Comptes Rendus
no. 11
On the instantaneous formation of cavitation in hydrodynamic lubrication
[Sur la formation instantanée de la cavitation en lubrification hydrodynamique]

Présenté par : Évariste Sanchez-Palencia

Jesús Ildefonso Díaz1  ; Sébastien Martin1, 2
1 Dept. de Matemática Aplicada, Fac. de Ciencias Matemáticas, Universidad Complutense Madrid, 28040 Madrid, Spain
2 Institut Camille-Jordan CNRS-UMR 5208, INSA Lyon, bâtiment Léonard-de-Vinci, 69621 Villeurbanne cedex, France
Comptes Rendus. Mécanique, Volume 334 (2006) no. 11, pp. 645-650.

Nous considérons le modèle d'Elrod–Adams, qui permet d'étendre l'équation de Reynolds (classiquement utilisée en lubrification) à la prise en compte de la cavitation (formation de bulles de gaz). Nous montrons que le comportement de la pression et de la saturation du lubrifiant dépend, de manière cruciale, du comportement de la distance entre les deux surfaces h(t,x,y). En particulier, nous établissons deux formulations simples pour lesquelles nous établissons (rigoureusement) qu'une zone cavitée se forme instantanément (y compris à partir d'une situation initiale totalement saturée). Des résultats numériques sont également présentés.

We consider the Elrod–Adams model extending the classical lubrication Reynolds equation to the case of the possible presence of a cavitation region. We show that the behaviour of the pressure and saturation depends crucially on the behaviour of the separation h(t,x,y) among the two surfaces. In particular, we exhibit some simple formulations for which we prove (rigorously) that a cavitation region is formed instantaneously (even for initially saturated flows). Some numerical experiences are also given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2006.08.003
Keywords: Lubrication, Cavitation, Elrod–Adams model, Reynolds equation
Mot clés : Lubrification, Cavitation, Modèle d'Elrod–Adams, Équation de Reynolds
Jesús Ildefonso Díaz 1 ; Sébastien Martin 1, 2

1 Dept. de Matemática Aplicada, Fac. de Ciencias Matemáticas, Universidad Complutense Madrid, 28040 Madrid, Spain
2 Institut Camille-Jordan CNRS-UMR 5208, INSA Lyon, bâtiment Léonard-de-Vinci, 69621 Villeurbanne cedex, France 
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Jesús Ildefonso Díaz; Sébastien Martin. On the instantaneous formation of cavitation in hydrodynamic lubrication. Comptes Rendus. Mécanique, Volume 334 (2006) no. 11, pp. 645-650. doi : 10.1016/j.crme.2006.08.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.08.003/

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[3] S.J. Alvarez; R. Oujja On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication, Nonlinear Anal., Volume 54 (2003) no. 5, pp. 845-872

[4] J.I. Díaz On the mathematical analysis of transient cavitation problems in hydrodynamics lubrification (G. Bayada; M. Chambat; J. Durany, eds.), Mathematical Modelling Lubrification, Univ. de Vigo, 1991, pp. 37-49

[5] S.J. Alvarez; J. Carrillo A free boundary problem in theory of lubrication, Comm. Partial Differential Equations, Volume 19 (1994) no. 11–12, pp. 1743-1761

[6] J. Carrillo; J.I. Díaz; G. Gilardi The propagation of the free boundary of the solution of the dam problem and related problems, Appl. Anal., Volume 4 (1993), pp. 255-276

[7] S.N. Antontsev; J.I. Díaz; S.I. Shmarev Energy methods for free boundary problems, Applications to Nonlinear PDEs and Fluid Mechanics, Series Progress in Nonlinear Differential Equations and Their Applications, vol. 48, Birkhäuser, Boston, 2002

[8] J.I. Díaz Nonlinear Partial Differential Equations and Free Boundaries, vol. I. Elliptic Equations, Research Notes in Mathematics, vol. 106, Pitman, Londres, 1985

[9] G. Bayada; M. Chambat; C. Vázquez Characteristics method for the formulation and computation of a free boundary cavitation problem, J. Comput. Appl. Math., Volume 98 (1998) no. 2, pp. 191-212

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