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We explore the scattering amplitudes of fluid quanta described by theNavier-Stokes equation and its non-Abelian generalization. These amplitudesexhibit universal infrared structures analogous to the Weinberg soft theoremand the Adler zero. Furthermore, they satisfy on-shell recursion relationswhich together with the three-point scattering amplitude furnish a pureS-matrix formulation of incompressible fluid mechanics. Remarkably, theamplitudes of the non-Abelian Navier-Stokes equation also exhibitcolor-kinematics duality as an off-shell symmetry, for which the associatedkinematic algebra is literally the algebra of spatial diffeomorphisms. Applyingthe double copy prescription, we then arrive at a new theory of a tensorbi-fluid. Finally, we present monopole solutions of the non-Abelian and tensorNavier-Stokes equations and observe a classical double copy structure.

Scattering Amplitudes and the Navier-Stokes Equation