  Starting from the linear sigma model with constituent quarks we derive the
chiral fluid dynamics where hydrodynamic equations for the quark fluid are
coupled to the equation of motion for the order-parameter field. In a static
system at thermal equilibrium this model leads to a chiral phase transition
which, depending on the choice of the quark-meson coupling constant, could be a
crossover or a first order one. We investigate the stability of the chiral
fluid in the static and expanding backgrounds by considering the evolution of
perturbations with respect to the mean-field solution. In the static background
the spectrum of plane-wave perturbations consists of two branches, one
corresponding to the sound waves and another to the sigma-meson excitations.
For large couplings these two branches "cross" and the excitation spectrum
acquires exponentially growing modes. The stability analysis is also done for
the Bjorken-like background solution by explicitly solving the time-dependent
differential equation for perturbations in the eta-space. In this case the
growth rate of unstable modes is significantly reduced.
