Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions.

*(English)*Zbl 1195.76406Summary: Pressure-driven flow of a noncolloidal suspension is studied in two-dimensional channel and axisymmetric circular pipe geometries at bulk solids fractions of \(0.2 \leq \phi \leq 0.5\). Flows are modeled by the “suspension balance” approach, consisting of mass and momentum balances for the bulk suspension and particle phase. For particles in Newtonian fluid, cross-stream motion is driven by spatial variation of particle phase normal stresses. The particle phase stress model is based strictly upon the computed rate of strain, with a nonlocal contribution to the normal stress. Two solution procedures for the suspension flow equations are described. The first is a “solve-evolve” scheme based upon a full two-dimensional solution of the unsteady, axially varying behavior using a conservative finite volume method to solve the bulk mass and momentum conservation equations. The flow solution is coupled to an explicit update (evolve) step of the particle conservation equation. The second is a nonconservative but efficient marching solution for the asymptotically steady, but axially varying, problem. Predicted axial variation of the particle fraction, velocity and pressure gradient, as well as the fully developed profiles in channel and pipe flows are presented. The rate of axial development is strongly dependent upon the ratio of particle size to channel half-width (or pipe radius), \(\epsilon \equiv a/B (or a/R)\). The agreement of marching method and full model solutions is very close for the cases studied; both agree quantitatively well with available experimental results, including axial development in the pipe flow, where the model predicts the second normal stress difference to influence the migration. Migration in a 2:1 contraction flow provides an illustration of a flow where the full solution is required.