Curvilinear flows of noncolloidal suspensions: The role of normal stresses

Abstract

The role of normal stresses in causing particle migration and macroscopic spatial variation of the particle volume fraction in a mixture of rigid neutrally buoyant spherical particles suspended in Newtonian fluid is examined for curvilinear shear flows. The problem is studied for monodisperse noncolloidal Stokes-flow suspensions, i.e., for conditions of low-Reynolds-number flow and infinite Péclet number, Pe ϭ O(␥ a 3 /kT), where is the suspending fluid viscosity, ␥ is the shear rate, a is the particle radius, and kT is the thermal energy. Wide-gap Couette, parallel-plate torsional, and cone-and-plate torsional flows are studied. The entire dependence of the compressive shear-induced normal stresses is captured by a ''normal stress viscosity'' n (), which vanishes ͑as 2) at ϭ 0 and diverges at maximum packing in the same fashion as does the shear viscosity s (). Anisotropy of the normal stresses arising from the presence of the particles is modeled as independent of , so that ratios of any two particle contributions to the bulk normal stress components are constants, ⌺ 22 p /⌺ 11 p ϭ 2 and ⌺ 33 p /⌺ 11 p ϭ 3 ; the standard convention of ͑1,2,3͒ denoting the ͑flow, gradient, vorticity͒ directions is used so that, for example, ⌺ 11 p is the normal component of the particle stress ⌺ p in the flow direction. Predictions for the steady and unsteady flows are presented to demonstrate the influence of variation of the normal stress anisotropy parameters 2 and 3 , the rheological functions s and n , and the sedimentation hindrance function used to represent the resistance to relative motions of the phases during migration. Comparison with available experimental data shows that a single set of parameters for the rheological model is able to describe all qualitative features of the observed migrations in the flows considered.

Rheology of sheared suspensions of rough frictional particles

“…Anisotropy parameter λ p 2 is experimentally noted to increase linearly with φ but with values slightly above the predictions. Conversely, experimental λ p 3 is found to be relatively constant and close to 0.5, which is the value introduced by Morris & Boulay (1999) in their model to represent the lack of migration in torsional flows. Figure 19 presents similar plots for the contact anisotropy parameters, defined as λ c 2 =Σ c yy /Σ c xx and λ c 3 =Σ c zz /Σ c xx .…”

“…Therefore there have been attempts to develop continuum models where the suspension is assimilated to an homogeneous fluid with some constitutive laws for the particle stress Σ p . A successful approach for shear flows was proposed by Morris & Boulay (1999) -referred to as the suspension balance model (SBM) -, in which where η p n is the normal stress viscosity and λ p 2 and λ p 3 are anisotropy parameters λ p 2 = Σ p yy /Σ p xx and λ p 3 =Σ p zz /Σ p xx .S u p e r s c r i p tp is here added to recall that those quantities are computed using the particle stress Σ p .…”

“…This possible flaw in the SBM might not impair the good prediction capabilities of this model, for two reasons. First, the anisotropy parameters have been chosen by Morris & Boulay (1999) to match experimental results on migration, so that their exact nature is unimportant ; and second, we have shown that in dense frictional suspensions, contact stress is the prevailing stress.…”

“…Due to the lack of experimental data, Morris & Boulay (1999) proposed constant anisotropy parameters λ p 2 ⇡ 0.8 and λ p 3 ⇡ 0.5. The SBM rests on the idea that the overall behaviour of the suspension is driven by the particle stress Σ p .…”

“…On the other hand, normal stresses are crucial since they determine the migration of particles in suspensions subjected to an inhomogeneous shear field (Nott & Brady 1994;Mills & Snabre 1995;Morris & Boulay 1999;Nott et al 2011;Lhuillier 2009). This paper therefore intends to address this issue by considering numerical simulations of suspensions accounting for friction and roughness.…”