Dynamics of the Nematic-Isotropic Sharp Interface for the Liquid Crystal

Fei, Mingwen; Wang, Wei; Zhang, Pingwen; Zhang, Zhifei

doi:10.1137/140994113

Cited by 15 publications

(20 citation statements)

References 27 publications

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“…A similar sharp interface limit also arises in the theory of liquid crystals [21]. In the low temperature regime, the Landau-De Gennes theory predicts the co-existence of an isotropic phase and a nematic phase.…”

Section: )supporting

confidence: 59%

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Sharp Interface Limit for a Stokes/Allen–Cahn System

Abels

Liu

2018

Arch Rational Mech Anal

111

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We consider the sharp interface limit of a coupled Stokes/Allen-Cahn system, when a parameter ε > 0 that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen-Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen-Cahn operator. Moreover, a careful treatment of the coupling terms is needed. Mathematics Subject Classification (2000): Primary: 76T99; Secondary: 35Q30, 35Q35, 35R35, 76D05, 76D45 Key words: Two-phase flow, diffuse interface model, sharp interface limit, Allen-Cahn equation, Free boundary problems 1 INTRODUCTION, MAIN RESULT, AND OVERVIEW is the jump of a function u : Ω × [0, T0] → R 2 at Γt in direction of nΓ t , HΓ t and VΓ t are the curvature and the normal velocity of Γt, both with respect to nΓ t . Furthermore, Dv = 1 2 (∇v + (∇v) T ) and σ = R θ ′ 0 (ρ) 2 dρ, where θ0 is the so-called optimal profile that is the unique solution ofIf the material derivative ∂tvε +vε ·∇vε is added to the right-hand side of (1.5) (i.e., the Navier-Stokes equations are considered), the system (1.5)-(1.9) was already suggested by Liu and Shen in [31] as an alternative approximation of a classical sharp interface model for a two-phase flow of viscous, incompressible, Newtonian fluids, which has advantages for numerical simulations since the Allen-Cahn equation is of second order and not of fourth order as the Cahn-Hilliard equation. On the other hand, for solutions of (1.5)-(1.9) the total mass Ω cε(x, t)dx is in general not preserved in time, in contrast to solutions of (1.1)-(1.4), which is a disadvantage if the model is used to approximate a two-phase flow without phase transitions. However, (1.5)-(1.9) can be considered as a simplified model for a two-phase flow with phase transitions. Such models can yield systems of Navier-Stokes/Allen-Cahn type, cf. e.g. Blesgen [11]. Finally, let us mention that in [31] a rigorous result on the sharp interface limes of (1.5)-(1.9) was announced, which was not published so far to the best of the author's knowledge.The limit system (1.11)-(1.16) was also studied by Liu, Sato, and Tonegawa in [30] if the Stokes equation on the right-hand side is replaced by a modified Navier-Stokes equation for a shear thickening non-Newtonian fluid of power-law type. They constructed weak solutions for this system using a Galerkin approximation by a corresponding Navier-Stokes/Allen-Cahn system. In the proof they pass to the limit in the Ga...

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Section: )supporting

confidence: 59%

“…By restricting (A.20) on Γ we can show that ∂ρṽ1,n(ρ, x, t) = 0, which implies v1,n(ρ, x, t) =ṽ1,n(x, t) on Γ, (A. 21) by showing that (ρ + h1)η ′ (ρ)v0,n(x, t) + divx (ṽ0(x, t) +v0(x, t)dΓη(ρ)) = 0 on Γ.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Sharp Interface Limit for a Stokes/Allen–Cahn System

Abels

Liu

2018

Arch Rational Mech Anal

111

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“…Hence, it becomes important to understand whether the anisotropic energy could affect the static or dynamic behaviors of liquid crystals. A typical example arises from the isotropic-nematic interface problem, in which it is found that whether the elastic energy is isotropic or anisotropic corresponds to different boundary conditions on the interface [6].…”

Section: Introductionmentioning

confidence: 99%

On equilibrium configurations of nematic liquid crystals droplet with anisotropic elastic energy

Wang

Zhang

2017

Res Math Sci

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We investigate the effect of anisotropic elastic energy on defect patterns of liquid crystals confined in a three-dimensional spherical domain within the framework of Landau-de Gennes model. Two typical strong anchoring boundary conditions, namely homeotropic and mirror-homeotropic anchoring conditions, are considered. For the homeotropic anchoring, we find three different configurations: uniaxial hedgehog, ring and split-core, in both cases with or without the anisotropic energy. For the mirror-homeotropic anchoring, there are also three analogue solutions: the uniaxial hyperbolic hedgehog, ring and split-core for the isotropic energy case. However, when the anisotropic energy is taken into account, the numerical results and rigorous analysis reveal that the uniaxial hyperbolic hedgehog is no longer a solution. Indeed, we find ring solution only for negative L 2 (the elastic coefficient of the anisotropic energy), while both split-core and ring solutions can be stable minimizers for positive L 2 . More precisely, the uniaxial hyperbolic hedgehog for L 2 = 0 bifurcates to a split-core solution when L 2 increases and to a ring solution when L 2 decreases. This example shows that the anisotropic energy may significantly affect the symmetry of point defects with degree −1 whenever it is introduced.

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“…which has been proved in [7](proved by a new method in [14]). And the next singular term 1 ε Ω f ′′′ (θ(z)) u (1) (x, t, z)v 2 dx will vanish in some sense due to the property Lemma 3.5 of u (1) .…”

Section: Introductionmentioning

confidence: 92%

“…. Setv = vJ 1 2 , from Lemma 5.8 in [14] (we show the proof in Appendix for completeness of this paper) one gets…”

Section: )mentioning

confidence: 99%

Sharp interface limit of a diffuse interface model for tumor-growth

Fei

Tao

Wang

2019

Communications in Mathematical Sciences

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we consider the asymptotic limit of a diffuse interface model for tumor-growth when a parameter ε proportional to the thickness of the diffuse interface goes to zero. An approximate solution which shows explicitly the behavior of the true solution for small ε will be constructed by using matched expansion method. Based on the energy method, a spectral condition in particular, we establish a smallness estimate of the difference between the approximate solution and the true solution.

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Dynamics of the Nematic-Isotropic Sharp Interface for the Liquid Crystal

Cited by 15 publications

References 27 publications

Sharp Interface Limit for a Stokes/Allen–Cahn System

Sharp Interface Limit for a Stokes/Allen–Cahn System

On equilibrium configurations of nematic liquid crystals droplet with anisotropic elastic energy

Sharp interface limit of a diffuse interface model for tumor-growth

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