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Published November 28, 2011 | Published
Journal Article Open

Oscillatory long-wave Marangoni convection in a layer of a binary liquid: Hexagonal patterns

Abstract

We consider a long-wave oscillatory Marangoni convection in a layer of a binary liquid in the presence of the Soret effect. A weakly nonlinear analysis is carried out on a hexagonal lattice. It is shown that the derived set of cubic amplitude equations is degenerate. A three-parameter family of asynchronous hexagons (AH), representing a superposition of three standing waves with the amplitudes depending on their phase shifts, is found to be stable in the framework of this set of equations. To determine a dominant stable pattern within this family of patterns, we proceed to the inclusion of the fifth-order terms. It is shown that depending on the Soret number, either wavy rolls 2 (WR2), which represents a pattern descendant of wavy rolls (WR) family, are selected or no stable limit cycles exist. A heteroclinic cycle emerges in the latter case: the system is alternately attracted to and repelled from each of three unstable solutions.

Additional Information

© 2011 American Physical Society. Received 19 June 2011; published 28 November 2011. This work is partially supported by the European Union via FP7 Marie Curie scheme Grant PITN-GA-2008-214919 (MULTIFLOW). A.O. was also partially supported by the Grant No. 2008038 from the Binational Science US–Israel Foundation and by Fund for Promotion of Research at the Technion.

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