On a non-isothermal model for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature , the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier–Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field d, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg–Landau type, with a suitable penalization term to relax the constraint |d| = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier's law, depending also on the director field d. The proposed model is shown to be compatible with first and second laws of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.

[1]  F. M. Leslie.  Some constitutive equations for liquid crystals , 1968 .

[2]  E. Feireisl,et al.  Singular Limits in Thermodynamics of Viscous Fluids , 2009 .

[3]  Zhong Tan,et al.  Global weak solution to the flow of liquid crystals system , 2009 .

[4]  Jie Shen,et al.  On liquid crystal flows with free-slip boundary conditions , 2001 .

[5]  Chun Liu,et al.  Existence of Solutions for the Ericksen-Leslie System , 2000 .

[6]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[7]  J. Ericksen.  Liquid crystals with variable degree of orientation , 1991 .

[8]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[9]  Eduard Feireisl,et al.  On the Navier-Stokes equations with temperature-dependent transport coefficients , 2006 .

[10]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[11]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[12]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[13]  J. Málek,et al.  A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients , 2009 .

[14]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[15]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[16]  Daniel Coutand,et al.  Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals , 2001 .

[17]  T. Blesgen,et al.  A generalization of the Navier-Stokes equations to two-phase flows , 1999 .

[18]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .