In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds partial differential equation (PDE) (that describes the flow) and the Rayleigh–Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.

In this paper we prove first the existence and uniqueness results for the weak solution, to the stationary equations for Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary condition; then we study the asymptotic analysis when one dimension of the fluid domain tend to zero. The strong convergence of the velocity is proved, a specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.

The growth of the size of cache and the number of processor cores in modern CPUs is the major factor in advancing the computing performance of modern machines. The effect of CPU cache size in multicore computers on performance, however, has attracted little attention in lubrication and engineering analyses. In this study, the effect of cache size on the computational performance of two parallel iterative methods in solving two Reynolds equations is examined. Four computers, with CPU cache size from 4 to 40 MB and the number of processor cores from 4 to 16, were used. The sizes of the numerical grid were selected to simulate large gridwork (256 × 256) to small gridwork (2048 × 2048) tasks. It is found that the size of CPU cache is a major factor influencing the parallel efficiency in using the RBSOR method. On the other hand, the SPSOR method obtains much higher parallel efficiency than the RBSOR for medium-grained tasks, regardless of the size of CPU cache. The use of the SPSOR can, therefore, provide a much better parallel computing performance than the RBSOR in the cases of having a large number of grids or in a system with limited CPU cache.

In this paper, a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier–Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids; thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model verifies a dissipative entropy inequality up to a second-order term. Moreover, we propose a correction of the model – by taking into account the second-order extension for the pressure – that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first test, we compare the numerical results with the viscous bilayer shallow water model proposed in Narbona-Reina et al. (Comput. Model. Eng. Sci., 2009, Vol. 43, pp. 27–71). In the second test, the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test, the influence of the friction coefficient on the coastal area affected by the pollutant is studied.

In an air foil bearing analysis the model is usually solved iteratively due in part to the nonlinearity of the modeling Reynolds equation and the compliance of the bearing surface. The solution procedure requires a multiple-level-deep nested iteration, which involves extended solution time and convergence difficulty. In this study, a simple air foil bearing model is used and the compressible-fluid Reynolds equation for modeling gas lubrication is linearized by Newton's method. The discretized equation is solved by one of the two parallel iterative methods, red-black or strip partition successive-over-relaxation (SOR) method. The parallel programming is conducted using OpenMP programming in an eight-core work-station. Then, a numerical damping scheme for the film-profile convergence is presented. Finally, a root-finding process is conducted to iteratively attain the eccentricity of the bearing for a given load. It is found that the numerical damping step is crucial, which allows the use of a larger relaxation factor to have a fast rate of convergence. Both the parallel SOR methods are easy to implement and the red-black SOR method exhibits better efficiency in the studied cases. This study presents a parallel computing scheme for analyzing air foil bearing of bump-type by today's shared-memory multicore platforms.

Ramp load/unload (L/UL) mechanisms are widely used to rest sliders in hard disk drives (HDDs). Loading/unloading a slider swiftly and smoothly is crucial in a HDD design. A novel, efficient simulation scheme is proposed to investigate the behaviors of a head disk interface (HDI) in ramp unloading processes. A dual scale model is enabled by decoupling the nano-meter scale change of an air bearing and the micro- or milli-meter scale deformation of a suspension. A modified Reynolds equation governing the air bearing was solved numerically. The slider design was characterized with performance functions. Three stages in an unloading process were analyzed with a lumped parameter suspension model. Key parameters for the model were estimated with a comprehensive finite element suspension model. Finally, simulation results are presented for a commercial HDI design.

We describe possible solutions for a stationary flow of two superposed fluids between two close surfaces in relative motion. Physically, this study is within the lubrication framework, in which it is of interest to predict the relative positions of the lubricant and the air in the device. Mathematically, we observe that this problem corresponds to finding the interface between the two fluids, and we prove that this interface can be viewed as a square root of a polynomial of degree at most 6. We solve this equation using an original method. First, we check that our results are consistent with previous work. Next, we use this solution to answer some physically relevant questions related to the lubrication setting. For instance, we obtain theoretical and numerical results, which can predict the occurrence of a full film with respect to physical parameters (fluxes, shear velocity, viscosities). In particular, we present a figure giving the number of stationary solutions depending on the physical parameters. Moreover, we give some indications for a better understanding of the multi-fluid case.

L' étude d'un joint à lèvre, basée sur une analyse élastohydrodynamique, s'appuie sur un modèle original très proche de la réalité. Elle permet de prédire la plupart des caractéristiques de fonctionnement d'un joint telles que l'épaisseur du film lubrifiant, le couple de frottement, les zones de contact et de cavitation, la pression hydrodynamique ainsi que le débit de pompage inverse.

The aim of this work is to deduce the existence of solutionof a coupled problem arising in elastohydrodynamiclubrication. The lubricant pressure and concentration aremodelled by Reynolds equation, jointly with the free-boundaryElrod-Adams model in order to take into account cavitationphenomena. The bearing deformation is solution of Koitermodel for thin shells. The existence of solution to thevariational problem presents some difficulties: the coupledcharacter of the equations, the nonlinear multivaluedoperator associated to cavitation and the fact of writing theelastic and hydrodynamic equations on two different domains.In a first step, we regularize the Heaviside operator.Additional difficulty related to the differentdomains is circumvented by means of prolongation andrestriction operators, arriving to a regularized coupledproblem. This one is decoupled into elastic and hydrodynamicparts, and we prove the existence of a fixed point for theglobal operator. Estimations obtained for theregularized problem allow us to prove the existence ofsolution to the original one. Finally, a numerical method is proposed in orderto simulate a real journal-bearing device and illustrate the qualitative andquantitative properties of the solution.