  In this work, we perform a phenomenological derivation of the first- and
second-order relativistic hydrodynamics of dissipative fluids. To set the
stage, we start with a review of the ideal relativistic hydrodynamics from
energy-momentum and particle number conservation equations. We then go on to
discuss the matching conditions to local thermodynamical equilibrium,
symmetries of the energy-momentum tensor, decomposition of dissipative
processes according to their Lorentz structure, and finally, the definition of
the fluid velocity in the Landau and Eckart frames. With this preparatory work,
we first formulate the first-order (Navier-Stokes) relativistic hydrodynamics
from the entropy flow equation, keeping only the first-order gradients of
thermodynamical forces. A generalized form of diffusion terms is found with a
matrix of diffusion coefficients describing the relative diffusion between
various flavors. The procedure of finding the dissipative terms is then
extended to the second order to obtain the most general form of dissipative
function for multiflavor systems up to the second order in dissipative fluxes.
The dissipative function now includes in addition to the usual second-order
transport coefficients of Israel-Stewart theory also second-order diffusion
between different flavors. The relaxation-type equations of second-order
hydrodynamics are found from the requirement of positivity of the dissipation
function, which features the finite relaxation times of various dissipative
processes that guarantee the causality and stability of the fluid dynamics.
These equations contain a complete set of nonlinear terms in the thermodynamic
gradients and dissipative fluxes arising from the entropy current, which are
not present in the conventional Israel-Stewart theory.
