  Suspensions of self-motile, elongated particles are a topic of significant
current interest, exemplifying a form of `active matter'. Examples include
self-propelling bacteria, algae and sperm, and artificial swimmers. Ericksen's
model of a transversely-isotropic fluid [J. L. Ericksen, Colloid Polym. Sci.
173(2):117-122 (1960)] treats suspensions of non-motile particles as a
continuum with an evolving preferred direction; this model describes fibrous
materials as diverse as extracellular matrix, textile tufts and plant cell
walls. Director-dependent effects are incorporated through a modified stress
tensor with four viscosity-like parameters. By making fundamental connections
with recent models for active suspensions, we propose a modification to
Ericksen's model, mainly the inclusion of self motility; this can be considered
the simplest description of an oriented suspension including
transversely-isotropic effects. Motivated by the fact that
transversely-isotropic fluids exhibit modified flow stability, we conduct a
linear stability analysis of two distinct cases, aligned and isotropic
suspensions of elongated active particles. Novel aspects include the
anisotropic rheology and translational diffusion. In general anisotropic
effects increase the instability of small perturbations, whilst translational
diffusion stabilises a range of wave-directions and, in some cases, a finite
range of wave-numbers, thus emphasising that both anisotropy and translational
diffusion can have important effects in these systems.
