Rational extended thermodynamics for non-Newtonian fluids with finite relaxation time

We present a thermodynamically consistent model for non-Newtonian fluids within rational extended thermodynamics, where stress evolves directly from local balance laws with production terms, eliminating the need for empirical constitutive laws and artificial derivatives. In the one-dimensional, isothermal case, the model reproduces power-law relaxation asymptotically and, in a particular case, exhibits nonlinear stress dynamics reminiscent of the Phan–Thien–Tanner model. The differential system is symmetric hyperbolic, and the Cauchy problem is well posed for both smooth and weak solutions, including shocks. Remarkably, for a quadratic viscous energy, it predicts finite-time extinction of viscous stress in the shear-thickening regime, a phenomenon largely unexplored in the literature. This framework provides a predictive, physically grounded alternative to empirical models and can be extended to multidimensional and thermal flows with applications in industry and biology.