Abstract
A solution to the problem of shallow laminar water flow above a porous surface is essential when modeling phenomena such as erosion, resuspension, and mass transfer between the porous media and the flow above it. Previous studies proposed theoretical, experimental, and numerical insight with no single general solution to the problem. Many studies have used the Brinkman equation, while others showed that it does not represent the actual interface flow conditions. In this paper we show that the interface macroscopic velocity can be accurately modeled by introducing a modification to the Brinkman equation. A moving average approach was proved to be successful when choosing the correct representative elementary volume and comparing the macroscopic solution with the average microscopic flow. As the size of the representative elementary volume was found to be equal to the product of the square root of the permeability and an exponential function of the porosity, a general solution is now available for any brush configuration. Given the properties of the porous media (porosity and permeability), the flow height and its driving force, a complete macroscopic solution of the interface flow is obtained.
Similar content being viewed by others
References
Adler, P. M. and Mills, P. M.: 1979, Motion and rupture of a porous sphere in a linear flow field, J. Rheol. 23, 25–37.
Basu, A. J. and Khalili, A.: 1999, Computation of flow through a fluid–sediment interface in a benthic chamber, Phys. Fluids 11(6), 1395–1405.
Beavers, G. S. and Joseph, D. D.: 1967, Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30, 197–207.
Beavers, G. S., Sparrow, E. M. and Magnuson, R. A.: 1970, Experiments on coupled parallel flows in a channel and a bounding porous medium, J. Basic Eng. 92D 843–848.
Beavers, G. S., Sparrow, E. M. and Masha, B. A.: 1974, Boundary condition at a porous surface which bounds a fluid flow, AIChE J. 20, 596–597.
Brinkman, H. C.: 1947, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. 1, 27–34.
Durlofsky, L. and Brady, J. F.: 1987, Analysis of the Brinkman equation as a model for flow in porous media, Phys. Fluids 30, 3329.
Gray, W. G. and O'Neill, K.: 1976, On the general equations for flow in porous media and their reduction to Darcy's law, Water Resour. Res. 12, 148–154.
Happel, J. and Brenner, H.: 1973, Low Reynolds Number Hydrodynamics, Noordhoff, Groningen, The Netherlands.
Howells, I. D.: 1974, Drag due to the motion of a Newtonian fluid through a sparse random array of small rigid fixed objects, J. Fluid Mech. 64, 449–475.
James, D. F. and Davis, A. M. J.: 2001, Flow at the interface of a model fibrous porous medium, J. Fluid Mech. 426, 47–72.
Jennings, A. A. and Pisipati, R.: 1999, The impact of Brinkman's extension of Darcy's law in the neighborhood of a circular preferential flow pathway, Environ. Model. Software 14, 427–435.
Kaviany, M.: 1995, Principles of Heat Transfer in Porous Media, Springer, New York.
Kim, S. and Russell, W. B.: 1985, Modeling of porous media by renormalization of the Stokes equations, J. Fluid Mech. 154, 269–286.
Koplik, J., Levine, H. and Zee, A.: 1983, Viscosity renormalization in the Brinkman equation, Phys. Fluids 26, 2864–2870.
Larson, R. E. and Higdon, J. J. L.: 1986, Microscopic flow near the surface of two-dimensional porous media. I. Axial flow, J. Fluid Mech. 166, 449–472.
Lundgren, T. S.: 1972, Slow flow through stationary random beds and suspensions of spheres, J. Fluid Mech. 51, 273–299.
Martys, N., Bentz, D. P. and Garboczi, E. J.: 1994, Computer simulation study of the effective viscosity in Brinkman's equation, Phys. Fluids 6(4), 1434–1439.
Nield, D. A. and Bejan, A.: 1992, Convection in Porous Media, Springer, New York, pp. 11–19.
Richardson, S.: 1971, A model for the boundary condition of a porous material. Part 2, J. FluidMech. 49, 327–336.
Saffman, P. G.: 1971, On the boundary condition at the surface of a porous medium, Stud. Appl. Math. 2, 93–101.
Sahraoui, M. and Kaviany, M.: 1992, Slip and no-slip velocity boundary-conditions at interface of porous, plain media, Int. J. Heat Mass Transfer 35(4), 927–943.
Shavit, U., Bar-Yosef, G., Rosenzweig, R. and Assouline, S.: 2002, Modified Brinkman equation for a free flow problem at the interface of porous surfaces: the Cantor–Taylor brush configuration case, Water Resour. Res. 38, 1320–1334.
Taylor, G. I.: 1971, A model for the boundary condition of a porous material. Part 1, J. Fluid Mech. 49, 319–326.
Vignes-Adler, M., Adler, P.M. and Gougat, P.: 1987, Transport processes along fractals. The Cantor–Taylor brush, PhysicoChemical Hydrodyn. 8(4), 401–422.
Whitaker, S.: 1999, The Method of Volume Averaging, Kluwer Academic Publishers, Dordrecht.
Zhou, D. and Mendosa, C. C.: 1993, Flow through porous bed of turbulent stream, J. Eng. Mech. ASCE 119 365–383.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shavit, U., Rosenzweig, R. & Assouline, S. Free Flow at the Interface of Porous Surfaces: A Generalization of the Taylor Brush Configuration. Transport in Porous Media 54, 345–360 (2004). https://doi.org/10.1023/B:TIPM.0000003623.55005.97
Issue Date:
DOI: https://doi.org/10.1023/B:TIPM.0000003623.55005.97