Lack of contact in a lubricated system
Authors:
Ionel Ciuperca and J. Ignacio Tello
Journal:
Quart. Appl. Math. 69 (2011), 357-378
MSC (2000):
Primary 35J20, 47H11, 49J10; Secondary 76D08
DOI:
https://doi.org/10.1090/S0033-569X-2011-01235-1
Published electronically:
March 10, 2011
MathSciNet review:
2729893
Full-text PDF Free Access
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Abstract: We consider the problem of a rigid surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The relative position of the surfaces is unknown except for the initial time $t=0$. The total load applied over the upper surface is a known constant for $t>0$. The mathematical model consists of a coupled system formed by the Reynolds variational inequality for incompressible fluids and Newton’s Second Law. We study the steady states of the problem and the global existence and uniqueness of the time-dependent problem. We assume one degree of freedom for the position of the surface. We consider different cases depending on the geometry of the upper surface.
References
- G. Buscaglia, I. Ciuperca, I. Hafidi, and M. Jai, Existence of equilibria in articulated bearings, J. Math. Anal. Appl. 328 (2007), no. 1, 24–45. MR 2285530, DOI https://doi.org/10.1016/j.jmaa.2006.04.042
- G. Buscaglia, I. Ciuperca, I. Hafidi, and M. Jai, Existence of equilibria in articulated bearings in presence of cavity, J. Math. Anal. Appl. 335 (2007), no. 2, 841–859. MR 2345504, DOI https://doi.org/10.1016/j.jmaa.2007.01.074
- Ionel Ciuperca, Imad Hafidi, and Mohammed Jai, Singular perturbation problem for the incompressible Reynolds equation, Electron. J. Differential Equations (2006), No. 83, 19. MR 2240831
- I. Ciuperca, M. Jai, and J. I. Tello, On the existence of solutions of equilibria in lubricated journal bearings, SIAM J. Math. Anal. 40 (2009), no. 6, 2316–2327. MR 2481296, DOI https://doi.org/10.1137/080724228
- Carlos Conca, Jorge San Martín H., and Marius Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25 (2000), no. 5-6, 1019–1042. MR 1759801, DOI https://doi.org/10.1080/03605300008821540
- B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal. 146 (1999), no. 1, 59–71. MR 1682663, DOI https://doi.org/10.1007/s002050050136
- J. I. Díaz and J. I. Tello, A note on some inverse problems arising in lubrication theory, Differential Integral Equations 17 (2004), no. 5-6, 583–591. MR 2054936
- Frêne J., Nicolas D., Deguerce B., Berthe D., Godet M.: Lubrification hydrodynamique. Eyrolles, Paris (1990).
- David Gérard-Varet and Matthieu Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 375–407. MR 2592281, DOI https://doi.org/10.1007/s00205-008-0202-9
- Hesla T.I.: Collision of smooth bodies in viscous fluids: a mathematical investigation. Ph.D. Thesis, University of Minnesota (2004).
- M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1345–1371. MR 2354496, DOI https://doi.org/10.1080/03605300601088740
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
References
- Buscaglia G., Ciuperca I., Hafidi I., Jai M.: Existence of equilibria in articulated bearings. J. Math. Anal. Appl. 328, pp. 24-45 (2007). MR 2285530 (2008c:74069)
- Buscaglia G., Ciuperca I., Hafidi I., Jai M.: Existence of equilibria in articulated bearings in presence of cavity. J. Math. Anal. Appl. 335, pp. 841-859 (2007). MR 2345504 (2008f:76068)
- Ciuperca I., Hafidi I., Jai M.: Singular perturbation problems for the incompressible Reynolds equation. Electronic Journal of Differential Equations, no. 83, pp. 1-19 (2006). MR 2240831 (2007f:76053)
- Ciuperca I., Jai M., Tello J.I.: On the existence of solutions of equilibria in lubricated journal bearings. SIAM, Mathematical Analysis, 40, pp. 2316-2327 (2009). MR 2481296
- Conca C., San Martin J.A., Tucsnak M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Partial Differential Equations, 25, pp. 1019-1042 (2000). MR 1759801 (2001c:35199)
- Desjardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146, pp. 59-71 (2000). MR 1682663 (2000b:76024)
- Díaz J.I., Tello J.I.: A note on some inverse problems arising in lubrication theory. Differential Integral Equations, 17, pp. 583–591 (2004). MR 2054936 (2005a:35278)
- Frêne J., Nicolas D., Deguerce B., Berthe D., Godet M.: Lubrification hydrodynamique. Eyrolles, Paris (1990).
- Gérard-Varet D., Hillairet M.: Regularity issues in the problem of fluid structure interaction. Arch. Ration. Mech. Anal. 195, pp. 375–407 (2010). MR 2592281
- Hesla T.I.: Collision of smooth bodies in viscous fluids: a mathematical investigation. Ph.D. Thesis, University of Minnesota (2004).
- Hillairet M.: Lack of collision between solid bodies in a 2D incompressible viscous flow. Communications on PDEs 32, pp. 1345-1366 (2007). MR 2354496 (2008i:76053)
- Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). MR 567696 (81g:49013)
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Additional Information
Ionel Ciuperca
Affiliation:
Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, Bat. Braconnier, 43, blvd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France
Email:
ciuperca@math.univ-lyon1.fr
J. Ignacio Tello
Affiliation:
Matemática Aplicada, E.U.I. Informática, Universidad Politécnica de Madrid, 28031 Madrid, Spain
Email:
jtello@eui.upm.es
Keywords:
Lubricated systems,
Reynolds variational inequality,
global solutions,
stationary solutions
Received by editor(s):
November 20, 2009
Published electronically:
March 10, 2011
Additional Notes:
The second author was partially supported by project MTM2009-13655 Ministerio de Ciencia e Innovación (Spain)
Article copyright:
© Copyright 2011
Brown University
The copyright for this article reverts to public domain 28 years after publication.
