Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-23T23:02:02.673Z Has data issue: false hasContentIssue false
  • English
  • Français

A mixed dirichlet-neumann problem for a nonlinear reynolds equation in elastohydrodynamic piezoviscous lubrication

Published online by Cambridge University Press:  20 January 2009

J. Durany ,
G. García  and
C. Vázquez
J. Durany
Affiliation:
Department of Applied MathematicsUniveristy of Vigo36280-Vioo, Spain
G. García
Affiliation:
Department of Applied MathematicsUniveristy of Vigo36280-Vioo, Spain
C. Vázquez
Affiliation:
Department of Applied MathematicsUniveristy of Vigo36280-Vioo, Spain
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this work is to study the existence of solutions for a mathematical model of the displacement of a piezoviscous lubricant between two elastic surfaces. As we deal with a rolling ball contact problem, the deformations are modelled by the linear Hertzian theory. The fluid pressure behaviour is governed by the classical Reynolds equation for thin film displacement. The relevant aspect of cavitation in lubrication is described by means of the Elrod Adams model which leads to a mathematical free boundary problem.

The two main original features of the model problem in relation to previous works are: the supply of lubricant coming from a groove that is transversal to the direction of fluid displacement and the consideration of a piezoviscous law of Barus. Mathematically, the first one leads to a mixed Dirichlet-Neumann problem for the Reynolds equation and the second one involves an additional nonlinearity in a diffusion type term.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 1 , February 1996 , pp. 151 - 162
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Bayada, G. and Chambat, M. Existence and uniqueness for a lubrication problem with nonregular conditons on the free boundary, Boll. Un. Mat. Ital. B 6 (3-B) (1984), 543–557.Google Scholar
2. Bayada, G. and Chambat, M., Sur quelques modelisations de la zone de cavitation en lubrication hydrodynamique, J. Theor. Appl. Mech. 5 (1986), 703–729.Google Scholar
3. Bayada, G., Durany, J. and Vázquez, C., Existence of solutions for a lubrication problem ir elastic journal bearing devices with thin bearing, Math. Methods Appl. Sci. 18 (1995), 255–266.CrossRefGoogle Scholar
4. Bayada, G., El Alaoui, M. and Vázquez, C., Existence of Solution for Elastohydrodynamic Piezoviscous Lubrication Problems with a New Model for Cavitation (Publication de l'Equipe d'Analyse Numérique Lyon-Saint Etienne, 163, 1994) and European J. Appl. Math. (1995), to appear.Google Scholar
5. Brezis, H., Kinderlehrer, H. and Stampacchia, G., Sur une nouvelle formulation du probléms de l'écoulement á travers une digue, C. R. Acad. Sci. Paris Sér A-B (1978), 711–714.Google Scholar
6. Cameron, A., Basic Lubrication Theory (Ellis Horwood Series, 1981).Google Scholar
7. Chipot, M., Variational Inequalities and Flow on Porous Media (Applied Math. Sciences Series 52, New York. Springer-Verlag, 1984).CrossRefGoogle Scholar
8. Dowson, D. and Higginson, G. R., Elastohydrodynamic Lubrication (Pergamon Press, 1977).Google Scholar
9. Durany, J. and Vázquez, C., Mathematical analysis of an elastohydrodynamic lubrication problem with cavitation, App. Anal. 53 (1994), 135–142.CrossRefGoogle Scholar
10. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (Berlin. Springer-Verlag, 1977).CrossRefGoogle Scholar
11. Hu, B., A quasivariational inequality arising in elastohydrodynamics, SIAM J. Math. Anal 21 (1990), 18–36.CrossRefGoogle Scholar
12. Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and their Applications (New York. Academic Press, 1980).Google Scholar
13. Oden, J. T. and Wu, S. R., Existence of solutions to the Reynolds equation of elastohydrodynamic lubrication, Internat. J. Engrg. Sci. 23 (1985), 207–215.CrossRefGoogle Scholar
14. Rodrígues, J. F., Remarks on the Reynolds problem of elastohydrodynamic lubrication, European J. Appl. Math. 4 (1993), 83–96.CrossRefGoogle Scholar
15. Vázquez, C., Existence and uniqueness of solution for a lubrication problem with cavitatior in a journal bearing with axial supply, Adv. in Math. Sci. and Appl. 4 (1994), 313–331.Google Scholar