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

Yield stress shear thinning/thickening fluids flow through flexible channels, tubes are widespread in the natural world with many technological applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In this paper, we have derived analytical formulae for the velocity profiles and flow rate using the Herschel-Bulkley rheological model in the rigid and deformable shallow channels under lubrication approximation. To represent deformable walls, we have utilized small displacement structural mechanics and perturbation theory presented by Gervais et al. [12] and Christov et al. [13], respectively. The newly derived formulae also facilitate the flow dynamics of Newtonian fluids, power-law fluids, and Bingham fluids as its limiting cases, which have been previously derived in the literature [12, 13, 14, 15]. We find that the deformability increases the effective channel height and the flow rate in the channel. We find many scalings for the flow rate under different regimes of applied pressure and the deformability parameter. We also find that increasing the yield stress leads to a decrease in the velocity in the plug flow as well as in the non-plug flow regions. Increasing yield stress also leads to increasing the yield surface height and the solid plug in the central region due to which decreasing in the flow rate. We also find that the shear thinning/thickening index does not affect the plug height, although as the index increases, the flow rate starts to decrease due to the corresponding increase in shear thickening of the material.