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Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions. (English) Zbl 1273.76073
Summary: We consider a family of solutions to the evolutionary Navier-Stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries. We give a complete description of the asymptotic limit by means of \(\Gamma \)-convergence arguments, and identify a general class of boundary conditions.

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
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[1] Amirat A.A., Bresch D., Lemoine J., Simon J.: Effect of rugosity on a flow governed by stationary Navier–Stokes equations. Quart. Appl. Math. 59, 768–785 (2001) · Zbl 1019.76014
[2] Amirat A.A., Climent E., Fernández-Cara E., Simon J.: The Stokes equations with Fourier boundary conditions on a wall with asperities. Math. Models. Methods Appl. 24, 255–276 (2001) · Zbl 1007.35058 · doi:10.1002/mma.206
[3] Ansini N., Garroni A.: \(\Gamma\)-convergence of functionals on divergence-free fields. ESAIM Control Optim. Calc. Var. 13(4), 809–828 (2007) (electronic) · Zbl 1127.49011 · doi:10.1051/cocv:2007041
[4] Basson A., Gérard-Varet D.: Wall laws for fluid flows at a boundary with random roughness. Preprint (2006) · Zbl 1179.35207
[5] Bogovskii M.E.: Solution of some vector analysis problems connected with operators div and grad (in Russian). Trudy Sem. S.L. Sobolev 80(1), 5–40 (1980)
[6] Bouchitté G., Seppecher P.: Cahn and Hilliard fluid on an oscillating boundary. Motion by Mean Curvature and Related Topics (Trento, 1992), de Gruyter, Berlin, 23–42, 1994 · Zbl 0807.76081
[7] Bucur D., Feireisl E., Nečasová Š., Wolf J.: On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries. J. Differ. Equ. 244, 2890–2908 (2008) · Zbl 1143.35080 · doi:10.1016/j.jde.2008.02.040
[8] Bucur D., Feireisl E., Nečasová Š.: On the asymptotic limit of flows past a ribbed boundary J. Math. Fluid Mech. 10(4), 554–568 (2008) · Zbl 1189.35219 · doi:10.1007/s00021-007-0242-1
[9] Casado-Díaz J., Fernández-Cara E., Simon J.: Why viscous fluids adhere to rugose walls: A mathematical explanation. J. Differ. Equ. 189, 526–537 (2003) · Zbl 1061.76014 · doi:10.1016/S0022-0396(02)00115-8
[10] Chenais D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52(2), 189–219 (1975) · Zbl 0317.49005 · doi:10.1016/0022-247X(75)90091-8
[11] Dal Maso G.: An introduction to \(\Gamma\)-convergence. Progress in Nonlinear Differential Equations and their Applications, Vol. 8. Birkhäuser Boston, Boston, 1993 · Zbl 0816.49001
[12] Dal Maso G., Defranceschi A., Vitali E.: Integral representation for a class of C 1-convex functionals. J. Math. Pures Appl. (9) 73(1), 1–46 (1994) · Zbl 0853.49013
[13] Defranceschi A., Vitali E.: Limits of minimum problems with convex obstacles for vector valued functions. Appl. Anal. 52(1–4), 1–33 (1994) · Zbl 0839.49011 · doi:10.1080/00036819408840221
[14] El Jarroudi M., Brillard A.: Relaxed Dirichlet problem and shape optimization within the linear elasticity framework. NoDEA Nonlinear Differ. Equ. Appl. 11(4), 511–528 (2004) · Zbl 1109.49041 · doi:10.1007/s00030-004-2025-1
[15] Galdi G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, I. Springer, New York (1994) · Zbl 0949.35004
[16] Henrot A., Pierre M.: Variation et optimisation de formes. Une analyse géométrique. Mathématiques & Applications (Berlin), Vol. 48. Springer, Berlin, 2005 · Zbl 1098.49001
[17] Hesla T.I.: Collision of smooth bodies in a viscous fluid: A mathematical investigation. Ph.D. Thesis, Minnesota, 2005
[18] Hillairet M.: Lack of collision between solid bodies in a 2D incompressible viscous flow. Preprint, ENS Lyon, 2006 · Zbl 1221.35279
[19] Jaeger W., Mikelić A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001) · Zbl 1009.76017 · doi:10.1006/jdeq.2000.3814
[20] Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969) · Zbl 0184.52603
[21] Málek J., Rajagopal K.R.: Mathematical issues concerning the Navier–Stokes equations and some of its generalizations. Evolutionary Equations Vol. II. Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam, pp. 371–459, 2005 · Zbl 1095.35027
[22] Moffat H.K.: Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1 (1964) · Zbl 0118.20501 · doi:10.1017/S0022112064000015
[23] Nitsche J.A.: On Korn’s second inequality. RAIRO Anal. Numer. 15, 237–248 (1981) · Zbl 0467.35019
[24] Priezjev N.V., Darhuber A.A., Troian S.M.: Slip behavior in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamics simulations. Phys. Rev. E 71, 041608 (2005) · doi:10.1103/PhysRevE.71.041608
[25] Priezjev N.V., Troian S.M.: Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions. J. Fluid Mech. 554, 25–46 (2006) · Zbl 1091.76016 · doi:10.1017/S0022112006009086
[26] Sohr H.: The Navier–Stokes equations: An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001) · Zbl 0983.35004
[27] Temam R.: Navier–Stokes equations. North-Holland, Amsterdam (1977) · Zbl 0383.35057
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