V
Figure 13—Dimensionless
numbers vs. descriptive parameters (in log scale) (I) L/Ly, (II)tP/tT, (III) tB/tT, (IV) Nd(b), (V) Nd(f).
Figure 13 demonstrates the functional relationship between the
descriptive parameters (I) L/Ly , (II) tP/tT , (III)tB/tT , (IV) Nd(b), (V) Nd(f) and
dimensionless numbers (Nca * ,Nca ** , M , Nca ,We , We* ).
(I) L/Ly vs. dimensionless scaling groups. Overall trends are
characterized by a decrease of L/Ly following an initial increase
as the dimensionless numbers increase. The slow decrease could also be
observed in the plots with Nca* andNca** , while fast decrease could be seen in the
case of M , and Nca . L/Ly plotted
with We* also displayed a relatively fast decrease compared to
that of other dimensionless numbers. L/Ly vs. We, however,
demonstrated a divergent trend from the rest of the plots characterized
by a steep increase in L/Ly following a sudden dip. Considering
that the magnitude of the considered viscosity term for each case is in
the ratio of 2:1:0 (Nca* ,Nca** : M , Nca :We , We* , respectively), viscosity force can be attributed
to being the leading factor in determining the normalized interfacial
length of fingers before their stability loss. With the larger
predominance of the viscosity effect, L/Ly development was shown
to be delayed. Such behavior could be due to the repressed shielding
effect as elaborated in the study by Nagatsu et al. (2007).
(II) tP/tT vs. dimensionless scaling groups. The general trends
for the dimensionless groups plotted with normalized production port
reaching time display an initial decrease followed by a stabilized phase
and subsequent increase. The length of the stagnant period, however, was
observed to vary and appeared to be shorter in the plots with Weand M . Considering that these two dimensionless groups do not
share any variables, several factors can be considered to be the cause
for the existence of the stagnant period. The chemicals responsible for
the stagnant behavior are chemical samples whose contact angle and
surface tension are relatively similar in range compared to that of
other chemical samples which indicates the sensitivity of tP/tTto the wettability and IFT effect.
(III) tB/tT vs. dimensionless scaling groups. The general
decreasing tendency can be observed in the plots except for the case oftB/tT vs. We . In the case of We , with the increase
in We , tB/tT increased due to the exclusive effect of
surface tension. However, its impact is immediately minimized when other
variables such as wettability and viscosity effect are considered in the
rest of the dimensionless numbers. In addition, the viscosity effect is
observed to be responsible for the long-stagnant period which exists in
the plots with Nca* ,Nca**, and M , following an initial steep
decrease. The combination of surface tension and contact angle effect
seems to be the responsible factor for the steep slope of decrease (with
the relatively shorter period of stagnation) observed in the case ofWe* and Nca .
(IV) Nd(b) vs. dimensionless scaling groups. The increasing
tendency could be observed in all cases except for We which
demonstrated that the number of droplets before the “finger break”
decreases with the decrease in surface tension (regardless of the
viscosity which has the function of stabilizing the hydrodynamic
instability). This is due to the hydrodynamic stability of fingers
associated with low surface tension. However, considering that the
opposite trend is observed to hold for the rest of the cases, it can be
concluded that while low IFT can maintain hydrostatic stability
of the fingers, it plays a minimal role in determining Nd(b) when
other forces such as viscosity and wettability effect are considered.
(V) Nd(f) vs. dimensionless scaling groups. The overall
decreasing tendency could be observed in the plots except in the case ofWe which again indicates that lower surface tension is associated
with an increase in the number of droplets after the finger break
(hydrodynamic stability loss). This is an interesting phenomenon which
indicates loss of hydrodynamic stability leads to the generation of a
number of droplets from the finger pinch off for the low IFT cases
(Figure 14 ). There are two well-established coarsening
mechanisms for emulsion: coalescence and Ostwald ripening. Coalescence
occurs due to the fusion of droplets while Ostwald ripening is caused by
the molecular exchange through the continuous phase. Visual data
analysis of the samples clearly demonstrated that the cause of emulsion
coarsening in the partial-miscibility fingering like state (brought on
by hydrostatic instability) is Ostwald ripening, rather than
coalescence.