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

Lobachevskii Journal of Mathematics volume 43, pages 1850–1865 (2022)Cite this article

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.

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Funding

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.

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Authors and Affiliations

  1. Mathematics Department, College of Engineering and Physical Sciences, Aston University, B4 7ET, Birmingham, United Kingdom

    G. Cartland Glover & S. C. Generalis

  2. School of Engineering, Aristotle University, 54124, Thessaloniki, Greece

    E. C. Aifantis

  3. Michigan Technological University, MI 49931, Houghton, USA

    E. C. Aifantis

  4. Friedrich-Alexander-University at Erlangen-Nurember, 90762, Fürth, Germany

    E. C. Aifantis

Authors
  1. G. Cartland Glover
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  2. S. C. Generalis
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  3. E. C. Aifantis
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Corresponding authors

Correspondence to G. Cartland Glover, S. C. Generalis or E. C. Aifantis.

Additional information

(Submitted by A. M. Elizarov)

APPENDIX

APPENDIX

In this Appendix data are presented, by plotting the amplitudes for hexagonal and roll states as the conductive laminar fluid state bifurcates to convective fluid states. Figures 9–16 support Fig. 6 of the main paper. The meaning of the various curves for the hexagons and rolls is as follows: thick lines—‘‘down’’ hexagons; long dashed lines—‘‘up’’ hexagons; short dashed lines—rolls.

Fig. 9
figure 9

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.025\). Amplitudes are indicated by \(\ell^{2}_{\textrm{norm}}\).

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Fig. 10
figure 10

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.250\). Amplitudes are indicated by \(\ell^{2}_{\textrm{norm}}\).

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Fig. 11
figure 11

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.255\). Amplitudes are indicated by \(\ell^{2}_{\textrm{norm}}\).

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Fig. 12
figure 12

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.260\). Amplitudes are indicated by \(\ell^{2}_{\textrm{norm}}\).

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Fig. 13
figure 13

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.270\). Amplitudes are indicated by \(\ell^{2}_{norm}\).

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Fig. 14
figure 14

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.310\). Amplitudes indicated by \(\ell^{2}_{\textrm{norm}}\).

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Fig. 15
figure 15

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=0.705\). Amplitudes indicated by \(\ell^{2}_{\textrm{norm}}\).

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Fig. 16
figure 16

Sequences of hexagonal and roll convections beyond the critical transition at \(Pr=7.000\). Amplitudes indicated by \(\ell^{2}_{\textrm{norm}}\).

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Cartland Glover, G., Generalis, S.C. & Aifantis, E.C. On the Convective Stability and Pattern Formation of Volumetrically Heated Flows with Asymmetric Boundaries. Lobachevskii J Math 43, 1850–1865 (2022). https://doi.org/10.1134/S1995080222100122

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  • DOI: https://doi.org/10.1134/S1995080222100122

Keywords:

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

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