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

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