  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-B\'enard 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.
