Abstract
We consider the system of equations that describes small non-stationary motions of viscous incompressible fluid with a large number of small rigid interacting particles. This system is a microscopic mathematical model of complex fluids such as colloidal suspensions, polymer solutions etc. We suppose that the system of particles depends on a small parameter $\varepsilon$ in such a way that the sizes of particles are of order $\varepsilon^{3}$, the distances between the nearest particles are of order $\varepsilon$, and the stiffness of the interaction force is of order $\varepsilon^{2}$. We study the asymptotic behavior of the microscopic model as $\varepsilon\rightarrow 0$ and obtain the homogenized equations that can be considered as a macroscopic model of diluted solutions of interacting colloidal particles.
Highlights
In the last years, in the literature on fluid mechanics we have seen an increasing number of papers devoted to the study of complex fluids
In this paper we study the case, when the diameters of particles are of order ε3, which is the well-known critical size of inclusions, leading to appearance of an additional potential in the homogenized equation
We find the homogenized variational functional and the system of Euler equations corresponding to this functional
Summary
In the literature on fluid mechanics we have seen an increasing number of papers devoted to the study of complex fluids. We construct this function in order to satisfy the following conditions: 1) for sufficiently small h and ε < ε(h), the function wεh(x) is close (in some sense) to the restriction of w(x) on the lattice {xiε}; 2) for h ≪ 1 and ε ≪ ε(h), the function wεh(x) is an “almost” minimizer of the functional (3.5) in every cube K(xα, h) when Tnp = enp[w(xα)].
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