Controlling of longwave oscillatory Marangoni patterns on a rhombic lattice^{*, **}

¹ Perm State University, 614990 Perm, Russia.
² Technion – Israel Institute of Technology, 32000 Haifa, Israel.

^*** Corresponding author: annsomeoil@gmail.com

Received: 24 November 2020
Accepted: 27 November 2020

Abstract

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.