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14 Modelling Dissipation

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 [53Jump To The Next Citation Point] 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 [39Jump To The Next Citation Point] and Landau and Lifshitz [66Jump To The Next Citation Point], which is based on a seemingly natural extension of the inviscid equations. However, a detailed analysis of Lindblom and Hiscock [54Jump To The Next Citation Point55] 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 [107Jump To The Next Citation Point57Jump To The Next Citation Point58Jump To The Next Citation Point]. Their description, the derivation of which was inspired by early work of Grad [50Jump To The Next Citation Point] and Müller [81Jump To The Next Citation Point] and results from relativistic kinetic theory, provides a framework that is generally accepted as meeting the key criteria for a relativistic model [53Jump To The Next Citation Point]. 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.

 14.1 The “standard” relativistic models
 14.2 The Israel–Stewart approach
 14.3 Carter’s canonical framework
 14.4 Remaining issues

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