scholarly journals The helicity and vorticity of liquid-crystal flows

2010 ◽  
Vol 467 (2128) ◽  
pp. 1197-1213 ◽  
Author(s):  
François Gay-Balmaz ◽  
Cesare Tronci
Keyword(s):  
Fluid Dynamics ◽  
Liquid Crystal ◽  
Ideal Fluid ◽  
Arbitrary Shape ◽  
Rotational Symmetry ◽  
Two Dimensions ◽  
Alignment Tensor ◽  
Minimal Coupling ◽  
Helicity Conservation ◽  
Nematic Liquid Crystal Flows

We present explicit expressions of the helicity conservation in nematic liquid-crystal flows, for both the Ericksen–Leslie and Landau–de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation for a modified vorticity involving both velocity and structure fields (e.g. director and alignment tensor). This equation for the modified vorticity shares many relevant properties with ideal fluid dynamics, and it allows for vortex-filament configurations, as well as point vortices, in two dimensions. We extend all these results to particles of arbitrary shape by considering systems with fully broken rotational symmetry.

scholarly journals An Elementary Approximation of Dwell Time Algorithm for Ultra-Precision Computer-Controlled Optical Surfacing

Micromachines ◽  
2021 ◽  
Vol 12 (5) ◽  
pp. 471
Author(s):  
Yajun Wang ◽  
Yunfei Zhang ◽  
Renke Kang ◽  
Fang Ji
Keyword(s):  
Dwell Time ◽  
Convergence Rates ◽  
Rotational Symmetry ◽  
Time Algorithm ◽  
Two Dimensions ◽  
Efficient Computation ◽  
Deconvolution Problem ◽  
Ultra Precision ◽  
Computer Controlled Optical Surfacing ◽  
Computer Controlled

The dwell time algorithm is one of the key technologies that determines the accuracy of a workpiece in the field of ultra-precision computer-controlled optical surfacing. Existing algorithms mainly consider meticulous mathematics theory and high convergence rates, making the computation process more uneven, and the flatness cannot be further improved. In this paper, a reasonable elementary approximation algorithm of dwell time is proposed on the basis of the theoretical requirement of a removal function in the subaperture polishing and single-peak rotational symmetry character of its practical distribution. Then, the algorithm is well discussed with theoretical analysis and numerical simulation in cases of one-dimension and two-dimensions. In contrast to conventional dwell time algorithms, this proposed algorithm transforms superposition and coupling features of the deconvolution problem into an elementary approximation issue of function value. Compared with the conventional methods, it has obvious advantages for improving calculation efficiency and flatness, and is of great significance for the efficient computation of large-aperture optical polishing. The flatness of φ150 mm and φ100 mm workpieces have achieved PVr150 = 0.028 λ and PVcr100 = 0.014 λ respectively.

scholarly journals Regularity Criterion for the 3D Nematic Liquid Crystal Flows

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Nematic Liquid Crystal ◽  
Regularity Criterion ◽  
Hydrodynamic Theory ◽  
Nematic Liquid Crystal Flows

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

INFLUENCE OF DIMENSIONALITY AND SURFACE ORDERING ON PHASE TRANSITIONS: STUDIES OF FREELY-SUSPENDED LIQUID-CRYSTAL FILMS

1994 ◽  
Vol 08 (22) ◽  
pp. 3051-3082 ◽  
Author(s):  
CHRISTIAN BAHR
Keyword(s):  
Phase Transitions ◽  
Liquid Crystal ◽  
Two Dimensions ◽  
Smectic Liquid Crystals ◽  
Liquid Crystal Films ◽  
Integral Number ◽  
Smectic Phases ◽  
Crystal Films ◽  
Freely Suspended ◽  
Freely Suspended Films

Freely-suspended films of smectic liquid crystals can be regarded as membranes consisting of a stack of an integral number of molecular (smectic) layers with the layer planes being parallel to the two free surfaces. Because of their variable thickness (between thousands and only two layers) and the large variety of phase transitions between different smectic phases, freely-suspended films are excellent systems to study the influence of the dimensional cross-over from three to two dimensions on phase transitions. Further, because the free surface of a liquid crystal has a strong ordering effect (contrary to solids which exhibit generally surface-induced disorder), freely-suspended films are well-suited for the study of the effect of enhanced surface order on phase transitions. A review of the corresponding experimental work is given.

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