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Stability analysis of a chemotaxis–convection–diffusion coupling system with the roles of deformed free surface and surface tension

Published online by Cambridge University Press:  27 July 2021

S. Chakraborty Open the ORCID record for S. Chakraborty [Opens in a new window]  and
T.W.-H. Sheu
S. Chakraborty
Affiliation:
Center for Advanced Study in Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei10617, Taiwan, R.O.C.
T.W.-H. Sheu*
Affiliation:
Center for Advanced Study in Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei10617, Taiwan, R.O.C. Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei10617, Taiwan, R.O.C.
*
†Email address for correspondence: [email protected]
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Abstract

We consider a three-dimensional chemotaxis–convection–diffusion coupled system with the effect of surface tension at the deformed free surface. The novelty of this research is to explore the impact of surface tension on bioconvection. Our aim is to determine the nature of the instability at the onset of bioconvection in a chemotaxis–convection–diffusion system involving surface tension by performing a detailed linear stability analysis of steady-state cell and oxygen concentration distributions. The influence of the surface tension on the accumulated chemotaxis cells at the deformed free surface is studied analytically to illustrate its effect on the stability of the system. The Froude number, ${{Fr_\tau }}$, and capillary number, ${{Ca_\tau }}$, are two additional parameters introduced here. A detailed parametric study is undertaken to investigate the roles of the critical Rayleigh number, ${{Ra_\tau }}_c$, as well as ${{Fr_\tau }}$ and ${{Ca_\tau }}$, in the chemotaxis system. Linear stability results revealed that an increasing value of ${{Ra_\tau }}$ would stabilize the chemotaxis system. At a higher value of ${{Fr_\tau }}$, the motion of the cells is faster towards the free surface, and as the surface tension force increases, less accumulated cells are found at the free surface. A cluster of the cells can be observed mostly at the trough rather than on the crest of the wave profile. While experimental results for the present model are not yet available, the results of the linear stability analysis provide useful information about the system's stability.

JFM classification

Biological Fluid Dynamics: Bioconvection Convection: Buoyancy-driven instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A14
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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