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 [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.

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

“…The above expression (38) is also derived by Gervais et al [13], Christov et al [14], Garg [15], Garg and Prasad [12] for β|∆p| ≪ 1.…”

“…The solid line, dashed line, and dotted lines show the data at yield stress values of τ y = 0 Pa, τ y = 2 Pa, and τ y = 4 Pa, respectively in both figures (a) and (b). In (a) λ = 0 (Taken from Garg and Prasad [12], please note that we can also calculate the same using the current model derived in this paper with λ = 0) and in (b) the slip-length λ = 0.1 m. We find that due to flexibility (β = 0.005 Pa −1 ) in the channel, the channel height increases by 30% for both slip lengths.…”

“…The solid line, dashed line, and the dotted lines show the data at shear thinning/thickening index n of n = 0.8, n = 1, and n = 1.2, respectively. In (a) λ = 0 (Taken from Garg and Prasad [12], please note that we can also calculate the same using the current model derived in this paper with λ = 0) and in (b) the slip-length λ = 0.1 m. The central region within the dashed blue line indicates the plug flow region. We find that due to non-zero yield stress in all predictions, the velocity profiles are divided into two parts, the plug velocity in the central region and the normal shearing velocity towards ----------------------------------------------------- 2) materials too.…”

“…Due to increased velocity, the flow rate also increases. We refer the flow profile discussion when the slip effects are absent from [12]. For λ = 0.1 m and n = 0.8, the centerline plug velocity increases from 0.65 m/s in the rigid channel to 1 m/s in the deformable channel with β = 0.005 Pa −1 as shown in Figure 3(b) with dashed black and red lines respectively.…”

“…Also, their research, unlike our research, was conducted on tubes rather than channels (as in our case). Recently Garg and Prasad [12] conducted studies on the Herschel-Bulkley fluid flow in deformable channels, but without wall-slip effect. As evident from the above literature, that the slip effects are important in the yield stress fluid flow.…”