  Topological defects in active polar fluids can organise spontaneous flows and
influence macroscopic density patterns. Both of them play, for example, an
important role during animal development. Yet the influence of density on
active flows is poorly understood. Motivated by experiments on cell monolayers
confined to discs, we study the coupling between density and polar order for a
compressible active polar fluid in presence of a +1 topological defect. As in
the experiments, we find a density-controlled spiral-to-aster transition. In
addition, biphasic orientational phases emerge as a generic outcome of such
coupling. Our results highlight the importance of density gradients as a
potential mechanism for controlling flow and orientational patterns in
biological systems.
