Numerical simulations of particle migration in suspension flows: Frame-invariant formulation of curvature-induced migration

The objective of this work is to investigate what mechanisms should be employed to qualitatively/quantitatively predict particle migration in a suspension flow. Based on the diffusive flux model originally proposed by Phillips et al. [R.J. Phillips, R.C. Armstrong, R.A. Brown, A.L. Graham, A constitutive equation for concentrated suspensions that account for shear-induced particle migration, Phys. Fluids A 4 (1992) 30-40], we survey the accuracy of three models including original Phillips model (Model I), modified Phillips model with curvature-induced migration mechanism (Model II), and finally the modified Model II with volume-fraction-dependent parameters (Model III). The empirical parameters which appear in the three models are determined by fitting to independent concentric Couette experiments. The accuracy of three models in concentric Couette problem is comparable except that Model III shows more improved predictions near the inner cylinder. However, the predictions of the three models are entirely different on a qualitative level for parallel plate problems and the existence and direction of particle migration are severely model-dependent. Models II and III predict no migration or very slight migration at high volume fraction, which is in good agreement with the previous experiments, whereas Model I predicts inward migration. We show that Model III accurately predicts a solid-free region near the center at low volume fraction, which was experimentally observed. In addition to a survey of migration mechanisms, we developed a frame-invariant curvature-induced migration model applicable to multi-dimensional flows. A transient 2D mixed-order finite element method (FEM) code was implemented to compare the predictions of the three models in a 2D problem. In this work, we considered the eccentric Couette problem, which is often used as a benchmarking problem. Though there is not much difference among the three models, Model III predicts that the particle migration is slightly retarded at high shear rate regions.