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In the past year, we focused on applying our diffuse-interface methodology and computational scheme to explore interfacial flow problems of scientific and industrial interest. These include four sub-projects: (a) Spreading and breakup of a compound drop on a partially wetting substrate; (b) Pressure driven gas-liquid two-phase flows through microchannels; (c) Wicking flows through microchannels; (d) Contact line dynamics using Cahn-Hilliard diffusion. Each is described in more detail below.

(a)   Spreading and breakup of a compound drop on a partially wetting substrate. Dynamics of compound drops are important to chemical and biological processes that use double emulsions. From a fundamental viewpoint, the interaction between a compound drop and a substrate involves multiple interfaces that may develop a rich array of morphologies depending on the geometry of the compound drop and the interfacial energies. We use a diffuse-interface method to simulate the spreading and breakup of a compound drop on a partially wetting solid substrate. Compared with a simple drop, the spreading of a compound drop exhibits much more complex behavior. Depending on the core-shell size ratio and the substrate wettability, various flow regimes are identified in which the interfacial morphology evolves in distinct ways. A phase-diagram is constructed in the parameter space of the core-shell size ratio and the wetting angle. For relatively small inner drops, the outer interface does not rupture during the spreading and the inner drop either remains suspended and encapsulated or attaches onto the substrate. Otherwise, the compound drop spontaneously breaks up and releases the inner drop into the ambient fluid. Several breakup scenarios are observed depending on the location of the initial rupture. In some regimes, the wetting of the substrate by one fluid can entrap secondary drops of the other, which can either attach to the substrate or stay suspended. The viscosity ratio mainly affects the spreading rate, and plays a minor role in the morphology evolution. This work has been submitted to Journal of Fluid Mechanics for publication.

(b)  Pressure-driven two-phase flow in microchannels. The motivation for this sub-project, as well as for the following one, comes from proton-exchange-membrane (PEM) fuel cells. A key component of the fuel cell is the gas diffusion medium (GDM), a porous layer that transports the water away from the catalyst layer (reaction sites) and draws oxygen toward it. There is a complex air-liquid two-phase flow problem in this porous medium, which we simplify to two-phase flows in microchannels with prototypical geometries, accounting for expansion, contraction, tortuosity and connectivity of the actual flow conduits in the porous layer. We further identify two different flow mechanisms. One is flow driven by an external pressure into a hydrophobic GDM, and the other is wicking flow driven by a surface wetting potential into a hydrophilic GDM. These are studied in two sub-projects, one described in the following paragraph, and the other in (c).

In pressure-driven flows, we have simulated gas-liquid flows in periodically patterned microchannels with grooves and ridges. The diffuse-interface algorithm is adapted to handle the interfacial motion and the three-phase contact line. A constant body force applies on both components to simulate the external pressure gradient driving the flow. Depending on the competition between the driving force and capillary force and the level of liquid saturation, several flow regimes have been observed in the microchannel, including slug flows with air bubbles, slug flows with water drops, water rivulets alongside air flow and driven sessile drops. We have investigated the critical conditions for the transition among the regimes as affected by substrate wettability, initial morphology of the interface, geometry of the microchannel and viscosity ratio.

(c)   Wicking flow in hydrophilic microchannels. Using the Cahn-Hilliard model and a finite-element algorithm, we have computed wicking flows in microfluidic channels of three types of geometries. The first type features axisymmetric tubes with contractions and expansions of the cross section. Both drainage and imbibition dynamics are studied, and we define critical conditions for the contact line to negotiate sharp corners on the wall. The second type consists of bifurcations in micro-channels where the competition between capillary pressure in the branches and viscous loss in the feeding tube produces different flow patterns. Finally, we examine tortuous channels in Z and U-shaped turns, where the effect of streamline on the flow rate is analyzed as a prototype for tortuosity in porous medium. The results show interesting competition between substrate-driven wicking and dissipation in the fluid bulk.

(d)  Contact line dynamics with diffuse-interface. The Cahn-Hilliard model uses diffusion between fluid components to regularize the stress singularity at a moving contact line. In addition, it represents the dynamics of the near-wall layer by the relaxation of a wall energy. In the first part of the project, we elucidated the role of the wall relaxation in a flowing system, with two main results. First, we showed that wall energy relaxation produces a dynamic contact angle that deviates from the static one, and derive an analytical formula for the deviation. Second, we demonstrated that wall relaxation competes with Cahn-Hilliard diffusion in defining the apparent contact angle, the former tending to "rotate" the interface at the contact line while the latter to "bend" it in the bulk. Thus, varying the two in coordination may compensate each other to produce the same macroscopic solution that is insensitive to the microscopic dynamics of the contact line. The second part of the study exploited this competition to develop a computational strategy for simulating realistic flows with microscopic slip length at a reduced cost. This consists in computing a moving contact line with a diffusion length larger than the real slip length, but using the wall relaxation to correct the solution to that corresponding to the small slip length. We derived an analytical criterion for the required amount of wall relaxation, and validated it by numerical results on dynamic wetting in capillary tubes and drop spreading. The result has been submitted to Physics of Fluids.