On the closure problem of the effective stress in the Eulerian-Eulerian and mixture modeling approaches for the simulation of liquid-particle suspensions

© 2019 Author(s). We address the closure problem of the phasic effective stress tensors in the Eulerian-Eulerian and mixture models, considering suspensions of identical particles dispersed in Newtonian liquids. First, after briefly describing the modeling approaches, we review the key mechanisms generating phasic stress and discuss the shortcomings of some constitutive expressions in reproducing important experimental observations. For dilute suspensions, these include the mixture viscosity rise with solid concentration whilst for dense suspensions, the occurrence of particle migration and the change of mixture rheology from Newtonian to non-Newtonian. We then use computational fluid dynamics simulations to compare results based on various stress tensor closures. In a first case study, the simulation results of a laminar flow in a horizontal pipe of a dilute suspension of particles dispersed in a Newtonian liquid are compared to experimental data obtained from the literature. We show that both the Eulerian-Eulerian and mixture models can predict pressure drops accurately but only if they are coupled with suitable experimental closures for the mixture rheology. In a second case study, we simulate the laminar flow of a dense suspension of identical particles dispersed in a Newtonian liquid through an abrupt expansion. We show that the particle concentration profile in the upstream tube, which develops owing to shear-induced particle migration, strongly affects the flow patterns downstream of the expansion. This migration must be modeled via an appropriate closure for the solid effective stress tensor; this allows capturing the sophisticated flow patterns in the expansion section