Yield–stress shear thinning and shear thickening fluid flows in deformable channels

Abstract

Yield stress shear thinning/thickening fluids flow through flexible channels, tubes are widespread in the natural world with many technological applications. In this paper, analytical formulae for the velocity profiles and flow rate are derived using the Herschel–Bulkley rheological model in both rigid and deformable shallow channels, employing the lubrication approximation. To account for deformable walls, the approach outlined by Gervais et al (2006, Lab on a Chip 6 500-7) and Christov et al (2018 J. Fluid Mech. 841 267-86) is utilized, applying small displacement structural mechanics and perturbation theory, respectively. The newly derived formulae also enable the analysis of flow dynamics in Newtonian fluids, power-law fluids, and Bingham fluids as their limiting cases, all of which have been previously described in the literature and also serves as the validation cases. It is observed that deformability increases the effective channel height and the flow rate within the channel. Multiple scaling relationships for the flow rate are identified under different applied pressure regimes and deformability parameters. Additionally, it is noted that increasing the yield stress results in decreased velocity in both the plug flow and non-plug flow regions. Higher yield stress also corresponds to an increase in the yield surface height and the solid plug within the central region, leading to a reduction in the flow rate. Furthermore, the shear thinning/thickening index is found to have no impact on plug height, although an increase in this index causes a reduction in the flow rate due to the corresponding increase in shear thickening of the material.