Evaluation of theapplicability of Schr\"odinger-variant equations for classical fluid dynamics

The artificial fluid model known as "Schr\"odinger flow" (SF) can represent rotational flow with dissipative effects, and has attracted attention despite its gap from real-world fluid behavior. To address the structural discrepancy arising from the incomplete transition from quantum hydrodynamics to classical fluid dynamics, we propose a variant of the hydrodynamic Schr\"odinger equation (HSE) that better aligns with classical formulations. We then revisit an alternative method for eliminating the quantum potential (referred to in this work as the "cancellation approach") in light of the known non-universality of the classical limit. Beyond examining conservation laws via Lagrangian field theory, we identify its deeper connection to a generalized Lagrangian field theory, which offers a more straightforward route to constructing a Lagrangian density in terms of the components of a two-component wave function for rotational inviscid flow. Finally, we discuss the completeness of Clebsch parameterization, concluding our evaluation of the applicability of the proposed SF-variant formulation.