zbMATH — the first resource for mathematics

Why viscous fluids adhere to rugose walls: A mathematical explanation. (English) Zbl 1061.76014
Summary: The main purpose is to justify rigorously the following assertion: A viscous fluid cannot slip on a wall covered by microscopic asperities because, due to the viscous dissipation, the surface irregularities bring to rest the fluid particles in contact with the wall. In mathematical terms, this corresponds to an asymptotic property established in this paper for any family of fields that slip on oscillating boundaries and remain uniformly bounded in the \(H^1\)-norm.

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
Full Text: DOI
[1] 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, 768-785, (2001) · Zbl 1019.76014
[2] Amirat, Y.; Climent, B.; Fernández-Cara, E.; Simon, J., The Stokes equations with Fourier boundary conditions on a wall with asperities, Math. models methods appl. sci., 24, 255-276, (2001) · Zbl 1007.35058
[3] Brenner, H.; Ganesan, V., Molecular effectsare conditions at the boundary “boundary conditions”?, Phys. rev., 61, 6, 6879-7547, (2000)
[4] Chu, D.C.; Karniadakis, G.E., A direct numerical simulation of laminar and turbulent flow over riblet-mounted surfaces, J. fluid mech., 250, 1-42, (1993)
[5] Cieplak, M.; Koplik, J.; Banavar, J.R., Boundary conditions at a fluid – solid interface, Phys. rev. lett., 86, 803-806, (2001)
[6] James, D.F.; Davis, A.M.J., Flow at the interface of a model fibrous porous medium, J. fluid mech., 426, 47-72, (2001) · Zbl 0981.76092
[7] Koplik, J.; Banavar, J.R., Continuum deductions from molecular hydrodynamics, Annu. rev. fluid mech., 27, 257-292, (1995)
[8] Marusić-Paloka, E., Average of the Navier’s law on the rapidly oscillating boundary, J. math. anal. appl., 259, 685-701, (2001) · Zbl 1121.35335
[9] A. Mikelić, Recent developments in multiscale problems coming from fluid mechanics, Publications du Laboratoire d’Analyse Numérique, Université Lyon 1, 2001, in Trends in Nonlinear Analysis, Springer, Berlin, to appear.
[10] Panton, R.L., Incompressible flow, (1984), Wiley-Interscience New York · Zbl 0623.76001
[11] Richardson, S., On the no-slip boundary condition, J. fluid mech., 59, 707-719, (1973) · Zbl 0265.76037
[12] Savill, A.M.; Truong, T.V.; Ryhming, I.L., Turbulent drag reduction by passive meansa review and report on the first European drag reduction meeting, J. Méc. théor. appl., 7, 353-378, (1988)
[13] Schlichting, H., Boundary-layer theory, (1968), McGraw-Hill New York
[14] Zhu, Y.; Granick, S., Limits of the hydrodynamic no-slip boundary condition, Phys. rev. lett., 88, 1061021-1061024, (2002)
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.