Sharp Residual-Work Criteria for Positive-Cone Oldroyd-B and FENE-P Reynolds States

Abstract

We prove sharp residual-work criteria for entropy-admissible Reynolds states in viscoelastic models whose elastic variables are constrained by a positive cone or by a finite-extensibility domain. The argument is formulated for an entropy-dual class of closures in which the entropy lever and the elastic stress satisfy a compatibility relation. For the stress-diffusion-free Oldroyd-B system, written in positive-cone variables A=e^B, we derive an exact defect-work identity and remove pressure and mean modes from the momentum residual. The resulting pressure-free admissibility condition is a signed work inequality coupling the conformation residual to the entropy-dual lever I-A^{-1}. The criterion gives the optimal pointwise cost, the unique aligned minimizing residual, a windowed three-channel alternative, and closed exclusion tests for structured families. We also prove the corresponding entropy-dual closure theorem and recover the FENE-P case as a finite-extensibility corollary. A concrete finite-thickness shear-layer construction shows that positive pressure-free work can outrun the available lever-residual-alignment budget, giving a gauge-invariant residual-level obstruction.