In the discussion of the preceding section viscous effects have been explicitly neglected. Otherwise, Euler’s equation would have to be replaced by the Navier–Stokes’ equation (no longer hyperbolic) and the energy equation would contain additional terms accounting for heat transfer and the dissipation of kinetic energy. While their numerical description still represents a challenging task, viscous effects due to the transfer of momentum along velocity gradients by turbulence and thermal motions should, in general, be considered in some representative astrophysical scenarios, namely accretion disks for which shearing motions or steep velocity gradients may appear. In addition, viscosity is also believed to play a significant role in the stability and dynamics of compact stars, destroying differential rotation in rapidly-rotating neutron stars or suppressing the bar-mode instability as well as gravitational-radiation driven instabilities. The equations of viscous hydrodynamics, the Navier–Stokes–Fourier equations, have been formulated in relativity in terms of causal dissipative relativistic fluids (see the Living Reviews article by Müller [274] and references therein). These extended fluid theories, however, remain largely unexplored, numerically, in astrophysical systems. The reason may be the lack of an appropriate formulation that is well suited to numerical studies. Work in this direction was done by Peitz and Appl [315] who provided a 3+1 coordinate-free representation of different types of dissipative relativistic-fluid theories, which possess, in principle, the potential of being adapted well to numerical applications.

The interaction between matter and radiation fields, present in different levels of complexity in all astrophysical systems, is described by the equations of radiation hydrodynamics. The Newtonian framework is highly developed (see, e.g., [257]; the special-relativistic transfer equation is also considered in that reference). Pons et al. [321] discuss a hyperbolic formulation of the radiative transfer equations, paying particular attention to the closure relations and to extending HRSC schemes to those equations. General-relativistic formulations of radiative transfer in curved spacetimes are considered in, e.g., [331], [432], and [314] (see also references therein). In [314] the equations of general-relativistic radiation hydrodynamics are derived from a tensor formalism and, therefore, can be applied to any spacetime or coordinate system. An explicit derivation of the radiation-moment equations is given for the case of the Schwarzschild spacetime.

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