The flow of an emulsion containing drops or bubbles through a porous medium has practical applications in biology, engineering, geology and oil recovery. Of fundamental importance are the relationships between pressure drop and flow rates of the dispersed and continuous phases, and the conditions where the drops or bubbles become trapped within the porous medium. These issues are particularly challenging when the drops or bubbles have sizes close to or larger than the pores, in which case an effective-medium approach fails.

This talk presents direct numerical simulations of emulsion flows through a porous medium modeled as a granular bed at low Reynolds number, using boundary-integral methods. A model problem is considered first, where a single drop squeezes between two or more solid obstacles. It is shown that the drop becomes trapped when the capillary number (representing the ratio of viscous and interfacial forces) is below a critical value and the drop is not able to deform sufficiently to pass through the constriction. Subsequently, an efficient, multipole-accelerated algorithm was used for dynamical simulations of many nonwetting deformable drops squeezing through a granular medium comprised of fixed spheres in a lattice formation or distributed randomly in a periodic box. A large number of boundary elements per surface is needed, because of the lubrication sensitivity of drop-solid interactions. Surprisingly, away from the critical condition, the dispersed phase may have higher average velocity than the continuous phase. This result is due to steric exclusion of the drops from the slow-moving streamlines near the solid surfaces. Near the critical condition, however, the average velocity of the dispersed phase is reduced as the drops become trapped (or nearly so) in the narrow spaces between the particles comprising the granular medium.