Modified Navier-Stokes model for nonequilibrium stationary states

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Abstract
In this paper we present a hydrodynamic model to study the features of the behavior of a fluid which is brought to a stationary state by the action of an external gradient. Two cases are considered, namely, the action of a thermal gradient and a constant shear rate. The hydrodynamic equations used are based on constitutive equations constructed from Curie's principle. The external gradients imposing the constraints as the steady state are regarded as quantities of zeroth order. We examine two features of the fluid's behavior, namely, the sound absorption and the light scattering. In the former case we show how the Stokes-Kirchhoff formula is modified by the presence of the gradients. This modification suggests an experimental verification of this model which is independent of the magnitude of the real wave vector. In the latter case we compute the Brillouin-Rayleigh spectra and obtain in both cases the same shift in the Brillouin peaks already predicted by others. However, we also predict a small charge in the intensity of the peaks because of the modification of the sound-absorption coefficient. The essential result in this calculation is the shift of the Rayleigh peak arising from the entropy flow in the case of the thermal gradient and an entropy-production term in the case of the constant rate of shear. In both cases the order of magnitude of this correction in terms of the wave vector is the same as the terms responsible for the shift in the Brillouin peaks. Finally, we also compare our results with those obtained by others.