Obstructed viscoplastic flow in a Hele-Shaw cell

Experiments are conducted exploring the flow of Carbopol past obstacles in a narrow slot and compared with predictions of a model based on the Herschel-Bulkley constitutive law and the conventional Hele-Shaw approximation. Although Carbopol is often assumed to be a relatively simple yield-stress fluid, the flow pattern around an obstacle markedly lacks the fore-aft symmetry expected theoretically. Such asymmetry has been observed previously for viscoplastic flows past obstacles in unconfined geometries, but the narrowness of the Hele-Shaw cell ensures that the stress state is very different, placing further constraints on the underlying origin. The asymmetry is robust, as demonstrated by varying the shape and number of the obstacles, the surfaces of the cell walls, and the steadiness of the flow rate. The results suggest that rheological hysteresis near the yield point may be the cause of the asymmetry.

Figure 1

Viscoplastic flow through a Hele-Shaw cell containing an obstacle. The plane of the cell is shown in (a), where the grid represents the mosaic pattern used to create composite images of the flow field. A velocity profile for flow down a uniform, unobstructed cell, with a central rigid plug, is shown in (b), while the corresponding profile through the centerline of the obstructed cell ($y=0$) is shown in (c), with a shaded pseudoplug that widens to fill the cell and become truly rigid near the obstacle.

Figure 2

Flow curves for the three different Carbopol suspensions used in this work [$0.075\%$ (black), $0.06\%$ (blue), and $0.055\%$ (red)]. The curves are obtained using a rheometer with roughened parallel plates (MCR501, Anton Paar) and correspond to ramps in shear stress, proceeding first up and then down with a ramp rate of $0.05\text{Pa/s}$. The dashed lines give the Herschel-Bulkley fits in Table 1.

Figure 3

Experimentally measured velocity profiles for 0.06% Carbopol in a uniform cell with chemically treated plates at the three Bingham numbers (flow rates) indicated (red). The solid black lines show corresponding theoretical predictions given the fluxes and the rheological parameters in Table 1. A profile from a cell with untreated plates for $B=1.2$ (green) is also shown.

Figure 4

PIV images showing speed, normalized by the incident speed $U_c=Q/WH$, and streamlines (all flowing left to right) along the midplane of the slot around various obstacles. (a) The flow around a disk for a glycerol-water solution (31 wt/wt%) at $Q=0.05$ ml/min; dashed lines show the corresponding potential-flow solution. (b) Flow field for a viscoelastic PEO solution (0.75 wt/wt%) with $Q=0.16$ ml/min. (c)–(k) Flow fields for a Carbopol solution (0.06 (wt/wt%), untreated walls) with $B=3.1$ (c), (g), (h) and $B=2.7$ (d), (e), (f), (i), (j), (k). The upper limit of the color bar was set at $X=2.8$ for (a), (b), $X=2$ for (j), (k), and $X=1.6$ for (c)–(i). (l) An average image near the plug at the front of a disk ($0.075\%$ Carbopol solution, $B=2.3$, treated walls). The solid lines indicate the edge of the plug according to edge detection (green) or a velocity threshold (red); the dashed line shows the estimated plug length ${\ell }_p$. White lines in (c)–(k) show the yield surfaces detected using a velocity threshold of $0.0005\text{mm}/\text{s}$.

Figure 5

Speed (shading; scaled by the incident speed $U_c$) and representative streamlines (solid black lines) along the midplane of the slot for computations of Herschel-Bulkley fluid around the gray-shaded obstacles, for $n=\frac{1}{2}$ and $B=4$. The range of the color bar matches that used in Figs. 4–4. The white lines show yield surfaces, given by the threshold $U/U_c=1\times {10}^{-3}$.

Figure 6

(a) Scaled theoretical (top) and experimental (bottom) speeds, $U/U_c$, for flow around a disk with $n=0.57$ and $B=2.3$, corresponding to panel (l) in Fig. 4 (and using the same scheme for the color map as that figure). In (b), the theoretical (solid lines) and PIV (dots) speeds are plotted along the sections indicated by the vertical dotted lines in (a). The notable disagreement in the velocity profiles past the top and bottom of the disk is a result of a local breakdown of the Hele-Shaw approximation over a distance of the order of the slot thickness from the disk, as discussed in the main text.

Figure 7

(a) Dimensional plug lengths ${\ell }_p$ of the configurations shown in Fig. 4, with the conventions indicated by the legend. The walls of the cell are treated to remove slip for the isolated disks, but for the squares, stadia, and multiple disks, the walls are untreated in view of the potentially destructive method of insertion or rearrangement. (b) Scaled plug lengths ${\ell }_p/{\ell }_c$, where ${\ell }_c$ is chosen as indicated in the main text for the three groupings of obstacles. The lines show theoretical results for isolated disks, calculated using the numerical method of Ref. [13] for the Herschel-Bulkley model with $n=0.5$ (black solid) and $n=1$ (red dashed). (c) A comparison of the plug size for flow of $0.06\%$ Carbopol around disks with treated and untreated acrylic sheets. In all three panels, we plot ${\ell }_p$ as positive for the plugs at the front, and ${\ell }_p<0$ for those at the back. The error bars indicate the standard deviation from three repetitions of each experiment.

Figure 8

Theoretical predictions of the plug lengths from numerical simulations for the obstacle arrangements indicated by the legends, and following the experimental results in Fig. 7, showing (a) ${\ell }_p/R$ and (b) ${\ell }_p/{\ell }_c$. These computations have $n=\frac{1}{2}$; in (c) data are shown for isolated disks and the three values of $n$ for the experimental fluids, as well as results for $n=\frac{1}{2}$ and for a Bingham fluid ($n=1$) [13], for reference.

Figure 9

Plug lengths for two disks aligned with the flow as a function of the center-to-center separation $L$, for experiments with untreated sheets and numerical computations; $B=2.7$ and $n=0.57$ ($0.06\%$ Carbopol), The images (i)–(iv) below correspond to the separations identified in the main panel, with the symbols corresponding to the plug locations indicated, and show scaled speed maps (with color bars as in Fig. 4) and streamlines. Given the lack of any apparent trend with $L$, the scatter in the experimental data for the leading plug length (filled red circles), with a standard deviation of about $15\%$, gives a sense of the overall uncertainty in the measurements.

Figure 10

Time dependency of the plug size on the front and back of an isolated disk subject to a flow rate $Q\left(t\right)$. The tests were performed with $0.06\%$ Carbopol solution and treated sheets. The solid and dashed horizontal lines represent the average and standard deviation of the steady-state values reported in Fig. 7.

Figure 11

(a) Thixotropic flow curves for an $0.1\%$ Carbopol suspension that was very highly sheared during mixing (measured using the same rheometer as in Fig. 2, but with an up-down ramp controlling the shear rate). (b) Time series of the plug length after a step change in flow rate behind a circular obstruction in the Hele-Shaw cell using the same fluid (with flow rate $Q=0.05$ ml/min after the step change).