A weighted residual method for two-layer non-Newtonian channel flows: steady-state results and their stability

Abstract

AbstractWe study buoyant displacement flows in a plane channel with two fluids in the long-wavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a first-order approximation to the interface height and flux functions in each layer. As the fluids are shear-thinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr–Sommerfeld equations are first derived for two generalized non-Newtonian fluids satisfying the Herschel–Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the long-wavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr–Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar two-layer flows, e.g. co-extrusion flows.