Relativistic and nonrelativistic description of fluids with anisotropic heat conduction

We formulate a set of Lorentz-covariant equations for an imperfect fluid with viscosity, dilatational viscosity, and anisotropic thermal conductivity that possess the full GENERIC structure of nonequilibrium thermodynamics. The GENERIC structure, which includes and goes beyond prior nonequilibrium generalizations of the second law of thermodynamics, is shown to provide a guideline for modifying previous phenomenological or kinetic-theory based equations of extended relativistic hydrodynamics. In the nonrelativistic limit, we discuss the form of the equations for viscous and viscoelastic fluids with anisotropic heat conduction