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Feedback control of Marangoni convection in a thin film heated from below

Published online by Cambridge University Press:  01 August 2019

Anna E. Samoilova Open the ORCID record for Anna E. Samoilova [Opens in a new window]  and
Alexander Nepomnyashchy
Anna E. Samoilova*
Affiliation:
Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel Department of Theoretical Physics, Perm State University, Perm 614990, Russia
Alexander Nepomnyashchy
Affiliation:
Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: annsomeoil@gmail.com
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Abstract

We use linear proportional control for the suppression of the Marangoni instability in a thin film heated from below. Our keen interest is focused on the recently revealed oscillatory mode caused by a coupling of two long-wave monotonic instabilities, the Pearson and deformational ones. Shklyaev et al. (Phys. Rev. E, vol. 85, 2012, 016328) showed that the oscillatory mode is critical in the case of a substrate of very low conductivity. To stabilize the no-motion state of the film, we apply two linear feedback control strategies based on the heat flux variation at the substrate. Strategy (I) uses the interfacial deflection from the mean position as the criterion of instability onset. Within strategy (II) the variable that describes the instability is the deviation of the measured temperatures from the desired, conductive values. We perform two types of calculations. The first one is the linear stability analysis of the nonlinear amplitude equations that are derived within the lubrication approximation. The second one is the linear stability analysis that is carried out within the Bénard–Marangoni problem for arbitrary wavelengths. Comparison of different control strategies reveals feedback control by the deviation of the free surface temperature as the most effective way to suppress the Marangoni instability.

JFM classification

Convection: Marangoni convection Flow Control: Instability control Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 573 - 590
Copyright
© 2019 Cambridge University Press 

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