Assessment of a general equilibrium assumption for development of algebraic viscoelastic models

Abstract

The recent development of algebraic explicit stress models (AESM) for viscoelastic fluids rests upon a general equilibrium assumption, by invoking a slow variation condition on the evolution of the viscoelastic anisotropy tensor (the normalized de-viatoric part of the extra-stress tensor [G. Mompean, R.L. Thompson, P.R. Souza Mendes, A general transformation procedure for differential viscoelastic models, J. Non-Newtonian Fluid Mech. 111 (2003) 151–174]). This equilibrium assumption can take various forms depending on the general objective derivative which is used in the slow variation assumption. The purpose of the present paper is to assess the validity of the equilibrium hypothesis in different flow configurations. Viscometric flows (pure shear and pure elongation) are first considered to show that the Harnoy derivative [A. Harnoy, Stress relaxation effect in elastico-viscous lubricants in gears and rollers, J. Fluid Mech. 76(3) (1976) 501–517] is a suitable choice as an objective derivative that allows the algebraic models to retain the viscometric properties of the differential model from which they are derived. A creeping flow through a 4:1 planar contraction then serves as a benchmark for testing the equilibrium assumption in a flow exhibiting complex kinematics. Results of numerical simulations with the differential Oldroyd-B constitutive model allow to evaluate a posteriori the weight of extra-stress terms in different regions of the flow. Computations show that the equilibrium assumption making use of the Harnoy derivative is globally well verified. The assumption is exactly verified in flow regions of near-viscometric kinematics, whereas some departures are observed in the very near region of the corner entrance.