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Oscillatory thermocapillary instability of a film heated by a thick substrate

Published online by Cambridge University Press:  14 June 2019

W. Batson Open the ORCID record for W. Batson [Opens in a new window] ,
L. J. Cummings Open the ORCID record for L. J. Cummings [Opens in a new window] ,
D. Shirokoff  and
L. Kondic Open the ORCID record for L. Kondic [Opens in a new window]
W. Batson*
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA Hemp Harvest Innovations, 6035 Longbow Drive Suite 105, Boulder, CO 80301-3294, USA
L. J. Cummings
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
D. Shirokoff
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
L. Kondic
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
*
†Email address for correspondence: wbatson@gmail.com
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Abstract

In this work we consider a new class of oscillatory instabilities that pertain to thermocapillary destabilization of a liquid film heated by a solid substrate. We assume the substrate thickness and substrate–film thermal conductivity ratio are large so that the effect of substrate thermal diffusion is retained at leading order in the long-wave approximation. As a result, the system dynamics is described by a nonlinear partial differential equation for the film thickness that is non-locally coupled to the full substrate heat equation. Perturbing about a steady quiescent state, we find that its stability is described by a non-self-adjoint eigenvalue problem. We show that, under appropriate model parameters, the linearized eigenvalue problem admits complex eigenvalues that physically correspond to oscillatory (in time) instabilities of the thin-film height. As the principal results of our work, we provide a complete picture of the susceptibility to oscillatory instabilities for different model parameters. Using this description, we conclude that oscillatory instabilities are more relevant experimentally for films heated by insulating substrates. Furthermore, we show that oscillatory instability where the fastest-growing (most unstable) wavenumber is complex, arises only for systems with sufficiently large substrate thicknesses. Finally, we discuss adaptation of our model to a practical setting and make predictions of conditions at which the reported instabilities can be observed.

JFM classification

Low-Reynolds-number flows: Lubrication theory Convection: Marangoni convection Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , pp. 928 - 962
Copyright
© 2019 Cambridge University Press 

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