Numerical models of flow patterns around a rigid inclusion in a viscous matrix undergoing simple shear: implications of model parameters and boundary conditions

Abstract

The hydrodynamic models that have recently been developed to investigate the nature of flow around coherent, rigid inclusions in simple shear reveal two contrasting patterns with eye-shaped and bow-tie shaped separatrix, even though all the models are based on Navier-Stokes law. In order to find the cause of this variance, this paper reviews the existing models in the light of different boundary conditions imposed on individual models. Scrutiny of the models reveals that inclusion-matrix systems, when considered infinitely extended in space, develop eye-shaped flows. However, those with finite dimensions essentially display bow-tie shaped flows. Using a finite element method (FEM), we advance the study to show the additional effects of model/inclusion dimension ratio (D_R) and model aspect ratio (A_R) under different boundary conditions. In the flow with bow-tie shaped separatrix, the regions of back flow define a nearly semi-circular geometry when D_R is low (<2). These regions assume a semi-elliptical shape with increasing D_R. The distance of stagnation points from the inclusion is found to increase non-linearly with D_R. Model results suggest that transformation of a flow with eye-shaped separatrix to that with bow-tie shaped separatrix can occur due to increasing AR under a specific boundary condition. Applying FEM results in geological situations thus requires the appropriate choice of dimensional parameters of the model as well as the kinematic conditions imposed at the model boundaries.