Boltzmann gain-loss-based non-Hookean spring model and sinh-type nonlinear coupled constitutive relation for viscoelastic fluids

Abstract

This study proposes a theoretical non-Hookean spring force model based on Boltzmann's 1872 gain-and-loss concept for the molecular-level description of polymer stresses in viscoelastic fluids. The resulting thermodynamically consistent second-order constitutive model, which is obtained from the Fokker-Planck kinetic equation of elastic dumbbells with a Boltzmann interaction term using the cumulant expansion-based balanced closure, takes the form of a hyperbolic sine function. The mean cumulant inside the hyperbolic sine factor represents a combination of the Rayleigh-Onsager dissipation function and a macromolecular property-dependent softening parameter. In parallel, the study presents a non-negative non-equilibrium distribution function in the exponential form, unlike conventional non-equilibrium distribution functions in polynomial-based perturbation form. The study also revealed similarities and differences between the present Boltzmann model of gain and loss of dumbbells and the Yamamoto model of reformation and breakage of network junctions. Furthermore, two phenomenological sinh-type models, the Ree-Eyring model for non-Newtonian flow and the Johnson-Tevaarwerk model for elastohydrodynamic lubrication, were found to be a subset of the new theoretical model. To validate the new model, it was applied to two benchmark problems of extensional and shear flows, and good agreement with experimental data was found for a wide range of Weissenberg numbers. The computational stability of the new model was also demonstrated based on the planar stability theory of autonomous dynamical systems. Finally, preliminary results using OpenFOAM-RheoTool solvers indicate that the new model is free of the high-Weissenberg-number problem for two-dimensional viscoelastic flow over a confined cylinder up to the maximum Weissenberg number considered, 1000.