On the Convective Stability and Pattern Formation of Volumetrically Heated Flows with Asymmetric Boundaries

G. Cartland Glover*, Sotos Generalis, Elias C Aifantis

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Non-linear solutions and their stability are presented for homogeneously heated fluids bounded by rigid conducting and insulating plates. In particular, we sought roll-type solutions emerging from the neutral stability curve for fluids with Prandtl numbers of 0.025, 0.25, 0.705, and 7. We determined the stability boundaries for the roll states in order to identify possible bifurcation points for the secondary flow in the form of regions that are equivalent to the Busse balloon. We also compared the stability exchange between ‘‘up’’ and ‘‘down’’ hexagons for a Prandtl number of $$0.25$$ obtained from weakly non-linear analysis in relation to the fully non-linear analysis, consistent with earlier studies. Our numerical analysis showed that there are potential bistable regions for both hexagons and rolls, a result that requires further investigations with a fully non-linear analysis.

Original languageEnglish
Pages (from-to)1850-1865
JournalLobachevskii Journal of Mathematics
Issue number7
Early online date11 Nov 2022
Publication statusPublished - 11 Nov 2022

Bibliographical note

Copyright © 2022, Pleiades Publishing, Ltd. This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use [https://www.springernature.com/gp/open-research/policies/accepted-manuscript-terms], but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1134/S1995080222100122.

Funding Information
The work presented here was funded by a Marie-Curie Intra-European Fellowship (Contract no. 274367) of the European Commission’s FP7 People Programme (GCG). We are grateful for the support of a visiting Professorship by the Leverhulme Trust, which resulted in many fruitful discussions and suggestions.


  • incompressible flow
  • bifurcation theory
  • homotopy method
  • stability
  • nonlinearity


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