Abstract
The momentum transfer between a homogeneous fluid and a porous medium in a system analogous to the one used by Beavers and Joseph (J Fluid Mech 30:197–207, 1967) is studied using volume averaging techniques. In this article, we present a closed generalized momentum transport equation (GTE) that is valid everywhere and is expressed in terms of position-dependent effective transport coefficients, which are computed from the solution of associated closure problems previously reported. A combination of the velocity profiles from the GTE in the definition of the excess terms that define the jump coefficients allows their computation using numerical techniques. The calculations are in concordance with those resulting from the work of Goyeau et al. (Int J Heat Mass Transf. 46:4071–4081, 2003), showing a strong dependence with the porosity. In addition, the effects of the roughness of the boundary on the computation of the position-dependent permeability tensor in the inter-region are also analyzed.
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Abbreviations
- \({\fancyscript{A}_{\eta \omega}}\) :
-
Area of the inter-region (m2)
- A j :
-
Surface of the j-region contained within the superficial area of V ∞, j = ω, η
- A βσ :
-
Interfacial surface contained within the averaging volume
- \({\left\langle {\textsf{\textbf{B}}_\beta}\right\rangle_s}\) :
-
Brinkman stress tensor (N/m2)
- b β :
-
Vector field that maps \({\mu_\beta \left\langle {{\bf v}_\beta}\right \rangle^{\beta}}\) onto \({\tilde {p}_\beta}\) (m−1)
- \({\textsf{\textbf{C}}_\beta}\) :
-
Second-order tensor field that maps \({\left\langle {{\bf v}_\beta}\right\rangle^\beta}\) onto (\({{\tilde{\bf v}}_\beta}\))
- g :
-
Gravity vector (m/s2)
- H :
-
Distance from the dividing surface to the lower wall (m)
- h :
-
Distance from the dividing surface to the upper wall (m)
- \({\textsf{\textbf{I}}}\) :
-
Identity tensor
- \({\textsf{\textbf{K}}_\beta}\) :
-
Permeability tensor valid everywhere (m2)
- \({{K}_\beta^{-1}}\) :
-
Tangential component of the global stress tensor valid everywhere (m2)
- \({\textsf{\textbf{K}}_{\beta \omega}}\) :
-
Darcy’s law permeability tensor in the ω-region (m2)
- K βω :
-
Norm of Darcy’s law permeability tensor in the ω-region (m2)
- l i :
-
Lattice vectors of the unit cell (i = 1, 2) (m)
- l j :
-
Characteristic length for the j-phase; j = β, σ (m)
- L :
-
Characteristic length associated with volume averaged quantities (m)
- L v :
-
Characteristic length associated with \({\left. {\left\langle {{\bf v}_\beta}\right\rangle}\right|_{\bf x}}\) (m)
- L v1 :
-
Characteristic length associated with \({\nabla \left. {\left\langle {{\bf v}_\beta}\right\rangle}\right|_{\bf x}}\) (m)
- n βσ :
-
Unit normal vector directed from the β-phase toward the σ-phase
- n ηω :
-
Unit normal vector directed from the η-region toward the ω-region
- p β :
-
Pressure in the β-phase (N/m2)
- \({\left\langle{p_\beta}\right\rangle^\beta}\) :
-
Intrinsic average pressure (N/m2)
- \({\left\langle {p_\beta}\right\rangle_{j}^\beta}\) :
-
Intrinsic average pressure in the j-region; j = ω, η (N/m2)
- r :
-
Position vector (m)
- r 0 :
-
Radius of the averaging volume (m)
- \({\left\langle \textsf{\textbf{T}}\right\rangle_s}\) :
-
Surface stress (N/m2)
- \({\left\langle {\textsf{\textbf{T}}_\beta}\right\rangle_s}\) :
-
Bulk stress tensor (N/m2)
- \({\fancyscript{V}_{j}}\) :
-
Volume of the j-phase contained within the local averaging volume, j = β, σ (m3)
- V :
-
averaging region
- \({\fancyscript{V}}\) :
-
Local averaging volume (m3)
- v β :
-
Velocity vector in the β-phase (m/s)
- \({\left\langle{{\bf v}_\beta}\right\rangle}\) :
-
Superficial average velocity vector (m/s)
- \({\left\langle {{\bf v}_\beta}\right\rangle_{j}}\) :
-
Superficial average velocity vector in the j-region; j = ω, η (m/s)
- \({\left\langle {v_\beta}\right\rangle_{j}}\) :
-
Tangential component of \({\left\langle{{\bf v}_\beta}\right\rangle_{j}}\) ; j = ω, η (m/s)
- x :
-
Position vector locating the centroid of the averaging volume (m)
- y β :
-
Position vector relative to the centroid of the averaging volume V (m)
- δ :
-
Half of the height of the unit cell (m)
- ε β (x):
-
Position-dependent volume fraction of the β-phase
- ε βω :
-
Volume fraction of the β-phase in the ω-region
- \({\phi }\) :
-
Jump coefficient associated with superficial effects
- η :
-
Roughness factor
- μ β :
-
Viscosity of the β-phase (Ns/m2)
- \({\nabla_s}\) :
-
\({\left({\textsf{\textbf{I}}-{\bf n}_{\eta \omega} {\bf n}_{\eta \omega}}\right)\cdot \nabla}\) Surface gradient operator
- ρ β :
-
Density of the β-phase (kg/m3)
- η :
-
Relative to the η-region
- ω :
-
Relative to the ω-region
- 0, s, b :
-
Relative to the dividing surface
- K :
-
Relative to the permeability
- GTE:
-
Generalized transport equations
- ODA:
-
One-domain approach
- TDA:
-
Two-domain approach
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Valdés-Parada, F.J., Alvarez-Ramírez, J., Goyeau, B. et al. Computation of Jump Coefficients for Momentum Transfer Between a Porous Medium and a Fluid Using a Closed Generalized Transfer Equation. Transp Porous Med 78, 439–457 (2009). https://doi.org/10.1007/s11242-009-9370-9
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DOI: https://doi.org/10.1007/s11242-009-9370-9