Liquid Crystal Flows Research Articles

Conservation-Dissipation Formalism for soft matter physics: II. Application to non-isothermal nematic liquid crystals.

For most existing non-equilibrium theories, the modeling of non-isothermal processes is a hard task. Intrinsic difficulties involve the non-equilibrium temperature, the coexistence of conserved energy and dissipative entropy, etc. In this paper, by taking the non-isothermal flow of nematic liquid crystals as a typical example, we illustrate that thermodynamically consistent models in either vectorial or tensorial forms can be constructed within the framework of the Conservation-Dissipation Formalism (CDF). And the classical isothermal Ericksen-Leslie model and Qian-Sheng model are shown to be special cases of our new vectorial and tensorial models in the isothermal, incompressible and stationary limit. Most importantly, from the above examples, it is known that CDF can easily solve the issues relating with non-isothermal situations in a systematic way. The first and second laws of thermodynamics are satisfied simultaneously. The non-equilibrium temperature is defined self-consistently as a partial derivative of the entropy function. Relaxation-type constitutive relations are constructed, which give rise to classical linear constitutive relations, like Newton's law and Fourier's law, in stationary limits. Therefore, CDF is expected to have a broad scope of applications in soft matter physics, especially under complicated situations, such as non-isothermal, compressible and nanoscale systems.

Conservation-Dissipation Formalism for soft matter physics: I. Augmentation to Doi's variational approach

In the first paper of this series, we prove that by choosing the proper variational function and variables, the variational approach proposed by Doi in soft matter physics is equivalent to the Conservation-Dissipation Formalism. To illustrate the correspondence between these two theories, several novel examples in soft matter physics, including the particle diffusion in dilute solutions, polymer phase separation dynamics and nematic liquid crystal flows, are carefully examined. Based on our work, a deep connection among the generalized Gibbs relation, the second law of thermodynamics and the variational principle in non-equilibrium thermodynamics is revealed.

Singularity formation to the two-dimensional compressible non-isothermal nematic liquid crystal flows in a bounded domain

We study the singularity formation of strong solutions to the two-dimensional (2D) compressible non-isothermal nematic liquid crystal flows in a bounded domain. Under a geometric condition of the initial orientation field, we show that the strong solution exists globally if the temporal integral of the maximum norm of the divergence of the velocity is bounded. Our method relies on critical Sobolev inequalities of logarithmic type and delicate energy estimates.

Blow-up criteria for the 3D liquid crystal flows involving two velocity components

This note is devoted to investigating blow-up criteria for the three dimensional nematic liquid crystal flows. More precisely, we mainly proved that the strong solution (u,d) can be extended beyond T, provided that the horizontal velocity components uh=(u1,u2) satisfy uh∈Ls(0,T;Lq(R3)),2s+3q≤12,6≤q≤∞.

Hydrodynamic interactions in polar liquid crystals on evolving surfaces

The flow of passive and active polar liquid crystals on evolving surfaces is considered. The models are derived as a thin-film limit, and a finite element method is used to study the effect of hydrodynamics on the interplay of topology, geometric properties, and defect dynamics.

Incompressible limit of the compressible nematic liquid crystal flows in a bounded domain with perfectly conducting boundary

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

L3,∞ solutions of the liquid crystals system

In this article, we study the simplified system of the original Ericksen–Leslie equations for the flow of liquid crystals. It is shown that the L3,∞ solutions of the Cauchy problem for the three-dimensional incompressible liquid crystal system are smooth.

On weak solution to the steady compressible flow of nematic liquid crystals

In this paper, we study the three‐dimensional‐simplified Ericksen‐Leslie system for the steady compressible flow of nematic liquid crystals in a bounded domain. It is proved that the existence of a weak solution for the adiabatic exponent γ > 1 provided the initial direction field in the upper hemisphere.

Time decay rate of global strong solutions to nematic liquid crystal flows in [formula omitted

In this paper, we obtain optimal time-decay rates in Lr(R+3) for r≥1 of global strong solutions to the nematic liquid crystal flows in R+3, provided the initial data has small L3(R+3)-norm.

