Numerical study of pattern formation following a convective instability in non-Boussinesq fluids.

1 We present a numerical study of a model of pattern formation following a con-vective instability in a non-Boussinesq fluid. It is shown that many of the features observed in convection experiments conducted on CO 2 gas can be reproduced by using a generalized two-dimensional Swift-Hohenberg equation. The formation of hexagonal patterns, rolls and spirals is studied, as well as the transitions and competition among them. We also study nucleation and growth of hexagonal patterns and find that the front velocity in this two dimensional model is consistent with the prediction of marginal stability theory for one dimensional fronts. 2 One of the most natural and intriguing behaviors of complex systems driven far from thermal equilibrium is their ability to undergo symmetry breaking instabilities that lead to the spontaneous formation of spatio-temporal structures. An excellent example is the Rayleigh-Bénard instability. Much of the earlier experimental work has been restricted to Oberbeck-Boussinesq type fluids, in which one observes various configurations of roll patterns. However, in a non-Oberbeck-Boussinesq system with, for example, temperature dependent transport coefficients, both roll and hexagonal patterns can exist. Very recently Bodenschatz et al. [1] have performed experiments on convection in CO 2 gas and studied the existence of and transitions between convec-tive patterns exhibiting different symmetries. They have observed the competition between a uniform conducting state, a convective state with hexagonal symmetry, and convecting patterns comprising parallel rolls. In this letter we present the results of a numerical solution of a two dimensional model equation for the case of a large aspect ratio cylindrical cell near onset, and qualitatively compare our results with the experiments of Bodenschatz et al. Our study is based on a numerical integration of a two dimensional generalized Swift-Hohenberg equation [2], ∂ψ(r, t) ∂t = ǫ(r) − ∇ 2 + 1 2 − ψ 2 ψ + g 2 ψ∇ 2 ψ + f (r), (1) with boundary conditions, ψ| B = ˆ n · ∇ψ| B = 0, (2) wherê n is the unit normal to the boundary of the domain of integration, B. This equation with g 2 = 0 reduces to the Swift-Hohenberg equation and has been extensively used to model convection in thin cells and near onset [3, 4, 5, 6]. The scalar order parameter y is related to the fluid temperature in the mid-plane of the convective cell. The quantity ǫ is the reduced Rayleigh number, ǫ = …