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We demonstrate that the Navier-Stokes equation can be covariantized under thefull infinite dimensional Galilean Conformal Algebra (GCA), such that itreduces to the usual Navier-Stokes equation in an inertial frame. Thecovariantization is possible only for incompressible flows, i.e when thedivergence of the velocity field vanishes. Using the continuity equation, wecan fix the transformation of pressure and density under GCA uniquely. We alsofind that when all chemical potentials vanish, $c_{s}$, which denotes the speedof sound in an inertial frame comoving with the flow, must either be afundamental constant or given in terms of microscopic parameters. We willdiscuss how both could be possible. In absence of chemical potentials, we alsofind that the covariance under GCA implies that either the viscosity shouldvanish or the microscopic theory should have a length scale or a time scale orboth. We further find that the higher derivative corrections to theNavier-Stokes equation, can be covariantized, only if they are restricted tocertain possible combinations in the inertial frame. We explicitly evaluate allpossible three derivative corrections. Finally, we argue that our analysishints that the parent relativistic theory with relativistic conformal symmetryneeds to be deformed before the contraction is taken to produce a sensible GCAinvariant dynamical limit.

A Covariant Form of the Navier-Stokes Equation for the Galilean Conformal Algebra