  We explore the scattering amplitudes of fluid quanta described by the
Navier-Stokes equation and its non-Abelian generalization. These amplitudes
exhibit universal infrared structures analogous to the Weinberg soft theorem
and the Adler zero. Furthermore, they satisfy on-shell recursion relations
which together with the three-point scattering amplitude furnish a pure
S-matrix formulation of incompressible fluid mechanics. Remarkably, the
amplitudes of the non-Abelian Navier-Stokes equation also exhibit
color-kinematics duality as an off-shell symmetry, for which the associated
kinematic algebra is literally the algebra of spatial diffeomorphisms. Applying
the double copy prescription, we then arrive at a new theory of a tensor
bi-fluid. Finally, we present monopole solutions of the non-Abelian and tensor
Navier-Stokes equations and observe a classical double copy structure.
