  We demonstrate that the Navier-Stokes equation can be covariantized under the
full infinite dimensional Galilean Conformal Algebra (GCA), such that it
reduces to the usual Navier-Stokes equation in an inertial frame. The
covariantization is possible only for incompressible flows, i.e when the
divergence of the velocity field vanishes. Using the continuity equation, we
can fix the transformation of pressure and density under GCA uniquely. We also
find that when all chemical potentials vanish, $c_{s}$, which denotes the speed
of sound in an inertial frame comoving with the flow, must either be a
fundamental constant or given in terms of microscopic parameters. We will
discuss how both could be possible. In absence of chemical potentials, we also
find that the covariance under GCA implies that either the viscosity should
vanish or the microscopic theory should have a length scale or a time scale or
both. We further find that the higher derivative corrections to the
Navier-Stokes equation, can be covariantized, only if they are restricted to
certain possible combinations in the inertial frame. We explicitly evaluate all
possible three derivative corrections. Finally, we argue that our analysis
hints that the parent relativistic theory with relativistic conformal symmetry
needs to be deformed before the contraction is taken to produce a sensible GCA
invariant dynamical limit.
