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The Tartar equation for homogenization of a model of the dynamics of fine-dispersion mixtures. (Russian, English) Zbl 0989.35105
Sib. Mat. Zh. 42, No. 6, 1375-1390 (2001); translation in Sib. Math. J. 42, No. 6, 1142-1155 (2001).
The author studies a mathematical model of the motion of a Stokes flow of fine-dispersion mixtures of viscous incompressible fluids in a bounded domain in \(\mathbb R^2\) with fast-oscillating initial date. It is supposed that the viscosity values \(\nu_{\varepsilon}\) are transferred along the trajectories of the motion of the fluid under the velocity \(\vec v_{\varepsilon}(\vec x,t,\lambda)\), where \(\varepsilon\), \(\lambda\) are arbitrary small positive parameters which characterize respectively the frequencies of oscillation for distributions of viscosity and velocity and amplitudes of deviation from a constant viscosity value \(a_0\) and a sufficiently smooth velocity field \(\vec v^{(0)}(\vec x,t)\) which defines a laminar nonperturbed flow of an average viscous homogeneous fluid. The correctness of this problem for fixed values of \(\varepsilon\), \(\lambda\) is guaranteed by the Navier–Stokes theory.
The aim of the article is to present and realize a method for approximate determination of the effective characteristics of fine-dispersion homogeneous mixtures without an ordered structure based on using the notion of an \(H\)-measure suggested by L. Tartar in [Proc. R. Soc. Edinb., Sect. A 115, No. 3/4, 193-230 (1990; Zbl 0774.35008)]. As a consequence, a system of approximate homogeneous equations is exposed for determining an \(H\)-measure. Finally, the author constructs a correct closed model whose solutions approximate the weak limits of solutions to an approximate model and, therefore, describe the motion of homogeneous mixtures with a sufficient accuracy.
MSC:
35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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