(314a) The Effect of Interfacial Slip On the Coalescence of Two Equal-Sized Drops in a Head-On Collision

A fundamental question that arises in the area of coalescence of two drops in a flow is the dependence on various parameters of the drainage time t_d, which is the time from the instant the drops collide to film rupture and coalescence.  In a recent experimental investigation from our group [Hsu et al. (2008)], the drainage time t_d for two equal-sized drops undergoing a head-on collision was found to scale with capillary number Ca as t_d ~ Ca ^m for 10^-4 < Ca < 10^-2, where the scaling exponent m was always found to lie between 1 for low capillary numbers to 4/3 for high capillary numbers.  Hsu et al. (2008) were successful in reasonably describing both limits of the exponent by scaling theories for a fixed viscosity ratio l between the droplet and suspending fluids. What they were unable to explain, however, was the curious dependence of the exponent m on l.  For drop radii smaller than 27 microns, the exponent was observed to increase in experiments as l was raised from 0.19 to 6.8, a change which is not captured by the scaling theories.  Coupling this dependence with an earlier observation (Park et al., 2004) that flow-induced coalescence between drops is facilitated by increasing the molecular weight of the polymeric suspending fluid, we speculate that these changes occur because of the phenomenon of interfacial slip between the droplet and suspending fluids.  Numerical simulations from our group have indicated that the droplets may come as close as a few nanometers before undergoing coalescence.  Such thin film dimensions are comparable to the thickness of the diffuse interface between the polymeric drop and suspending fluids where variations in concentration and entanglement densities occur.  Obviously, this invalidates our assumptions of zero-interfacial thickness and no-slip applied in both scaling theory and simulations.  In this work, we examine the effect of interfacial slip on the dependence of the drainage time with capillary number for different viscosity ratios via boundary integral simulations.  Interfacial slip is modeled via the Navier-slip condition, and the slip parameter employed in the simulations is predicted using the work of Goveas and Frederickson (1998).  We show the boundary integral equations with the slip boundary condition can be converted to two Fredholm integral equations of the second kind, which can be solved iteratively to yield the stress and velocity fields on the interface at every instant of time.  For the droplet coalescence problem, our results indicate that drainage time does decrease with the introduction of slip as expected.  Also, the exponent m relating the dependence of drainage time with capillary number is greater in the presence of slip.  These results are verified against the scaling analysis presented by Hsu et al. (2008).  However, with the current parameters we have employed to describe the slip phenomenon, the changes are overpredicted, i.e. drainage times are shorter and the exponent m is greater than experimentally observed values.   Possible reasons are explored.

References

Goveas, J. L. and G. H. Fredrickson, “Apparent slip at a polymer-polymer interface,” Eur. Phys. J. B 2, 79–92 (1998).

Hsu A. S., A. Roy and L. G.  Leal, “Drop-size effects on coalescence of two equal-sized drops in a head-on collision,” J. Rheol. 52 (6), 1291-1310 (2008).

Park, C. C., F. Baldessari, and L. G. Leal, “Study of molecular weight effects on coalescence: Interface slip layer,” J. Rheol. 47, 911–942 (2003).