Space–Time Decay Estimates for the Incompressible Flow of Liquid Crystals

In this paper, we study decay in space and time for derivatives of solutions to the incompressible flow of liquid crystals. Based on the known time decay results, we further obtain space–time decay rates agreed with those of the heat equation. More precisely, we firstly derive weighted bounds of the director field by weighted energy estimate, and then by regarding the nonlinear term $$\mathrm{div}{(}\nabla {d}\odot \nabla {d}{)}$$ as a forcing term in the velocity equation, we could get the weighted bounds for the vorticity, which implies the corresponding estimates for the velocity field. Our results show that the weighted $$L^p$$ decay for the velocity is not as good as that of both the director field and the vorticity. This is basically due to the presence of the pressure term in the velocity equation, which leads to the restriction on the exponent of the weight.

Low Mach number limit of global solutions to 3‐D compressible nematic liquid crystal flows with Dirichlet boundary condition

In this paper, we consider the uniform estimates of strong solutions in the Mach number ϵ and t ∈ [0,∞) for the compressible nematic liquid crystal flows in a 3‐D bounded domain , provided the initial data are small enough and the density is close to the constant state. Here, we consider the case that the velocity field satisfies the Dirichlet boundary condition. Based on the uniform estimates, we obtain the global convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.

Existence and uniqueness of mild solutions to the incompressible nematic liquid crystal flow

In this paper, we consider the incompressible nematic liquid crystal flow in the whole space, and obtain the existence of global mild solutions with small initial data in the scaling invariant space. The main tool we have used is the implicit function theorem. Furthermore, we establish the asymptotic stability of solutions when the time goes to infinity. And in view of our construction of solutions in the weak Lp-spaces, we finally derive the existence of self-similar solutions provided the initial data are small homogeneous functions.

Shear flows of low molecular weight nematic liquid crystals between parallel circular plates

Shear flows of low molecular weight nematic liquid crystals between parallel circular plates

Incompressible limit for compressible nematic liquid crystal flows in a bounded domain

ABSTRACTIn this paper, we consider the incompressible limit of strong solution for the compressible nematic liquid crystal flows in a 3D bounded domain where the the velocity field satisfies the Dirichlet boundary condition. We establish the uniform estimates with respect to the Mach number for the strong solutions with large initial data in a short time interval. Consequently, we obtain the convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.

Uniqueness of hydrodynamic flow of nematic liquid crystals in $\boldsymbol{L^n}$

In this paper, we consider the Cauchy problem of hydrodynamic flow of nematic liquid crystals. We get the uniqueness of the mild solution in $C([0,T);L^n)$.

Global Existence of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Compressible Nematic Liquid Crystal Flows

This paper is concerned with a simplified hydrodynamic flow modeling the three-dimensional motion of compressible, nematic liquid crystal materials. The authors establish the global existence of classical solution to the Cauchy problem with smooth initial data which are of small energy but possibly large oscillations with constant state as far-field condition which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. As a byproduct, the large-time behavior of the solution is also studied.

On some large global solutions to the incompressible inhomogeneous nematic liquid crystal flows

ABSTRACTIn this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully using the Fourier localization technique and the advantage of weighted function generated by heat kernel. The meanings of this kind of initial data are that we can provide an example of large highly oscillating data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large.

On the regularity criteria for liquid crystal flows

In this paper, we show a regularity criteria for three-dimensional nematic liquid crystal flows. More precisely, we prove that the weak solution (u, d) can be extended beyond T, provided $$u_{3}, \nabla _{h}d\in {L^{p}(0,T; L^{q}(R^{3}))}$$ , $$\frac{2}{p}+\frac{3}{q}\le {\frac{3}{4}+\frac{1}{2q}}(q>{\frac{10}{3}})$$ .

Global well-posedness and decay for a [formula omitted] tensor model of incompressible nematic liquid crystals in [formula omitted

We consider the motion of a viscous incompressible liquid crystal flow in the N-dimensional space RN. We study the global well-posedness of strong solutions in the Lp–Lq maximal regularity class.

Dynamics and Flow Effects in the Beris-Edwards System Modeling Nematic Liquid Crystals

We consider the Beris-Edwards system modelling incompressible liquid crystal flows of nematic type. This couples a Navier-Stokes system for the fluid velocity with a parabolic reaction-convection-diffusion equation for the Q-tensors describing the direction of liquid crystal molecules. In this paper, we study the effect that the flow has on the dynamics of the Q-tensors, by considering two fundamental aspects: the preservation of eigenvalue-range and the dynamical emergence of defects in the limit of high Ericksen number