TY - JOUR
T1 - Oscillatory thermocapillary instability of a film heated by a thick substrate
AU - Batson, William
AU - Cummings, L. J.
AU - Shirokoff, D.
AU - Kondic, L.
N1 - Funding Information:
The research was supported by a fellowship from the New Jersey Institute of Technology Department of Mathematical Sciences (W.B.); by NSF CBET–1604351 (W.B., L.K.); by NSF DMS–1815613 (L.J.C., L.K.); and by NSF DMS–1719693 (D.S.). D.S. was supported by a grant from the Simons Foundation (no. 359610).
Publisher Copyright:
© 2019 Cambridge University Press.
PY - 2019/8/10
Y1 - 2019/8/10
N2 - 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.
AB - 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.
KW - Marangoni convection
KW - lubrication theory
KW - thin films
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U2 - 10.1017/jfm.2019.417
DO - 10.1017/jfm.2019.417
M3 - Article
AN - SCOPUS:85073165768
SN - 0022-1120
VL - 872
SP - 928
EP - 962
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
ER -