On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication

We study an evolution free boundary problem issued from hydrodynamic lubrication. The cavitation phenomenon takes place and is described by the Elrod-Adams model. This model is suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The considered model is governed by the parabolic Reynolds equation and the unknows are the pressure of the lubricant contained in the narrow gap between two circular cylinders and its percentage in an elementary volume. The main result of this paper is the uniqueness of the solution for the parabolic free boundary problem. The proof of this result is based on a comparison principle permitting compare two solutions of the problem when their initial values and their values on the boundary can be compared. Previously, we state a continuity result and some monotonicity properties of the solution

Micro-Roughness Effects in (Elasto)Hydrodynamic Lubrication Including a Mass-Flow Preserving Cavitation Model

27 pagesAn average Reynolds equation is proposed for predicting the effects of deterministic periodic roughness, taking JFO mass flow preserving cavitation model and elastohydrodynamic effects into account. For this, the asymptotic model is based upon double scale analysis approach. The average Reynolds equation can be used both for microscopic interasperity cavitation and macroscopic one. The validity of such a model is verified by numerical experiments

Viscoelastic fluids in thin domains: a mathematical proof

The present paper deals with non Newtonian viscoelastic flows of Oldroyd-B tye 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 sytem towards the asymptotic model

Homogenization of a nonlocal elastohydrodynamic lubrication problem: a new free boundary model

36 pagesThe present paper deals with the homogenization of a lubrication problem, using two-scale convergence techniques and periodic unfolding methods. 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

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

25 pagesAn average Reynolds equation for predicting the effects of deterministic periodic roughness, taking JFO mass flow preserving cavitation model into account, is introduced based upon 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

About a generalized Buckley-Leverett equation and lubrication multifluid flow

30 pagesIn 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

From the Phan-Thien Tanner / Oldroyd-B non-Newtonian model to the double shear thining Rabinowisch thin film model

International audienceIn this paper, an asymptotic expansion is used to derive a description of Phan-Tien Tanner~/ Oldroyd-B flows in the thin film situation without the classical "Upper Convective Maxwell'' assumption. We begin with a short presentation of the Phan-Thien Tanner~/ Oldroyd-B models which introduce viscoelastic effects in a solute-solvant mixture. The three dimensional flow is described using five parameters, namely the Deborah number (or the relaxation parameter), the viscosity ratio, the bulk fluid viscosity, the material slip parameter related to the "convected derivative'' and an elongation number. Then we focus on the thin film assumption and the related asymptotic analysis that allows us to derive a reduced model. A perturbation procedure for "not too small'' values of the elongation number allows us to obtain further results such as an asymptotic "effective viscosity / shear rate'' law which appears to be a perturbation of the double Rabinowisch model whose parameters are completely defined by those of the original three dimensional model. And last a numerical procedure is proposed based on a penalized Uzawa method, to compute the corresponding solution. This algorithm can also be used for any generalized double Newtonian shear thinning Carreau law

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

An unconditional existence result for the quasi-variational elastohydrodynamic free boundary value problem

AbstractIn this paper a two-dimensional quasi-variational inequality arising in elastohydrodynamic lubrication is studied for non-constant viscosity. So far, existence results for such piezo–viscous problems require an L∞ property for an auxiliary problem. For the usual pressure–viscosity relations, this property needs small data assumptions which are not observed in experimental conditions. In the present work, such small data assumptions are proved unnecessary for existence results. Besides well-established monotonicity behavior for the viscosity–pressure relation, the only condition used here is on the asymptotic behavior for this law as the pressure tends to infinity. If the procedure used here, namely the introduction of a reduced pressure by Grubin transform followed by a regularization procedure, appears somewhat classical, the way in which an upper bound is obtained is completely new