13,147 research outputs found
Rayleigh-Benard Convection with a Radial Ramp in Plate Separation
Pattern formation in Rayleigh-Benard convection in a large-aspect-ratio
cylinder with a radial ramp in the plate separation is studied analytically and
numerically by performing numerical simulations of the Boussinesq equations. A
horizontal mean flow and a vertical large scale counterflow are quantified and
used to understand the pattern wavenumber. Our results suggest that the mean
flow, generated by amplitude gradients, plays an important role in the roll
compression observed as the control parameter is increased. Near threshold the
mean flow has a quadrupole dependence with a single vortex in each quadrant
while away from threshold the mean flow exhibits an octupole dependence with a
counter-rotating pair of vortices in each quadrant. This is confirmed
analytically using the amplitude equation and Cross-Newell mean flow equation.
By performing numerical experiments the large scale counterflow is also found
to aid in the roll compression away from threshold but to a much lesser degree.
Our results yield an understanding of the pattern wavenumbers observed in
experiment away from threshold and suggest that near threshold the mean flow
and large scale counterflow are not responsible for the observed shift to
smaller than critical wavenumbers.Comment: 10 pages, 13 figure
Finite Size Scaling of Domain Chaos
Numerical studies of the domain chaos state in a model of rotating
Rayleigh-Benard convection suggest that finite size effects may account for the
discrepancy between experimentally measured values of the correlation length
and the predicted divergence near onset
Structure of Stochastic Dynamics near Fixed Points
We analyze the structure of stochastic dynamics near either a stable or
unstable fixed point, where force can be approximated by linearization. We find
that a cost function that determines a Boltzmann-like stationary distribution
can always be defined near it. Such a stationary distribution does not need to
satisfy the usual detailed balance condition, but might have instead a
divergence-free probability current. In the linear case the force can be split
into two parts, one of which gives detailed balance with the diffusive motion,
while the other induces cyclic motion on surfaces of constant cost function.
Using the Jordan transformation for the force matrix, we find an explicit
construction of the cost function. We discuss singularities of the
transformation and their consequences for the stationary distribution. This
Boltzmann-like distribution may be not unique, and nonlinear effects and
boundary conditions may change the distribution and induce additional currents
even in the neighborhood of a fixed point.Comment: 7 page
Pattern formation outside of equilibrium
A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Bénard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future
Extensive chaos in Rayleigh-Bénard convection
Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size
Traveling waves in rotating Rayleigh-Bénard convection: Analysis of modes and mean flow
Numerical simulations of the Boussinesq equations with rotation for realistic no-slip boundary conditions and a finite annular domain are presented. These simulations reproduce traveling waves observed experimentally. Traveling waves are studied near threshhold by using the complex Ginzburg-Landau equation (CGLE): a mode analysis enables the CGLE coefficients to be determined. The CGLE coefficients are compared with previous experimental and theoretical results. Mean flows are also computed and found to be more significant as the Prandtl number decreases (from sigma=6.4 to sigma=1). In addition, the mean flow around the outer radius of the annulus appears to be correlated with the mean flow around the inner radius
Power-Law Behavior of Power Spectra in Low Prandtl Number Rayleigh-Benard Convection
The origin of the power-law decay measured in the power spectra of low
Prandtl number Rayleigh-Benard convection near the onset of chaos is addressed
using long time numerical simulations of the three-dimensional Boussinesq
equations in cylindrical domains. The power-law is found to arise from
quasi-discontinuous changes in the slope of the time series of the heat
transport associated with the nucleation of dislocation pairs and roll
pinch-off events. For larger frequencies, the power spectra decay exponentially
as expected for time continuous deterministic dynamics.Comment: (10 pages, 6 figures
Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion
In the limit of large diffusivity ratio, spot-like solutions in the
two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion are
studied. It is shown analytically that such spots undergo an instability as the
diffusivity ratio is decreased. An instability threshold is derived. For spots
of small radius, it is shown that this instability leads to a spot splitting
into precisely two spots. For larger spots, it leads to deformation, fingering
patterns and space-filling curves. Numerical simulations are shown to be in
close agreement with the analytical predictions.Comment: To appear, PR
Weakly Nonlinear Analysis of Electroconvection in a Suspended Fluid Film
It has been experimentally observed that weakly conducting suspended films of
smectic liquid crystals undergo electroconvection when subjected to a large
enough potential difference. The resulting counter-rotating vortices form a
very simple convection pattern and exhibit a variety of interesting nonlinear
effects. The linear stability problem for this system has recently been solved.
The convection mechanism, which involves charge separation at the free surfaces
of the film, is applicable to any sufficiently two-dimensional fluid. In this
paper, we derive an amplitude equation which describes the weakly nonlinear
regime, by starting from the basic electrohydrodynamic equations. This regime
has been the subject of several recent experimental studies. The lowest order
amplitude equation we derive is of the Ginzburg-Landau form, and describes a
forward bifurcation as is observed experimentally. The coefficients of the
amplitude equation are calculated and compared with the values independently
deduced from the linear stability calculation.Comment: 26 pages, 2 included eps figures, submitted to Phys Rev E. For more
information, see http://mobydick.physics.utoronto.c
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