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Onset of patterns in an oscillated granular layer: continuum and molecular dynamics simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:021301. [PMID: 15783318 DOI: 10.1103/physreve.71.021301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2004] [Indexed: 05/24/2023]
Abstract
We study the onset of patterns in vertically oscillated layers of frictionless dissipative particles. Using both numerical solutions of continuum equations to Navier-Stokes order and molecular dynamics (MD) simulations, we find that standing waves form stripe patterns above a critical acceleration of the cell. Changing the frequency of oscillation of the cell changes the wavelength of the resulting pattern; MD and continuum simulations both yield wavelengths in accord with previous experimental results. The value of the critical acceleration for ordered standing waves is approximately 10% higher in molecular dynamics simulations than in the continuum simulations, and the amplitude of the waves differs significantly between the models. The delay in the onset of order in molecular dynamics simulations and the amplitude of noise below this onset are consistent with the presence of fluctuations which are absent in the continuum theory. The strength of the noise obtained by fit to Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in fluid convection experiments, and is comparable to the noise found in experiments with oscillated granular layers and in recent fluid experiments on fluids near the critical point. Good agreement is found between the mean field value of onset from the Swift-Hohenberg fit and the onset in continuum simulations. Patterns are compared in cells oscillated at two different frequencies in MD; the layer with larger wavelength patterns has less noise than the layer with smaller wavelength patterns.
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Crucial role of sidewalls in velocity distributions in quasi-two-dimensional granular gases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:040301. [PMID: 15600385 DOI: 10.1103/physreve.70.040301] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2004] [Indexed: 05/24/2023]
Abstract
Our experiments and three-dimensional molecular dynamics simulations of particles confined to a vertical monolayer by closely spaced frictional walls (sidewalls) yield velocity distributions with non-Gaussian tails and a peak near zero velocity. Simulations with frictionless sidewalls are not peaked. Thus interactions between particles and their containers are an important determinant of the shape of the distribution and should be considered when evaluating experiments on a constrained monolayer of particles.
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Noise, coherent fluctuations, and the onset of order in an oscillated granular fluid. PHYSICAL REVIEW LETTERS 2004; 92:174302. [PMID: 15169154 DOI: 10.1103/physrevlett.92.174302] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2003] [Indexed: 05/24/2023]
Abstract
We study fluctuations in a vertically oscillated layer of grains below the critical acceleration for the onset of ordered standing waves. As onset is approached, transient disordered waves with a characteristic length scale emerge and increase in power and coherence. The scaling behavior and the shift in the onset of order agrees with the Swift-Hohenberg theory for convection in fluids. However, the noise in the granular system is an order of magnitude larger than the thermal noise in the most sensitive convecting fluid experiments to date; the effect of the granular noise is observable 20% below the onset of order.
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Role of friction in pattern formation in oscillated granular layers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:031301. [PMID: 15089280 DOI: 10.1103/physreve.69.031301] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2003] [Indexed: 05/24/2023]
Abstract
Particles in granular flows are often modeled as frictionless (smooth) inelastic spheres; however, there exist no frictionless grains, just as there are no elastic grains. Our molecular dynamics simulations reveal that friction is essential for realistic modeling of vertically oscillated granular layers: simulations of frictionless particles yield patterns with an onset at a container acceleration about 30% smaller than that observed in experiments and simulations with friction. More importantly, even though square and hexagonal patterns form for a wide range of the oscillation parameters in experiments and in our simulations of frictional inelastic particles, only stripe patterns form in the simulations without friction, even if the inelasticity is increased to obtain as much dissipation as in frictional particles. We also consider the effect of particle friction on the shock wave that forms each time the granular layer strikes the container. While a shock wave still forms for frictionless particles, the spatial and temporal dependence of the hydrodynamic fields differ for the cases with and without friction.
