  We report numerical results for the hydrodynamics of inhomogeneous lyotropic
and extensile active nematic gels. By simulating the coupled Cahn-Hilliard,
Navier-Stokes, and Beris-Edwards equation for the evolution of the composition,
flow and orientational order of an active nematic, we ask whether composition
variations are important to determine its emergent physics. As in active gels
of uniform composition, we find that increasing either activity or nematic
tendency (e.g., overall active matter concentration) triggers a transition
between an isotropic passive phase and an active nematic one. We show that
composition inhomogeneities are important in the latter phase, where we find
three types of possible dynamical regimes. First, we observe regular patterns
with defects and vortices: these exist close to the passive-active transition.
Second, for larger activity, or deeper in the nematic phase, we find active
turbulence, as in active gels of uniform composition, but with exceedingly
large composition variation. In the third regime, which is uniquely associated
with inhomogeneity and occurs for large nematic tendency and low activity, we
observe spontaneous microphase separation into active and passive domains. The
microphase separated regime is notable in view of the absence of an explicit
demixing term in the underlying free energy which we use, and we provide a
theoretical analysis based on the common tangent construction which explains
its existence. We hope this regime can be probed experimentally in the future.
