Resonant interactions in Bénard-Marangoni convection in cylindrical containers

Echebarria, Blas; Krmpotic, D.; Pérez–García, C.

doi:10.1016/s0167-2789(96)00167-4

Cited by 14 publications

(7 citation statements)

References 18 publications

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“…This bifurcation arises in a variety of applications, including Rayleigh-Bénard convection in a plane layer [6,17], Marangoni convection in a cylindrical container [9], cellular flame patterns on a circular porous plug burner [15], turbulent boundary layer breakdown [8], and the von Kármán flow between two exactly counter-rotating disks [18]. In each of these cases imperfections may be present that break the assumed O(2) symmetry, i.e., the symmetry generated by rotations and reflection of a circle, or translations and reflections along the real line, modulo an imposed spatial period.…”

Section: Resultsmentioning

confidence: 99%

Dynamics in the 1:2 spatial resonance with broken reflection symmetry

Porter

Knobloch

2005

Physica D: Nonlinear Phenomena

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The dynamics near the 1:2 spatial resonance in systems with broken O(2) symmetry are described. The system is assumed to remain SO(2) invariant, and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the O(2)-symmetric system. The symmetry-breaking destroys these cycles, and introduces a variety of new bifurcations into the system, some which are associated with complex behavior. The possible scenarios with increasing symmetry-breaking are described and possible applications of the results are discussed.

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Section: Resultsmentioning

confidence: 99%

Dynamics in the 1:2 spatial resonance with broken reflection symmetry

Porter

Knobloch

2005

Physica D: Nonlinear Phenomena

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“…This relation between Rayleigh and Marangoni number has been addressed in a previous work. 23 On the other hand, we do not vary R freely since we are interested only in the values it takes for our particular physical situation, therefore it is determined by ⌬T. These parameters are equivalent to the set of physical ones, d, ⌬T, l, ␦T c , and h. In this work we use any of these sets to define the physical state.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Instabilities in a laterally heated liquid layer

Mancho

Herrero

2000

Physics of Fluids

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We study a convection problem in a free-surface container with lateral walls heated at different temperatures. The effects of buoyancy and thermocapillarity are taken into account. A basic convective state appears as soon as a temperature gradient with nonzero horizontal component is applied. This state bifurcates to new convective solutions for further values on the imposed temperature gradient. Our main contribution is to consider this situation in a container finite not only in the vertical coordinate, but also in the direction of the gradient. The third dimension is kept infinite. We determine the basic state, compare it with the usual one of parallel flow approach, and study its stability. When the lateral heating walls are considered new results are found. The boundary conditions on the top surface are no longer restricted to those that allow analytical solutions for the basic state, and we have considered for the heat interchange with the atmosphere the Newton law with constant ambient temperature. Due to this boundary condition, two control parameters related to the temperature field appear. One is the temperature difference between lateral walls as in previous research, and the new one is the temperature difference between the atmosphere and the cold wall. After a stationary bifurcation a three-dimensional structure which along the infinite direction consists of longitudinal rolls grows. On the vertical plane along the gradient direction this structure is nonhomogeneous but located near the hot side. These features coincide with observations of recent experiments.

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“…Proctor and Jones [1988] analyzed a two-layer system using the normal form (4), locating parameter regimes with structurally stable heteroclinic cycles, but did not study the dynamics in the original pdes. Perhaps the best system for further study of the dynamical behavior studied here is provided by the Bénard-Marangoni problem in a cylinder [Echebarría et al, 1997] where heteroclinic cycles were also identified. Simulations of the relevant pdes as well as further experiments on this system are therefore of great interest.…”

Section: Discussionmentioning

confidence: 99%

Robust Heteroclinic Cycles in Two-Dimensional Rayleigh–bénard Convection Without Boussinesq Symmetry

Mercader

Prat

Knobloch

2002

Int. J. Bifurcation Chaos

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The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton's law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually defined is shown to be inappropriate as a bifurcation parameter since the temperature difference across the layer depends on the amplitude of convection and hence changes as convection evolves at fixed external parameter values. A modified Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one. Near the 1 : 2 spatial resonance in low Prandtl number fluids these effects open up intervals of Rayleigh number with no stable solutions in the form of steady convection or steadily traveling waves. Direct numerical simulations in two dimensions show that in such intervals the dynamics typically take the form of a nearly heteroclinic modulated traveling wave. This wave may be quasiperiodic or chaotic.

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Resonant interactions in Bénard-Marangoni convection in cylindrical containers

Cited by 14 publications

References 18 publications

Dynamics in the 1:2 spatial resonance with broken reflection symmetry

Dynamics in the 1:2 spatial resonance with broken reflection symmetry

Instabilities in a laterally heated liquid layer

Robust Heteroclinic Cycles in Two-Dimensional Rayleigh–bénard Convection Without Boussinesq Symmetry

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