Relativistic Hydrodynamics 2026, 10: Relativistic Non-Perfect Fluid

When viscosity effects and heat conduction are present in a fluid, the definition of local rest frames become ambiguous, and the rest-mass density current, energy-momentum tensor and entropy current become primary field variables. A kinetic theory may resolve this uncertainty, though a "hydrodynamic" approach can be attempted using the equations of motion and the maximum entropy principle. Non-perfect fluids has a 4-velocity u, separable into a timelike vector parallel to J, and another parallel to T. In perfect fluids these two overlap, giving a unique 4-velocity. In general, either of these two options may be chosen for the 4-velocity, which defines the "Landau frame" or "energy frame" (adapt the vector parallel to J), where there is no net energy flux. The rest-mass density current is not parallel to the 4-velocity. In the "Eckart frame" or "particle frame" there is no dissipative contribution to the rest-mass density current.

The energy-momentum tensor for non-perfect fluids decomposes into that of energy fluxes, and that for the viscous contributions, which itself defines the the anisotropic stress tensor π and viscous bulk pressure Π.

In the Eckart frame, the total rest-mass density is proportional to the 4-velocity only. The momentum equation is derived again by projecting the energy equation of motion into the into the orthogonal space to u.

When irreversible processes are present, entropy is no longer conserved, and the second law of thermodynamics holds. The dissipative part in the entropy is assumed to be a function of the thermodynamic fluxes. For a perfect fluid, is vanishes, as those fluxes themselves vanish at equilibrium.

Classical irreversible TD would assume a linear dependence of the dissipative contribution of entropy on the thermodynamic fluxes. The entropy current built from the TD fluxs and the 4-velocity, it is

where f are TD functions of the number density n and energy density e. The derived entropy density requires a maximum in the equilibrium state. The fluxes can be written in terms of the transport coefficients as the constitutive equations of CIT, known as Eckart's theory of Relativistic irreversible TD.