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Thermocapillary instabilities in a horizontal liquid layer under partial basal slip

Published online by Cambridge University Press:  20 September 2018

Katarzyna N. Kowal Open the ORCID record for Katarzyna N. Kowal [Opens in a new window] ,
Stephen H. Davis  and
Peter W. Voorhees
Katarzyna N. Kowal*
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Trinity College, University of Cambridge, Cambridge CB2 1TQ, UK
Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Peter W. Voorhees
Affiliation:
Department of Materials Science and Engineering, Northwestern University, 2225 Campus Drive, Evanston, IL 60208, USA
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the onset of three-dimensional hydrothermal waves in a low-capillary-number liquid layer of arbitrary depth, bounded by a free liquid–gas interface from above and a partial slip, rigid surface from below. A selection of two- and three-dimensional hydrothermal waves, longitudinal rolls and longitudinal travelling waves, form the preferred mode of instability, which depends intricately on the magnitude of the basal slip. Partial slip is destabilizing for all modes of instability. Specifically, the minimal Marangoni number required for the onset of instability follows $M_{m}\sim a(\unicode[STIX]{x1D6FD}^{-1}+b)^{-c}$ for each mode, where $a,b,c>0$ and $\unicode[STIX]{x1D6FD}^{-1}$ is the slip parameter. In the limit of free slip, longitudinal travelling waves disappear in favour of longitudinal rolls. With increasing slip, it is common for two-dimensional hydrothermal waves to exchange stability in favour of longitudinal rolls and oblique hydrothermal waves. Two types of oblique hydrothermal waves appear under partial slip, which exchange stability with increasing slip. The oblique mode that is preferred under no slip persists and remains near longitudinal for small slip parameters.

JFM classification

Convection: Convection Interfacial Flows (free surface): Interfacial Flows (free surface) Convection: Marangoni convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 839 - 859
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Copyright
© 2018 Cambridge University Press 

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