Gradient-dependent transport coefficients in the Navier–Stokes–Fourier system

In the engineering praxis, Newton's law of viscosity and Fourier's heat conduction law are applied to describe thermomechanical processes of fluids. Despite several successful applications, there are some obscure and unexplored details, which are partly answered in this paper using the methodology of irreversible thermodynamics. Liu's procedure is applied to derive the entropy production rate density, in which positive definiteness is ensured via linear Onsagerian equations; these equations are exactly Newton's law of viscosity and Fourier's heat conduction law. The calculations point out that, theoretically, the transport coefficients (thermal conductivity and viscosity) can also depend on the gradient of the state variables in addition to the well-known dependence of the state variables. This gradient dependency of the transport coefficients can have a significant impact on the modeling of such phenomena as welding, piston effect or shock waves.