×

Homogenization of the Stokes system in a thin film flow with rapidly varying thickness. (English) Zbl 0675.76033

Summary: We study a problem with two small parameters, that models a fluid flow between two close rough surfaces. We study the convergence by the energy method of the 3-dimensional Stokes system solution when the ratio \(\lambda =\eta /\epsilon\) is constant (\(\eta\) is linked to the fluid thickness and \(\epsilon\) to the size of the roughness period). Then making \(\lambda\) tend to infinity (resp. to zero) we show that the case in which the thickness is greater (resp. smaller) than the period is an asymptotic limit of the intermediate case.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
76D08 Lubrication theory

Citations:

Zbl 0675.76036
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] G. BAYADA et M. CHAMBAT, The transition between the Stokes equations and the Reynolds equation: a mathematical proof, Appl. Math. Opt. 14 (1986)73-93. Zbl0701.76039 MR826853 · Zbl 0701.76039 · doi:10.1007/BF01442229
[2] G. BAYADA et M. CHAMBAT, On the various aspects of the thin film equation in hydrodynamic lubrification when the roughness occurs, in < Applications of multiple scaling in mechanics >, Masson, Paris 1987. Zbl0649.76013 MR901987 · Zbl 0649.76013
[3] M. CHAMBART, G. BAYADA et J. B. FAURE, Some effects of the boundary roughness in a thin film flow, in < Boundary variation and boundary control >, Proc. IFIP Conf. Nice, 1986, Lect. Notes Control Inf. Sciences, 100, Springer Verlag, Berlin 1988. Zbl0651.76013 MR942450 · Zbl 0651.76013
[4] G. BAYADA et J. B. FAURE, Application des techniques d’homogénéisation à des phénomènes de rugosité en lubrification hydrodynamique, Publications U.A. 740 CNRS Lyon St Etienne, 32 (1984) 1-52.
[5] A. BENSOUSSAN, J. L. LIONS et G. PAPANICOLAOU, Asymptotic analysis for periodic structure, North Holland, Amsterdam 1978. Zbl0404.35001 MR503330 · Zbl 0404.35001
[6] [6] D. CAILLERIE, Homogénéisation des équations de la diffusion stationnaire dans des domaines cylindriques aplatis, RAIRO Anal. Num.,15 (4) (1981) 295-319. Zbl0483.35003 MR642495 · Zbl 0483.35003
[7] D. CAILLERIE, Thin Elastic and Periodic Plates, Math. Meth. in the Appl. Sci., 6 (1984) 159-191. Zbl0543.73073 MR751739 · Zbl 0543.73073 · doi:10.1002/mma.1670060112
[8] K. K. CHEN et D. C. SUN, On the statistical treatment of rough surface in hydrodynamic lubrication problems, Proc. of the 4th Leeds-Lyon Symp. on surface roughness in lubrication, IME (1977) 41-45.
[9] H. CHRISTENSEN et K. TONDER, The hydrodynamic lubrication of rough bearing surfaces of finite width, J. of Lub. Tech., Trans. ASME, F, 93 (3) (1971) 324-330.
[10] V. GIRAULT et P. A. RAVIART, Finite element approximation of the Navier-Stokes equations, Springer Verlag, Berlin, 1981. Zbl0441.65081 MR548867 · Zbl 0441.65081
[11] R. KOHN et M. VOGELIUS, A new model for thin plates with rapidly varying thickness, Int. J. Solid and Struct., 20 (1984) 333-350. Zbl0532.73055 · Zbl 0532.73055 · doi:10.1016/0020-7683(84)90044-1
[12] R. KOHN et M. VOGELIUS, A new model for thin plates with rapidly varying thickness II. a convergence proof, Quart. of Appl Math. 18 (1985) 1-22. Zbl0565.73046 MR782253 · Zbl 0565.73046
[13] J. L. LIONS, Perturbations singulières dans les problèmes aux limites en contrôle optimal, Springer Verlag, Berlin, 1973. Zbl0268.49001 MR600331 · Zbl 0268.49001
[14] N. PATIR et H. S CHENG, An average flow model for determining effects of three dimensional roughness in partial hydrodynamic roughness, J of Lub. Tech., Trans. ASME, F 100 (1978) 12-17.
[15] N. PHAN THIEN, On the effects of the Reynolds and Stokes roughness in a two dimensional slider hearing, Proc. R. Soc. London, A377 (1981) 349-362. Zbl0481.76045 · Zbl 0481.76045 · doi:10.1098/rspa.1981.0128
[16] N. PHAN THIEN, Hydrodynamic lubrication of rough surfaces, Proc. R. Soc. London, A383 (1982) 439-446. Zbl0535.76043 · Zbl 0535.76043 · doi:10.1098/rspa.1982.0139
[17] [17] J. SAINT JEAN PAULIN, Homogénéisation et perturbations dans un problème lié à l’échauffement d’un câble électrique, Ann Fac. Sc., Toulouse 5 (1983) 43-59. Zbl0531.35010 MR709810 · Zbl 0531.35010 · doi:10.5802/afst.588
[18] E. SANCHEZ-PALENCIA, Non homogeneous media and vibration theory, Lecture Notes in Physics 127, Springer Verlag, Berlin, 1978. Zbl0432.70002 MR578345 · Zbl 0432.70002
[19] D. C. SUN and K. K CHEN, First effects of Stokes roughness on hydrodynamic lubrication, J. of Lub Tech., Trans. ASME, F, 99 (1977) 2-5.
[20] L. TARTAR, Cours Peccot, Collège de France, 1977.
[21] J. L. TEALE et A. O. LEBECK, An évaluation of the average flow model for surface roughness effects in lubrication, J. of Lub. Tech., Trans. ASME, F, 102 (1980) 360-367.
[22] R. TEMAM, Navier Stokes equation, North Holland, Amsterdam, 1979.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.