https://doi.org/10.1140/epjp/s13360-023-04055-5
Regular Article
Nonlinear stability for convection with temperature dependent viscosity in a Navier–Stokes–Voigt fluid
Department of Mathematical Sciences, University of Durham, Stockton Road, DH1 3LE, Durham, UK
a
brian.straughan@durham.ac.uk
Received:
10
March
2023
Accepted:
2
May
2023
Published online:
22
May
2023
We present a model for thermal convection in a Navier–Stokes–Voigt fluid when the viscosity is a quadratic function of the temperature. Three different approaches to deriving fully nonlinear stability bounds are discussed. These are based on utilizing a maximum principle for the temperature field. The structure of the Navier–Stokes–Voigt equations is very important to deriving nonlinear stability thresholds, and the Kelvin–Voigt term plays an important role in the analysis. The nonlinear stability thresholds for two of the procedures are optimal in that the critical Rayleigh number is the same as the one of linear instability theory. Due to the very nonlinear nature of the equations with a variable viscosity, the nonlinear stability thresholds depend on the size of the initial temperature perturbation. The nonlinear terms which arise due to the viscosity dependence upon temperature are very different from those which appear in Bénard convection with a constant viscosity.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
