Instabilities and Spatio-temporal Chaos of Long-wave Hexagon Patterns in Rotating Marangoni Convection

A. M. Mancho, Hermann Rieck, and Fil Sain

Abstract

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.

We derive a long-wave equation from the Navier-Stokes equations for a thin layer of liquid heated from below in a rotating system. Convection is driven by surface tension gradients due to temperature gradients. We neglect surface deformation and assume poorly conducting boundary conditions. The latter are needed to have convection set in at long wavelengths.

F_T = - (b₁+b₂) F - mD_H F - lD_H² F - c₁ Ñ_H ·| Ñ_H F|² Ñ_H F - c₂ Ñ_H ·D_H F Ñ_H F - c₃ D_H | Ñ_H F|² -c₄ e_z ·Ñ_H F ×Ñ_H {DF} +c₅ e_z ·Ñ_H F ×Ñ_H {(Ñ_H F)²}.
(1)
Here m is the control parameter related to the heating. The constants l, c₁, c₂, c₃, c₄ and c₅ are complicated functions of the rotation rate W.

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On 19 Oct 2001, 07:17.