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Squeezing of Deformable Drops through Granular Materials

Alexander Z. Zinchenko, University of Colorado at Boulder, Campus Box 424, Boulder, CO 80309-0424 and Robert H. Davis, Department of Chemical and Biological Engineering, University of Colorado at Boulder, 424 UCB, Boulder, CO 80309-0424.

The flow of emulsion drops through a granular material is a problem of great fundamental importance with many applications (oil filtration through underground reservoirs, etc.). Among fundamental issues is the pressure gradient-flow rate relationship and determining the conditions when the emulsion flow effectively stops due to drop blockage in the pores by capillary forces. We have developed an efficient, multipole-accelerated algorithm capable of three-dimensional dynamical simulations for many non-wetting deformable drops squeezing through a granular material. The material skeleton is modeled as a random arrangement of N>>1 solid spherical particles rigidly-held in a periodic box at arbitrary volume fraction (including the case of close solid-solid contact). The problem is solved in the "constant flow rate" formulation, with the prescribed volume-averaged fluid velocity over the periodic box. The particle boundary-integral contribution is used in the Hebeker form as a proportional combination of a single- and double-layer potentials with the Hasimoto periodic Green function. The problem is reduced to a well-behaved system of second-kind integral equations for the fluid velocity on drop and Hebeker density on solid surfaces. Success of squeezing simulations through tight constrictions, especially for conditions close to critical for trapping to occur, crucially depends on novel double-layer desingularizations [1]. In this problem, Hebeker form is much favored over other boundary-integral formulations for particle contribution, since it allows robust simulations both in the supercritical and the subcritical (when trapping occurs) regimes. To facilitate large system simulations, multipole acceleration tools, similar to those for drop-drop interactions [2-3], are incorporated. However, much higher resolution is required in the present problem, with 5000-6000 boundary elements per surface, since drop-solid close interactions are much more lubrication sensitive. Calculations are made with 50-100 drops (of non-deformed diameter bigger than the diameter of interparticle constrictions) and 50-100 solid particles in a periodic box. We study the time- and volume-averaged emulsion flow rate as a function of the capillary number and other parameters to delineate the critical conditions when the emulsion can no longer squeeze through a granular material. The distinction between the flow-induced and gravity-induced squeezing is also discussed. For gravity-induced squeezing, non-wetting drops do not reach a trapped steady state in the interparticle constrictions (for subcritical conditions), but rather asymptotically approach the solid boundaries in an infinite time, due to the absence of the pumping mechanism [4]. The results are compared to those for a single drop interacting with several solids in an unbounded medium [1].

[1] Zinchenko A.Z., Davis R.H. 2006 A boundary-integral study of drop squeezing through interparticle constrictions. J. Fluid Mech. (in press)

[2] Zinchenko A.Z., Davis R.H. 2000 An efficient algorithm for hydrodynamical interaction of many deformable drops. J. Comput. Phys., vol. 157, pp. 539-587.

[3] Zinchenko A.Z., Davis R.H. 2003 Large-scale simulations of concentrated emulsion flows. Phil. Trans. Roy. Soc. Lond. A, vol. 361, pp. 813-845.

[4] Nemer, M.B., Chen, X., Papadopoulos, D.H., Blawzdziewicz,J. & Loewenberg, M. 2004 Hindered and enhanced coalescence of drops in Stokes flow. Phys. Rev. Lett., vol.92, p. 114501