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December  2008, 3(4): 831-862. doi: 10.3934/nhm.2008.3.831

The homogenized model of small oscillations of complex fluids

M. Berezhnyi 1, , L. Berlyand 2, and  Evgen Khruslov 3,

1. 

Institute of Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Lenin Ave 47, Kharkiv 61164, Ukraine
2. 

Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA-16802-6401, 218 Mc Allister Building, United States
3. 

Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Lenin Ave 47, Kharkiv 61164, Ukraine

Received  June 2007 Revised  February 2008 Published  October 2008

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.
Keywords: triangulization condition, non-Newtonian fluids, interacting particles, Homogenization, interaction energy, Brinkman's law.
Mathematics Subject Classification: Primary: 35B27, 35Q30, 74Q10; Secondary: 76M30, 76M50.
Citation: M. Berezhnyi, L. Berlyand, Evgen Khruslov. The homogenized model of small oscillations of complex fluids. Networks & Heterogeneous Media, 2008, 3 (4) : 831-862. doi: 10.3934/nhm.2008.3.831
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