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Instability triggered by mixed convection in a thin fluid layer

Published online by Cambridge University Press:  01 December 2025

Florian Rein Open the ORCID record for Florian Rein [Opens in a new window] ,
Keaton J. Burns Open the ORCID record for Keaton J. Burns [Opens in a new window] ,
Stefan G. Llewellyn Smith Open the ORCID record for Stefan G. Llewellyn Smith [Opens in a new window] ,
William R. Young Open the ORCID record for William R. Young [Opens in a new window] ,
Benjamin Favier Open the ORCID record for Benjamin Favier [Opens in a new window]  and
Michael Le Bars Open the ORCID record for Michael Le Bars [Opens in a new window]
Florian Rein*
Affiliation:
Aix Marseille Université, CNRS, Centrale Med, IRPHE, Marseille, France Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093, USA
Keaton J. Burns
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
Stefan G. Llewellyn Smith
Affiliation:
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093, USA Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California San Diego, La Jolla, CA 92093, USA
William R. Young
Affiliation:
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093, USA
Benjamin Favier
Affiliation:
Aix Marseille Université, CNRS, Centrale Med, IRPHE, Marseille, France
Michael Le Bars
Affiliation:
Aix Marseille Université, CNRS, Centrale Med, IRPHE, Marseille, France
*
Corresponding author: Florian Rein, florian.rein@protonmail.com
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Abstract

We investigate the convective stability of a thin, infinite fluid layer with a rectangular cross-section, subject to imposed heat fluxes at the top and bottom and fixed temperature along the vertical sides. The instability threshold depends on the Prandtl number as well as the normalized flux difference ($f$) and decreases with the aspect ratio ($\epsilon$), following a $\epsilon f^{-1}$ power law. Using a three-dimensional (3-D) initial value and two-dimensional eigenvalue calculations, we identify a dominant 3-D mode characterized by two transverse standing waves attached to the domain edges. We characterize the dominant mode’s frequency and transverse wavenumber as functions of the Rayleigh number and aspect ratio. An analytical asymptotic solution for the base state in the bulk is obtained, valid over most of the domain and increasingly accurate for lower aspect ratios. A local stability analysis, based on the analytical base state, reveals oscillatory transverse instabilities consistent with the global instability characteristics. The source term for this most unstable mode appears to be interactions between vertical shear and horizontal temperature gradients.

JFM classification

Convection: Convection Instability: Instability

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 1024 , 10 December 2025 , A23
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© The Author(s), 2025. Published by Cambridge University Press

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