A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium

Anna Marciniak-Czochra; Andro Mikelić

doi:10.3934/dcdss.2014.7.1065

Article Contents

2014, Volume 7, Issue 5: 1065-1077. Doi: 10.3934/dcdss.2014.7.1065

This issue Previous Article Stokes and Navier-Stokes equations with perfect slip on wedge type domains Next Article Approximate solutions to a model of two-component reactive flow

A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium

Anna Marciniak-Czochra^1, and
Andro Mikelić^2,

1.
Institute of Applied Mathematics, Interdisciplinary Center of Scientic Computing and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg
2.
Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex

Received: February 28, 2013

Published: September 2014

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Abstract

We present modeling of an incompressible viscous flow through a fracture adjacent to a porous medium. A fast stationary flow, predominantly tangential to the porous medium is considered. Slow flow in such setting can be described by the Beavers-Joseph-Saffman slip. For fast flows, a nonlinear filtration law in the porous medium and a non- linear interface law are expected. In this paper we rigorously derive a quadratic effective slip interface law which holds for a range of Reynolds numbers and fracture widths. The porous medium flow is described by the Darcy law. The result shows that the interface slip law can be nonlinear, independently of the regime for the bulk flow. Since most of the interface and boundary slip laws are obtained via upscaling of complex systems, the result indicates that studying the inviscid limits for the Navier-Stokes equations with linear slip law at the boundary should be rethought.

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Mathematics Subject Classification: Primary: 35B27, 35Q30, 35Q35, 74Q15, 76D07, 76M50, 76S.

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