Thermal convection with a Cattaneo heat flux model

Gentile, M.; Straughan, B.

Abstract

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

Citation

Gentile, M., & Straughan, B. (2024). Thermal convection with a Cattaneo heat flux model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 480(2282), Article 20230771. https://doi.org/10.1098/rspa.2023.0771

• Journal Article Type: Article
• Acceptance Date: Jan 5, 2024
• Online Publication Date: Jan 31, 2024
• Publication Date: 2024-01
• Deposit Date: Feb 8, 2024
• Publicly Available Date: Feb 8, 2024
• Journal: Proceedings of the Royal Society A
• Print ISSN: 1364-5021
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 480
• Issue: 2282
• Article Number: 20230771
• DOI: https://doi.org/10.1098/rspa.2023.0771
• Keywords: Mariano flux, Cattaneo heat flux, nonlinear stability, thermal convection
• Public URL: https://durham-repository.worktribe.com/output/2192720