Using the suspension balance model in a finite-element flow solver

A suspension balance model (SBM) is implemented to describe the shear-driven migration of particles in noncolloidal suspensions in the context of a finite element (FE) solver. Before developing the FE methodology, the SBM is analyzed in the context of a rigorous two-phase averaging procedure, in which the traditional SBM model can be thought of as an approximate closure relationship for the rigorous two-phase equations. It is shown that the standard SBM equations are inconsistent, which is demonstrated analytically for the case of Couette flow. A FE model is developed using a corrected set of SBM equations, and the numerical techniques needed to handle the anisotropic Q-tensor and FE stabilization are detailed. The resultant FE-SBM method is tested using a Couette geometry and compared with existing models and experiments. A high level of sensitivity of the particle migration to the chosen viscosity model is noted, as well as the influence of excess diffusion caused by the FE stabilization procedure. The FE-SBM method is also used in conjunction with an arbitrary Lagrangian–Eulerian formulation to simulate the deflection of the free surface in a Couette cell. Secondary flows are observed in the free-surface simulation results, and the underlying mechanisms driving these secondary flows are explored.