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Steady-state velocity distributions of an oscillated granular gas. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:011301. [PMID: 14995608 DOI: 10.1103/physreve.69.011301] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2003] [Indexed: 05/24/2023]
Abstract
We use a three-dimensional molecular dynamics simulation to study the single particle distribution function of a dilute granular gas driven by a vertically oscillating plate at high accelerations (15g-90g). We find that the density and the temperature fields are essentially time-invariant above a height of about 40 particle diameters, where typically 20% of the grains are contained. These grains form the nonequilibrium steady-state granular gas with a Knudsen number unity or greater. In the steady-state region, the probability distribution function of the horizontal velocity c(x) (scaled by the local horizontal temperature) is found to be nearly independent of height, even though the hydrodynamic fields vary with height. We find that the high energy tails of the distribution function are described by a stretched exponential approximately exp(-Bcalphax), where alpha depends on the restitution coefficient e and falls in the range 1.2<alpha<1.6. However, alpha does not vary significantly for a wide range of friction coefficient values. We find that the distribution function of a frictionless inelastic hard sphere model can be made similar to that of a frictional model by adjusting e. However, there is no single value of e that mimics the frictional model over a range of heights.
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Kink-induced transport and segregation in oscillated granular layers. PHYSICAL REVIEW LETTERS 2003; 91:134301. [PMID: 14525310 DOI: 10.1103/physrevlett.91.134301] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2003] [Indexed: 05/24/2023]
Abstract
We use experiments and molecular dynamics simulations of vertically oscillated granular layers to study horizontal particle segregation induced by a kink (a boundary between domains oscillating out of phase). Counterrotating convection rolls carry the larger particles in a bidisperse layer along the granular surface to a kink, where they become trapped. The convection originates from avalanches that occur inside the layer, along the interface between solidified and fluidized grains. The position of a kink can be controlled by modulation of the container frequency, making possible systematic harvesting of the larger particles.
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Lattice dynamics and melting of a nonequilibrium pattern. PHYSICAL REVIEW LETTERS 2003; 90:104302. [PMID: 12688998 DOI: 10.1103/physrevlett.90.104302] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2002] [Indexed: 05/24/2023]
Abstract
We present a new description of nonequilibrium square patterns as a harmonically coupled crystal lattice. In a vertically oscillating granular layer, different transverse normal modes of the granular square-lattice pattern are observed for different driving frequencies (f(d)) and accelerations. The amplitude of a mode can be further excited by either frequency modulation of f(d) or reduction of friction between the grains and the plate. When the mode amplitude becomes large, the lattice melts (disorders), in accord with the Lindemann criterion for melting in two dimensions.
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Shocks in vertically oscillated granular layers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:051301. [PMID: 12513479 DOI: 10.1103/physreve.66.051301] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2002] [Indexed: 05/24/2023]
Abstract
We study shock formation in vertically oscillated granular layers, using both molecular dynamics simulations and numerical solutions of continuum equations to Navier-Stokes order. A flat layer of grains is thrown up from an oscillating plate during each oscillation cycle and collides with the plate later in the cycle. The collisions produce layer compaction near the plate and a high temperature shock front that rapidly propagates upward through the layer. The shock is highly time dependent, propagating through the layer in only a quarter of the cycle. We compare numerical solutions of the continuum equations to molecular dynamics simulations that assume binary, instantaneous collisions between frictionless, inelastic hard spheres. The two simulations yield results for the shock position, shape, and speed that agree well. An investigation of the effect of inelasticity shows that the shock velocity increases continuously with decreasing inelasticity; the elastic limit is not singular.
