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Substantiation of the Darcy law for a porous medium with condition of partial adhesion. (English. Russian original) Zbl 0932.35162

Sb. Math. 189, No. 12, 1871-1888 (1998); translation from Mat. Sb. 189, No. 12, 135-153 (1998).
The author shows in this paper that the phenomenological Darcy law holds for an incompressible fluid in a porous medium with suitable structure. Darcy law says that the mean velocity of the flow is proportional to the gradient of the pressure; they are linked by the so-called permeability tensor.
The fluid obeys a stationary Stokes system on a periodically perforated domain. The conditions on the boundary are of mixed type. At the exterior boundary there is a condition of full adhesion, that is, a homogeneous Dirichlet condition is imposed. The conditions imposed on the walls of the pores are of impermeability and of partial adhesion. The first condition is realized imposing that the normal component of the flow is zero on the walls. The partial adhesion condition means that the tangential motion of the fluid is allowed, but there is a friction term which is proportional to the velocity.
The problem is analyzed by means of homogenization method, and the asymptotic behaviour of the solutions is studied; here the perforation period, which represents the small-scale structure of the periodic medium, plays the role of small parameter.
The main result of the paper is that the velocity field \(u^\varepsilon\) and the pressure \(p^\varepsilon\) have the same order, as \(\varepsilon\) goes to zero, as the same problem with full adhesion condition on the boundary: the order depends only on the impermeability hypothesis.
The homogenization of the original Stokes problem gives \[ u_i=\sum^{d}_{j=1}K_{ij}\left(f_j-{{\partial p}\over{\partial x_j}}\right)\quad \text{in }\Omega,\qquad \text{div }u=0 \quad \text{in }\Omega, \qquad u\cdot n=0 \quad \text{on }\Omega, \] as the limit problem, where \(f\) is the external forcing term and \(K\) is the permeability tensor. The first equation is the Darcy law. The permeability tensor depends only on the geometry of the periodic structure.
Reviewer: M.Romito (Firenze)

MSC:

35Q30 Navier-Stokes equations
76S05 Flows in porous media; filtration; seepage
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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