Diffusion-driven transition between two regimes of viscous fingering

Viscous fingering patterns can form at the interface between two immiscible fluids confined in the gap between a pair of flat plates; whenever the fluid with lower viscosity displaces the one of higher viscosity the interface is unstable. For miscible fluids the situation is more complicated due to the formation of interfacial structure in the thin dimension spanning the gap. Here we study the effect of the inherent diffusion between the two miscible fluids on this structure and on the viscous fingering patterns that emerge. We discover an unexpected transition separating two distinct regimes where the pattern morphologies and mode of onset are different. This transition is marked by a regime of transient stability as the structure of the fingers evolves from having three-dimensional structure to being quasi-two-dimensional. The presence of diffusion allows an instability to form where it was otherwise forbidden.

Figure 1

(a) Schematic of both a radial and rectilinear Hele-Shaw cell showing two large flat plates, of size $L$, separated by a small gap of width $b$. (b) Fingering patterns for different injection rates. One of the fluids is dyed and the gray level indicates the local concentration of inner-fluid. Both scale bars are 2.5 cm. For the radial cell (top row of images) $b=205\mu \mathrm{m}$ and for linear cell (bottom row of images) $b=356\mu \mathrm{m}$, both have ${\eta }_{\text{in}}/{\eta }_{\text{out}}=0.2$. (Note that the images for the linear cell had the outer fluid dyed; here we invert the colors for comparison.)

Figure 2

Profiles of gap-averaged inner fluid concentrations, $C$, from (a) high and (b) low injection-rate experiments taken along the lines $L$ and $W$. Tip-splitting occurs when a fingers width, denoted by $w$, reaches a value of $2{\lambda }_c$.

Figure 3

(a) Finger growth rate ${\mathrm{\Gamma }}_{\text{init}}$, (b) normalized most unstable wavelength ${\lambda }_c/b$, and (c) normalized onset length, $L_{\text{onset}}/b$ are shown versus ${\mathrm{Pe}}_{\text{onset}}$. Inset of (b) shows ${\lambda }_c/b$ measured away from onset. At a critical value, ${\mathrm{Pe}}^{*}$ (the dotted line), there is a smooth decrease in ${\mathrm{\Gamma }}_{\text{init}}$, and a sharp jump in both ${\lambda }_c/b$ and $L_{\text{onset}}/b$.

Figure 4

Comparison of data from the radial and linear cells with ${\eta }_{\text{in}}/{\eta }_{\text{out}}=0.2$. (a) ${\mathrm{\Gamma }}_{\text{init}}$, (b) ${\lambda }_c/b$, and (c) $L_{\text{onset}}/b$, versus ${\mathrm{Pe}}_{\text{onset}}$. The red line denotes the same Pe number as in Fig. 3. This shows quantitative agreement between the two geometries, except for a slight shift of ${\mathrm{Pe}}^{*}$ for the linear cell.

Figure 5

(a) Images from experiments in the high- and low-Pe regimes and corresponding measurements of ${\lambda }_c/b$ versus Pe for ${\eta }_{\text{in}}/{\eta }_{\text{out}}=1.1\times {10}^{-3},3.2\times {10}^{-2},1.9\times {10}^{-1}$. Images taken when the outer radius of inner fluid is 5 cm. In the ${\lambda }_c/b$ data, all but the lowest ${\eta }_{\text{in}}/{\eta }_{\text{out}}$ show a clear jump in wavelength ${\lambda }_c/b$. The red shaded region shows the bounds on the critical Pe number, ${\mathrm{Pe}}^{*}$. (b) ${\mathrm{Pe}}^{*}$ versus ${\eta }_{\text{in}}/{\eta }_{\text{out}}$. Filled and open circles are the upper and lower bounds for ${\mathrm{Pe}}^{*}$ found in ${\lambda }_c/b$ data. Line shows power-law fit with exponent 0.55. Inset: Magnitude of the jump in the unstable wavelength normalized by the plate spacing, $\mathrm{\Delta }\lambda /b$, versus ${\eta }_{\text{in}}/{\eta }_{\text{out}}$. Line shows power-law fit with exponent 0.30.

