Flow-index universality and critical regime transitions of shock thickness in a power-law fluid model within rational extended thermodynamics

The shock-wave structure of isothermal non-Newtonian hyperbolic fluids is studied in Lagrangian variables within the framework of rational extended thermodynamics. We consider a power-law relaxation model with flow index m>0 and with a general pressure law p(F) satisfying p′<0 and p″>0, where F is the deformation gradient. The shock-structure problem is reduced to a single first-order ordinary differential equation for F. We first prove an existence theorem for continuous monotone shock profiles and identify an explicit critical Mach number M0*: such profiles exist for subcritical shocks, 1<M0<M0*, whereas beyond this threshold a sub-shock forms. We then analyze, in the subcritical regime, the shock thickness Δ and obtain two distinct asymptotic classifications governed by the flow index m. In the weak-shock limit M0→1+, the thickness diverges for m<2, tends to a finite non-zero limit for m=2, and vanishes for m>2. In the near-subshock limit M0→M0*−, the thickness tends to a finite positive limit for 0<m≤1, whereas it vanishes for m>1. The critical flow-index thresholds m=2 and m=1 are universal in the sense that they are independent of the nonlinear equation of state. In contrast, the finite limiting values and scaling amplitudes depend in general on the pressure law. Numerical calculations for the analytically tractable quasi-incompressible law p∝1/F illustrate the two transitions and the distinct behavior of the shock thickness on the two sides of the flow-index thresholds.