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On the roughness-induced effective boundary conditions for an incompressible viscous flow. (English) Zbl 1009.76017
The authors study the steady flow of a viscous incompressible fluid through a channel having roughly a square cross-section. The real boundary is obtained by perturbing a flat surface with periodic irregularities whose characteristic length and amplitude are given by some small parameter \(\varepsilon\). The flow is driven by a constant pressure gradient. The problem with no-slip boundary condition is considered first with the aim of getting information on the limit \(\varepsilon \to 0\). Through a series of estimates derived under the restriction that the pressure gradient is sufficiently small, the authors obtain a precise description of how the tangential velocity and its normal derivative become related as \(\varepsilon\) approaches zero. Their result provides a rigorous justification of the so-called Navier’s slip condition.

76D05 Navier-Stokes equations for incompressible viscous fluids
76M50 Homogenization applied to problems in fluid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
Full Text: DOI
[1] Achdou, Y.; Pironneau, O., Domain decomposition and wall laws, C.R. acad. sci. Paris Sér. I, 320, 541-547, (1995) · Zbl 0834.76014
[2] Achdou, Y.; Pironneau, O.; Valentin, F., Shape control versus boundary control, (), 1-18 · Zbl 0920.35020
[3] Achdou, Y.; Pironneau, O.; Valentin, F., Effective boundary conditions for laminar flows over periodic rough boundaries, J. comput. phys., 147, 187-218, (1998) · Zbl 0917.76013
[4] Amirat, Y.; Simon, J., Influence de la rugosité en hydrodynamique laminaire, C.R. acad. sci. Paris Sér. I, 323, 313-318, (1996) · Zbl 0858.76020
[5] Amirat, Y.; Simon, J., Riblet and drag minimization, (), 9-17 · Zbl 0899.76107
[6] Antontsev, S.N.; Kazhikhov, A.V.; Monakhov, V.N., Boundary value problems in mechanics of nonhomogeneous fluids, (1990), Elsevier Amsterdam · Zbl 0696.76001
[7] Beavers, G.S.; Joseph, D.D., Boundary conditions at a naturally permeable wall, J. fluid mech., 30, 197-207, (1967)
[8] Conca, C., On the applications of the homogenization theory to a class of problems arising in fluid mechanics, J. math. pures appl., 64, 31-75, (1985) · Zbl 0566.35080
[9] Conca, C., Étude d’un fluide traversant une paroi perforée. I. comportement limite près de la paroi, J. math. pures appl., 66, 1-44, (1987) · Zbl 0622.35061
[10] Conca, C.; Murat, F.; Pironneau, O., The Stokes and navier – stokes equations with boundary conditions involving the pressure, Japan J. math., 20, 263-318, (1994) · Zbl 0826.35093
[11] Jäger, W.; Mikelić, A., On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. scuola norm. super. Pisa cl. fis. mat. (IV), 23, 403-465, (1996) · Zbl 0878.76076
[12] Jäger, W.; Mikelić, A., On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness, Comm. pure appl. math., 51, 1073-1121, (1998) · Zbl 0938.76101
[13] Jäger, W.; Mikelić, A., On the interface boundary conditions by Beavers, Joseph and Saffman, SIAM J. appl. math., 60, 1111-1127, (2000) · Zbl 0969.76088
[14] Lions, J.L., Some methods in the mathematical analysis of systems and their control, (1981), Gordon and Breach New York · Zbl 0542.93034
[15] Mohammadi, B.; Pironneau, O.; Valentin, F., Rough boundaries and wall laws, Int. J. numer. methods fluids, 27, 169-177, (1998) · Zbl 0904.76031
[16] Navier, C.L.M.H., Sur LES lois de l’équilibre et du mouvement des corps élastiques, Mem. acad. R. sci. inst. France, 6, 369, (1827)
[17] Panton, R.L., Incompressible flow, (1984), Wiley New York · Zbl 0623.76001
[18] Pedlosky, J., Geophysical fluid dynamics, (1979), Springer-Verlag Heidelberg · Zbl 0429.76001
[19] Saffman, P.G., On the boundary condition at the interface of a porous medium, Stud. appl. math., 1, 77-84, (1971) · Zbl 0271.76080
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