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40430-AC9
Emulsion Flow through Porous Media
During the 1-year no-cost extension, the work was continued on developing and applying the simulation methods for emulsion flow through granular materials based on rigorous microstructural modelling. The multipole-accelerated code for multiparticle-multidrop long-time simulations in a periodic box, with prescribed average pressure gradient, was improved and applied to systematic calculations for a periodic emulsion squeezing through a cubic lattice of solid spheres (at 50% solid volume fraction), drop concentration in the available space being 36% and 50%, at drop-to-medium viscosity ratios of 1 and 4 for a series of capillary numbers, with particular emphasis on the conditions near-critical for squeezing to occur. In this regime, even tens of thousands of boundary elements per surface and about 10000 time steps are required to properly resolve lubrication and accurately describe the squeezing dynamics, which our code makes feasible owing to multipole acceleration. The calculations include the case when drops are large and can not be accommodated as spheres between the solids, so a new special algorithm was developed for a start-up configuration of deformed drops. While, for a 36% emulsion, relaxation to a periodic regime requires drops to pass the constrictions just once, the approach can be much slower for 50% emulsions requiring several cycles. The average drop-phase velocity is considerably larger than the average continuous-phase velocity for ALL attainable capillary numbers Ca, although our data clearly indicate that this trend would reverse (as expected) for conditions very close to critical. In contrast, for 40% concentrated emulsions flowing through a random densely-packed granular material of solid spheres (up to 14 solids and 40 drops in a periodic cell), both trends are observed: drops move faster than the continuous phase for the homoviscous case, but slower than the continuous phase for viscosity ratio of 4, even away from the critical regime. Drop clustering is never observed in these simulations, the system always attains a statistical steady state; the ability of drops to break is severely limited by geometrical constraints imposed by solid particles.
In a parallel study, a graduate student, T. Ratcliffe is working on the axisymmetrical problem of a gravity-induced drop squeezing through a toroidal constriction by both boundary integral and physical experiments. Theoretical solution in this case has allowed us to come much closer to the critical squeezing regime with higher accuracy, explore drop-solid spacing (which can reach 0.01-0.001 of the drop size) and extract scaling for the squeezing time T_sq vs. Bond number from the simulations. These results suggest that T_sq scales like (Bo-Bo_crit)^(-1/3) as the critical Bo is approached. This scaling is confirmed by our 3D calculations for flow-induced squeezing of a periodic emulsion (above) at different viscosity ratios and concentrations (with Ca instead of Bo), indicating its universality. This scaling (although empirical at this stage) is very useful for finding critical Bo or Ca by extrapolations. Experiments are in progress to verify our theoretical squeezing times for a toroidal constriction and thereby evaluate the effect of surface roughness on the squeezing mechanism. Torus geometry is ideally suited for maximum control of experimental conditions. This work is accepted for presentation at the Fluid Dynamics 2007 APS Fall Meeting.
Another graduate student, A. Griggs, concentrated during the no-cost extension on experiments to compare with our boundary-integral solution for a deformable drop sliding down an inclined wall. The agreement is quite good, confirming, in particular, a generally non-monotonic behavior of the steady-state drop velocity vs. Bo, most pronounced for 30 deg. inclination angle.
Overall, three papers were published on this PRF sponsored project, and three more submitted [1-3]. The results were presented at AIChE 2005,2006 and international MFGM2006 meetings, and at the IUTAM 2007 Symposium. One Ph.D. thesis is forthcoming (A. Griggs, Fall 2007), another one (T. Ratcliffe) is in progress.
[1] Griggs A.J., Zinchenko A.Z. & Davis R.H. Gravity-driven motion of a deformable drop or bubble near an inclined plane at low Reynolds number. Int. J. Multiphase Flow, accepted.
[2] Zinchenko A.Z. & Davis R.H. Algorithm for direct numerical simulation of emulsion flow through a granular material. J. Comput. Phys., submitted.
[3] Zinchenko A.Z. & Davis R.H. Squeezing of a periodic emulsion of deformable drops through a cubic lattice of spheres. Phys. Fluids, submitted.
