Wavenumber selection and Eckhaus instability in Couette-Taylor flow

Ahlers, Guenter; Cannell, David S.; Dominguez-Lerma, M. A.; Heinrichs, Richard

doi:10.1016/0167-2789(86)90129-6

Cited by 78 publications

(72 citation statements)

References 40 publications

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“…A key issue in these systems is the selection of patterns from many coexisting metastable states with different geometries and wave numbers [2,3]. The answer to this fundamental question still remains a mystery even after decades of explorations.…”

Section: Introductionmentioning

confidence: 99%

Minimum-action paths for wave-number selection in nonequilibrium systems

2016

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The problem of wave-number selections in nonequilibrium pattern-forming systems in the presence of noise is investigated. The minimum-action method is proposed to study the noise-induced transitions between the different spatiotemporal states by generalizing the traditional theory previously applied in low-dimensional dynamical systems. The scheme is shown as an example in the stabilized Kuramoto-Sivashinsky equation. The present method allows us to conveniently find the unique noise selected state, in contrast to previous work using direct simulations of the stochastic partial differential equation, where the constraints of the simulation only allow a narrow band to be determined.

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Section: Introductionmentioning

confidence: 99%

Minimum-action paths for wave-number selection in nonequilibrium systems

2016

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“…One of the successes in the study of pattern-forming systems is the agreement between the calculated and measured k E (ǫ) for TVF which has been found with three different ratios of the cylinder radii. [4,7,9,10,14] Theoretically, phase pinning (and thus the Eckhaus instability) at small ǫ is expected when the boundary conditions at the system ends z = 0, L correspond to a large amplitude A(0) = A(L) = A 0 of the velocity field, say A 0 = O(1), while in the system interior the amplitude A(z) is small, say O(ǫ 1/2 ). This situation closely corresponds to the TVF system with rigid ends where the influence of the Ekman vortex can be approximated by this boundary condition.…”

Section: Introductionmentioning

confidence: 99%

“…The (more narrow) stable band of states is limited by the long-wavelength Eckhaus instability. [3,4,5,6,7,8,9,10,11,12,13,14] The Eckhaus instability is a bulk instability which manifests itself in the system interior where one pattern wavelength (one pair of Taylor vortices) is either gained or lost, depending on whether k is larger or smaller than the stable band limited by k E (ǫ). One of the successes in the study of pattern-forming systems is the agreement between the calculated and measured k E (ǫ) for TVF which has been found with three different ratios of the cylinder radii.…”

Section: Introductionmentioning

confidence: 99%

“…This situation closely corresponds to the TVF system with rigid ends where the influence of the Ekman vortex can be approximated by this boundary condition. [15,7,10] The Eckhaus-stable band then has the same width as the band of stable states for the infinite system, [16] but the number of states is finite since only discrete states with an integer number of vortices N = 2L/λ = Lk/π can occur.…”

Section: Introductionmentioning

confidence: 99%

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Boundary limitation of wave numbers in Taylor-vortex flow

Linek

Ahlers

1998

Phys. Rev. E

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We report experimental results for a boundary-mediated wavenumber-adjustment mechanism and for a boundary-limited wavenumber-band of Taylor-vortex flow (TVF). The system consists of fluid contained between two concentric cylinders with the inner one rotating at an angular frequency Ω. As observed previously, the Eckhaus instability (a bulk instability) is observed and limits the stable wavenumber band when the system is terminated axially by two rigid, non-rotating plates. The band width is then of order ǫ 1/2 at small ǫ (ǫ ≡ Ω/Ωc − 1) and agrees well with calculations based on the equations of motion over a wide ǫ-range. When the cylinder axis is vertical and the upper liquid surface is free (i.e. an air-liquid interface), vortices can be generated or expelled at the free surface because there the phase of the structure is only weakly pinned. The band of wavenumbers over which Taylor-vortex flow exists is then more narrow than the stable band limited by the Eckhaus instability. At small ǫ the boundary-mediated band-width is linear in ǫ. These results are qualitatively consistent with theoretical predictions, but to our knowledge a quantitative calculation for TVF with a free surface does not exist.

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“…Figure 5 shows his results. [100,101] Although the Eckhaus boundary had been investigated before in a number of systems, [102 − 104] really only the essentially simultaneous work by Lowe and Gollub [104] on electroconvection in a nematic liquid crystal can be regarded as quantitive. Marco's measurements were in excellent agreement with calculations based on the NS equations and a Galerkin method by Riecke and Paap [105] which are shown by the solid line in Fig.…”

Section: The 1980'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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Wavenumber selection and Eckhaus instability in Couette-Taylor flow

Cited by 78 publications

References 40 publications

Minimum-action paths for wave-number selection in nonequilibrium systems

Minimum-action paths for wave-number selection in nonequilibrium systems

Boundary limitation of wave numbers in Taylor-vortex flow

Over two decades of pattern formation, a personal perspective

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