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The past year has seen three major accomplishments in this project: (1) development of a diffuse-interface numerical package for computing three-phase contact line motion; (2) application of this code to two interesting problems; (3) simulation and computation of interfacial deformation due to selective withdrawal. These are outlined below.

(a) Development of a diffuse-interface code. Previously we have built a numerical toolkit AMPHI (Adaptive Meshing with Phase Field Φ) for computing interfacial flows in complex fluids. For the encapsulation problem at hand, we need to generalize the package in two aspects. First, we need to introduce interfacial energies between each of two immiscible fluids and a solid substrate. This makes it possible to specify the wetting properties of the solid with respect to the fluids, especially the wetting angle. This is done by building algebraic formulations for the energies in terms of the phase-field variable, which are then incorporated into the governing equations via wall boundary conditions. Second, we need to generalize the code to full 3D. This is necessitated by the nature of two-phase transport and extrusion; the interface deforms in the plane of the cross-section but the axial gradient cannot be ignored. In fact, it is the driving force behind the interfacial evolution. We have generalized the finite-element solver of AMPHI, using unstructured grids, to 3D, and then integrated it into a 3D adaptive meshing algorithm which controls the spatial distribution of tetrahedral elements. The phase-field variable provides a convenient way to compute and control the spatial variation of grid sizes. This new toolkit has been validated by comparing solutions of benchmark problems.

(b) Simulation of interfacial deformation. We have applied the new AMPHI3D code to two interesting problems in interfacial flow and transport. The first is the apparent slip on micro-textured hydrophobic substrates. We have uncovered four different regimes depending on air entrapment in the microgrooves and cavities. At a critical capillary number, the air pockets are deformed to such a degree that the downstream contact line depins from the edge of the cavity and shifts downward. At even higher capillary number, the air pocket coalesces with the downstream pocket to form a continuous gas film that completely insulates the liquid flow from the solid substrate. This brings about a tremendous amount of slip and drag reduction. These results have just been published in the Physics of Fluids. The second application deals with the motion of a semi-submerged solid on an air-water interface, with applications in propulsion and locomotion. We have found that the propulsion comes chiefly from the capillary forces on the interface, and is only mildly dependent on the vortical structures near the interface. This answers an outstanding question in the literature. Furthermore, our results show that the hydrophobicity of the solid surface is important for the object to stay afloat, but not for the amount of propulsion.

(c) Selective withdrawal. Selective withdrawal refers to the process of drawing one or both components of stratified fluids through a tube placed near their interface. We carried out computational and experimental studies of selective withdrawal of viscous and viscoelastic liquids under air. The key mechanism of interest is how the viscoelasticity in the bulk liquid affects the evolution of the free surface. Experimentally, this is investigated by comparing the interfacial behavior between a Newtonian silicone oil and two dilute polymer solutions. While the surface undergoes smooth and gradual deformation for Newtonian liquids, for the polymer solutions there is a critical transition where the surface forms a cusp from which an air jet emanates toward the suction tube. This transition shows a hysteresis when the flow rate or location of the tube is varied. In the subcritical state, the surface of polymer solutions deform much more than its Newtonian counterpart but the deformation is more localized. Finite-element simulations based on an arbitrary Lagrangian-Eulerian scheme showed that the novel interfacial behavior of the polymer solutions can be attributed to the large polymer stress that develops under the surface because of predominantly extensional deformation.