On the Phase Dynamics of Hexagonal Patterns

Lauzeral, J.; Métens, Stephane; Walgraef, Daniel

doi:10.1209/0295-5075/24/9/002

Cited by 38 publications

(29 citation statements)

References 6 publications

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“…In a perfect hexagonal structure the most stable defect is realized by means of a pair of dislocations (with opposite topological charge and same spatial position), that occur in two of the three components of the order parameter, corresponding to the two independent geometrical phases of a hexagon [Echebarria, 1998;Lauzeral et al, 1993].…”

Section: Resultsmentioning

confidence: 99%

Defect Dynamics During a Quench in a Bénard–marangoni Convection System

González-Viñas

Casado

Burguete

et al. 2001

Int. J. Bifurcation Chaos

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We report experimental evidence of defect formation and dynamics in a symmetry breaking transition for a conduction-convection Bénard-Marangoni system. As opposite to the behavior of perfect patterns, defects appear to interact in a spatial region, responsible for the formation of bounded states that survive much longer than the characteristic time scales. The analysis of the transient defect dynamics allows to define this defect interaction region in the space, giving rise to penta-hepta-like defects on top of the hexagonal pattern. Other defect configurations are shown to disappear rapidly either through dislocations moving toward the boundaries or through dislocation-dislocation annihilation. This evidence suggests that the scaling law of defects in the final structure versus quench time might be investigated by analyzing the probability of two or more dislocations to appear in the same interaction region.

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Section: Resultsmentioning

confidence: 99%

Defect Dynamics During a Quench in a Bénard–marangoni Convection System

González-Viñas

Casado

Burguete

et al. 2001

Int. J. Bifurcation Chaos

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“…We first review and discuss the results for infinite Prandtl number [25,29,27]. The long-wave analysis of sec.3 revealed two phase instabilities associated with the longitudinal and the transverse phase mode, respectively.…”

Section: General Linear Stability Analysismentioning

confidence: 99%

Mean flow in hexagonal convection: stability and nonlinear dynamics

Young¹,

Riecke²

2002

Physica D: Nonlinear Phenomena

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Weakly nonlinear hexagon convection patterns coupled to mean flow are investigated within the framework of coupled Ginzburg-Landau equations. The equations are in particular relevant for non-Boussinesq Rayleigh-Bénard convection at low Prandtl numbers. The mean flow is found to (1) affect only one of the two longwave phase modes of the hexagons and (2) suppress the mixing between the two phase modes. As a consequence, for small Prandtl numbers the transverse and the longitudinal phase instability occur in sufficiently distinct parameter regimes that they can be studied separately. Through the formation of penta-hepta defects, they lead to different types of transient disordered states. The results for the dynamics of the penta-hepta defects shed light on the persistence of grain boundaries in such disordered states.

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“…For hexagons, slight perturbations can be written as Ai = (if + rj)exp(iq-Xi+ </>j). Far from the Hopf bifurcation the dynamics is governed by two independent phase components <f> = (<f) x , <p y ) related to the translationally invariant modes in a hexagonal lattice by <f) x = -(02 + 03) and 4>y = -(02 -03)/v / 3-Without rotation, they evolve according to the expression [Lauzeral et al, 1993;Hoyle, 1995;Echebarria & Perez-Garcia, 1998]…”

Section: Sideband Instabilitiesmentioning

confidence: 99%

Hexagonal Patterns in a Model for Rotating Convection

Madruga

Pérez–García

2004

Int. J. Bifurcation Chaos

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We study a model equation that mimics convection under rotation in a fluid with temperaturedependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Kiippers-Lortz instability [Kuppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a weakly nonlinear analysis to find out explicitly the coefficients in the amplitude equation as functions of the rotation rate. These equations describe hexagons and oscillating hexagons quite well, and include the Busse-Heikes (BH) model [Busse & Heikes, 1980] as a particular case. The sideband instabilities as well as short wavelength instabilities of such hexagonal patterns are discussed and the threshold for oscillating hexagons is determined.

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On the Phase Dynamics of Hexagonal Patterns

Cited by 38 publications

References 6 publications

Defect Dynamics During a Quench in a Bénard–marangoni Convection System

Defect Dynamics During a Quench in a Bénard–marangoni Convection System

Mean flow in hexagonal convection: stability and nonlinear dynamics

Hexagonal Patterns in a Model for Rotating Convection

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