  Out-of-equilibrium effects may play an important role in the dynamics of
neutron star mergers and in heavy-ion collisions. Bemfica, Disconzi, Noronha
and Kovtun (BDNK) recently derived a causal, locally well-posed, and modally
stable relativistic fluid model that incorporates the effects of viscosity and
heat diffusion. We study the non-relativistic limit of this fluid model and
show that causality for relativistic motion restricts the transport
coefficients in the non-relativistic limit. This restriction provides an upper
bound on the ratio of the shear viscosity to the entropy density for fluids
that can be described as relativistic within the BDNK model and can be exactly
modeled using the Navier-Stokes equation in the non-relativistic limit.
Furthermore, we show that the Fourier law of heat conduction must be modified
by higher gradient corrections for such fluids. We also show that the
non-relativistic limit of the BDNK equations of motion are never hyperbolic, in
contrast to the non-relativistic limit of extended variable models, whose
non-relativistic equations of motion can be hyperbolic or not depending on the
scaling of the transport coefficients present in the auxiliary equations for
viscous degrees of freedom.
