Figure 11—\({\mathbf{t}_{\mathbf{D}}}^{\mathbf{e}}\mathbf{s(x,t)}\) vs.
( \({\mathbf{x/}\mathbf{t}_{\mathbf{D}}}^{\mathbf{1+e}}\))
using DB.
The saturation profiles are
expected not to scale linearly with position and time when using
traditional linear-flow models. With the collapse of curves when\({t_{D}}^{e}s(x,t)\) was plotted with (x/\({t_{D}}^{1+e}\)), however,
self-similarity is expected to be observed (in accordance with the
results from the previous literature). We plotted time vs. saturation
for time steps before the finger break and the time steps after the
finger break (e.g., t>600s). Time steps after the finger
break are indicated in a red rectangle in the legend box of Figures 5-11
and their corresponding saturation was recorded. As can be seen,
complete collapsed saturation profiles could not be obtained regardless
of the adequacy of the range of fractal numbers (for both D andDB) . However, saturation profiles allowed us to
observe that there exists a significant difference between the fingering
behavior prior to the finger break (which often results in droplet
generation—this will be elaborated further in the next section) and
after the finger break. And they tend to follow a similar trend of each
section; in the earlier stages of experiments, fingers did not reach
significant portions of the Hele-Shaw sample resulting in a lack of data
points and accordingly smooth curves for most samples. After the finger
break, however, significant fluctuations in the graphs can be observed
due to the expansion of saturated profiles (in the form of fingers or
droplets). Therefore, it can be inferred that hydrodynamic instability
(in the form of droplet generation) is the determining factor in
generating different types of “self-similar” fingering behaviors.
Figure 12 displays the relationship between fractal dimension
(DB ) and classic capillary number (Equation 1). Since Dvalues for Experiment I and Experiment V were observed to be inordinate,
(while the rest were consistent with DB) ,DB were selected for further analysis. As seen,
with the capillary number increase, fractal dimensions increase until
reaching a peak followed by a relatively slow decrease. Such behavior
can be explained by the interplay between surface tension and the
viscosity effect. As the surface tension decreases, the fractal number
increases (due to the high tip-splitting), however, after the capillary
number reaches a certain point
(10-3~10-3), the
viscosity becomes the dominant force in the establishment of finger
morphology (smoothing the frontal mobility line) which leads to a
reduction in fractal number.