3,962 research outputs found
Hexagonal patterns in a model for rotating convection
We study a model equation that mimics convection under rotation in a fluid with temperature- dependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Kuppers-Lortz instability [Kuppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a weakly nonlinear analysis to find out explicitly the coefficients in the amplitude equation as functions of the rotation rate. These equations describe hexagons and os- cillating hexagons quite well, and include the Busse?Heikes (BH) model [Busse & Heikes, 1980] as a particular case. The sideband instabilities as well as short wavelength instabilities of such hexagonal patterns are discussed and the threshold for oscillating hexagons is determined
Instabilities and Spatio-temporal Chaos of Long-wave Hexagon Patterns in Rotating Marangoni Convection
We consider surface-tension driven convection in a rotating fluid layer. For
nearly insulating boundary conditions we derive a long-wave equation for the
convection planform. Using a Galerkin method and direct numerical simulations
we study the stability of the steady hexagonal patterns with respect to general
side-band instabilities. In the presence of rotation steady and oscillatory
instabilities are identified. One of them leads to stable, homogeneously
oscillating hexagons. For sufficiently large rotation rates the stability
balloon closes, rendering all steady hexagons unstable and leading to
spatio-temporal chaos.Comment: 26 pages, 9 jpeg figures. Postscript file with all figures included
available at http://www.esam.northwestern.edu/~riecke/lit/lit.html Movies
available at
http://www.esam.northwestern.edu/~riecke/research/Marangoni/marangoni.htm
Whirling Hexagons and Defect Chaos in Hexagonal Non-Boussinesq Convection
We study hexagon patterns in non-Boussinesq convection of a thin rotating
layer of water. For realistic parameters and boundary conditions we identify
various linear instabilities of the pattern. We focus on the dynamics arising
from an oscillatory side-band instability that leads to a spatially disordered
chaotic state characterized by oscillating (whirling) hexagons. Using
triangulation we obtain the distribution functions for the number of pentagonal
and heptagonal convection cells. In contrast to the results found for defect
chaos in the complex Ginzburg-Landau equation and in inclined-layer convection,
the distribution functions can show deviations from a squared Poisson
distribution that suggest non-trivial correlations between the defects.Comment: 4 mpg-movies are available at
http://www.esam.northwestern.edu/~riecke/lit/lit.html submitted to New J.
Physic
Reentrant and Whirling Hexagons in Non-Boussinesq convection
We review recent computational results for hexagon patterns in non-Boussinesq
convection. For sufficiently strong dependence of the fluid parameters on the
temperature we find reentrance of steady hexagons, i.e. while near onset the
hexagon patterns become unstable to rolls as usually, they become again stable
in the strongly nonlinear regime. If the convection apparatus is rotated about
a vertical axis the transition from hexagons to rolls is replaced by a Hopf
bifurcation to whirling hexagons. For weak non-Boussinesq effects they display
defect chaos of the type described by the two-dimensional complex
Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf
bifurcation becomes subcritical and localized bursting of the whirling
amplitude is found. In this regime the coupling of the whirling amplitude to
(small) deformations of the hexagon lattice becomes important. For yet stronger
non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly
disordered states characterized by whirling and lattice defects are obtained.Comment: Accepted in European Physical Journal Special Topic
Instabilities of Hexagonal Patterns with Broken Chiral Symmetry
Three coupled Ginzburg-Landau equations for hexagonal patterns with broken
chiral symmetry are investigated. They are relevant for the dynamics close to
onset of rotating non-Boussinesq or surface-tension-driven convection. Steady
and oscillatory, long- and short-wave instabilities of the hexagons are found.
For the long-wave behavior coupled phase equations are derived. Numerical
simulations of the Ginzburg-Landau equations indicate bistability between
spatio-temporally chaotic patterns and stable steady hexagons. The chaotic
state can, however, not be described properly with the Ginzburg-Landau
equations.Comment: 11 pages, 7 figures, submitted to Physica
On buoyant convection in binary solidification
We consider the problem of nonlinear steady buoyant convection in horizontal mushy layers during the solidification of binary alloys. We investigate both cases of zero vertical volume flux and constant pressure, referred to as impermeable and permeable conditions, respectively, at the upper mush???liquid interface. We analyze the effects of several parameters of the problem on the stationary modes of convection in the form of either hexagonal cells or non-hexagonal cells, such as rolls, rectangles and squares. [More ...]published or submitted for publicationis not peer reviewe
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