Wall-slip effects on the Yield-stress fluid flows in the rigid and deformable channel

Yield stress fluids flow through deformable conduits and are prevalent in nature and have numerous technological applications [1-11]. In this paper, we focus on investigating the impact of many factors such as the deformability of the channel-wall, yield stress, shear-thinning, and shear-thickening index in the presence of slip and compared it with flow dynamics with no-slips as predicted by Garg and Prasad [12]. Using lubrication theory, we have derived a model for the velocity profiles and flow rate using the Herschel-Bulkley rheological model in rigid and deformable shallow channels with slip-walls. To model deformable walls, we have utilized small displacement structural mechanics and perturbation theory presented by Gervais et al. [13] and Christov et al. [14], respectively. Our newly developed model encompasses the flow characteristics of Newtonian fluids, power-law fluids, and Bingham fluids, both with and without wall-slip, as observed in previous literature [13-16]. We find that the deformability increases the same effective channel height with and without wall-slip but the flow rate is increased more when slips are present within the channel. We find many scalings for the flow rate under different regimes of applied pressure and the deformability parameter. A threshold inlet pressure is required for the onset of yield-stress fluid flow in the channels unlike in the case of the Newtonian or power-law fluids. Garg and Prasad [12] finds that below this threshold, the flow is choked in the channels with plug height the same as the channel height: we find the same observations in the presence of slips. Although in case of deformable channels an early onset of flow with the pressure is found in comparison to the rigid channel. We observe the back flow due to deformability in the channel when the yield surface is between $H_o/2 < H_p < (H_o+\delta)/2$, where $H_o$ and $H_p$ represents the initial height of the channel without deformability and the yield surface’s height, respectively. $\delta$ is the increase in channel's height due to deformability. Beyond choked flow, the plug height decreases for both the rigid and the deformable channels with the pressure. We also observe that for any given applied pressure and yield stress, the $(H_p)_{\text{deformable}} < (H_p)_{\text{rigid}}$. This suggests that deformable walls decrease the plug region in comparison to the rigid channel. We also find that the wall-slip has no effect on the plug region and the onset of flow. In the presences of wall-slip, 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 there is decrease in the flow rate similar to as found by Garg and Prasad [12]. Further, 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.