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Simulating gravity in rotational flow

We consider classical fluids in non-relativistic framework. The flow is considered to be barotropic, inviscid and rotational. We study the linear perturbations over a steady state background flow. We find the acoustic metric from the conservation equation of a current constructed from linear perturbation of first order derivatives (in position and time coordinate) of Bernoulli's constant (scalar field) and vorticity (a vector field). We have rather shown that the conservation equation of current reduces to a massless scalar field equation in the high frequency limit. In contrast to the contemporary works, our work shows that even if we can not find a wave equation (in rotational flow) which is structurally similar to a massless scalar field equation in curved space-time, but still an analogue space-time exists through a conservation equation. Considering velocity potential and Clebesch coefficients, we find that only for some specific systems current conservation equation can be found yielding the same analogue space-time. We conclude that for rotational flows, it is wise to study linear perturbation of Bernoulli's constant over the velocity potential and Clebsch coefficients.