Liquid Crystal Flows Research Articles

Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes

In this paper, we consider numerical approximations for a model of smectic-A liquid crystal flows in its weak flow limit. The model, derived from the variational approach of the de Gennes free energy, is consisted of a highly nonlinear system that couples the incompressible Navier-Stokes equations with two nonlinear order parameter equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop an unconditionally energy stable, linear and decoupled time marching numerical scheme for the reduced model in the weak flow limit. We also rigorously prove that the numerical scheme obeys the energy dissipation law at the discrete level. Various numerical simulations are presented to demonstrate the accuracy and the stability of the scheme.

On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

This work is concerned with the solvability of a Navier-Stokes/Q-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

Hydrodynamic instabilities at decay flow of nematic liquid crystals through the plane capillary of a variable gap

In this paper we consider the possibility to study hydrodynamic instabilities (HDI) in a specific type of a decay flow existing at a rise of a nematic liquid crystal (NLC) in the flat capillary of a variable gap. In this case the capillary can be replaced by a number of parallel channels of different gaps with different instant values of a pressure gradient, which induces a flow. The last feature makes such flow to be different from the previously studied decay flow [1] with a constant value of a pressure gradient, applied to the channels of different thickness.

Nonuniqueness of nematic liquid crystal flows in dimension three

For suitable initial and boundary data, we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three. These solutions are in the axisymmetric class with bounded energy and “backward bubbling” at a large time.

Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy is small. In particular, the initial density may contain vacuum states and even have compact support. Moreover, the large time behavior of the solution is also investigated.

Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains

We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish $\mathrm{L}^{p}$ estimates. For the whole space and the half space case we use a reduction of the Stokes equations to $(n-1)$ heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of $\mathcal{R}$-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.

Blow-up criterion of classical solutions for the incompressible nematic liquid crystal flows

In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in R3. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if (u,d) is smooth up to time T provided that ∫0T||∇ × u(t,⋅)||BMO(R3)+||∇ d(t,⋅)||L4(R3)8dt < ∞.

Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals

The Cauchy problem for the compressible flow of nematic liquid crystals in the framework of critical spaces is considered. We first establish the existence and uniqueness of global solutions provided that the initial data are close to some equilibrium states. This result improves the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), pp. 2678-2699] through relaxing the regularity requirement of the initial data in terms of the director field. We then consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to the compressible flow of liquid crystals converges to the solution to the corresponding incompressible model in some function spaces. Moreover, the accurate converge rates are obtained.

The global well-posedness of the Cauchy problem for a liquid crystal system

ABSTRACTThis paper concerns the existence and asymptotic behavior for a system modeling the nematic liquid crystal flow, which consists of a subsystem of compressible Navier–Stokes equations coupling with heat flow for harmonic mapping. Under some assumptions on initial data, we established global existence and asymptotic behavior of solutions in of the Cauchy problem for liquid crystal equations.

Time‐periodic solution to the compressible nematic liquid crystal flows in periodic domain

In this paper, we consider the time‐periodic solution to a simplified version of Ericksen‐Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals with a time‐periodic external force in a periodic domain in . By using an approach of parabolic regularization and combining with the topology degree theory, we establish the existence of the time‐periodic solution to the model under some smallness and symmetry assumptions on the external force. Then, we give the uniqueness of the periodic solution of this model.

On the interior regularity criteria for liquid crystal flows

In this paper, we provide some new regularity criteria for suitable weak solutions to the 3D nematic liquid crystal flow in terms of the velocity field. More precisely, we prove that the suitable weak solution (u,d) is regular provided that either the scaled Lp,q norm of u, or the scaled Lp,q norm of ∇×u is sufficiently small.

