Descriptive parameters and their relationship to dimensionless
numbers
To enhance our understanding of the influence that each parameter has on
droplet formation in emulsion flow and explore the corresponding
mechanism, we have selected dimensionless scaling groups and parameters
and investigated the functional relationship between them. The
normalized indicators studied include \(\frac{L}{L_{y}}\) , which is the
normalized value of the interfacial length of fingers before the finger
break and \(L_{y}\) is the vertical length of the Hele-Shaw samples.\(Nd(b)\) is the number of droplets before the complete finger stability
loss. It should be noted that complete finger loss is defined as the
loss of an “initiator,” equivalent to the equilateral triangle forming
an “island” per description by Mandelbrot (1983). \(Nd(f)\) is the
number of droplets at the final stage of the injection.\(\frac{t_{B}}{t_{T}}\) is the normalized time to reach finger break,\(t_{T}\) is the total time of the injection period.\(\frac{t_{P}}{t_{T}}\) is the normalized time for fingers to reach the
production port, where \(t_{T}\) is the normalized total injectant
volume. Both the structural terms and time-dependent terms are
considered in the description as functions of chemical parameters as
suggested by De Wit (2001). In the past, finger-width was widely
suggested as an indicator to determine dynamic surface tension and
viscosity in a complex colloidal fluid fingering morphology (Bonn et al.
1995). Association of the chemical reaction with the finger width was
also shared by De Wit (2001), Sastry et al. (2001), and Fernandez and
Homsy (2003). While the legitimacy of such observations continues to
hold, proper finger width analysis does not include droplet presence and
therefore, was not considered in this study. The magnitude of the
viscous and capillary forces in their influence of flow morphologies
leading to the fingering loss stage was investigated using (1) classic
capillary number Nca , (2) modified capillary
number Nca* , which considers an extra viscosity
ratio term (\(\frac{\mu_{s}}{\mu_{o}}\)) to the original capillary
number, (3) modified capillary number (Nca**) ,
which considers both the contact angle and an extra viscosity ratio
term, and (4) viscosity number (M ) which considers viscosity
ratio and contact angle. Droplet break up behavior dependent on the
velocity was also investigated using the (5) Weber number (W )
which describes the rate of inertial forces to capillary forces, and (6)
modified Weber number which includes the wettability effect.
Nca =\(\frac{\mu_{s}\text{\ ν}}{\sigma}\) (1)
Nca* =\(\frac{{\mu_{s}}^{2}\nu}{\sigma}\frac{1}{\mu_{o}}\) (2)
Nca** =\(\frac{{\mu_{s}}^{2}\nu}{\text{σcosθ}}\frac{1}{\mu_{o}}\) (3)
M = \(\frac{\frac{\mu_{s}}{\mu_{o}}}{\cos\theta}\) (4)
We =\(\ \frac{\rho v^{2}l}{\sigma}\) (5)
We* = \(\frac{\rho v^{2}l}{\text{σcosθ}}\) (6)
(\(\mu\)s: viscosity of the solution,\(\mu\)o: viscosity of oil, v: velocity,\(\sigma\): surface tension, \(\theta\): contact angle, \(\rho\):
solution density, \(l\): characteristic length, viscosity ratio\(\mu^{s}/\mu^{o}\) ).