The effective theory of rotating pion superfluid in the presence of topological defects will be considered. We study the anomaly induced effects and the interplay between domain-wall and superfluid vortex under rotation. A non-uniform rotation leads to new effects in the domain-wall and vortex system. It will be shown that the effective theory predicts radial current flows of charges in the system whereas the previously studied cases dealt with induced static charges. The main observation is that the radial flow consists of two parts which are related to the presence of gauge and gravitational anomalies. The microscopic picture of fermionic zero modes propagating along the vortices will be used to justify the chiral effective theory results and clarifies the mechanism for the current flow. Then gravitoelectromagnetic formalism is used to redrive the gauge anomaly related part of radial flow. Finally, as our main observation, we discuss the $(1+1)$-dimensional gravitational anomaly on the vortex ring which entails an energy-momentum flow on the domain-wall. Again the results will be confirmed from a microscopic point of view.
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