Droplet Breakup in Flow Past an Obstacle: A Capillary Instability Due to Permeability Variations

In multiphase flow in confined geometries an elementary event concerns the interaction of a droplet with an obstacle. As a model of this configuration we study the collision of a droplet with a circular post that spans a significant fraction of the cross section of a microfluidic channel. We demonstrate that there exist conditions for which a drop moves completely around the obstacle without breaking, while for the same geometry but higher speeds the drop breaks. Therefore, we identify a critical value of the capillary number above which a drop will break. We explain the results with a one-dimensional model characterizing the flow in the narrow gaps on either side of the obstacle, which identifies a surface-tension-driven instability associated with a variation in the permeability in the flow direction. The model captures the major features of the experimental observations. Introduction. – Microfluidic technologies as a platform for manipulating droplets are providing new approaches for designer emulsions [1], encapsulation of molecules and cells [2], high-throughput biological assays [3], kinetic analyses [4], and detection of surface contamination [5]. Continued developments of such multiphase flows requires understanding and control of droplet traffic [6], drop breakup [7], and drop coalescence [8]. For example, new insights into the latter topic have come from microfluidic studies of controlled coalescence using extensional flows [9] and electric fields [10]. The motion of drops in confined geometries is also relevant to traditional subjects such as multiphase flow in porous media, the motion of drops in networks of channels [11], and for understanding physiological flows, which are relevant to airway re-opening [12, 13]. One direction for improved understanding of these flows is to quantify breakup in confined flows, e.g. past studies have reported and rationalized drop breakup at T-junctions [14–16] or in a constricted cylindrical capillary [17]. In this study we investigate the breakup of drops flowing past an obstacle. Previous work showed it was possible to (a)present address: Institut Jean le Rond d’Alembert, CNRS/UPMC UMR 7190, 75252 Paris Cedex 05, France break drops in such flows, but no quantitative measurements were provided [14]. Some features of the influence of various obstruction shapes, including experiments and numerics, have been reported recently [18]. Here we describe the case where a drop that is significantly larger than the gap or channel can squeeze past the occlusion without breaking; we focus on understanding the conditions for breakup. We identify a critical capillary number below which drops do not break and above which drops break. Finally, we explain this transition in terms of an instability of the menisci located in the narrow gaps on either side of the obstacle and provide a one-dimensional theory that rationalizes the results qualitatively. Experimental setup. – We disperse droplets of hexadecane (viscosity ηdrop = 8× 10−3 Pa.s) in a continuous phase of distilled water (viscosity ηcont = 1× 10−3 Pa.s). A surfactant, sodium dodecyl sulfate (SDS) at 1 wt. % is added to the water to stabilize the droplets against coalescence. The interfacial tension between the surfactant solution and hexadecane is γ ≈ 5×10−3 N/m, as measured using a ring tensiometer. Syringe pumps are used to control the flow rates of the fluids in the microfluidic devices, which are fabricated in poly(dimethylsiloxane) (PDMS)