American Institute of Mathematical Sciences

Advanced
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 and Heterogeneous Media, 2008, 3 (4) : 831-862. doi: 10.3934/nhm.2008.3.831
[1]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255
[2]

Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations and Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331
[3]

Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565
[4]

Pitágoras Pinheiro de Carvalho, Juan Límaco, Denilson Menezes, Yuri Thamsten. Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021043
[5]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[6]

Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037
[7]

Xin Liu, Yongjin Lu, Xin-Guang Yang. Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids. Evolution Equations and Control Theory, 2021, 10 (2) : 365-384. doi: 10.3934/eect.2020071
[8]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096
[9]

Yukun Song, Yang Chen, Jun Yan, Shuai Chen. The existence of solutions for a shear thinning compressible non-Newtonian models. Electronic Research Archive, 2020, 28 (1) : 47-66. doi: 10.3934/era.2020004
[10]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[11]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[12]

Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic and Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361
[13]

Changli Yuan, Mojdeh Delshad, Mary F. Wheeler. Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework. Networks and Heterogeneous Media, 2010, 5 (3) : 583-602. doi: 10.3934/nhm.2010.5.583
[14]

Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719
[15]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[16]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[17]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
[18]

Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231
[19]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
[20]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy