H.I. ENE, E. SANCHEZ-PALENCIA , Equations et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux, J. M'ecan. , 14 (1975), pp. 73-108.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the law at physical level of rigor, by ensemble averaging. His argument involves some ad hoc assumptions, e.g. the representation of the averaged interfacial forces as an integral functional of the seepage velocity, with an unknown kernel. Then the problem was reconsidered in the fundamental paper [6] and the pressure continuity was proposed as an alternative to the law of Beavers and Joseph. We note that this approach leads to some mathematical difficulties in solving the effective equations. In the contrary, the law by Beavers and Joseph leads to the well posed boundary value problem in the ....

....order O( p ) We refer to Theorem 3 from [9] for the details. The interface condition (18) connecting the pressures is not very intuitive since it involves the contribution from the geometry in the constant C bl . For a general periodic geometry the pressure continuity cannot be expected. In [6] the pressure continuity is proposed as an alternative to the law of Beavers and Joseph and the second term on the 6 On the condition by Beavers, Joseph and Saffman 7 right hand side of (18) does not appear. It should be noted that in [6] the boundary layer determining C bl was not ....

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H.I. ENE, E. SANCHEZ-PALENCIA , Equations et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux, J. M'ecan. , 14 (1975), pp. 73-108.

....approach a Brinkman type approximation in the transition layer was derived. A matching argument allowed to obtain the formula u eff Delta = 1 ff K 1=2 r u eff Delta Delta O(K) 1. 2) Interested reader can also consult the lecture note [4] Different considerations can be found in [5] and [11] They distinguish two cases: a) The pressure gradient on the side of the porous solid of the interface is normal to it. Consequently, we have a balanced flow on both sides of the interface. Then using an asymptotic point of view the following laws were obtained in [11] u eff ....

.... p = const. 1.3) on the interface. This case describes the flows in cavities. The mathematical justification is in [9] We shall not consider it in this paper. b) The pressure gradient on the side of porous solid at the interface is not normal. This case was considered in the fundamental paper [5]. After discussing the orders of magnitude of the unknowns it was found that on the interface the velocity of the free fluid was zero, and the pressure was continuous. All results cited above are not mathematically rigorous. Furthermore, different approaches gave different results and two natural ....

H. I. Ene and E. Sanchez-Palencia, Equations et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux, J. M'ecan., 14 (1975), pp. 73--108.

....order O( p ) We refer to theorem 3 from [9] for the details. The interface condition (18) connecting the pressures is not very intuitive since it involves the contribution from the geometry in the constant C bl . For a general periodic geometry the pressure continuity cannot be expected. In [6] the pressure continuity is proposed as an alternative to the law of Beavers and Joseph and the second term on the right hand side of (18) does not appear. It should be noted that in [6] the boundary layer determining C bl was not constructed. In the view of the problem setting in [9] the ....

....in the constant C bl . For a general periodic geometry the pressure continuity cannot be expected. In [6] the pressure continuity is proposed as an alternative to the law of Beavers and Joseph and the second term on the right hand side of (18) does not appear. It should be noted that in [6] the boundary layer determining C bl was not constructed. In the view of the problem setting in [9] the Beavers and Joseph s law corresponds to taking into the account the next order corrections for the velocity. Then formally we get u = u 0 H(x 2 ) Gamma fi bl ( x ) u 01 x 2 ....

Ene H.I., Sanchez-Palencia E. : Equations et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux. J. M'ecan. , Vol. 14 (1975), p. 73-108.

....complications. We derive rigorously the laws governing the flow at the interface by constructing the corresponding boundary layers. Furthermore, we compare our results with the well known results at the physical level of rigour by Beavers and Joseph [4] Saffman [23] Ene and Sanchez Palencia [8] and Levy and Sanchez Palencia [15] and find a partial agreement, depending on the choice of fl. For example the Beavers Joseph s slip condition is not obtained in the first step, but it is an additional property of the solution for the homogenized problem. Our main results are presented in the ....

....Remark : As one expects the velocity of the free fluid is dominant for fl = 0: The free fluid flow behaves as being in contact with a rigid wall i.e. in the leading order of approximation we have the no slip condition on Sigma. This agrees with the results from the paper Ene Sanchez Palencia [8], derived by the formal asymptotics. However, the relation (1.152) between the pressures is far away from any physical intuition. Since Sigma ae fx 2 = 0g we get p 0 (x 1 ; Gamma0) 0 (x 1 ; 0) C bl (u 0 ) 1 x 2 (x 1 ; Gamma0) on Sigma (1:169) and it involves the contribution ....

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Ene H.I., Sanchez-Palencia E. : Equations et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux. J. M'ecan. , Vol. 14 (1975), p. 73-108.

....; H 1=2 ( T ) n ) 8 2 H ; Z Omega u r dx = 0; 8 2 H : Classical results give the existence of a unique solution of these problems. We are interested to give the asymptotic behaviour of (u ; p ) when 0. We are here in the classical homogenization framework. In [10] H. I. Ene and E. Sanchez Palencia study the Stokes flow in a periodic porous medium with Dirichlet conditions on the boundary of the holes. The limit law describing the homogenized medium is a Darcy s law. In [5] D. Cioranescu and P. Donato consider the Laplace equation with non homogeneous ....

H. I. ENE and E. SANCHEZ- PALENCIA, Equation et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux. Journal M'ecanique, 14 (1975), 73108.

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H.I.Ene, E.Sanchez-Palencia : Equations et ph'enom`enes de surface pour l"ecoulement dans un mod`ele de milieu poreux, J. M'ecan. , 14 (1975), pp. 73-108.

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