Effect of rugosity on a flow governed by stationary Navier-Stokes equations
Authors:
Youcef Amirat, Didier Bresch, Jérôme Lemoine and Jacques Simon
Journal:
Quart. Appl. Math. 59 (2001), 769-785
MSC:
Primary 76D05; Secondary 76M45
DOI:
https://doi.org/10.1090/qam/1866556
MathSciNet review:
MR1866556
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Abstract: We study the stationary flow of a fluid occupying a 3D infinite horizontal domain bounded by a rough wall that is at rest and by a plane that moves with a constant velocity. The rough wall is a plane covered with periodically distributed asperities of size $\varepsilon$. We prove that, outside a neighbourhood of the rough region, the flow behaves asymptotically as a Couette flow, as $\varepsilon \to 0$, up to an exponentially small error.
- Yves Achdou and Olivier Pironneau, Domain decomposition and wall laws, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 5, 541–547 (English, with English and French summaries). MR 1322334, DOI https://doi.org/10.1016/S0045-7825%2897%2900118-7
- Yves Achdou, O. Pironneau, and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys. 147 (1998), no. 1, 187–218. MR 1657773, DOI https://doi.org/10.1006/jcph.1998.6088
- Youcef Amirat, Didier Bresch, Jérôme Lemoine, and Jacques Simon, A problem of exponential decay for Navier-Stokes equations arising in the analysis of rugosity, Navier-Stokes equations and related nonlinear problems (Palanga, 1997) VSP, Utrecht, 1998, pp. 1–14. MR 1690698
- Youcef Amirat, Didier Bresch, Jérôme Lemoine, and Jacques Simon, Existence of semi-periodic solutions of steady Navier-Stokes equations in a half space with an exponential decay at infinity, Rend. Sem. Mat. Univ. Padova 102 (1999), 341–365. MR 1739546
- Youcef Amirat and Jacques Simon, Riblets and drag minimization, Optimization methods in partial differential equations (South Hadley, MA, 1996) Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1997, pp. 9–17. MR 1472284, DOI https://doi.org/10.1090/conm/209/02755
- A. G. Belyaev, A. L. Pyatnitskiĭ, and G. A. Chechkin, Asymptotic behavior of the solution of a boundary value problem in a punctured domain with an oscillating boundary, Sibirsk. Mat. Zh. 39 (1998), no. 4, 730–754, i (Russian, with Russian summary); English transl., Siberian Math. J. 39 (1998), no. 4, 621–644. MR 1654152, DOI https://doi.org/10.1007/BF02673049
A. Carrau, Modèlisation numérique d’un écoulement sur paroi rugueuse, Thesis, Université de Bordeaux, 1992
- Gregory A. Chechkin, Avner Friedman, and Andrey L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl. 231 (1999), no. 1, 213–234. MR 1676697, DOI https://doi.org/10.1006/jmaa.1998.6226
J. Cousteix, Couche limite laminaire, Cepadues Editions, 1988
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205
O. A. Ladyshenskaya and V. A. Solonnikov, Determination of solutions of boundary value problems for steady-state Stokes and Navier-Stokes equations in domains having an unbounded Dirichlet integral, J. Soviet Math. 21, 728–761 (1983)
R. Lewandowski, Analyse mathématique et océanographie, Masson, 1997
J.-L. Lions, Quelques méthodes de résolutions des problémes aux limites non linéaires, Dunod, Paris, 1969
- J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR 0600331
- J.-L. Lions, Some methods in the mathematical analysis of systems and their control, Kexue Chubanshe (Science Press), Beijing; Gordon & Breach Science Publishers, New York, 1981. MR 664760
Y. Achdou and O. Pironneau, Analysis of Wall laws, C. R. Acad. Sci. Paris, Série I, 320, 541–547 (1995)
Y. Achdou, O. Pironneau, and F. Valentin, Effective boundary conditions for laminar flows over rough boundaries, J. Comput. Phys. 147, 187–218 (1998)
Y. Amirat, D. Bresch, J. Lemoine, and J. Simon, A problem of exponential decay for Navier-Stokes equations arising in the analysis of rugosity, In Navier-Stokes equations and related nonlinear problems (Palanga, 1997), J. Pileckas, ed., H. Amann et al., VSP/TEV, Utrecht, 1998, pp. 1–14
Y. Amirat, D. Bresch, J. Lemoine, and J. Simon, Existence of semi-periodic solutions of steady Navier-Stokes equations in a half space with an exponential decay at infinity, Rend. Sem. Mat. Univ. Padova 102, 341—365 (1999)
Y. Amirat and J. Simon, Riblets and drag minimization, In Optimization Methods in Partial Differential Equations, S. Cox and I. Lasiecka, eds., Contemporary Mathematics, vol. 209, Amer. Math. Soc., Providence, RI, 1997, pp. 9–17
A. G. Belyaev, G. A. Chechkin, and A. L. Piatniski, Asymptotic behavior of the solutions of a boundary value problem in a perforated domain with oscillating boundary, Preprint, Université de Paris VI, 1998
A. Carrau, Modèlisation numérique d’un écoulement sur paroi rugueuse, Thesis, Université de Bordeaux, 1992
G. A. Chechkin, A. Friedman, and A. L. Piatniski, The boundary value problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl. 231, 213–234 (1999)
J. Cousteix, Couche limite laminaire, Cepadues Editions, 1988
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vols. 1 and 2, Springer-Verlag, New York, 1994
O. A. Ladyshenskaya and V. A. Solonnikov, Determination of solutions of boundary value problems for steady-state Stokes and Navier-Stokes equations in domains having an unbounded Dirichlet integral, J. Soviet Math. 21, 728–761 (1983)
R. Lewandowski, Analyse mathématique et océanographie, Masson, 1997
J.-L. Lions, Quelques méthodes de résolutions des problémes aux limites non linéaires, Dunod, Paris, 1969
J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics 323, Springer-Verlag, New York, 1973
J.-L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control, Science Press, Beijing, China, Gordon and Breach, Inc. New York, 1981
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