Modelling with viscous and viscoplastic materials under combining and separating flow configurations

Combining and separating incompressible flow of Newtonian and inelastic Herschel–Bulkley fluids is studied numerically employing a semi-implicit Taylor–Galerkin pressure-correction algorithm, where steady solutions are obtained through a transient finite element procedure. The influence of inertia and fluid rheology is analysed on flow patterns, velocity fields and pressure drops for various flow configurations, with fixed geometric gap width that stimulates the merging and splitting in the flow. For Newtonian fluids and at larger levels of inertia, the appearance of vortices was observed, with an increase in velocity differences and pressure drops across the channel. In this case, the numerical procedure was verified with good agreement against previous numerical and experimental observations. To extend the consideration to non-Newtonian inelastic materials, the material rheological characteristics were approximated with the use of the Herschel–Bulkley fluid model, incorporating the Ostwald–de Waele power-law model and viscoplastic yield stress. Findings for unyielded power-law fluids reveal slight increase in the size of the vortices as power index (m) was decreased. Variation of the consistency index (k) shows strong influence on the streamline patterns with a rapid increase in the vortex formation as k was decreased. For Bingham model solutions, devoid of shear-thinning and increasing yield stress, a higher value of Reynolds number is required for equivalent levels of vortex formation; also one observes the appearance of yielded and unyielded regions. Under Herschel–Bulkley modelling, there was little change noted in the kinematics, but some was apparent in rheological response. Once more, observations reveal the tendency to eliminate vortices at larger yield stress levels, with the appearance of unyielded regions.