  We discuss the nonrelativistic limit of the relativistic
Navier-Fourier-Stokes (NFS) theory. The next-to-leading order relativistic
corrections to the NFS theory for the Landau-Lifshitz fluid are obtained. While
the lowest order truncation of the velocity expansion leads to the usual NFS
equations of nonrelativistic fluids, we show that when the next-to-leading
order relativistic corrections are included, the equations can be expressed
concurrently with two different fluid velocities. One of the fluid velocities
is parallel to the conserved charge current (which follows the Eckart
definition) and the other one is parallel to the energy current (which follows
the Landau-Lifshitz definition). We compare this next-to-leading order
relativistic hydrodynamics with bivelocity hydrodynamics, which is one of the
generalizations of the NFS theory and is formulated in such a way to include
the usual mass velocity and also a new velocity, called the volume velocity. We
find that the volume velocity can be identified with the velocity obtained in
the Landau-Lifshitz definition. Then, the structure of bivelocity
hydrodynamics, which is derived using various nontrivial assumptions, is
reproduced in the NFS theory including the next-to-leading order relativistic
corrections.