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Phase bubbles and spatiotemporal chaos in granular patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:011301. [PMID: 11800687 DOI: 10.1103/physreve.65.011301] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2001] [Indexed: 05/23/2023]
Abstract
We use inelastic hard sphere molecular dynamics simulations and laboratory experiments to study patterns in vertically oscillated granular layers. The simulations and experiments reveal that phase bubbles spontaneously nucleate in the patterns when the container acceleration amplitude exceeds a critical value, about 7 g, where the pattern is approximately hexagonal, oscillating at one-fourth the driving frequency (f/4). A phase bubble is a localized region that oscillates with a phase opposite (differing by pi) to that of the surrounding pattern; a localized phase shift is often called an arching in studies of two-dimensional systems. The simulations show that the formation of phase bubbles is triggered by undulation at the bottom of the layer on a large length scale compared to the wavelength of the pattern. Once formed, a phase bubble shrinks as if it had a surface tension, and disappears in tens to hundreds of cycles. We find that there is an oscillatory momentum transfer across a kink, and the shrinking is caused by a net collisional momentum inward across the boundary enclosing the bubble. At increasing acceleration amplitudes, the patterns evolve into randomly moving labyrinthian kinks (spatiotemporal chaos). We observe in the simulations that f/3 and f/6 subharmonic patterns emerge as primary instabilities, but that they are unstable to the undulation of the layer. Our experiments confirm the existence of transient f/3 and f/6 patterns.
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Velocity distributions and correlations in homogeneously heated granular media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:031303. [PMID: 11580328 DOI: 10.1103/physreve.64.031303] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2001] [Indexed: 05/23/2023]
Abstract
We compare the steady state velocity distributions from our three-dimensional inelastic hard sphere molecular dynamics simulation for homogeneously heated granular media, with the predictions of a mean field-type Enskog-Boltzmann equation for inelastic hard spheres [T. P. C. van Noije and M. H. Ernst, Granular Matter 1, 57 (1998)]. Although we find qualitative agreement for all values of density and inelasticity, the quantitative disagreement approaches approximately 40% at high inelasticity or density. By contrast the predictions of the pseudo-Maxwell molecule model [J. A. Carrillo, C. Cercignani, and I. M. Gamba, Phys. Rev. E, 62, 7700 (2000)] are both qualitatively and quantitatively different from those of our simulation. We also measure short-range and long-range velocity correlations exhibiting nonzero correlations at contact before the collision, and being consistent with a slow algebraic decay over a decade in the unit of the diameter of the particle, proportional to r(-(1+alpha)), where 0.2<alpha<0.3. The existence of these correlations implies the failure of the molecular chaos assumption and the mean field approximation, which is responsible for the quantitative disagreement of the inelastic hard sphere kinetic theory.
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Suppression of dripping from a ceiling. PHYSICAL REVIEW LETTERS 2001; 86:1203-1206. [PMID: 11178044 DOI: 10.1103/physrevlett.86.1203] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2000] [Indexed: 05/23/2023]
Abstract
An isothermal layer suspended from a surface is gravitationally (Rayleigh-Taylor) unstable. We find that, when a vertical temperature difference DeltaT above a critical value (DeltaT)(c) is imposed across the liquid-gas layer system (heated from below), the restoring force provided by the temperature-dependent surface tension (thermocapillarity) can stabilize the layer. Our measurements of the most unstable wave number for DeltaT<(DeltaT)(c) agree well with our linear stability analysis. The instability occurs at long wavelengths: the most unstable wavelength at (DeltaT)(c) is infinite.
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Planform selection in two-layer Benard-marangoni convection. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:6540-6553. [PMID: 11101991 DOI: 10.1103/physreve.62.6540] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2000] [Indexed: 05/23/2023]
Abstract
Benard-Marangoni convection in a system of two superimposed liquids is investigated theoretically. Extending previous studies, the complete hydrodynamics of both layers is treated and buoyancy is consistently taken into account. The planform selection problem between rolls, squares, and hexagons is investigated by explicitly calculating the coefficients of an appropriate amplitude equation from the parameters of the fluids. The results are compared with recent experiments on two-layer systems in which squares at onset have been reported.
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Rayleigh-Benard convection in a vertically oscillated fluid layer. PHYSICAL REVIEW LETTERS 2000; 84:87-90. [PMID: 11015841 DOI: 10.1103/physrevlett.84.87] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/1999] [Indexed: 05/23/2023]
Abstract
We report on the first quantitative observations of convection in a fluid layer driven by both heating from below and vertical sinusoidal oscillation. Just above onset, convection patterns are modulated either harmonically or subharmonically to the drive frequency. Single-frequency patterns exhibit nearly solid-body rotations with harmonic and subharmonic states always rotating in opposite directions. Flows with both harmonic and subharmonic responses are found near a codimension-two point, yielding novel coexisting patterns with symmetries not found in either single-frequency states. Predictions from linear stability analysis of the onset Rayleigh and wave numbers compare well with experiment, and phase boundaries for coexisting patterns track single-frequency marginal stability curves.
