Emulsion flows through confined geometries (i.e. packed beds or porous media) have many important applications, including food and pharmaceutical manufacturing, oil recovery, and fixed-bed catalytic reactors. When the emulsion drops have a comparable size to the typical constriction diameter, continuum models fail, because the complex phenomena associated with emulsion flow through confined geometries are not considered, including pore blockage by emulsion drops, circuitous emulsion flow pathways, and drop squeezing through constrictions. In this case, a microphysical description is needed, which includes the detailed interactions of the liquid drops with the solid pores. The key objectives for modeling emulsion flows through a confined geometry are to determine the relationship between pressure drop and flow rate of each phase and to determine the critical conditions when emulsion drops become trapped in the throats of the constriction pathways.

In the present work, we consider the model problem of the axisymmetric motion of a deformable emulsion drop moving due to gravity through a torus, using both experiments and a boundary-integral method. For the latter, the problem is reduced to a system of well-behaved, second-kind integral equations for the fluid velocity on the drop surface and the Hebeker representation for the solid-particle contribution. The trapping mechanisms and the conditions close to critical, below which drop trapping occurs, are of particular importance. The Hebeker representation is used instead of the normal boundary-integral representation for a solid particle, because the representation allows for calculation of both slow-squeezing and trapping cases. Successful simulations, especially for conditions near critical squeezing, require high-order desingularization techniques, to accurately calculate the fluid gap between the drop and solid during squeezing¹.

During squeezing, the average drop velocity decelerates between 10²-10⁴ times and the drop-solid spacing becomes only 0.1-1% of the non-deformed drop radius as the drop passes through the constriction. The critical Bond number, below which trapping occurs, is calculated for different squeezing conditions (drop-to-torus size ratio, drop-to-opening size ratio, and the drop-to-surrounding fluid viscosity ratio), where the Bond number is a dimensionless ratio of gravitational and interfacial forces. As expected from considerations of the static case, the critical Bond number is shown to be essentially independent of viscosity ratio. The drop exit time, which is the time for the drop to pass through the constriction when the Bond number is supercritical, is shown to increase with increasing viscosity ratio, increasing torus size with fixed hole radius, decreasing hole size, and decreasing Bond number. Simulation results indicate that the drop exit time scales with the difference between the Bond number and its critical value to the negative third power.

The critical Bond numbers obtained from the boundary-integral simulations are compared to analytical scaling estimates from a force balance, numerical values from the static drop shapes using the Young-Laplace equation, and experimentally determined values. Below the critical Bond number, when trapping occurs, the steady-state drop shape determined from boundary-integral calculations is compared to the drop shape obtained from the Young-Laplace equation.

[1] Zinchenko A.Z., Davis R.H. 2006 A boundary-integral study of drop squeezing through interparticle constrictions, J. Fluid Mech. 564:227-266.