  We analyze one of the simplest active suspensions with complex dynamics: a
suspension of immotile "Extensor" particles that exert active extensile dipolar
stresses on the fluid in which they are immersed. This is relevant to several
experimental systems, such as recently studied tripartite rods that create
extensile flows by consuming a chemical fuel. We first describe the system
through a Doi-Onsager kinetic theory based on microscopic modeling. This theory
captures the active stresses produced by the particles that can drive
hydrodynamic instabilities, as well as the steric interactions of rod-like
particles that lead to nematic alignment. This active nematic system yields
complex flows and disclination defect dynamics very similar to phenomenological
Landau-deGennes Q-tensor theories for active nematic fluids, as well as by more
complex Doi-Onsager theories for polar microtubule/motor-protein systems. We
apply the quasi-equilibrium Bingham closure, used to study suspensions of
passive microscopic rods, to develop a non-standard Q-tensor theory. We
demonstrate through simulation that this "BQ-tensor" theory gives an excellent
analytical and statistical accounting of the suspension's complex dynamics, at
a far reduced computational cost. Finally, we apply the BQ-tensor model to
study the dynamics of Extensor suspensions in circular and bi-concave domains.
In circular domains, we reproduce previous results for systems with weak
nematic alignment, but for strong alignment find novel dynamics with
activity-controlled defect production and absorption at the boundaries of the
domain. In bi-concave domains, a Fredericks-like transition occurs as the width
of the neck connecting the two disks is varied.
