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\(\mu^{s}/\mu^{o}\) ).