Pattern Selection in Faraday Waves

Chen, Peilong; Viñals, Jorge

doi:10.1103/physrevlett.79.2670

Cited by 91 publications

(74 citation statements)

References 14 publications

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“…A weakly nonlinear theory based on corresponding amplitude equations has been recently developed to study the stability of this unavoidable hexagonal pattern [20]. Unfortunately, for Turing pattern selection, a nonlinear theory foreseeing different symmetries similar to that found for Faraday wave patterns [21] does not exist up to date. In the strategy of the present paper, we apply simple linear theory and the geometric frustration concept to reduce the usual hexagonal symmetry of the Turing patterns and to produce the biologically important fivefold patterns in small-size disks.…”

Section: Discussionmentioning

confidence: 99%

Size-dependent symmetry breaking in models for morphogenesis

Barrio

Maini

Aragón

et al. 2002

Physica D: Nonlinear Phenomena

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A general property of dynamical systems is the appearance of spatial and temporal patterns due to a change of stability of a homogeneous steady state. Such spontaneous symmetry breaking is observed very frequently in all kinds of real systems, including the development of shape in living organisms. Many nonlinear dynamical systems present a wide variety of patterns with different shapes and symmetries. This fact restricts the applicability of these models to morphogenesis, since one often finds a surprisingly small variation in the shapes of living organisms. For instance, all individuals in the Phylum Echinodermata share a persistent radial fivefold symmetry. In this paper, we investigate in detail the symmetry-breaking properties of a Turing reaction-diffusion system confined in a small disk in two dimensions. It is shown that the symmetry of the resulting pattern depends only on the size of the disk, regardless of the boundary conditions and of the differences in the parameters that differentiate the interior of the domain from the outer space. This study suggests that additional regulatory mechanisms to control the size of the system are of crucial importance in morphogenesis.

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Section: Discussionmentioning

confidence: 99%

Size-dependent symmetry breaking in models for morphogenesis

Barrio

Maini

Aragón

et al. 2002

Physica D: Nonlinear Phenomena

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“…Indeed, Keolian et al (1981) showed the appearance of capillary waves in shallow water at subharmonic frequencies with other ratios of f=i, where i ¼ 2, 4, 12, 14, 16, 18, 20, 22, 24, 28, and 35. This, and more complex wave patterns [see, e.g., Binks and van de Water (1997)], is made possible, as shown by Chen and Viñals (1997), by the interaction of triad (three-wave) resonant interactions of the waves on the free surface that does not rely on the quadratic nonlinearity presumed by the weakly nonlinear form that results in the Mathieu equation form. Beyond discrete modes of capillary waves, the appearance of turbulent cascades in the wave interaction for larger systems where gravity is relevant has been shown to exist by .…”

Section: =3mentioning

confidence: 99%

“…4(b)] only came into popular usage in telecommunications about a decade ago (Hode et al, 1995), probably due to the challenges of properly analyzing the electrical and acoustic fields present in the piezoelectric substrate (Chen et al, 1992). By providing internally placed electrodes to reflect the electroacoustic wave and induce constructive interference, nearly all of the acoustic energy generated via transduction by the SPUDT propagates out from the electrode in a single direction.…”

Section: Ultrasonic Devicesmentioning

confidence: 99%

Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics

2011

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This article reviews acoustic microfluidics: the use of acoustic fields, principally ultrasonics, for application in microfluidics. Although acoustics is a classical field, its promising, and indeed perplexing, capabilities in powerfully manipulating both fluids and particles within those fluids on the microscale to nanoscale has revived interest in it. The bewildering state of the literature and ample jargon from decades of research is reorganized and presented in the context of models derived from first principles. This hopefully will make the area accessible for researchers with experience in materials science, fluid mechanics, or dynamics. The abundance of interesting phenomena arising from nonlinear interactions in ultrasound that easily appear at these small scales is considered, especially in surface acoustic wave devices that are simple to fabricate with planar lithography techniques common in microfluidics, along with the many applications in microfluidics and nanofluidics that appear through the literature.

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“…A cutoff at n = N (in the present work N = 100) leads to a self-consistent equation for the acceleration amplitude a [18,19],…”

Section: Linear Stability Analysismentioning

confidence: 99%

“…The nonlinear behavior of the system has been studied theoretically in a great number of papers (see [16,17,18,19,20] and references therein). An amplitude equation for an infinitely deep layer of a viscous fluid is derived by Chen and Viñals [19]. A good agreement between the predicted frequencies for the transition between regions of different symmetry [19] and the experimental observations [5] is found.…”

Section: Introductionmentioning

confidence: 99%

Faraday instability on viscous ferrofluids in a horizontal magnetic field: Oblique rolls of arbitrary orientation

Mekhonoshin

Lange

2002

Phys. Rev. E

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A linear stability analysis of the free surface of a horizontally unbounded ferrofluid layer of arbitrary depth subjected to vertical vibrations and a horizontal magnetic field is performed. A nonmonotonic dependence of the stability threshold on the magnetic field is found at high frequencies of the vibrations. The reasons of the decrease of the critical acceleration amplitude caused by a horizontal magnetic field are discussed. It is revealed that the magnetic field can be used to select the first unstable pattern of Faraday waves. In particular, a rhombic pattern as a superposition of two different oblique rolls can occur. A scaling law is presented which maps all data into one graph for the tested range of viscosities, frequencies, magnetic fields and layer thicknesses.

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Pattern Selection in Faraday Waves

Abstract: We present a systematic nonlinear theory of pattern selection for parametric surface waves (Faraday waves), not restricted to fluids of low viscosity. Typeset using REVT E X 1

Cited by 91 publications

References 14 publications

Size-dependent symmetry breaking in models for morphogenesis

Size-dependent symmetry breaking in models for morphogenesis

Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics

Faraday instability on viscous ferrofluids in a horizontal magnetic field: Oblique rolls of arbitrary orientation

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