Although the inviscid model provides a natural starting point for any investigation of the dynamics of a fluid system, the effects of dissipative mechanisms are often key to the construction of a realistic model. Consider, for example, the case of neutron star oscillations and possible instabilities. While it is interesting from the conceptual point of view to establish that an instability (such as the gravitational-wave driven instability of the fundamental f-mode or the inertial r-mode discussed above) may be present in an ideal fluid, it is crucial to establish that the instability actually has opportunity to grow on a reasonably short timescale. To establish this, one must consider the most important damping mechanisms and work out whether they will suppress the instability or not. A recent discussion of these issues in the context of the r-mode instability can be found in [4].
From the point of view of relativistic fluid dynamics, it is clear already from the outset that we are
facing a difficult problem. After all, the Fourier theory of heat conduction leads to instantaneous
propagation of thermal signals. The fact that this non-causality is built into the description is unattractive
already in the context of the Navier–Stokes equations. After all, one would expect heat to propagate
at roughly the mean molecular speed in the system. For a relativistic description non-causal
behavior would be truly unacceptable. As work by Lindblom and Hiscock [53] has established,
there is a deep connection between causality, stability, and hyperbolicity of a dissipative model.
One would expect an acceptable model to be hyperbolic, not allowing signals to propagate
superluminally.
Our aim in this section is to discuss the three main models that exist in the literature. We first consider
the classic work of Eckart [39] and Landau and Lifshitz [66
], which is based on a seemingly natural
extension of the inviscid equations. However, a detailed analysis of Lindblom and Hiscock [54
, 55] has
demonstrated that these descriptions have serious flaws and must be considered unsuitable for practical use.
However, having discussed these models it is relatively easy to extend them in the way proposed by Israel
and Stewart [107
, 57
, 58
]. Their description, the derivation of which was inspired by early work of
Grad [50
] and Müller [81
] and results from relativistic kinetic theory, provides a framework that is
generally accepted as meeting the key criteria for a relativistic model [53
]. Finally, we describe Carter’s
more recent approach to the problem. This model is elegant because it makes maximal use of a variational
argument. The construction is also more general than that of Israel and Stewart. In particular,
it shows how one would account for several dynamically independent interpenetrating fluid
species. This extension is important for, for example, the consideration of relativistic superfluid
systems.
http://www.livingreviews.org/lrr-2007-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |