General-relativistic hydrodynamics of non-perfect fluids: 3+1 conservative formulation and application to viscous black-hole accretion

TL;DR

This paper develops a fully general, causal 3+1 conservative formulation of relativistic hydrodynamics for non-perfect fluids, suitable for numerical simulations in curved spacetime, and applies it to viscous black-hole accretion.

Contribution

It introduces a novel, general formulation of relativistic dissipative hydrodynamics that is compatible with numerical relativity codes and demonstrates its application to viscous accretion onto black holes.

Findings

The formulation is fully conservative and causal.

It successfully models viscous accretion with deviations from inviscid solutions.

The approach is validated through standard tests and a new black-hole accretion scenario.

Abstract

We consider the relativistic hydrodynamics of non-perfect fluids with the goal of determining a formulation that is suited for numerical integration in special-relativistic and general-relativistic scenarios. To this end, we review the various formulations of relativistic second-order dissipative hydrodynamics proposed so far and present in detail a particular formulation that is fully general, causal, and can be cast into a 3+1 flux-conservative form, as the one employed in modern numerical-relativity codes. As an example, we employ a variant of this formulation restricted to a relaxation-type equation for the bulk viscosity in the general-relativistic magnetohydrodynamics code $\texttt{BHAC}$. After adopting the formulation for a series of standard and non-standard tests in 1+1-dimensional special-relativistic hydrodynamics, we consider a novel general-relativistic scenario, namely, the stationary, spherically symmetric, viscous accretion onto a black hole. The newly developed solution $-$ which can exhibit even considerable deviations from the inviscid counterpart $-$ can be used as a testbed for numerical codes simulating non-perfect fluids on curved backgrounds.