Existence of solutions for elastohydrodynamic piezoviscous lubrication problems with a new model of cavitation

Abstract

The purpose of this paper is to study a mathematical model of lubricating flow between elastic surfaces obeying the linear Hertzian theory when cavitation takes place. Cavitation is a free boundary phenomenon that is described in this paper by the New Elrod–Adams model. This model introduces the concentration of fluid as well as the pressure as unknown functions and is suggested in preference to the classical variational inequality due to its ability to describe inflow and outflow. This leads to a nonlinear variational and nonlocal equation. Herein, an existence theorem is proved by means of two different techniques.

Existence and uniqueness for several non-linear elliptic problems arising in lubrication theory

“…L ∞ -estimates (see [19]) are then used to show that the resulting fixed point solves the original problem. The technique is similar to that used in elastohydrodynamic problems by Bayada et al [1]. For Models 1, 2 and 5 we then show under some additional regularity assumptions on the coefficients that the solution is bounded from below by a positive constant.…”

Some aspects of lubrication in heavy regimes: Thermal effects, stability and turbulence

“…Moreover, U represents the free boundary between the lubricated region (£2 + ) and the cavitated one (X2°), n a normal vector to S, i the unitary normal vector pointing to 0-direction, A) the boundary where fluid is supplied through (located at 6 = 0), and t F the final time. The mathematical model (2)-(6) has been widely and thoroughly analyzed by Bayada et al Several results of existence and uniqueness of solution to stationary and evolutionary problems, for different boundary conditions, the asymptotic derivation of this model and some of its variants, along with many other properties related to the referred model, can be seen in [1][2][3][4] (just to mention some references) and references therein (precisely, the authors thank professor Bayada for his great scientific advice throughout these years).…”

“…For numerical purposes, the solution of the free boundary problem associated to the Elrod-Adams cavitation model (see ( [1,2], for details) is obtained by means of a duality method applied to a maximal monotone operator, and a finite element spatial discretization ( [3,6], and references therein). The energy equation in the lubricant film is solved with first order finite element schemes due to the simple upwinding used for convective terms.…”

An elastohydrodynamic coupled problem between a piezoviscous Reynolds equation and a hinged plate model

“…When the viscosity Barus law is used another non-linearity appears in the hydrodynamic problem and a generalized version of the mentioned resuit is not trivial. Recently, in [8] it has been demonstrated the existence of solution for a similar problem but replacing the elastic part, équations (2.8), (2.9), by an hertzian contact law. It is also in the hertzian contact domain that, in [20] and [18] it has been proved the existence of solution for elastohydrodynamic piezoviscous problems with mixed Dirichlet-Neumann boundary conditions.…”