Geodesic
Controlling of longwave oscillatory Marangoni patterns on a rhombic lattice
Anna Samoilova1 ; Alexander Nepomnyashchy2
1 Perm State University, 614990 Perm, Russia.
2 Technion – Israel Institute of Technology, 32000 Haifa, Israel.
Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 1.

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We apply nonlinear feedback control to govern the stability of long-wave oscillatory Marangoni patterns. We focus on the patterns caused by instability in thin liquid film heated from below with a deformable free surface. This instability emerges in the case of substrate of low thermal conductivity, when two monotonic long-wave instabilities, Pearson’s and deformational ones, are coupled. We provide weakly nonlinear analysis within the amplitude equations, which govern the evolution of the layer thickness and the temperature deviation. The action of the nonlinear feedback control on the nonlinear interaction of two standing waves is investigated. It is shown that quadratic feedback control can produce additional stable structures (standing rolls, standing squares and standing rectangles), which are subject to instability leading to traveling wave in the uncontrolled case.
DOI : 10.1051/mmnp/2020054

Anna Samoilova 1 ; Alexander Nepomnyashchy 2

1 Perm State University, 614990 Perm, Russia.
2 Technion – Israel Institute of Technology, 32000 Haifa, Israel. 
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Anna Samoilova; Alexander Nepomnyashchy. Controlling of longwave oscillatory Marangoni patterns on a rhombic lattice. Mathematical modelling of natural phenomena, Tome 16 (2021), article  no. 1. doi : 10.1051/mmnp/2020054. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020054/

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[4] A.E. Samoilova, A. Nepomnyashchy Marangoni patterns on a rhombic lattice in a thin film heated from below Phys. Fluids 2021 014101

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[6] M. Silber, E. Knobloch Hopf bifurcation on a square lattice Nonlinearity 1991 1063 1106

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