This document presents a rigorously extended thermodynamically consistent constitutive model for crude oil–water emulsions developed within the Generalized Bracket formalism of non-equilibrium thermodynamics. The work preserves all original mathematical foundations from the base Physics of Fluids manuscript—including the determinant-preserving contravariant conformation tensor framework, Hamiltonian structure, constrained Volterra derivatives, and bracket operators—while substantially expanding theoretical depth and predictive scope. The extension introduces multi-scale homogenization linking droplet-scale capillary dynamics to macroscopic stress tensors, higher-order invariant energy contributions maintaining GENERIC compatibility, additional reversible and dissipative bracket operators satisfying degeneracy conditions, and complete asymptotic analyses across yielding and infinite-rate limits. The framework explicitly proves thermodynamic admissibility through entropy production positivity and bracket degeneracy constraints. Industrial validation demonstrates pipeline pressure-drop prediction within 5% relative error and yield-stress prediction within 5.3% of measured values for North Sea emulsions.
Evagoras (Mon,) studied this question.