Delayed onset and the transition to late time growth in viscous fingering

Viscous fingering patterns form in confined geometries at the interface between two fluids as the lower-viscosity fluid displaces the one with higher viscosity. Previous studies have examined the most unstable wavelength of the patterns that form using both linear-stability analysis and the dynamics of finger growth in the nonlinear regime. Interesting differences in dynamics have been seen between rectilinear and radial geometries as well as between fluid pairs that are immiscible (with interfacial tension) or miscible (with negligible interfacial tension). This paper reports measurements of how all of these systems transition from the linearly unstable regime to their late time, nonlinear dynamics. In all four cases there is a region of stable or slow growth characterized by an onset length scale before fingers enter the late-time regime. For immiscible fluids in a rectilinear geometry this onset length is consistent with linear-stability analyses. All other cases are not adequately described by existing theory. In radial geometries, the onset length predicted from theory is an order of magnitude smaller than what is experimentally observed and has the incorrect scaling with dimensionless numbers. For miscible fluids in a rectilinear geometry the onset length is related to the development of steady-state structures within the confining dimension and cannot be explained by quasi-two-dimensional theories. By combining the onset length with the finger growth rate measured after onset, the global patterns that form well into the late-time dynamics can be predicted.

Figure 1

Contours at different times for radial and rectilinear geometries with both miscible and immiscible fluids. The red curve is the interface where fingering is first detected. $R/b$ and $L/b$ are radial and linear lengths rescaled by the gap dimension, $b$. $W/b$ is the distance along the width of the interface for the rectilinear geometry. Starting with the top left and going clockwise the fluid properties for these experiments are: ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}=[0.35,0.13,0.10,0.59]$ and $\mathrm{\Delta }\eta =[460,587,350,534]$ cP, with $b=[205,485,660,660]\mu \mathrm{m}$, respectively.

Figure 2

Panels (a) and (b) show measurements of the finger length $L_{\text{finger}}$ and $R_{\text{finger}}$ against the pattern size, $L_{\text{out}}$ or $R_{\text{out}}$, for immiscible fluids in rectilinear and radial geometries, respectively. Each colored line is a measurement from a different section of the interface, either a small width for the rectilinear cell or a small angular section for the radial cell. The white lines just behind the fingers are artifacts from imaging due to the index of refraction difference between the two fluids and a slight curvature to the interface between them. Note that $L_{\text{out}}$ and $R_{\text{out}}$ has been shifted by the transition point $L_{\text{onset}}$ and $R_{\text{onset}}$. All lengths are nondimensionalized by the plate spacing, $b$. Panels (c) and (d) show the same measurements for miscible fluids. Fluid properties for the experiments from left to right are: ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}$=$[0.50,0.33,0.11,0.20]$ and $\mathrm{\Delta }\eta =[183,551,347,345]$ cP, with $b=[660,203,660,305]\mu \mathrm{m}$.

Figure 3

Panels (a) and (b) show the growth of $L_{\mathrm{finger}}$ at early times in a rectilinear geometry for the immiscible or miscible case, respectively. Note that the finger size is shown on a logarithmic scale. The fluid parameters are ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}=[0.50,0.21]$, $\mathrm{\Delta }\eta =[183,306]$ cP, with $b=[660,660]\mu \mathrm{m}$, respectively. (c) For the radial immiscible case the comparison of the arc length of a disturbance, ${\lambda }_s$, and the allowed most unstable wavelength, ${\lambda }_c$, based on the local interface velocity. Fluid properties are ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}=0.33$ and $\mathrm{\Delta }\eta =551$ cP, with $b=203\mu \mathrm{m}$. The shaded area around these curves represents a measurement error. For ${\lambda }_c$ there is error associated with the interfacial velocity while for ${\lambda }_s$ the error comes from determining the average angular width of a finger during growth.

Figure 4

Onset length versus viscosity ratio, ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}$. For the radial geometry (a) $R_{\text{onset}}/b$ is shown while in the rectilinear geometry (b) $L_{\text{onset}}/{\lambda }_c$ is shown. (c) shows the growth rate, ${\mathrm{\Lambda }}_l$, versus ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}$ for all four cases. ${\lambda }_c$ used for rescaling has been measured in experiment; for the immisicble case ${\lambda }_c$ matches theory well [9, 10] and for the miscible case I find ${\lambda }_c/b=2.97\pm 0.33$. (d) shows the measured finger length, $L_{\mathrm{finger}}$ or $R_{\mathrm{finger}}$, at the transition point. All systems show a finger length of $(0.30\pm 0.07){\lambda }_c$ at the transition. As shown in the legend, circle or square symbols denote radial and linear geometries, respectively, while filled or open symbols denote immiscible or miscible fluids, respectively.

Figure 5

Panel (a) shows the Ca dependence of $L_{\text{onset}}$ for the rectilinear immiscible system. Both $L_{\mathrm{onset}}/b$ (red) and $L_{\mathrm{onset}}/{\lambda }_c$ (black) are shown. Panel (b) shows the Atwood dependence of $L_{\mathrm{onset}}/{\lambda }_c$, the black line has a slope of $-1$.

Figure 6

Panels (a) show the evolution of an interface for miscible fluids in a rectilinear geometry. Below show the evolution of the thickness profile. The red line denotes the point where the bump at the tip first starts to form, the dotted red line is where $L_{\text{onset}}$ is measured from linear interpolation. In the images the lighter band at the interface in all but the first panel corresponds to this region of increased thickness. In panels (b) are results from COMSOL simulations showing the evolution of the inner fluid tongue for ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}=1,0.125,0.05$. Each simulation is shown for the same amount of injection time. For the two lower viscosity ratios the tip of the inner fluid tongue reaches a steady state shape. (c) The change of the fit parameters $\beta$ and $\alpha$ as the profile evolves for ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}=0.1$. After the vertical red dashed line the fit values become constant. The inset shows a comparison of the fit to the inner fluid tongue contour. Panel (d) shows a comparison between measured experimental onset lengths for miscible fluids based on the appearance of the bump at the tip and the onset of a steady state shape for the tip of the inner fluid in the simulations.

Figure 7

Onset in the radial geometry with immiscible fluids. The data is for silicone oil invading a water-glycerol mixture with $\mathrm{\Delta }\eta =828\mathrm{cP}$ and ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}=0.38$. (a) The measured onset radius, $R_{\mathrm{onset}}/b$ versus Ca. (b) The number of fingers at onset, $N_{\text{fingers}}$, is plotted versus Ca. (c) $R_{\text{onset}}/b$ is plotted versus the viscosity ratio. The solid lines in each graph are the marginal stability curves of the linear stability analysis of Paterson [10]. The curve in (c) was calculated for $\mathrm{Ca}=0.01$ and has been scaled by a factor of 4 to compare the functional form with the experimental data.

Figure 8

In panel (a) is a schematic of the flow profile within the gap for the calculation of the modified mobility. $\beta$ is the width of the inner fluid tongue. (b) Experimental measurements of the jump in viscosity contrast, $\beta$, for miscible fluids in a radial geometry over a range of ${\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}$. The black line shows a best fit power law to a functional form of $({\beta }^{-1}-{\beta }_0^{-1})\propto {(M-M_c)}^{-\alpha }$ $[M={\eta }_{\mathrm{out}}/{\eta }_{\mathrm{in}}]$ where $1/M_c=0.51\pm 0.08$, ${\beta }_0=0.60\pm 0.01$, and $\alpha =1.1\pm 0.3$; this form is seen in the inset. The dotted line is the prediction for $\beta$ given a kinematic-wave theory for the gap dynamics [27]. In panel (c) is the onset radius for miscible fluids compared to the linear-stability analysis of Paterson [10] using the modified mobility from Eq. (4). The blue line uses values of $\beta$ taken from experiment, in panel (b), and the red curves use $\beta$ derived from kinematic-wave theory [27]. The dotted line is from the Paterson linear-stability analysis without modifying the mobility.

Figure 9

(a) Comparisons of the size ratio from previous experimental data (ref. [18] for immiscible fluids (left) and Ref. [17] for miscible fluids (right)) (open circles) and the expected size ratio using my measured onset data and Eq. (5) (black points). $R_{\mathrm{onset}}=315b$ is used in the calculation, corresponding to the reported pattern size. In panel (b) is the measured growth rates at onset plotted against ${\mathrm{At}}_{\mathrm{\Omega }}$. The dotted line has a slope of 1 and is a guide for the eye. The legend is the same as in Fig. 4.

Figure 10

Panel (a) is the accumulated amplitude in each mode, $n$, from whenever the first nonzero mode appears. This is done from radii of $R_{\mathrm{onset}}/b$ to $10R_{\mathrm{onset}}/b$. The transition of color from black to red corresponds to an increasing radius. Panel (b) shows the mode, $n_c$, that has the largest accumulated amplitude as a function of radius. The start of all of these curves is the point where the interface first becomes unstable and scales with $P$. The inset shows the unscaled data. Last, panel (c) shows the maximum value of accumulated amplitude against the rescaled radius. The inset shows the unscaled data. The transition of color from black to red in the insets corresponds to increasing $P$.

Figure 11

This figure compares the value of the onset length for the radial geometry measured by linear interpolation to the delay length measured by looking at the finger size, $R_t$ (the white points), and from the $\lambda$ crossover measurement, $R_{\lambda }$ (the black points). This is done for both immiscible (top) and miscible (bottom) systems.