Dynamic equations and bulk viscosity near the gas-liquid critical point

We derive dynamic equations of the mass, energy, and momentum densities to describe dynamics of highly compressible fluids near the gas-liquid critical point. In particular, the complete expression for the stress tensor is presented, which is nonlinear with respect to the fluctuations of the mass and energy densities. The most dominant nonlinearity in the stress tensor can then be identified and a very simple and systematic theory can be constructed on the enhanced bulk viscosity near the critical point with no adjustable parameter. We introduce the frequency-dependent adiabatic compressibility and constant-volume specific heat. Our theory is essentially equivalent to Kawasaki's theory [Phys. Rev. A 1, 1750 (1970)] at low frequencies and reproduces Ferrell and Bhattacharjee's phenomenology [Phys. Lett. A 36, 109 (1981); 88, 77 (1982); Phys. Rev. A 31, 1788 (1985)] at high frequencies. We explicitly calculate strikingly slow decay of the time correlation function of the diagonal part of the stress tensor. As proposed experiments we examine how the density changes adiabatically in two situations with a fixed volume or pressure. As a by-product we also derive some relations among the critical amplitudes of the constant-volume specific heat above and below ${\mathrm{T}}_{\mathrm{c}}$. It is shown to correspond to the specific heat at constant magnetization in Ising spin systems.

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