Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-19T17:40:51.356Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batson, W., Cummings, L. J., Shirokoff, D. & Kondic, L. 2019 Oscillatory thermocapillary instability of a film heated by a thick substrate. J. Fluid Mech. 872, 928–962.Google Scholar
Bau, H. H. 1999 Control of Marangoni–Bénard convection. Intl J. Heat Mass Transfer 42, 1327–1341.Google Scholar
Benz, S., Hintz, P., Riley, R. & Neitzel, G. 1998 Instability of thermocapillary–buoyancy convection in shallow layers. Part 2. Suppression of hydrothermal waves. J. Fluid Mech. 359, 165–180.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
Howle, L. E. 1997 Linear stability analysis of controlled Rayleigh–Bénard convection using shadowgraphic measurement. Phys. Fluids 9, 3111–3113.Google Scholar
Nepomnyashchy, A., Simanovskii, I. & Legros, J. C. 2012 Interfacial Convection in Multilayer Systems. Springer.Google Scholar
Or, A. C., Kelly, R. E., Cortelezzi, L. & Speyer, J. L. 1999 Control of long-wavelength Marangoni–Bénard convection. J. Fluid Mech. 387, 321–341.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Scriven, L. E. & Sternling, C. V. 1964 On cellular convection driven surface tension gradients: effects of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321–340.Google Scholar
Shklyaev, S., Alabuzhev, A. A. & Khenner, M. 2015 Marangoni convection in a thin film on a vertically oscillating plate. Phys. Rev. E 92, 013019.Google Scholar
Shklyaev, S., Khenner, M. & Alabuzhev, A. A. 2012 Long-wave Marangoni convection in a thin film heated from below. Phys. Rev. E 85, 016328.Google Scholar
Shklyaev, S. & Nepomnyashchy, A. 2017 Longwave Instabilities and Patterns in Fluids. Birkhäuser.Google Scholar
Smith, K. A. 1966 On convection instability induced by surface tension gradient. J. Fluid Mech. 24, 401–414.Google Scholar
Tang, J. & Bau, H. H. 1994 Stabilization of the no-motion state in the Rayleigh–Bénard problem. Proc. R. Soc. A 447, 587–607.Google Scholar
Tang, J. & Bau, H. 1998 Experiments on the stabilization of the no-motion state of a fluid layer heated from below and cooled from above. J. Fluid Mech. 363, 153–171.Google Scholar