Liquid Crystal Flows Research Articles

Global Weak Solutions of 3D Compressible Nematic Liquid Crystal Flows with Discontinuous Initial Data and Vacuum

In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic nematic liquid crystal flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in L2, and the initial density, the gradients of initial velocity/liquid crystal director field are bounded in L?, L2 and H1, respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.

Global Existence of Weak Solutions of the Nematic Liquid Crystal Flow in Dimension Three

AbstractFor any bounded smooth domain , we establish the global existence of a weak solution of the initial boundary value (or the Cauchy) problem of the simplified Ericksen‐Leslie system LLF modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data , with (the upper hemisphere). Furthermore, (u,d) satisfies the global energy inequality.© 2016 Wiley Periodicals, Inc.

Transitions in Poiseuille flow of nematic liquid crystal

Recent experiments by Sengupta et al. (Phys. Rev. Lett. 2013) [9] revealed interesting transitions that can occur in flow of nematic liquid crystal under carefully controlled conditions within a long microfluidic channel of width much larger than height, and homeotropic anchoring at the walls. At low flow rates the director field of the nematic adopts a configuration that is dominated by the surface anchoring, being nearly parallel to the channel height direction over most of the cross-section; but at high flow rates there is a transition to a flow-dominated state, where the director configuration at the channel centerline is aligned with the flow (perpendicular to the channel height direction). We analyze simple channel-flow solutions to the Leslie–Ericksen model for nematics. We demonstrate that two solutions exist, at all flow rates, but that there is a transition between the elastic free energies of these solutions: the anchoring-dominated solution has the lowest energy at low flow rates, and the flow-dominated solution has lowest energy at high flow rates.

Three-dimensional coating flow of nematic liquid crystal on an inclined substrate

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

On Global Behavior of Weak Solutions of Compressible Flows of Nematic Liquid Crystals

In this article, we investigate the global behavior of weak solutions of a simplified Ericksen-Leslie system for compressible flows of nematic liquid crystals in time in a bounded three-dimension domain-arbitrary forces. By adapting the arguments for the compressible Navier–Stokes equations, and carefully analyzing the direction field of liquid crystals in the equations of angular momentum, we show the existence of bounded absorbing sets, global bounded trajectories, and global attractors to weak solutions of compressible flows of nematic liquid crystals with the adiabatic constant γ>5/3.

On the well-posedness for the compressible nematic liquid crystal flow

The Cauchy problem for the compressible flow of nematic liquid crystals in dimension ( N ≧ 2) is considered. Employing the Littlewood-Paley theory and Schauder-Tychonoff fixed point theorem, we prove the local well-posedness of the system for large initial data in critical Besov spaces based on the L p framework under the sole natural assumption that the initial density is bounded away from zero.

Uniform well-posedness and low Mach number limit to the compressible nematic liquid crystal flows in a bounded domain

We prove global-in-time and uniform-in-ϵ of the strong solutions to the 3D compressible nematic liquid crystal flows in a bounded domain, where ϵ is the Mach number. Consequently, we obtain the strong solution of compressible nematic liquid crystal model that converges to that of incompressible nematic liquid crystal model. This is the first result on the low Mach number limit for compressible nematic liquid crystal flows in 3D bounded domain. Our proof relies on the dedicated estimates on the solutions and the subtle use of the boundary conditions.

A Splitting in Time Scheme and Augmented Lagrangian Method for a Nematic Liquid Crystal Problem

We study the numerical approximation of nematic liquid crystal flows governed by a Ericksen–Leslie problem. This problem couples the incompressible Navier–Stokes dynamic with a gradient flow system related to the orientation unitary vector of molecules. First, a two sub-step viscosity-splitting time scheme is proposed. The first sub-step couples diffusion and convection terms whereas the second one is concerned with diffusion terms and constraints (divergence free and unit director field). Then, in the first sub-step we use a Gauss–Seidel decoupling algorithm, and in the second sub-step, we use Uzawa type algorithms on augmented Lagrangian functionals to overcome the divergence free and the unit director field constraints. From the computational point of view, it is a fully decoupled linear scheme (where all systems to solve are for scalar variables). Some numerical experiments in 2D domains are carried out by using only linear finite elements in space, confirming at least numerically the viability and the convergence of our scheme.

