(541e) Flow and Drop Breakup in a Highly Concentrated Emulsion Squeezing through a Narrow Constriction

The present computational study has been motivated by experiments [1] demonstrating how a highly-concentrated monodisperse emulsion (with 92% drop volume fraction) behaves as a monolayer between two tight parallel walls when it enters a constriction, smaller in size than the non-deformed drop diameter. They found in [1] that drop breakup in such a geometry is due to pairs of drops entering the constriction almost simultaneously, so that one drop can pancake the other one and cause that one to break. The goal of the present study is to develop a realistic and accurate simulation method for such flows, and understand trapping and breakup mechanisms. The Stokes flow assumption is justified by experimental conditions [1] where the constriction height and micro-channel depth were ~ 30 micron. The problem is also simplified assuming equal viscosities for the emulsion drops and the carrier fluid. This assumption greatly speeds up the algorithm and is not much different than the experimental conditions [1], which had a viscosity ratio of ~0.8. Flow is pressure-driven, and the micro-channel consists of four panels: front and back panels parallel to each other, and top and bottom panels with constriction. The channel is continued periodically along the flow direction, which is essential for simulating a batch operation with a limited number of 3D drops in a periodic cell (40-100 drops in the present study). Accordingly, the fluid velocity and pressure gradient are 1-periodic. Specifying the average pressure gradient along the flow direction makes it a well-defined problem. The algorithm is for general geometries, but the applications herein are for the case when the drop non-deformed diameter is about the same as the channel depth (between parallel front and back walls) and the same as the channel constriction height. This minimum height, in turn, is much smaller than the maximum height, producing a strong constriction.

The boundary-integral (BI) formulation is based on the 1-periodic Green function (Stokeslet) and related stresslet. The fluid velocity in the channel is sought as the inhomogeneous term F, plus double-layer contribution from solid panels, with yet unknown density function q. The inhomogeneous term is the sum of capillary contributions from all drop surfaces, which include surface tension, local curvature, normal vector and periodic Stokeslet. An additional contribution due to average pressure gradient is simply the flux of the periodic Stokeslet through the whole inlet/outlet section of the periodic cell - one of the crucial advantages of using 1-periodic kernels in the present BI formulation. No-slip conditions on the solid walls give an integral equation for the density function q. The benefit of this BI formulation for matching viscosities is that drop-wall interactions need to be handled only before and after the solution of the BI equation and are, therefore, decoupled from the BI iterations.

A challenge of this geometry, especially at extreme emulsion concentrations, is that in addition to high drop surface triangulations, ultra-high resolution is required on solid walls to capture strong drop-wall interactions and avoid divergence of BI iterations. High-order, near-singularity subtraction, in the spirit of [2], greatly improves the solution quality and alleviates the problem of numerical drop-solid overlapping (severe at extreme emulsion concentrations), but high wall discretizations are still unavoidable, with 36K-90K boundary elements used in the present simulations. So are high drop surface triangulations (with 6K-9K mesh triangles per drop used herein). Multipole acceleration is a crucial tool in the present work to handle such large systems in long-time dynamical simulations. Roughly, a two-orders of magnitude gain (over standard BI coding) is achieved by partitioning all mesh nodes into compact blocks/patches, and handling block/patch to block/patch contributions mostly by multipole expansions/re-expansions, and only rarely by direct summations. A special swelling algorithm is used to generate a random highly–compressed configuration of many drops in the domain, as an initial condition for hydrodynamic simulations.

Most simulations are for smooth (albeit strong) constrictions of cosine shape, with the neck height being four times smaller than the channel height away from the constriction. In such geometries, a short length of the constriction has a strong inhibiting effect on drop breakup at high emulsion concentrations c=0.6-0.9, even at large capillary numbers. Animations show that, for every pair of drops entering the constriction almost simultaneously, the drops attain quite large elongations but are unable to break, since, upon exit from the constriction, they are blocked from the front by another, largely deformed drop that develops an orientation almost orthogonal to the pair orientation. It is also observed that all the drops that were initially tight to the top and bottom channel walls, become noticeably separated from these walls as time proceeds, due to the diverging character of the flow on the exit side of the constriction, even at extreme drop volume fractions of c= 0.9. Additional flow properties to be presented include the volume-averaged fluid velocities of the carrier fluid and of the droplet phase versus the capillary number. Those are observed to reach a statistical steady state long before predictions about individual drop breakup can be made. We currently study other constriction profiles, closer to experimental [1]. Those have different entry and exit angles, and a longer middle section. An increased constriction length greatly promotes drop breakup.

[1] L. Rosenfeld et. al. Break-up of droplets in a concentrated emulsion flowing through a narrow constriction. Soft Matter, 2014, 10, 421-430.

[2] A.Z. Zinchenko & R.H. Davis. Emulsion flow through a packed bed with multiple drop breakup. J. Fluid Mech.,2013,725, 611-663.