Abstract
Flow in a three-layer channel is modeled analytically. The channel consists of a transition layer sandwiched between a porous medium and a fluid clear of solid material. Within the transition layer, the reciprocal of the permeability varies linearly across the channel. The Brinkman model is used for the momentum equations for the porous medium layer and the transition layer. The velocity profile is obtained in closed form in terms of Airy, exponential, and polynomial functions. The overall volume flux and boundary friction factors are calculated and compared with values obtained with a two-layer model employing the Beavers–Joseph condition at the interface between a Darcy porous medium and a clear fluid.
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Abbreviations
- c f :
-
Friction factor, defined as −du 1/dy at y = ξ
- Da:
-
Darcy number, K 0 /H 2
- G :
-
Applied pressure gradient
- H :
-
Channel width
- K :
-
Permeability
- K 0 :
-
Reference permeability defined in Eqs. 1a– 1c
- M 2 :
-
Viscosity ratio in region 2, μ e2/μ
- M 3 :
-
Viscosity ratio in region 3, μ e3/μ
- u :
-
Dimensionless filtration velocity, μ u*/GH 2
- u D :
-
Dimensionless filtration velocity for Darcy flow model with the Beavers–Joseph boundary condition
- u*:
-
Filtration velocity
- ū :
-
Dimensionless mean velocity defined in Eq. 19
- y :
-
Dimensionless transverse coordinate, y*/H
- y*:
-
Transverse coordinate
- α :
-
Beavers–Joseph boundary coefficient
- β :
-
\({\alpha /Da^{\frac{1}{2}}}\)
- η :
-
Position of the interface between region 2 and region 3
- λ 2 and λ 3 :
-
Parameters defined in Eqs. 7a,b
- μ :
-
Fluid viscosity
- μ e2 :
-
Effective viscosity in region 2
- μ e3 :
-
Effective viscosity in region 3
- ξ :
-
Position of the interface between region 1 and region 2
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Nield, D.A., Kuznetsov, A.V. The effect of a transition layer between a fluid and a porous medium: shear flow in a channel. Transp Porous Med 78, 477–487 (2009). https://doi.org/10.1007/s11242-009-9342-0
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DOI: https://doi.org/10.1007/s11242-009-9342-0