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- W2004015873 abstract "This paper deals with the homogenization of the Stokes equations in a cylinder with varying viscosity and with Dirichlet boundary condition. The viscosity is equal to αε⪢1 in a ε-periodic lattice of unidirectional cylinders of radius εrε where rε⪡1, and is equal to 1 elsewhere. In the critical regime defined by limε→0ε2|lnrε|∈]0,+∞[ and limε→0αεrε2∈]0,+∞], the limit problem is a coupled Stokes system satisfied by the limit velocity and the limit of the rescaled velocity in the cylinders, which can be read as a nonlocal law of Brinkman type. Moreover, if limε→0αεrε2=+∞, the limit of the rescaled velocity is equal to 0 and the Brinkman law is derived as in [G. Allaire, Arch. Rational Mech. Anal. 13 (1991) 209–259]. In the other regimes the homogenization leads either to classical Stokes problems or to a zero limit velocity. In the critical case the pressure is not bounded in L2 but only in H−1. Moreover, the pressure of the limit problem is not equal to the weak limit of the pressure in H−1. Dans cet article, on étudie l'homogénéisation des équations de Stokes dans un cylindre, avec une viscosité variable et une condition de Dirichlet au bord. La viscosité est égale à αε⪢1 dans un réseau ε-périodique de cylindres unidirectionnels de rayon εrε où rε⪡1, et est égale à 1 sinon. Sous le régime critique défini par limε→0ε2|lnrε|∈]0,+∞[ et limε→0αεrε2∈]0,+∞], le problème limite est un système couplé de type Stokes satisfait par la limite de la vitesse et celle de la vitesse remise à l'échelle dans les cylindres, qui s'écrit comme une loi de Brinkman non locale. Si, de plus, limε→0αεrε2=+∞, on obtient la loi de Brinkman comme dans [G. Allaire, Arch. Rational Mech. Anal. 13 (1991) 209–259]. Sous les autres régimes, l'homogénéisation conduit soit à des problèmes de Stokes classiques soit à une vitesse limite nulle. Dans le cas critique, la pression n'est pas bornée dans L2 mais seulement dans H−1. En outre, la pression du problème limite est différente de la limite faible de la pression dans H−1." @default.
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- W2004015873 date "2003-07-01" @default.
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- W2004015873 title "Homogenization of the Stokes equations with high-contrast viscosity" @default.
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- W2004015873 doi "https://doi.org/10.1016/s0021-7824(03)00012-6" @default.
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