Measurement of pressure gradients near the interface in the viscous fingering instability

The viscous fingering instability, which forms when a less-viscous fluid invades a more-viscous one within a confined geometry, is an iconic system for studying pattern formation. For both miscible and immiscible fluid pairs the growth dynamics change after the initial instability onset and the global structures, typical of late-time growth, are governed by the viscosity ratio. Here we introduce an experimental technique to measure flow throughout the inner and outer fluids. This probes the existence of a new length scale associated with the local pressure gradients around the interface and allows us to compare our results to the predictions of a previously proposed model for late-time finger growth.

Figure 1

Viscous fingering patterns imaged with alternate injections of dyed and undyed fluid rings. The low-viscosity inner fluid is dyed in alternate rings of blue. The outer fluid is transparent. The scale bar represents 1 cm, $b=420\mu \mathrm{m}$, and $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})=0.067$.

Figure 2

(a) Schematic of a radial Hele-Shaw cell consisting of two circular glass plates separated by a thin uniform gap spacing, $b$. The less-viscous fluid moves radially, displacing the more-viscous one. (b) Finger patterns of late-time growth viewed from above. The inner fluid is dyed and the outer fluid is left transparent for contrast. Characteristic length scales $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$ are labeled.

Figure 3

Length scales of viscous fingering patterns. (a) One pattern quadrant is shown that has been imaged using alternate dyed-ring injection in both the inner and outer fluids. A region surrounding the interface over which local pressure gradients perturb flow is estimated with the bounds $r_{{\mathrm{dp}}_{\mathrm{in}}}$ and $r_{{\mathrm{dp}}_{\mathrm{out}}}$. (b) Dyed rings are analyzed in local pattern segments pertaining to a finger and an adjacent valley. Length scales $r_{\mathrm{finger}}$ and $r_{\mathrm{valley}}$ along a single dyed ring are labeled in the inner and outer fluids. The scale bars represent 0.5 cm.

Figure 4

Undulation magnitude versus the negative distance behind the interface for a single dyed ring. The interface, $R_{\mathrm{in}}$, is marked by the dashed vertical line. Deformations are measured behind four different fingers. Experimental parameters are $b=420\mu \mathrm{m}$ and $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})=0.104$.

Figure 5

Characterization and scaling of perturbation growth in the inner fluid. (a) The magnitude of undulations versus distance behind the interface for five dyed rings labeled in order of their relative injection times. Curves are fit by an exponential function to determine a characteristic decay length, ${\ell }_{\mathrm{in}}$. The inset shows ${\ell }_{\mathrm{in}}$ increasing as a function of ring injection sequence (time). (b) The $x$ and $y$ axes are rescaled such that the deformation curves from sequential rings collapse onto a single curve. Experimental parameters are: $b=420\mu \mathrm{m}$ and $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})=0.104$.

Figure 6

Scaled collapse of inner fluid experiments with varied control parameters. (a) Scaled curves from six experiments for $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})=0.06$ and varied gap widths, $305\le b\le 610\mu \mathrm{m}$. (b) The $y$ axis is scaled by $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})$ to collapse data from experiments with different viscosity ratios. Here, $b=420\mu \mathrm{m}$.

Figure 7

Perturbation decay in the outer fluid. (a) Deformation curves from three sequential dyed rings (color coded as in inset) ahead of four different fingers (represented by square, diamond, circle, and triangle) collapse when the $x$ axis is scaled by the local finger tip diameter, $d_{\mathrm{t}}$. The scaled master curve is fit to an exponential form. The inset shows individual fits for $|{\ell }_{\mathrm{out}}|$ as a function of ring injection sequence before scaling. Experimental parameters are $b=420\mu \mathrm{m}$ and $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})=0.107$. (b) The normalized decay length $|{\ell }_{\mathrm{out}}|/d_{\mathrm{t}}$ from different experiments is plotted versus gap width, $b$. Different color-symbol combinations represent different viscosity ratios. Individual experimental values agree with the overall fit (represented by the horizontal line).

Figure 8

Two independent radial flow channels corresponding to a finger and adjacent valley. The interface is denoted by $R_{\mathrm{out}}$ in the finger and by $R_{\mathrm{in}}$ in the adjacent valley. The net pressure drop across each channel is the same; however, the local pressure gradients are different around the interface: flow is identical in both channels up to $r_{{\mathrm{dp}}_{\mathrm{in}}}$ and beyond $r_{{\mathrm{dp}}_{\mathrm{out}}}$.

Figure 9

Comparison of the measured and predicted velocity ratios. (a) The velocity ratio, ($V_{\mathrm{in}}/V_{\mathrm{out}}$), for a single finger-valley pair is shown over the course of an experiment and is compared to model predictions. Model data (connected plus symbols) are shown to differ significantly from experimental data (circles) unless modified to account for flow structure in the gap (line with the shaded region). Values for the $\beta$ model assume that the equal-pressure point behind and ahead of the interface occurs where ring undulations decay to 90% of their maximum value; the bounds of the shaded region correspond to 75% (upper) and 95% (lower) thresholds. Data are shown for $b=420\mu \mathrm{m}, ({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})=0.067$. (b) The average plateau value for ($V_{\mathrm{out}}/V_{\mathrm{in}}$) [such as that seen in (a)] is plotted versus $({\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}})$. The experiment and model outputs are marked with the same symbols as in (a). For the model, we plot an average of outputs for 3–4 fingers with the bar denoting the range from smallest to largest values. The averaged $\beta$ adjusted model output is marked by a cross. Uncertainties due to finger variation and parameter definitions are smaller than the symbol size for the two larger viscosity ratios and are thus omitted. The error bars shown correspond to the bounds of the shaded region in (a). All experiments were done with $b=420\mu \mathrm{m}$.

Figure 10

An illustration of a small volume of dyed fluid viewed between two glass plates stretched by the steady-state Poiseuille flow profile. The black arrows in the gap denote the velocity as a function of height. Viewed from the side, the $r$ location of the highest gap averaged concentration corresponds to the boundary between the back of the dyed ring and the tip of the incoming undyed fluid, which lies in the middle of the gap. Here the aspect ratio is highly exaggerated, as $b\ll r$. Viewed from above, the highest concentration of dye appears around the back of a dyed ring.

Figure 11

Confocal imaging of Poiseuille flow profile. (a) Fluorescently dyed fluid is injected alternately with undyed fluid between two glass plates separated by gap width $b\approx 533\mu \mathrm{m}$. The fluid is imaged with a confocal microscope which samples slices at 12 heights, $y/b$, within the gap. Slices are vertically stacked to reconstruct the fluid profile. The aspect ratio has been exaggerated as $y\ll r$. (b) The mean intensity is averaged over gap height, $y/b$, for a radial segment in the image in (a) as marked by the red bracket and tick marks. The highest mean pixel intensity is located at the center of the gap where dyed fluid is most concentrated, as expected of Poiseuille flow. The fluid used has a viscosity of $\eta =31.0\mathrm{cP}$.