About thin film micropolar asymptotic equations
Authors:
G. Bayada, M. Chambat and S. R. Gamouana
Journal:
Quart. Appl. Math. 59 (2001), 413-439
MSC:
Primary 76A05; Secondary 35B40, 35Q35, 76D08, 76T20
DOI:
https://doi.org/10.1090/qam/1848526
MathSciNet review:
MR1848526
Full-text PDF Free Access
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Abstract: We study from a mathematical point of view the asymptotic behaviour of micropolar fluids in lubrication theory. Different assumptions are made on the dependence of the micropolar characteristic numbers and of the Dirichlet boundary conditions for the velocity and the rotation, with respect to the small gap parameter. Various generalized Reynolds equations are rigorously obtained and second-order estimates are also given.
E. L. Aero, A. N. Bulygin, and E. V. Kuvshinskii, Asymmetric hydromechanics, J. Appl. Math. Mech. 2, 297–308 (1965) (in Russian)
- Ahiko Assemien, Guy Bayada, and Michèle Chambat, Inertial effects in the asymptotic behavior of a thin film flow, Asymptotic Anal. 9 (1994), no. 3, 177–208. MR 1295293
- Guy Bayada and Michèle Chambat, The transition between the Stokes equations and the Reynolds equation: a mathematical proof, Appl. Math. Optim. 14 (1986), no. 1, 73–93. MR 826853, DOI https://doi.org/10.1007/BF01442229
- Guy Bayada and Grzegorz Łukaszewicz, On micropolar fluids in the theory of lubrication. Rigorous derivation of an analogue of the Reynolds equation, Internat. J. Engrg. Sci. 34 (1996), no. 13, 1477–1490. MR 1423717, DOI https://doi.org/10.1016/0020-7225%2896%2900031-6
N. M. Bessonov, A new generalization of the Reynolds equation for a micropolar fluid and its application to bearing theory, J. Tribology, Trans. AMSE 116, 654–657 (1994)
- A. Cemal Eringen, Simple microfluids, Internat. J. Engrg. Sci. 2 (1964), 205–217 (English, with French, German, Italian and Russian summaries). MR 0169468, DOI https://doi.org/10.1016/0020-7225%2864%2990005-9
J. Frene, D. Nicolas, B. Degueurce, D. Berthe, and M. Godet, Lubrification Hydrodynamique, Collec. DER-EDF, Eyrolles, Paris, 1990
S. R. Gamouana, Analyse asymptotique de fluide micropolaire en film mince et milieu poreux, Thèse-Mathématiques, Univ. Claude Bernard Lyon I, no. 170–99, 1999
G. Łukaszewicz, Micropolar fluids—Theory and Applications, Birkhäuser - Modeling and Simulation in Science, Boston, MA, 1999
G. Łukaszewicz, On stationary flows of asymmetric fluids, Rendiconti, Academia naz. delle scienze detta dei XL, Memorie di mathematica 106, Vol. XII, fasc. 3, 35–44 (1988)
M. S. Mostefai, Deduction rigoureuse de l’équation de Reynolds à partir d’un système modelisant l’écoulement à faible épaisseur d’un fluide micropolaire et étude de deux problèmes à frontières libres. Hele-Shaw généralisé et Stefan à deux phases pour un fluide non Newtonien. Thèse - Mathématiques, Univ. Jean Monnet, Saint-Etienne, 1997
L. G. Pestrosyan, Some problems of mechanics of fluids with antisymmetric stress tensor, Erevan University Press, Erevan, 1984 (in Russian)
J. Prakash and P. Sinha, Lubrication theory for micropolar fluid and its application to a journal bearing, Internat. J. Engrg. Sci. 13, 217–232 (1975)
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1979
E. L. Aero, A. N. Bulygin, and E. V. Kuvshinskii, Asymmetric hydromechanics, J. Appl. Math. Mech. 2, 297–308 (1965) (in Russian)
A. Assemien, G. Bayada, and M. Chambat, Inertial effects in the asymptotic behaviour of a thin film flow, Asymptotic Anal. 9, 177–208 (1994)
G. Bayada and M. Chambat, The transition between the Stokes equation and the Reynolds equation: A mathematical proof, Appl. Math. Optim. 14, 73–93 (1986)
G. Bayada and G. Lukaszewicz, On micropolar fluids in the theory of lubrication. Rigorous derivation of an analogue of the Reynolds equation, Internat. J. Engrg. Sci. 34, 1477–1490 (1996)
N. M. Bessonov, A new generalization of the Reynolds equation for a micropolar fluid and its application to bearing theory, J. Tribology, Trans. AMSE 116, 654–657 (1994)
A. C. Eringen, Simple microfluids, Internat. J. Engrg. Sci. 2, 205–217 (1964)
J. Frene, D. Nicolas, B. Degueurce, D. Berthe, and M. Godet, Lubrification Hydrodynamique, Collec. DER-EDF, Eyrolles, Paris, 1990
S. R. Gamouana, Analyse asymptotique de fluide micropolaire en film mince et milieu poreux, Thèse-Mathématiques, Univ. Claude Bernard Lyon I, no. 170–99, 1999
G. Łukaszewicz, Micropolar fluids—Theory and Applications, Birkhäuser - Modeling and Simulation in Science, Boston, MA, 1999
G. Łukaszewicz, On stationary flows of asymmetric fluids, Rendiconti, Academia naz. delle scienze detta dei XL, Memorie di mathematica 106, Vol. XII, fasc. 3, 35–44 (1988)
M. S. Mostefai, Deduction rigoureuse de l’équation de Reynolds à partir d’un système modelisant l’écoulement à faible épaisseur d’un fluide micropolaire et étude de deux problèmes à frontières libres. Hele-Shaw généralisé et Stefan à deux phases pour un fluide non Newtonien. Thèse - Mathématiques, Univ. Jean Monnet, Saint-Etienne, 1997
L. G. Pestrosyan, Some problems of mechanics of fluids with antisymmetric stress tensor, Erevan University Press, Erevan, 1984 (in Russian)
J. Prakash and P. Sinha, Lubrication theory for micropolar fluid and its application to a journal bearing, Internat. J. Engrg. Sci. 13, 217–232 (1975)
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1979
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Article copyright:
© Copyright 2001
American Mathematical Society
