  Taylor's swimming sheet is a classical model of microscale propulsion and
pumping. Many biological fluids and substances are fibrous, having a preferred
direction in their microstructure; for example cervical mucus is formed of
polymer molecules which create an oriented fibrous network. Moreover,
suspensions of elongated motile cells produce a form of active oriented matter.
To understand how these effects modify viscous propulsion, we extend Taylor's
classical model of small-amplitude zero-Reynolds-number propulsion of a
'swimming sheet' via the transversely-isotropic fluid model of Ericksen, which
is linear in strain rate and possesses a distinguished direction. The energetic
costs of swimming are significantly altered by all rheological parameters and
the initial fibre angle. Propulsion in a passive transversely-isotropic fluid
produces an enhanced mean rate of working, independent of the initial fibre
orientation, with an approximately linear dependence of energetic cost on the
extensional and shear enhancements to the viscosity caused by fibres. In this
regime the mean swimming velocity is unchanged from the Newtonian case. The
effect of the constant term in Ericksen's model for the stress, which can be
identified as a fibre tension or alternatively a stresslet characterising an
active fluid, is also considered. This stress introduces an angular dependence
and dramatically changes the streamlines and flow field; fibres aligned with
the swimming direction increase the energetic demands of the sheet. The
constant fibre stress may result in a reversal of the mean swimming velocity
and a negative mean rate of working if sufficiently large relative to the other
rheological parameters.