Figure 6

(a) ${\tau }_{\text{onset}}$ versus $D$ at $b=205\mu \mathrm{m}$. Dotted line shows $t_{\text{onset}}\propto D^{-1}$. (b) ${\tau }_{\text{onset}}$ versus $b$ with $D=0.90\times {10}^{-6}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$. Dotted line shows $t_{\text{onset}}\propto b^2$. (c) ${\tau }_{\text{onset}}$ versus $L_{\text{onset}}/b$. High-Pe regime has constant $L_{\text{onset}}/b$ while low-Pe regime has constant ${\tau }_{\text{onset}}$. Legend is the same as Fig. 3. The inset shows a comparison between linear cell (open squares) and radial cell data. (d) ${\tau }_{\text{onset}}$ versus ${\eta }_{\text{in}}/{\eta }_{\text{out}}$. The red and black lines show stability thresholds from Ref. [17] and Ref. [16], respectively.

Figure 7

Concentration profiles spanning the transition between the two regimes of fingering. Profile on the left are from a radial geometry while those on the right are from a linear one. In each set of curves the red (—) curve denotes the profile at the onset of fingering while the blue (- -) curve is the profile at the average onset length for the high-Pe regime. The labels for the red and blue curves are the Pe numbers for the corresponding profiles. (a), (b), (f), and (g) have fingers in the high-Pe regime while (c)–(e) and (h)–(j) are in the low-Pe regime. Radial profiles here are taken with $b=205\mu \mathrm{m}$ and $\mathrm{\Delta }\eta =150\mathrm{cP}$, linear profiles are taken with $b=356\mu \mathrm{m}$ and $\mathrm{\Delta }\eta =319\mathrm{cP}$. All profiles have ${\eta }_{\text{in}}/{\eta }_{\text{out}}=0.2$.

Figure 8

(a) The profiles of concentration ${\varphi }_{\text{glycerol}}$ at three different points in time from the numerics based on Eq. (A1). The inset shows the derivative of the concentration profile to highlight the asymmetry in the shape. (b) Plot of the effective diffusion constant versus the weight fraction of glycerol. The red circles are our simulated $D$ values and the blue circles are experimental values reported in Ref. [41].

Figure 9

The images show different methods of visualizing the fluids in our experiments at different flow rates. The rows show, from top to bottom, flow rates of $2\mathrm{ml}/\min ,0.1\mathrm{ml}/\min ,0.01\mathrm{ml}/\min$. From left to right, the columns depict final patterns seen with (i) schlieren imaging and standard imaging with (ii) inner and (iii) outer fluid dyed. The schlieren images have been divided by the background to account for non-uniform lighting. Dyed images were taken when the outer extent of the dyed patterns reach a radius of 3.5 cm, the schlieren images were taken so that the total time of injection matches that of the corresponding experiments with dyed fluids. A final radius is not considered for schlieren since the diffuse boundary is not clearly visible for lower injection rates. All experiments were done with $b=205\mu \mathrm{m}, {\eta }_{\text{in}}=24.7\mathrm{cP}$, and ${\eta }_{\text{out}}=116\mathrm{cP}$.

Figure 10

$L_{\text{onset}}$ (blue) and $t_{\text{onset}}$ (red), are shown for radial cell experiments with different time-varying injection rates $Q\left(t\right)$, shown schematically in the insets. The average onset time, $\langle t_{\text{onset}}\rangle$, is taken from experiments that use a single, nonvarying injection rate. (a) After injection at a constant rate to a radius of 3 cm, injection stops for a time $t_{\text{wait}}$ before continuing. (b) The injection rate ($Q$) is low for a time $t_{Q_1}$ before $Q$ is increased.