Curvilinear flows of noncolloidal suspensions: The role of normal stresses

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Abstract
The role of normal stresses in causing particle migration and macroscopic spatial variation of the particle volume fraction φ 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=O(ηγ̇a3/kT), where η is the suspending fluid viscosity, γ̇ 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 φ dependence of the compressive shear-induced normal stresses is captured by a “normal stress viscosity” ηn(φ), which vanishes (as φ2) at φ=0 and diverges at maximum packing in the same fashion as does the shear viscosity ηs(φ). Anisotropy of the normal stresses arising from the presence of the particles is modeled as independent of φ, so that ratios of any two particle contributions to the bulk normal stress components are constants, Σ22p/Σ11p=λ2 and Σ33p/Σ11p=λ3; the standard convention of (1,2,3) denoting the (flow, gradient, vorticity) directions is used so that, for example, Σ11p is the normal component of the particle stress Σp 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 λ2 and λ3, the rheological functions ηs and ηn, 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.