  The present paper is based on a previous work (involving two of the present
authors) where a generalized fluid dynamical model was proposed. The underlying
symplectic structure of the Lagrangian discrete degrees of freedom obeyed a
Non-Commutative algebra, generated by Berry curvature correction. In an Euler
(or Hamiltonian) framework, this is manifested as an extended algebra between
the fluid variables, leading to the extended fluid model. Here we study the
dynamics of sonic fluctuations that live in this effective analogue gravity
spacetime. Interestingly enough, the effective metric resembles that of a
spinning Black Hole; the spin is induced by the underlying Non-Commutative
structure. The effective mass and spin parameters of the Black Hole, in terms
of fluid parameters, are also identified. The connection of our model with
anomalous Hall systems may lead to observable signatures of the analogue black
hole in physical systems.
