  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.
