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The effective boundary conditions for vector fields on domains with rough boundaries: applications to fluid mechanics. (English) Zbl 1224.35340
Summary: The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.

35Q35 PDEs in connection with fluid mechanics
Full Text: DOI EuDML
[1] Y. Amirat, D. Bresch, J. Lemoine, J. Simon: Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Q. Appl. Math. 59 (2001), 768–785. · Zbl 1019.76014
[2] D. Bucur, E. Feireisl, Š. Nečasová: On the asymptotic limit of flows past a ribbed boundary. J. Math. Fluid Mech. 10 (2008), 554–568. · Zbl 1189.35219 · doi:10.1007/s00021-007-0242-1
[3] D. Bucur, E. Feireisl, Š. Nečasová: Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions. Arch. Ration. Mech. Anal. 197 (2010), 117–138. · Zbl 1273.76073 · doi:10.1007/s00205-009-0268-z
[4] D. Bucur, E. Feireisl, Š. Nečasová, J. Wolf: On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries. J. Differ. Equations 244 (2008), 2890–2908. · Zbl 1143.35080 · doi:10.1016/j.jde.2008.02.040
[5] M. Bulíček, J. Málek, K.R. Rajagopal: Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. 56 (2007), 51–85. · Zbl 1129.35055 · doi:10.1512/iumj.2007.56.2997
[6] J. Casado-Díaz, E. Fernández-Cara, J. Simon: Why viscous fluids adhere to rugose walls: A mathematical explanation. J. Differ. Equations 189 (2003), 526–537. · Zbl 1061.76014 · doi:10.1016/S0022-0396(02)00115-8
[7] W. Jaeger, A. Mikelić: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equations 170 (2001), 96–122. · Zbl 1009.76017 · doi:10.1006/jdeq.2000.3814
[8] H. Koch, V.A. Solonnikov: L p estimates for a solution to the nonstationary Stokes equations. J. Math. Sci. 106 (2001), 3042–3072. · doi:10.1023/A:1011375706754
[9] J. A. Nitsche: On Korn’s second inequality. RAIRO, Anal. Numér. 15 (1981), 237–248.
[10] H. Sohr: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser, Basel, 2001. · Zbl 0983.35004
[11] J. Wolf: Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9 (2007), 104–138. · Zbl 1151.76426 · doi:10.1007/s00021-006-0219-5
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