Statistics of Emulsion Flow Through Granular Materials with Multiple Drop Breakups

We study the low Reynolds number flow of an emulsion of three-dimensional, immiscible deformable drops through a  randomly-packed,  granular material by numerical simulations. This problem is relevant to many applications, including  oil filtration through underground reservoirs. Of particular interest is to  obtain the pressure-gradient/flow-rate relationships and delineate the  conditions when the emulsion can no longer squeeze through the material due  to drop blockage in the pores by capillary forces. It is also important to explore  the effects of drop breakup in this process. If the emulsion drops were much  smaller than the  pores, an effective-medium model could be assumed for the emulsion. However,  in many cases, the drops are comparable in size with the particles and pores,  and the system has to be handled as a three-phase one, by direct numerical  simulation, with distinct velocities for the drop and continuous phases. A  recent multipole-accelerated, boundary-integral algorithm [1] has   been extended and systematically used in this work to study emulsion flow  through granular materials.  The material skeleton is modeled as a random arrangement of many monodisperse solid spheres rigidly held in a  periodic cell in mechanical equilibrium under the action of contact forces.  Two extreme cases are of interest (i) frictionless spheres forming a "random  close packing" (RCP), with the average density of 0.637 and the coordination  number of six, and (ii) highly-frictional particles in a “random loose  packing” (RLP), with the density of about 0.555 and coordination number close to  four.   Due to numerical difficulties, we focus on the second case. A novel algorithm is   developed to simulate such particle arrangements of perfectly-rigid spheres, and  it turns out that the mechanical equilibrium is maintained   through a percolating cluster containing ~ 91% of the particles, while other spheres are stress-free (although in contact with neighbors).  This feature is in  qualitative  agreement with classical physical experiments on 2D systems of disks, but  has never been simulated for frictional arrays of spheres. 

  We consider an emulsion of many non-wetting drops flowing under a specified volume-averaged pressure gradient. The problem presents severe computational difficulties, because of necessary very high resolution  (~ 10 000 boundary elements per surface) due to tight squeezing, and a large number (50K-100K) of  time steps to allow each  drop pass through  constrictions at least several times and to evaluate long-time averages for the phase velocities. Our largest systems include 36 solid particles and 100 drops in a periodic cell; the effects of the system size are carefully explored.   Multipole  acceleration,  built into our algorithm, has a two-order-of-magnitude  advantage over the standard boundary-integral coding at each time step, and appears to be, at present, the only way to perform resolved simulations for this problem. Compared to the original algorithm [1],  the present version incorporates topological mesh changes  on drop surfaces (with the help of some techniques from [2]) to simulate a cascade of multiple drop breakups observed in this problem at sufficiently large capillary  numbers .  A novel fragmentation algorithm is used to continue simulations  after each pinch-off.  Due to geometrical constraints imposed by solid  particles, the drop shapes are typically compact (but quite intricate) at breakup, and the daughter drops are usually comparable in size;  small satellite drops are occasionally observed  after primary breakups.  Our movies demonstrate how, after a cascade of breakups (typically, 10-20 events),  an initially monodisperse system attains a statistically steady (polydisperse) state. The phase velocities reach this state much faster than the drop size distribution does.

  Another significant extension, compared to [1], is ensemble  averaging of the results over many realizations of a granular material. We  found that using the empirical Carman-Kozeny correlation for the pure-fluid  velocity  in the definition for the capillary number significantly reduces the  dispersion of data, to facilitate ensemble averaging.  For drop-to-medium  viscosity ratio of unity, drop volume fraction of 40% in the space between the solids,  and non-deformed drop-to-particle size ratio of ~0.5,  the average drop and continuous phase permeabilities are studied as functions  of the capillary number. For strong deformations,  the drop-phase velocity is larger than the continuous-phase velocity (somewhat akin to the phenomenon observed  in  blood flow in capillaries), but the trend  changes as the drops become less deformable; the critical capillary number for the drop phase squeezing to stop (so that the drops become  trapped)  is accurately evaluated. The results are compared to those for  dilute emulsions (one or several drops in a periodic box, with the same drop-to-particle size ratio) travelling  through a dense RLP granular material.

[1] Zinchenko A.Z., Davis R.H. 2008 Algorithm for direct numerical simulation  of emulsion flow through a granular material. J. Comput. Phys., vol. 227,  pp.7841-7888.

[2] V. Cristini, J. Blawzdziewicz, M. Loewenberg 2001 An adaptive mesh
  algorithm for evolving surfaces: simulations of drop breakup and coalescence.
  J. Comput. Phys., vol. 168, pp. 445-463.