Abstract
In relativistic fluid mechanics, positive entropy production is known to be insufficient for guaranteeing stability. Much stronger criteria for thermodynamic admissibility have become available in nonequilibrium thermodynamics. We here perform a linear stability analysis for a model of relativistic hydrodynamics that is based on the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Assuming a quadratic entropy function near equilibrium, we find stability for the entire range of physically meaningful model parameters for relativistic fluid dynamics based on GENERIC. The search for thermodynamic admissibility moreover reveals a fundamental difference between liquids and gases in relativistic fluid dynamics.
Highlights
Relativistic fluid dynamics plays an important role in astrophysics and cosmology as, for example, it allows us to describe the collapse of stars into neutron stars, flows around black holes, jets with relativistic speeds originating from the core of active galactic nuclei, the formation of galaxies, or the expansion of the entire universe
In the present paper we considered the thermodynamically consistent model for relativistic fluid dynamics developed within the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework [29,30], where the treatment of nonequilibrium phenomena is described by means of additional structural variables to prevent the problem of infinite propagation speeds
We performed a linear stability analysis of the equilibrium rest state and we analytically proved that, in the entire range of the physically meaningful parameters, the model is unconditionally stable
Summary
Relativistic fluid dynamics plays an important role in astrophysics and cosmology as, for example, it allows us to describe the collapse of stars into neutron stars, flows around black holes, jets with relativistic speeds originating from the core of active galactic nuclei, the formation of galaxies, or the expansion of the entire universe. The first attempts to generalize the laws of fluid mechanics and thermodynamics to the relativistic context were based on an extended Fourier law for heat conduction [7,8] These formulations suffered from two fundamental flaws, which are closely related to each other [9,10]: they are unstable [11,12], and the parabolic nature of the corresponding differential equation leads to instantaneous propagation of heat, violating causality [13]. In the limit of slow phenomena, these equations reduce to the classical constitutive laws, but they are suitable to describe fast phenomena, since they lead to hyperbolic equations with finite speeds of propagation for thermal and viscous perturbations Such models are compatible with the kinetic theory of Grad’s 14-moment method [28]; they are compatible with causality and stable under specific constraints [10]. For the indexes of four-vectors and tensors, we use Greek letters to denote components from 0 to 3 (i.e., time and space components) and Latin letters to denote components from 1 to 3 (i.e., space components only)
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