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Linear stability analysis of a vertically oscillated granular layer. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:7210-6. [PMID: 11970664 DOI: 10.1103/physreve.60.7210] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/1999] [Indexed: 04/18/2023]
Abstract
We present a linear stability analysis of an oscillating granular layer, treating it as an isothermal incompressible fluid with zero surface tension, which undergoes periodic collisions with and separations from an oscillating plate. Because the viscosity of the granular layer is unknown, we use the experimental value of the critical acceleration for the transition from a flat to patterned layer as input for the theory, and use the analysis to calculate the granular viscosity and the wavelength of the most unstable mode. The wavelength compares favorably with the experimental pattern wavelength. Further, we find that the wavelengths are controlled by the viscosity of the granular layer.
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Transport coefficients for granular media from molecular dynamics simulations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:4340-51. [PMID: 11970288 DOI: 10.1103/physreve.60.4340] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/1999] [Indexed: 04/18/2023]
Abstract
Under many conditions, macroscopic grains flow like a fluid; kinetic theory predicts continuum equations of motion for this granular fluid. In order to test the theory, we perform event-driven molecular simulations of a two-dimensional gas of inelastic hard disks, driven by contact with a heat bath. Even for strong dissipation, high densities, and small numbers of particles, we find that continuum theory describes the system well. With a bath that heats the gas homogeneously, strong velocity correlations produce a slightly smaller energy loss due to inelastic collisions than that predicted by kinetic theory. With an inhomogeneous heat bath, thermal or velocity gradients are induced. Determination of the resulting fluxes allows calculation of the thermal conductivity and shear viscosity, which are compared to the predictions of granular kinetic theory, and which can be used in continuum modeling of granular flows. The shear viscosity is close to the prediction of kinetic theory, while the thermal conductivity can be overestimated by a factor of 2; in each case, transport is lowered with increasing inelasticity.
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Instability of the Kolmogorov flow in a soap film. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:715-21. [PMID: 11969812 DOI: 10.1103/physreve.60.715] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/1998] [Indexed: 04/18/2023]
Abstract
We examine the instability of a soap film flow driven by a time-independent force that is spatially periodic in the direction perpendicular to the forcing (Kolmogorov flow). Linear stability analysis of an idealized model of this flow predicts a critical Reynolds number R(c) is approximately equal to the square root of 2. In our soap film experiment, we find a critical value R(c) is approximately equal to 70. This discrepancy can be ascribed to frictional effects from viscous coupling of gas to the film, which is neglected in the idealized model. The kinematic viscosity of the surrounding gas and the thickness of gas layers on each side of the soap film are varied in the experiments to better understand these frictional effects. Our observations indicate that flow in the soap film cannot be decoupled from flow in the surrounding gas.
