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Derivation of microstructure models of fluid flow by homogenization. (English) Zbl 0927.35081
The author deals with the case of flow in a medium composed of two materials with very different properties by homogenization. The main result is as follows. Let \(\Omega\) be an open subset of \(\mathbb{R}^N\) and \(Y= [0,1]^N\). Following G. Allaira [SIAM J. Math. Anal. 23, No. 6, 1482-1518 (1992; Zbl 0770.35005)]. \(Y\) is divided into two parts \(Y_1\) and \(Y_2\). \(\Omega\) is assumed to have an \(\varepsilon Y\) periodic structure. The coefficient functions \(a_{ij}(t,y)\in C([0, T]\times Y_1)\) and \(A_{ij}(t, y)\in C([0, T]\times Y_2)\) are uniformly positive definite and symmetric. Off their respective domains they are defined to be zero. Then, there exists a couple of functions \(u(x,t)\) and \(U(x,t,y)\) which is the unique solution of the microstructure model: \[ {\partial u\over\partial t}- {\partial\over\partial x_i} \Biggl(a^h_{ij}{\partial u\over\partial x_j}\Biggr)+ \int_{\partial Y_2} A_{ij}{\partial U\over\partial y_j} \nu_i ds=f\quad\text{in }\Omega\times(0, T), \]
\[ {\partial U\over\partial t}-{\partial\over\partial y_j} \Biggl(A_{ij}{\partial U\over\partial y_j}\Biggr)= 0\quad\text{in }\Omega\times Y_2\times (0,T), \]
\[ U(x,t,x)= u(x,t)\quad\text{on }\partial Y_2, \]
\[ U(x,0,y)= U_0(x,y)\quad\text{in }\Omega\times Y_2, \]
\[ u(x,0)= u_0(x)\quad\text{in }\Omega, \] where \(a^h_{ij}(t)\) is the homogenized coefficient of \(a_{ij}(t,y)\).

35Q35 PDEs in connection with fluid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
76S05 Flows in porous media; filtration; seepage
Zbl 0770.35005
Full Text: DOI
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