Mohammed Jai

A Mass-Conserving Algorithm for Dynamical Lubrication Problems With Cavitation

A numerical algorithm for fully dynamical lubrication problems based on the Elrod― Adams formulation of the Reynolds equation with mass-conserving boundary conditions is described. A simple but effective relaxation scheme is used to update the solution maintaining the complementarity conditions on the variables that represent the pressure and fluid fraction. The equations of motion are discretized in time using Newmarks scheme, and the dynamical variables are updated within the same relaxation process just mentioned. The good behavior of the proposed algorithm is illustrated in two examples: an oscillatory squeeze flow (for which the exact solution is available) and a dynamically loaded journal bearing. This article is accompanied by the ready-to-compile source code with the implementation of the proposed algorithm.

The Effect of Periodic Textures on the Static Characteristics of Thrust Bearings

Given a bearing of some specified macroscopic shape, what is the effect of texturing its surfaces uniformly? Experimental and numerical investigations on this question have recently been pursued, which we complement here with a mathematical analysis based on a seemingly novel combination of homogenization techniques and perturbation analysis. The flow is assumed governed by the Reynolds equation, with cavitation effects disregarded, and the texture length is assumed much smaller than the bearings length. The results, which hold true for small-amplitude periodic textures and in the limit of vanishing period, can be summarized as follows: (a) The texture that maximizes the load for a given minimum clearance is no texture at all (i.e., the untextured shape); and (b) the texture that minimizes the friction coefficient is again the untextured shape.

A new numerical scheme for non uniform homogenized problems: Application to the non linear Reynolds compressible equation

A new numerical approach is proposed to alleviate the computational cost of solving non-linear non-uniform homogenized problems. The article details the application of the proposed approach to lubrication problems with roughness effects. The method is based on a two-parameter Taylor expansion of the implicit dependence of the homogenized coefficients on the average pressure and on the local value of the air gap thickness. A fourth-order Taylor expansion provides an approximation that is accurate enough to be used in the global problem solution instead of the exact dependence, without introducing significant errors. In this way, when solving the global problem, the solution of local problems is simply replaced by the evaluation of a polynomial. Moreover, the method leads naturally to Newton-Raphson nonlinear iterations, that further reduce the cost.

Sensitivity analysis and Taylor expansions in numerical homogenization problems

Summary. Two-scale numerical homogenization problems are addressed, with particular application to the modified compressible Reynolds equation with periodic roughness. It is shown how to calculate sensitivities of the homogenized coefficients that come out from local problems. This allows for significant reduction of the computational cost by two means: The construction of accurate Taylor expansions, and the implementation of rapidly convergent nonlinear algorithms (such as Newtons) instead of fixed-point-like ones. Numerical tests are reported showing the quantitative accuracy of low-order Taylor expansions in practical cases, independently of the shape and smoothness of the roughness function.

Optimization tools in the analysis of micro-textured lubricated devices

We address the problem of optimizing the performance of lubricated devices by means of artificial texturing. We consider a slider (or equivalently a thrust bearing) and minimize the friction using optimization tools such as sensitivity analysis and genetic algorithms. We show that textures that perform significantly better than the smooth (untextured) one can be found, and that the optimized texture depends on the working conditions (load, velocity). The GENESIS code that we used, with no fine tuning of algorithmic variables, proved a valuable tool in the identification of improved shapes.

Approximate Transmission Conditions through a rough thin layer. The case of the periodic roughness

We study the behavior of the steady-state voltage potentials in a material composed by a bidimensional medium surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be

Texture-induced cavitation bubbles and friction reduction in the Elrod–Adams model

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Mass-conserving cavitation model for dynamical lubrication problems. Part I: Mathematical analysis

--periodic,

Existence, uniqueness, and homogenization of the second order slip Reynolds equation

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On the Existence of Solutions of Equilibria in Lubricated Journal Bearings

beeing the small thickness of the layer. We build approximate transmission conditions in order to replace the rough thin layer by these conditions on the boundary of the interior material. This paper extends previous works of the third author, in which the layer had constant or weakly oscillating thickness.