Global well-posedness of the non-isothermal model for incompressible nematic liquid crystals

In this paper, a macromolecular non-isothermal model for the incompressible hydrodynamics flow of nematic liquid crystals on mathbb {T}^{3} is considered. By a Galerkin approximation, we prove the local existence of a unique strong solution if the initial data u_{0}, d_{0} and theta_{0} satisfy some natural conditions and provided that the viscosity coefficients μ and the heat conductivity κ, h, which are temperature dependent, are properly differentiable and bounded. Moreover, with small initial data, we can extend the local strong solution to be a global one by the argument of contradiction. In this case, the exponential time decay rate is also established.

Large‐time behavior of solutions to a 1D liquid crystal system

In this paper, we prove the large‐time behavior, as time tends to infinity, of solutions in and for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier‐Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.

Global solution to the nematic liquid crystal flows with heat effect

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain Ω⊂RN (N=2,3) are studied in this paper. Following Danchin's method in [7], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of a unique strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu–Wang's result in [21] to the non-isothermal case.

Optimal Error Estimates of Semi-implicit Galerkin Method for Time-Dependent Nematic Liquid Crystal Flows

This paper focuses on the optimal error estimates of a linearized semi-implicit scheme for the nematic liquid crystal flows, which is used to describe the time evolution of the materials under the influence of both the flow velocity and the microscopic orientation configurations of rod-like liquid crystal flows. Optimal error estimates of the scheme are proved without any restriction of time step by using an error splitting technique proposed by Li and Sun. Numerical results are provided to confirm the theoretical analysis and the stability of the semi-implicit scheme.

BKM's criterion for the 3D nematic liquid crystal flows in Besov spaces of negative regular index

BKM's criterion for the 3D nematic liquid crystal flows in Besov spaces of negative regular index

Collisional dissipation rate in shearing flows of granular liquid crystals.

We make use of discrete-element-method numerical simulations of inelastic frictionless cylinders in simple shearing at different length-to-diameter ratios and solid volume fractions to analyze the rate of collisional dissipation of the fluctuation kinetic energy. We show that the nonspherical geometry of the particles is responsible, by inducing rotation, for increasing the dissipation rate of the fluctuation kinetic energy with respect to that for frictionless spheres. We also suggest that the partial alignment of the cylinders induced by shearing concurs with the particle inelasticity in generating correlation in the velocity fluctuations and thus affecting the collisional dissipation rate as the solid volume fraction increases. Finally, we propose simple phenomenological modifications to the expression of the collisional dissipation rate of kinetic theory of granular gases to take into account our findings.

A 2D numerical study of polar active liquid crystal flows in a cavity

We study systematically dynamics of polar active liquid crystals in a 2D cavity flow geometry using a continuum model based on the polarity vector. We devise a numerical scheme based on the projection method and discretized using finite difference methods to solve the model equations and to investigate various patterns and their dependence on the active stresses, the self-propelled speed, the geometry of the active liquid crystal molecule, and the imposed shear rate subject to physical boundary conditions. In addition to the already known 2D spatial temporal patterns obtained using periodic boundary conditions by various groups, we discover new emergent out-of-plane structures both in steady state and spatial-temporal patterns. We then qualitatively categorize the various patterns with respect to the active parameters as well as the geometric parameter of the active liquid crystal molecules into three types: spatially inhomogeneous steady state, periodic patterns, and irregularly oscillatory patterns in both space and time.

A projection‐based time‐splitting algorithm for approximating nematic liquid crystal flows with stretching

AbstractA numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time‐splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection‐based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal‐order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.

A representation formula of incompressible liquid crystal flow and its applications

In this paper, we give a representation formula for the solutions of incompressible liquid crystal flow in arbitrary dimensions. As a direct application, one can obtain a strong global solution in two dimensions, as long as the image of the initial orientation field of the liquid crystal lies in a plane, or is a nontrivial curve on the unit sphere. The results also hold for inhomogeneous incompressible liquid crystals. This generalizes the results in Gong et al (2015 Nonlinearity 28 3677–94) and Dong and Lei (2012 (arXiv:org/abs/1205.3697)).