American Institute of Mathematical Sciences

Advanced
February  2009, 23(1&2): 455-475. doi: 10.3934/dcds.2009.23.455

On energetic variational approaches in modeling the nematic liquid crystal flows

Chun Liu 1, and  Huan Sun 1,

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  February 2008 Revised  April 2008 Published  September 2008

In this paper we present results for the existence of classicalsolutions of a hydrodynamical system modeling the flow of nematicliquid crystals. The system consists of a coupled system ofNavier-Stokes equations and various kinematic transport equationsfor the molecular orientations. A formal physical derivation of theinduced elastic stress using least action principle reflects thespecial coupling between the transport and the induced stress terms.The derivation and the analysis of the system falls into a generalenergetic variational framework for complex fluids with elasticeffects due to the presence of nontrivial microstructures.
Keywords: Navier-Stokes Equations, Non-newtonian fluids, Existence.
Mathematics Subject Classification: Primary: 76D05, 35Q35; Secondary: 76D03, 76A05.
Citation: Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
[1]

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 & Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[2]

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

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
[4]

Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565
[5]

Pitágoras Pinheiro de Carvalho, Juan Límaco, Denilson Menezes, Yuri Thamsten. Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021043
[6]

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
[7]

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
[8]

Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427
[9]

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
[10]

Xin Liu, Yongjin Lu, Xin-Guang Yang. Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids. Evolution Equations & Control Theory, 2021, 10 (2) : 365-384. doi: 10.3934/eect.2020071
[11]

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 & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096
[12]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[13]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[14]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[15]

Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126
[16]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17
[17]

Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021146
[18]

Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785
[19]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[20]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (214)
  • HTML views (0)
  • Cited by (54)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy