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

G. Cartland Glover; Sotos Generalis; Elias C Aifantis

doi:10.1134/S1995080222100122

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 journal › Article › peer-review

Abstract

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 language	English
Pages (from-to)	1850-1865
Journal	Lobachevskii Journal of Mathematics
Volume	43
Issue number	7
Early online date	11 Nov 2022
DOIs	https://doi.org/10.1134/S1995080222100122
Publication status	Published - 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.

Keywords

incompressible flow
bifurcation theory
homotopy method
stability
nonlinearity

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10.1134/S1995080222100122

Cite this

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title = "On the Convective Stability and Pattern Formation of Volumetrically Heated Flows with Asymmetric Boundaries",

abstract = "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 {\textquoteleft}{\textquoteleft}up{\textquoteright}{\textquoteright} and {\textquoteleft}{\textquoteleft}down{\textquoteright}{\textquoteright} 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.",

keywords = "incompressible flow, bifurcation theory, homotopy method, stability, nonlinearity",

author = "{Cartland Glover}, G. and Sotos Generalis and Aifantis, {Elias C}",

note = "Copyright {\textcopyright} 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{\textquoteright}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{\textquoteright}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.",

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AU - Generalis, Sotos

AU - Aifantis, Elias C

N1 - 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.

PY - 2022/11/11

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N2 - 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.

AB - 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.

KW - incompressible flow

KW - bifurcation theory

KW - homotopy method

KW - stability

KW - nonlinearity

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VL - 43

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ER -

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

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