A Causal Anti-Calculus Memory-Kernel Model for Viscoelastic NonNewtonian Turbulence

Non-Newtonian turbulence in polymeric and viscoelastic liquids involves an interaction between the velocity field, constitutive memory, elastic energy storage, dissipation, and multi-scale transfer. This manuscript develops a causal Anti-Calculus-inspired constitutive extension (AC-NNNS) and tests its principal mechanisms and an exploratory relaxation/crossover scaling against external experimental literature. The mathematical starting point is the supplied Anti-Calculus framework, which defines reverse accumulation through sign-reversed derivative/integral operations. A new causal extension is introduced here by applying a history operator to the rate-of-strain tensor. For a normalized kernel K(tau), the memory field is A_K[D](t)=∫K(tau)[D(t-tau)-D(t)]d tau, and the extra stress is taken as tau_A=2 eta_A A_K[D]. The contribution -tau_A:D to the kinetic-energy equation is sign-indefinite, permitting the same constitutive channel to act locally either as an energy sink
or as a return pathway from stored elastic energy to fluid motion. This feature is consistent with external experiments showing both polymeric energy dissipation and random release of stored elastic energy into velocity fluctuations. A two-relaxation exponential kernel is then introduced to represent multiple molecular/
time scales. Quantitative comparison uses two microchannel experiments. The conclusions are limited to the stated model, dataset, and assumptions; the analysis does not establish the proposed mechanism as unique or experimentally confirmed.

Keywords: Non-Newtonian Turbulence, Viscoelastic Turbulence, Elastic Turbulence, Polymer Solutions, Constitutive Memory.