Comptes Rendus
no. 7
Rigorous justification of the Reynolds equations for gas lubrication
[Justification rigoureuse d'équation de Reynolds pour lubrification par gaz]

Présenté par : Évariste Sanchez-Palencia

Eduard Marušić-Paloka1  ; Maja Starčević1
1 Department of Mathematics, University of Zagreb, Bijenička 30, Zagreb, Croatia
Comptes Rendus. Mécanique, Volume 333 (2005) no. 7, pp. 534-541.

Le but de cette Note est de donner une justification de modèle de Reynolds compressible via une analyse asymptotique. À partir des équations de mouvement d'un fluide visqueux compressible dans un domaine mince nous étudions la limite lorsque l'épaisseur du domaine tend vers 0. À la limite nous trouvons un modèle de Reynolds compressible. La clé de la preuve est la convergence forte de la pression obtenue par la décomposition.

The goal of this Note is to give a rigorous justification of the compressible Reynolds model for gas lubrication, via asymptotic analysis. We start from the equations of motion of compressible viscous fluid in a thin domain and study the limit as the domain thickness tends to zero. At the limit we find the known engineering model. The key of the proof is the strong convergence for the pressure obtained by its decomposition.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2005.06.006
Keywords: Lubrication, Compressible fluid, Gas lubrication, Magnetic hard disk, Rigorous asymptotic analysis
Mot clés : Lubrification, Fluide compressible, Lubrication par gaz, Disque dur magnétique, Analyse asymptotique rigoureuse
Eduard Marušić-Paloka 1 ; Maja Starčević 1

1 Department of Mathematics, University of Zagreb, Bijenička 30, Zagreb, Croatia 
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Eduard Marušić-Paloka; Maja Starčević. Rigorous justification of the Reynolds equations for gas lubrication. Comptes Rendus. Mécanique, Volume 333 (2005) no. 7, pp. 534-541. doi : 10.1016/j.crme.2005.06.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.06.006/

[1] A.Z. Szeri Fluid Film Lubrication, Cambridge University Press, 1998

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[3] J.S. Ausman Gas-lubricated bearings (E.E. Bisson; W.J. Anderson, eds.), Advanced Bearing Technology, NASA, Washington, DC, 1966

[4] H.G. Elrod; A. Burgdorfer Refinement of the theory of gas lubricated journal bearing of infinite length, Proc. 1st Int. Symp. Gas-Lub. Bearings, ACR-49, ONR, Washington, DC, 1959, pp. 93-118

[5] M. Chipot; M. Luskin Existence and uniqueness of solutions to the compressible Reynolds lubrication equation, SIAM J. Math. Anal., Volume 17 (1986) no. 7, pp. 1390-1399

[6] G. Bayada; M. Chambat The transition between the Stokes equations and the Reynolds equation: a mathematical proof, Appl. Math. Optimization, Volume 14 (1986), pp. 73-93

[7] N. Masmoudi Homogenization of the compressible Navier–Stokes equations in a porous medium, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 885-906

[8] P.L. Lions Mathematical Topics in Fluid Mechanics, vol. 2: Compressible Models, Oxford University Press, 1998

[9] S. Marušić; E. Marušić-Paloka Two-scale convergence for thin domains and its applications to some lower dimensional models in fluid mechanics, Asymptotic Anal., Volume 23 (2000) no. 1, pp. 23-58

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