Hydrodynamics with spacetime-dependent scattering length

Hydrodynamics provides a concise but powerful description of long-time and long-distance physics of correlated systems out of thermodynamic equilibrium. Here we construct hydrodynamic equations for nonrelativistic particles with a spacetime-dependent scattering length and show that it enters constitutive relations uniquely so as to represent the fluid expansion and contraction in both normal and superfluid phases. As a consequence, we find that a leading dissipative correction to the contact density due to the spacetime-dependent scattering length is proportional to the bulk viscosity ($\zeta_2$ in the superfluid phase). Also, when the scattering length is slowly varied over time in a uniform system, the entropy density is found to be produced even without fluid flows in proportion to the bulk viscosity, which may be useful as a novel probe to measure the bulk viscosity in ultracold-atom experiments.