The phase-field method, also known as the diffuse interface method, has gained popularity in simulating interfacial flows. The interface between two immiscible fluids is treated as a diffuse layer governed by a phase-field variable that obeys the Cahn-Hilliard equation. In my talk, I will first describe two recent contributions to this class of methods: a generalization of the theoretical framework to account for complex rheology of non-Newtonian fluids, and a finite-element implementation with adaptive meshing for highly accurate interfacial resolution. These have allowed us to simulate interfacial dynamics of complex fluids with a refined understanding of the microscopic physics. Then I will discuss numerical simulations of the unique partial coalescence process. This refers to the phenomenon that a drop falling onto a fluid-fluid interface does not merge with it completely but leaves a smaller daughter drop behind, thus forming a cascade of partial coalescence. With a wide range of length scales, this phenomenon highlights the advantages and limitations of the numerical method. The numerical results show qualitative and sometime quantitative agreement with experiments. Furthermore, we develop general criteria for the occurrence of partial coalescence, which are very difficult to explore experimentally.

Experiments and numerical simulations show that energy dissipation in incompressible fluid turbulence tends to a positive value in the inviscid limit (infinite Reynolds number). Lars Onsager (1949) proposed an explanation for this phenomenon in terms of energy cascade for certain
singular solutions of Euler equations. We shall review current ideas on the nature of turbulent energy cascade and their status within rigorous
theory of PDE's. In particular, we shall discuss a classical picture of Geoffrey Taylor (1937) on the role of vortex line-stretching in generating
turbulent energy dissipation. Taylor's argument was based on a statistical hypothesis that material lines in a turbulent flow will tend to elongate, on average, and appealed to the Kelvin Theorem (1869) on conservation of circulations. For smooth solutions the Kelvin Theorem for all loops is equivalent to the Euler equations of motion, but we shall present rigorous results which suggest that the theorem breaks down in turbulent flow due to nonlinear effects. This turbulent "cascade of circulations" has been verified by high-Reynolds-number numerical simulations. We propose another conjecture, that circulations on material loops may be martingales of a generalized Euler flow (in the sense of Brenier and Shnirelman). We shall
show that this property has a close analogue in the "Kraichnan model" of random advection, which accounts for anomalous scalar dissipation in that model. The "Kraichnan model" is also known to probabilists as a generalized stochastic flow and its basic features have been put on a rigorous footing by Le Jan and Raimond (2002, 2004). We propose a geometric treatment of this model, formally as a diffusion process on an infinite-dimensional semi-group of volume-preserving maps.

The interaction of flowing fluids with free bodies -- sometimes
compliant, sometimes active, sometimes multiple -- constitutes a class
of beautiful dynamic boundary problems that are central to biology and
engineering. Examples range from how organisms locomote in fluids
(which depends strongly on scale) to how non-Newtonian stresses
develop in complex liquids (strongly dependent on the nature of
fluidic microstructure). I will discuss several interesting examples,
emphasizing how they are formulated mathematically so as to yield
models tractable for analysis or simulation, and show how this work
has interacted with experimental studies.