On the theory of Rayleigh-Bénard convection in homeotropic nematic liquid crystals

Feng, Quanyuan; Decker, Werner; Pesch, Werner; Kramer, Lorenz

doi:10.1051/jp2:1992202

Cited by 14 publications

(12 citation statements)

References 39 publications

(59 reference statements)

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“…A sub-critical bifurcation is signaled by a negative ͉c r ͉ 2 and higher powers in A would have to be included. By the use of a similar ansatz for perturbations ␦A, which contains ␦-functions peaked at qϭ͑Ϯq 0 Ϫs͒, one can investigate stability of the pattern with respect to modulations s. 17 Many features become more transparent when formulated in real ͑position͒ space in terms of amplitude ͑enve-lope͒ or Ginzburg-Landau equations ͑GLE͒. Then one sees that the important information is really condensed in a few parameters and the universal aspects of the systems become apparent.…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

Complex spatiotemporal convection patterns

Pesch

1996

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This paper reviews recent efforts to describe complex patterns in isotropic fluids (Rayleigh-Benard convection) as well as in anisotropic liquid crystals (electro-hydrodynamic convection) when driven away from equilibrium. A numerical scheme for solving the full hydrodynamic equations is presented that allows surprisingly well for a detailed comparison with experiments. The approach can also be useful for a systematic construction of models (order parameter equations). (c) 1996 American Institute of Physics.

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Section: Weakly Nonlinear Analysismentioning

confidence: 99%

Complex spatiotemporal convection patterns

Pesch

1996

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…[12,13] In that case, concentration gradients oppose convection, and concentration diffusion has the slow and heat diffusion the fast time scale. As in the binary mixtures, the Hopf bifurcation in the NLC case is subcritical [9,10]. For the NLC the fully developed nonlinear state no longer is time periodic.…”

Section: Introductionmentioning

confidence: 96%

“…At a critical temperature difference ∆T c (H) the fluid will undergo a transition from conduction to convection. The precise value of ∆T c (H), the nature of the bifurcation at ∆T c (H), and the pattern-formation phenomena beyond ∆T c (H) are expected to depend in interesting ways upon H [7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

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Rayleigh-Bénard convection in a homeotropically aligned nematic liquid crystal

Thomas¹,

Pesch²,

Ahlers³

1998

Phys. Rev. E

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We report experimental results for convection near onset in a thin layer of a homeotropically aligned nematic liquid crystal heated from below as a function of the temperature difference ⌬T and the applied vertical magnetic field H. When possible, these results are compared with theoretical calculations. The experiments were done with three cylindrical cells of aspect ratios ͓͑radius͒/͑height͔͒ ⌫ϭ10.6, 6.2, and 5.0 over the field range 8ՇhϵH/H F Շ80 (H F ϭ20.9, 12.6, and 9.3 G are the Fréedericksz fields for the three cells͒. We used the Nusselt number N ͑effective thermal conductivity͒ to determine the critical Rayleigh number R c and the nature of the transition. We analyzed digital images of the flow patterns to study the dynamics and to determine the mean wave numbers of the convecting states. For h less than a codimension-two field h ct Ӎ46 the bifurcation is subcritical and oscillatory, with traveling-and standing-wave transients. Beyond h ct the bifurcation is stationary and subcritical until a tricritical field h t ϭ57.2 is reached, beyond which it is supercritical. We analyzed the patterns to obtain the critical wave number k c and, for hϽh ct , the Hopf frequency c . In the subcritical range we used the early transients towards the finite-amplitude state for this purpose. The bifurcation sequence as a function of h found in the experiment confirms the qualitative aspects of the theoretical predictions. Even quantitatively the measurements of R c , k c , and ⍀ c are reproduced surprisingly well considering the complexity of the system. However, the value of h ct is about 10% higher than the predicted value and the results for k c are systematically below the theory by about 2% at small h and by as much as 7% near h ct . At h ct , k c is continuous within the experimental resolution whereas the theory indicates a 7% discontinuity. The theoretical tricritical field h t th ϭ51 is somewhat below the experimental one. The fully developed flow above R c for hϽh ct has a very slow chaotic time dependence that is unrelated to the Hopf frequency. For h ct ϽhϽh t the subcritical stationary bifurcation also leads to a chaotic state. The chaotic states persist upon reducing the Rayleigh number below R c , i.e., the bifurcation is hysteretic. Above the tricritical field h t , we find a bifurcation to a time independent pattern which within our resolution is nonhysteretic. However, in this field range, there is a secondary hysteretic bifurcation that again leads to a chaotic state observable even slightly below R c . We discuss the behavior of the system in the high-field limit, and show that at the largest experimental field values R c and k c are within 6% and 1%, respectively, of their values for an infinite field.

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“…In spite of these complexities, it has been possible to carry out quantitative stability analyses of these equations, and under some conditions predictions in the weakly nonlinear regime have been made. [128,155,156] The system is very rich since the director (average alignment direction of the molecules) can be horizontal (known as parallel alignment) or vertical (known as homeotropic alignment). In the homeotropic case convection can occur when heating from below or above.…”

Section: The 1990'smentioning

confidence: 99%

Over two decades of pattern formation, a personal perspective

Ahlers

25 Years of Non-Equilibrium Statistical Mechanics

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Patterns are ubiquitous in the world that surrounds us. They can form via bifurcations, for instance from the spatially uniform state, as a control parameter is varied. Their nature generally is determined by nonlinear terms in the relevant equations of motion, and thus their elucidation is a non-trivial goal in nonlinear physics. In the early 1970's, there was a revival of interest in the condensed-matter physics community in chaos and pattern formation in nonlinear dissipative systems. Experimentalists and theorists brought the tools of their field to bear on these challenging problems. Mostly in terms of his own experiences, the author of this paper reviews some of the issues that have been addressed, some of the techniques that have been applied, and some of the progress that has been made by experimentalists during the two-and-a-half decades since then, and the relationship which these results have to our present-day understanding of nonlinear systems.

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On the theory of Rayleigh-Bénard convection in homeotropic nematic liquid crystals

Cited by 14 publications

References 39 publications

Complex spatiotemporal convection patterns

Complex spatiotemporal convection patterns

Rayleigh-Bénard convection in a homeotropically aligned nematic liquid crystal

Over two decades of pattern formation, a personal perspective

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