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Linear Rayleigh–Bénard stability of a transversely isotropic fluid

Published online by Cambridge University Press:  25 June 2018

C. R. HOLLOWAY ,
D. J. SMITH  and
R. J. DYSON
C. R. HOLLOWAY
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK email: Craig.Holloway@tessella.com, d.j.smith@bham.ac.uk, R.J.Dyson@bham.ac.uk
D. J. SMITH
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK email: Craig.Holloway@tessella.com, d.j.smith@bham.ac.uk, R.J.Dyson@bham.ac.uk Institute for Metabolism and Systems Research, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
R. J. DYSON
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK email: Craig.Holloway@tessella.com, d.j.smith@bham.ac.uk, R.J.Dyson@bham.ac.uk
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Abstract

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Suspended fibres significantly alter fluid rheology, as exhibited in for example solutions of DNA, RNA and synthetic biological nanofibres. It is of interest to determine how this altered rheology affects flow stability. Motivated by the fact thermal gradients may occur in biomolecular analytic devices, and recent stability results, we examine the problem of Rayleigh–Bénard convection of the transversely isotropic fluid of Ericksen. A transversely isotropic fluid treats these suspensions as a continuum with an evolving preferred direction, through a modified stress tensor incorporating four viscosity-like parameters. We consider the linear stability of a stationary, passive, transversely isotropic fluid contained between two parallel boundaries, with the lower boundary at a higher temperature than the upper. To determine the marginal stability curves the Chebyshev collocation method is applied, and we consider a range of initially uniform preferred directions, from horizontal to vertical, and three orders of magnitude in the viscosity-like anisotropic parameters. Determining the critical wave and Rayleigh numbers, we find that transversely isotropic effects delay the onset of instability; this effect is felt most strongly through the incorporation of the anisotropic shear viscosity, although the anisotropic extensional viscosity also contributes. Our analysis confirms the importance of anisotropic rheology in the setting of convection.

Keywords

76E0676A0576D99
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 4 , August 2019 , pp. 659 - 681
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2018

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