American Institute of Mathematical Sciences

Advanced
May  2006, 6(3): 449-470. doi: 10.3934/dcdsb.2006.6.449

Convergence to equilibrium of a multiscale model for suspensions

Eric Cancès 1, and  Claude Le Bris 2,

1. 

CERMICS, École Nationale des Ponts et Chaussées, 6 et 8, avenue Blaise Pascal, Cité Descartes - Champs sur Marne, F-77455 Marne La Vallée Cedex 2, France
2. 

CERMICS, École Nationale des Ponts et Chaussées, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France

Received  April 2005 Revised  December 2005 Published  February 2006

We consider a multiscale model describing the flow of a concentrated suspension. The model couples the macroscopic equation of conservation of momentum with a nonlinear nonlocal kinetic equation describing at the microscopic level the rheological behaviour of the fluid. We study the long-time limit of the time-dependent solution. For this purpose, we use the entropy method to prove the convergence to equilibrium of the kinetic equation.
Keywords: long-time limit, non-Newtonian fluid, Couette flow, concentrated suspension, partial differential equation, entropy method., convergence to equilibrium.
Mathematics Subject Classification: 35G25, 35K55 ; Secondary: 76A0.
Citation: Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449
[1]

Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic & Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361
[2]

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 & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[3]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[4]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010
[5]

Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041
[6]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
[7]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[8]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[9]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
[10]

Changli Yuan, Mojdeh Delshad, Mary F. Wheeler. Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework. Networks & Heterogeneous Media, 2010, 5 (3) : 583-602. doi: 10.3934/nhm.2010.5.583
[11]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[12]

Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations & Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331
[13]

Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025
[14]

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

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068
[16]

Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231
[17]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
[18]

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 & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[19]

Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002
[20]

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 & Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy