(108h) Momentum Stress Jump Condition at the Fluid-Porous Boundary: Prediction of the Jump Coefficient

Momentum stress jump condition at the fluid-porous boundary: Prediction of the jump coefficient.

Francisco J. Valdés-Parada^(a), Benoît Goyeau^(b) and J. Alberto Ochoa-Tapia^(a)*

^(a)División de Ciencias Básicas e Ingeniería; Universidad Autónoma Metropolitana-Iztapalapa; México, D.F., Mexico. E-mail: (Ochoa-Tapia) jaot@xanum.uam.mx; (Valdés-Parada) iqfv@xanum.uam.mx

^(b)FAST, UMR-CNRS 7608, Universités Paris VI et XI, Campus Universitaire, Bât 502, 91405 Orsay, Cedex, France. E-mail : goyeau@fast.u-psud.fr

Introduction

Transport phenomena at the boundary between a porous medium and a fluid are the subject of intense research activity since they are involved in many industrial or environmental applications. To study this kind of problems, two modeling approaches can be identified. In one hand, the one domain approach allows obtaining the transition between both fluid and porous regions through continuous spatial variations of properties. On the other hand, in the two-domain approach the problem lies on coupling the conservation equations in both regions through the use of appropriate boundary conditions at the inter-region. The problem of momentum transport (Figure 1) was first studied by Beavers and Joseph¹, who proposed a semi-empirical slip boundary condition in order to couple the Stokes and Darcy equations. Later, using the method of volume averaging Ochoa-Tapia and Whitaker² (OT-W from here on) derived a jump boundary condition to couple the Darcy-Brinkman and Stokes equations obtaining good agreement with the experimental results in ref. [1]. This condition was expressed in terms of an adjustable jump coefficient (β) which was determined to be of the order of the unity. Many attempts have been made to estimate the jump coefficient, however few have studied its physical meaning, furthermore an expression for β depending on the microstructure of the inter-region still has to be obtained.

The objective of this work is to derive a stress jump boundary condition at the inter-region free of adjustable coefficients. Associated local closure problems, including the microstructure of the inter-region, are derived for the determination of a mixed stress tensor which is found to be the responsible of the jump.

Methodology

The physical configuration considered here is the parallel fluid flow (η- region) over a homogeneous porous layer (ω- region) saturated by the same fluid¹ (Figure 1). The flow is assumed to be stationary, incompressible and inertial effects in both regions are neglected. In Figure 1, the β- phase refers to the fluid phase while the solid phase is referred as the σ- phase. The local continuity and momentum conservation equations are the following

with a no-slip boundary condition at the fluid-solid interface. Averaging the above equations over a volume V   yields after some manipulations

These equations are free of length-scale constraints and are thus valid everywhere in the system.

 Figure 1: Averaging volume in both homogeneous regions and at the inter-region

In addition, when appropriate length scale constraints are satisfied (l_β « r₀ « L), Eqs. (3) and (4) reduce to the corresponding effective-medium equations of the homogeneous regions; i.e., the continuity, Stokes and Darcy-Brinkman equations.

Due to spatial variations of the micro-structure, the effective medium equations for the homogeneous regions are not valid at the inter-region. This difficulty can be overcome by the introduction of jump boundary conditions. Following the work of OT-W leads to the condition of continuity of the normal component of the velocity while, for the stress the following jump boundary condition is obtained following the recently reported methodology of Valdés-Parada et al. ³

In Eq. (5) we have introduced a mixed stress tensor

which is actually a combination of the global and Brinkman stresses. Such combination was previously used by OT-W to define a drag vector d, whose tangent component is the adjustable parameter β.  Let us note that both terms in K^-1 are non-local quantities involving spatial variations of macroscopic properties.

which indicates that the jump coefficient β can be predicted by the determination of the tangential component (K^-1) of K^-1.

On the other hand, the Brinkman contribution is evaluated from available expressions for the spatial variations of the porosity⁴

while the global stress tensor at the inter-region can be expressed in terms of  the spatial deviations of pressure and velocity as

therefore, the determination of the permeability tensor at the inter-region can be achieved by solving the associated boundary value problems for the spatial deviations of pressure and velocity in a representative portion of the inter-region (Figure 2). These problems are the result of subtracting Eqs. (3) and (4) to the local conservation equations (1) and (2).

Due to the complexity of these problems, an approximate methodology has been proposed in order to quantify this contribution and estimate the mixed stress tensor. The idea is to extrapolate the use of available expressions for the estimation of permeability in the bulk of the porous medium to the inter-region leading to,

where γ = γ ₁ = (180)^-1, provides the well-known Carman-Kozeny equation while g = γ ₂ =2x10^-3/(1?ε_βω)^2/3 leads to the results obtained by Larson-Higdon⁵.Comparing the tangential component of Eq. (5) with the jump condition of OT-W and using Eqs. (7) and (9) gives the following expression for β

Figure 2: Representative zone of the inter-region.

In Figure 3 we show the effect of the porosity on β, note that the intersection of the curves corresponds to ε_βω ≈0.79, which is the porosity of the Foametal samples used in ref. [1]. Furthermore, while the behavior of both models is the same as ε_βω→0, they provide different limits for  ε_βω→1. However in the latter case, since the permeability of the porous medium tends to infinity, the R.H.S. of the tangential component of Eq.(5)  tends to zero, which corresponds to the continuity of the velocity derivatives.

Figure 3: Dependence of the jump coefficient with the porosity using γ = γ_1, γ₂.

Moreover, the results for the jump coefficient (for Aloxite materials) using model γ₂ are found to be in good agreement with predictions previously reported in ref. [6].

Conclusions

In this study, a jump stress boundary condition free of adjustable jump coefficients has been derived using the volume averaging method. This jump condition involves a so-called mixed stress tensor which combines the Brinkman and global stress at the inter-region. The Brinkman contribution is evaluated from available expressions for the spatial variations of the porosity. Associated local closure problems, including the microstructure of the inter-region, have been derived for the determination of the global stress tensor. Due to the complexity of these problems, an approximate methodology has been proposed in order to quantify this contribution and estimate the tangential component of the mixed stress tensor. Furthermore, this component has been related to the stress jump coefficient in the boundary condition proposed by OT-W. The results for the jump coefficient (for Aloxite materials) are found to be in good agreement with predictions previously reported in ref. [6].

References

[1]     G.S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 1967; 30: 197-207.

[2]     J.A. Ochoa-Tapia,
S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I: Theoretical development. Int. J. Heat and mass Trans. 1995; 38(14): 2635-2646.

[3]     F.J. Valdés-Parada, B. Goyeau, J.A. Ochoa-Tapia, Diffusive mass transfer between a microporous medium and an homogeneous fluid: Jump boundary conditions. Chem. Eng. Sci. 2006; 61: 1692-1704.

[4]     H. Pérez Córdova, J.A. Ochoa-Tapia, Cálculo de variables y propiedades promedio en la región entre un fluido y un medio poroso a partir de valores puntuales.  Avances en Ingeniería Química. 1995; 5: 43-49.

[5]     R.E. Larson, J.L. Higdon, A periodic grain consolidation model of porous media. Physics of Fluids. 1989; A1: 38-46.

[6]     B. Goyeau, D. Lhuillier, D. Gobin, M.G. Velarde, Momentum transport at a fluid-porous interface. International Journal of Heat and Mass Transfer. 2003; 46: 4071-4081.