The onset of Marangoni bio-thermal convection in a layer of fluid containing gyrotactic microorganisms

Latifa I. Khayyat; Abdullah A. Abdullah; Latifa I. Khayyat; Abdullah A. Abdullah

doi:10.3934/math.2021787

AIMS Mathematics

2021, Volume 6, Issue 12: 13552-13565. doi: 10.3934/math.2021787

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The onset of Marangoni bio-thermal convection in a layer of fluid containing gyrotactic microorganisms

Latifa I. Khayyat ¹,
Abdullah A. Abdullah ^{2
,
,}

1.
Department of Biology, College of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
2.
Department of Mathematical Sciences, College of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 17 April 2021 Accepted: 15 September 2021 Published: 24 September 2021
MSC : 76E06

The problem of the onset of Marangoni bio-thermal convection is investigated for a horizontal layer of fluid containing motile gyrotactic microorganisms. The fluid layer is assumed to rest on a rigid surface with fixed temperature and the top boundary of the layer is assumed to be a free non deformable surface. The resulting equations of the problem constitute an eigenvalue problem which is solved using the Chebyshev tau numerical method. The critical values of the thermal Marangoni number are calculated for several values of the bioconvection Péclet number, bioconvection Marangoni number, bioconvection Lewis number and gyrotaxis number. The results of this study showed that the existence of gyrotactic microorganisms increases the critical thermal Marangoni numbers. Moreover, the critical eigenvalues obtained were real-valued indicating that the mode of instability is via a stationary mode, however oscillatory mode is possible for some ranges of the parameters values.
- linear stability,
- bio-thermal convection,
- Marangoni,
- gyrotactic microorganisms,
- Chebyshev tau method
Citation: Latifa I. Khayyat, Abdullah A. Abdullah. The onset of Marangoni bio-thermal convection in a layer of fluid containing gyrotactic microorganisms[J]. AIMS Mathematics, 2021, 6(12): 13552-13565. doi: 10.3934/math.2021787

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Abstract

The problem of the onset of Marangoni bio-thermal convection is investigated for a horizontal layer of fluid containing motile gyrotactic microorganisms. The fluid layer is assumed to rest on a rigid surface with fixed temperature and the top boundary of the layer is assumed to be a free non deformable surface. The resulting equations of the problem constitute an eigenvalue problem which is solved using the Chebyshev tau numerical method. The critical values of the thermal Marangoni number are calculated for several values of the bioconvection Péclet number, bioconvection Marangoni number, bioconvection Lewis number and gyrotaxis number. The results of this study showed that the existence of gyrotactic microorganisms increases the critical thermal Marangoni numbers. Moreover, the critical eigenvalues obtained were real-valued indicating that the mode of instability is via a stationary mode, however oscillatory mode is possible for some ranges of the parameters values.

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