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Abstract
We have studied turbulent convection in a vertical thin (Hele-Shaw) cell at very high Rayleigh numbers (up to 7x10(4) times the value for convective onset) through experiment, simulation, and analysis. Experimentally, convection is driven by an imposed concentration gradient in an isothermal cell. Model equations treat the fields in two dimensions, with the reduced dimension exerting its influence through a linear wall friction. Linear stability analysis of these equations demonstrates that as the thickness of the cell tends to zero, the critical Rayleigh number and wave number for convective onset do not depend on the velocity conditions at the top and bottom boundaries (i.e., no-slip or stress-free). At finite cell thickness delta, however, solutions with different boundary conditions behave differently. We simulate the model equations numerically for both types of boundary conditions. Time sequences of the full concentration fields from experiment and simulation display a large number of solutal plumes that are born in thin concentration boundary layers, merge to form vertical channels, and sometimes split at their tips via a Rayleigh-Taylor instability. Power spectra of the concentration field reveal scaling regions with slopes that depend on the Rayleigh number. We examine the scaling of nondimensional heat flux (the Nusselt number, Nu) and rms vertical velocity (the Peclet number, Pe) with the Rayleigh number (Ra(*)) for the simulations. Both no-slip and stress-free solutions exhibit the scaling NuRa(*) approximately Pe(2) that we develop from simple arguments involving dynamics in the interior, away from cell boundaries. In addition, for stress-free solutions a second relation, Nu approximately nPe, is dictated by stagnation-point flows occurring at the horizontal boundaries; n is the number of plumes per unit length. No-slip solutions exhibit no such organization of the boundary flow and the results appear to agree with Priestley's prediction of Nu approximately Ra(1/3). (c) 1997 American Institute of Physics.
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Nonlinear Control of Remote Unstable States in a Liquid Bridge Convection Experiment. PHYSICAL REVIEW LETTERS 1996; 77:3779-3782. [PMID: 10062306 DOI: 10.1103/physrevlett.77.3779] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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20
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Long-wavelength instability in surface-tension-driven Bénard convection. PHYSICAL REVIEW LETTERS 1995; 75:4397-4400. [PMID: 10059898 DOI: 10.1103/physrevlett.75.4397] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Metastability in fluctuation-driven first-order transitions: Nucleation of lamellar phases. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:1828-1845. [PMID: 9963602 DOI: 10.1103/physreve.52.1828] [Citation(s) in RCA: 46] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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23
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Onset of traveling waves in isothermal double diffusive convection. PHYSICAL REVIEW LETTERS 1994; 72:218-221. [PMID: 10056089 DOI: 10.1103/physrevlett.72.218] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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24
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Effects of additive noise at the onset of Rayleigh-Bénard convection. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:4773-4785. [PMID: 9908696 DOI: 10.1103/physreva.46.4773] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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25
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Theory of acoustic properties near the cholesteric-twisted-grain-boundary-A phase transition in liquid crystals. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:940-945. [PMID: 9908194 DOI: 10.1103/physreva.46.940] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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26
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Stochastic Landau equation with time-dependent drift. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:6572-6580. [PMID: 9905008 DOI: 10.1103/physreva.43.6572] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Influence of asymmetry on multifractal properties of maps. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:4095-4099. [PMID: 9905507 DOI: 10.1103/physreva.43.4095] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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28
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Scaling properties of multifractal functions at an attractor-repeller transition. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:6615-6620. [PMID: 9903074 DOI: 10.1103/physreva.41.6615] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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29
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Onset of secondary flow in the modulated Taylor-Couette system. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 40:7197-7201. [PMID: 9902135 DOI: 10.1103/physreva.40.7197] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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30
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Rayleigh-Bénard convection with time-dependent boundary conditions. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 39:4132-4136. [PMID: 9901739 DOI: 10.1103/physreva.39.4132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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31
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Finite-size effects on the f( alpha ) spectrum of the period-doubling attractor. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 39:2734-2737. [PMID: 9901552 DOI: 10.1103/physreva.39.2734] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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32
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Pattern competition in temporally modulated Rayleigh-Bénard convection. PHYSICAL REVIEW LETTERS 1988; 61:947-950. [PMID: 10039475 DOI: 10.1103/physrevlett.61.947] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Comment on "Initial stages of pattern formation in Rayleigh-Bénard convection". PHYSICAL REVIEW LETTERS 1988; 60:75. [PMID: 10037875 DOI: 10.1103/physrevlett.60.75] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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34
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Modulated convection at high frequencies and large modulation amplitudes. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 36:4870-4875. [PMID: 9898746 DOI: 10.1103/physreva.36.4870] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hexagons and rolls in periodically modulated Rayleigh-Bénard convection. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 35:3855-3873. [PMID: 9898613 DOI: 10.1103/physreva.35.3855] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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