Space–time derivative estimates of the Koch–Tataru solutions to the nematic liquid crystal system in Besov spaces

In recent paper [7], Y. Du and K. Wang (2013) proved that the global-in-time Koch–Tataru type solution (u,d) to the n-dimensional incompressible nematic liquid crystal flow with small initial data (u0,d0) in BMO−1×BMO has arbitrary space–time derivative estimates in the so-called Koch–Tataru space norms. The purpose of this paper is to show that the Koch–Tataru type solution satisfies the decay estimates for any space–time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch–Tataru type solution (u,d) to the nematic liquid crystal flow with initial data (u0,d0)∈BMO−1×BMO and ‖u0‖BMO−1+[d0]BMO≤ε for some small enough ε>0, and for any positive integers k and m, one has‖tk+m2(∂tk∇mu,∂tk∇m∇d)‖L˜∞(R+,B˙∞,∞−1)∩L˜1(R+;B˙∞,∞1)≤ε. Furthermore, we shall give that the solution admits a unique trajectory which is Hölder continuous with respect to space variables.

Global well-posedness for the dynamical Q-tensor model of liquid crystals

We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.

Global existence of solutions to the 2D incompressible liquid crystal flow with fractional diffusion

In this paper, we consider the global regularity for 2D liquid crystal model with fractional diffusion. Global regularity is proved for two cases: 0<β<1, α+β=2 or β=1, α>0.

Global existence and temporal decay for the nematic liquid crystal flows

This paper is concerned with the Cauchy problem of the three-dimensional nematic liquid crystals. We establish the global well-posedness and time decay rates of the classical solutions with smooth initial data which are of small energy by using Fourier splitting technique and pure energy method.

Incompressible Limit for the Compressible Flows of Nematic Liquid Crystals in the Whole Space

We consider the compressible flows of liquid crystals arising in a variety of scientific examples. We here study the incompressible limit of weak solutions of the compressible flows of nematic liquid crystals in the whole spaceR2.

On Strong Dynamics of Compressible Nematic Liquid Crystals

Given a uniform $W^{3-1/q}_q$-domain $\Omega \subseteq \mathbb R^N$ for $N<q<\infty$, we consider a simplified Ericksen--Leslie system modeling the flow of compressible nematic liquid crystals based on Lin and Liu [Comm. Pure Appl. Math., 48 (1995), pp. 501--537]. We show the unique existence of local-in-time strong solutions. Furthermore, if $\Omega$ is bounded and initial data are chosen suitably small, we obtain global-in-time strong solutions. Our approach is based on maximal regularity estimates of the compressible Navier--Stokes equation by Enomoto, von Below, and Shibata [Ann. Univ. Ferrara, 60 (2014), pp. 55--89] and maximal regularity estimates for the Neumann problem as a consequence of Weis's 2001 vector-valued Fourier multiplier theorem.

A Time-Splitting Finite-Element Stable Approximation for the Ericksen--Leslie Equations

In this paper we propose an unconditional energy-stable time-splitting finite-element scheme for approximating the Ericksen--Leslie equations governing the flow of nematic liquid crystals. These equations are to be solved for a velocity vector field and a scalar pressure as well as a director vector field representing the direction along which the molecules of the liquid crystal are oriented. The algorithm is designed at two levels. First, at the variational level, the velocity, pressure, and director are computed separately, but the director field has to be computed together with an auxiliary variable (associated to the equilibrium equation for the director) in order to deduce a priori energy estimates. Second, at the algebraic level, one can avoid computing such an auxiliary variable if this is approximated by a piecewise constant finite-element space. Therefore, these two steps give rise to a numerical algorithm that computes separately only the primary variables: velocity, pressure, and director vecto...

The effect of anchoring on the nematic flow in channels.

Understanding the flow of liquid crystals in microfluidic environments plays an important role in many fields, including device design and microbiology. We perform hybrid lattice-Boltzmann simulations of a nematic liquid crystal flowing under an applied pressure gradient in two-dimensional channels with various anchoring boundary conditions at the substrate walls. We investigate the relationship between the flow rate and the pressure gradient and the corresponding profile of the nematic director, and find significant departures from the linear Poiseuille relationship. We also identify a morphological transition in the director profile and explain this in terms of an instability in the dynamical equations. We examine the qualitative and quantitative effects of changing the type and strength of the anchoring. Understanding such effects may provide a useful means of quantifying the anchoring of a substrate by measuring its flow properties.

A regularity criterion for the 3D density-dependent incompressible flow of liquid crystals with vacuum

A regularity criterion for the 3D density-dependent incompressible flow of liquid crystals with vacuum

Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow

Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow

Global Finite Energy Weak Solutions to the Compressible Nematic Liquid Crystal Flow in Dimension Three

In this paper, we consider the initial and boundary value problem of a simplified compressible nematic liquid crystal flow in $\Omega\subset\mathbb R^3$. We establish the existence of global weak solutions, provided the initial orientational director field $d_0$ lies in the hemisphere $\mathbb S^2_+$.

Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry

Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry