zbMATH — the first resource for mathematics

The Navier wall law at a boundary with random roughness. (English) Zbl 1176.35127
Summary: We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size \({\varepsilon\ll 1}\). In the parent paper [A. Basson and D. Gérard-Varet, Commun. Pure Appl. Math. 61, No. 7, 941–987 (2008; Zbl 1179.35207)], we derived a homogenized boundary condition of Navier type as \({\varepsilon\to 0}\). We show here that for a large class of boundaries, this Navier condition provides a \({O(\varepsilon^{3/2} |\ln \varepsilon|^{1/2})}\) approximation in \(L ^2\), instead of \({O(\varepsilon^{3/2})}\) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35A35 Theoretical approximation in context of PDEs
Full Text: DOI arXiv
[1] Achdou, Y., Le Tallec, P., Valentin, F., Pironneau, O.: Constructing wall laws with domain decomposition or asymptotic expansion techniques. In: Symposium on Advances in Computational Mechanics, Vol. 3 (Austin, TX, 1997) Comput. Methods Appl. Mech. Engrg. 151, 1–2, 215–232 (1998) · Zbl 0920.76063
[2] Achdou, Y., Mohammadi, B., Pironneau, O., Valentin, F.: Domain decomposition & wall laws. In: Recent developments in domain decomposition methods and flow problems (Kyoto, 1996; Anacapri, 1996), Vol. 11 of GAKUTO Internat. Ser. Math. Sci. Appl. Tokyo: Gakkōtosho 1998, pp. 1–14 · Zbl 0931.76077
[3] Achdou Y., Pironneau O., Valentin F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147(1), 187–218 (1998) · Zbl 0917.76013 · doi:10.1006/jcph.1998.6088
[4] Amirat Y., Bresch D., Lemoine J., Simon J.: Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Quart. Appl. Math. 59(4), 769–785 (2001) · Zbl 1019.76014
[5] Avellaneda M., Lin F.-H.: Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. 40(6), 803–847 (1987) · Zbl 0632.35018 · doi:10.1002/cpa.3160400607
[6] Avellaneda M., Lin F.-H.: L p bounds on singular integrals in homogenization. Comm. Pure Appl. Math. 44(8–9), 897–910 (1991) · Zbl 0761.42008 · doi:10.1002/cpa.3160440805
[7] Baladi, V.: Decay of correlations. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), Vol. 69 of Proc. Sympos. Pure Math. Providence, RI: Amer. Math. Soc. 2001, pp. 297–325 · Zbl 0993.37003
[8] Basson A., Gérard-Varet D.: Wall laws for fluid flows at a boundary with random roughness. Comm. Pure Applied Math. 61(7), 941–987 (2008) · Zbl 1179.35207 · doi:10.1002/cpa.20237
[9] Bechert D., Bartenwerfer M.: The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206(1), 105–129 (1989) · doi:10.1017/S0022112089002247
[10] Bresch D., Gérard-Varet D.: Roughness-induced effects on the quasi-geostrophic model. Comm. Math. Phys. 253(1), 81–119 (2005) · Zbl 1149.76680 · doi:10.1007/s00220-004-1173-9
[11] Bresch, D., Milisic, V.: Higher order boundary layer correctors and wall laws derivation: a unified approach. http://arXiv.org./list/math/0611083 , 2006
[12] De Bouard, A., Craig, W., Daz-Espinosa, O., Guyenne, P., Sulem, C.: Long wave expansions for water waves over random topography. http://arXiv.org./list/math/07100389 , 2007 · Zbl 1228.76029
[13] Durrett R.: Probability: theory and examples. Second ed. Belmont, CA: Duxbury Press, 1996 · Zbl 1202.60001
[14] Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Vol. 38 of Springer Tracts in Natural Philosophy. New York: Springer-Verlag, 1994
[15] Gérard-Varet, D.: Highly rotating fluids in rough domains. J. Math. Pures Appl. (9) 82, 11, 1453–1498 (2003) · Zbl 1033.76008
[16] Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, vol. 105 of Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 1983 · Zbl 0516.49003
[17] Giaquinta M., Modica G.: Nonlinear systems of the type of the stationary Navier-Stokes system. J. Reine Angew. Math. 330, 173–214 (1982) · Zbl 0492.35018
[18] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Berlin: Springer-Verlag, 2001 (Reprint of the 1998 edition) · Zbl 1042.35002
[19] Jäger W., Mikelić A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Eq. 170(1), 96–122 (2001) · Zbl 1009.76017 · doi:10.1006/jdeq.2000.3814
[20] Jäger W., Mikelić A.: Couette flows over a rough boundary and drag reduction. Comm. Math. Phys. 232(3), 429–455 (2003) · Zbl 1062.76012
[21] Jäger, W., Mikelić, A., Neuss, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22, 6, 2006–2028 (2000) (electronic) · Zbl 0980.35124
[22] Jikov V.V., Kozlov S.M., Oleĭnik O.A.: Homogenization of differential operators and integral functionals. Berlin: Springer-Verlag, 1994. (Translated from the Russian by G. A. Yosifian [G. A. Iosif’ yan])
[23] Luchini P.: Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces. European J. Mech. B Fluids 14(2), 169–195 (1995) · Zbl 0835.76024
[24] Varadhan S., Zygouras N.: Behavior of the solution of a random semilinear heat equation. Comm. Pure Applied Math. 61(9), 1298–1329 (2008) · Zbl 1152.60077 · doi:10.1002/cpa.20256
[25] Varadhan, S.R.S.: Stochastic processes. Notes based on a course given at New York University during the year 1967/68. New York: Courant Institute of Mathematical Sciences New York University, 1968
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.