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Networks and Heterogeneous Media

2008, Volume 3, Issue 4: 831-862. doi: 10.3934/nhm.2008.3.831
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The homogenized model of small oscillations of complex fluids

  • 1.
    Institute of Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Lenin Ave 47, Kharkiv 61164
  • 2.
    Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA-16802-6401, 218 Mc Allister Building
  • 3.
    Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Lenin Ave 47, Kharkiv 61164
  • Received: 01 June 2007 Revised: 01 February 2008

  • Primary: 35B27, 35Q30, 74Q10; Secondary: 76M30, 76M50

  • 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 ε in such a way that the sizes of particles are of order ε3, the distances between the nearest particles are of order ε, and the stiffness of the interaction force is of order ε2.
    We study the asymptotic behavior of the microscopic model as ε→0 and obtain the homogenized equations that can be considered as a macroscopic model of diluted solutions of interacting colloidal particles.

    Citation: M. Berezhnyi, L. Berlyand, Evgen Khruslov. The homogenized model of small oscillations of complex fluids[J]. Networks and Heterogeneous Media, 2008, 3(4): 831-862. doi: 10.3934/nhm.2008.3.831

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  • 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 ε in such a way that the sizes of particles are of order ε3, the distances between the nearest particles are of order ε, and the stiffness of the interaction force is of order ε2.
    We study the asymptotic behavior of the microscopic model as ε→0 and obtain the homogenized equations that can be considered as a macroscopic model of diluted solutions of interacting colloidal particles.


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