highly oscillating roughness effects occur. The difficulty arises when considering cavitation free boundary phenomena, leading to highly nonlinear and nonlocal problems.

Viscoelastic fluids in thin domains: A mathematical proof

The present paper deals with non-Newtonian viscoelastic flows of Oldroyd-B type in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier-Stokes/Oldroyd-B system towards the asymptotic model.

About a generalized Buckley-Leverett equation and lubrication multifluid flow

In this paper, we analyse the asymptotic system corresponding to a thin film flow with two different fluids, from theoretical and numerical point of view. We also compare this model to the Elrod-Adams one, which allows to consider cavitation phenomena in lubrication theory.

TWO-SCALE HOMOGENIZATION STUDY OF A REYNOLDS-ROD ELASTOHYDRODYNAMIC MODEL

In this paper, the existence of solution for a coupled system of variational inequalities with highly oscillating coefficients modelling a micro-elastohydrodynamic lubrication problem is stated. Moreover, by means of two-scale convergence techniques, the limit coupled models when the frequency of oscillation tends to zero are established for different amplitude-frequency rates. The effect of roughness appears to be limited to one of the coupled nonlinear equations, either the elastic one or the hydrodynamic one. Finally, some numerical examples to illustrate both the qualitative behavior of the solution and the theoretical convergence results are presented.