  We present a dissipative hydrodynamic theory of "s-wave dipole superfluids"
that arise in phases of translation-invariant and dipole-symmetric models in
which the U(1) symmetry is spontaneously broken. The hydrodynamic description
is subtle on account of an analogue of dangerously irrelevant operators, which
requires us to formalize an entirely new derivative counting scheme suitable
for these fluids. We use our hydrodynamic model to investigate the linearized
response of such a fluid, characterized by sound modes $\omega \sim \pm k -
ik^2$, shear modes $\omega\sim-ik^2$, and magnon-like propagating modes $\omega
\sim \pm k^2 - ik^4$ that are the dipole-invariant version of superfluid
"second sound" modes. We find that these fluids can also admit equilibrium
states with "dipole superflow" that resemble a polarized medium. Finally, we
couple our theory to slowly varying background fields, which allows us to
compute response functions of hydrodynamic operators and Kubo formulas for
hydrodynamic transport coefficients.
