We perform fundamental studies of the behavior of two or more fluids
subject to external and internal forces. We are particularly interested
in the effect of surface tension on the flow morphology.

Our approach follows two directions:

** (1). ** The development of new,
physically-motivated and experimentally validated multiphase flow models;

** (2). ** The development of
state-of-the-art adaptive numerical methods;

** A. Phase-field models. ** We have been developing new physically based models capable of
simulating flows with two or more components with varying miscibilities.
A new system of equations, the Navier-Stokes-Cahn-Hilliard System, has
been derived. In this approach, related to phase-field models more
traditionally used in phase-transformations, sharp interfaces are replaced
by narrow transition regions. Concentration fields and corresponding
Helmholtz free energies are introduced that characterize the miscibility
properties of the components. The concentration fields satisfy a fourth
order nonlinear advection-diffusion equation (Cahn-Hilliard) and are
coupled to the Navier-Stokes equations through extra reactive stresses
that mimic surface tension. Examples of such flows are shown below.

A derivation of the two-phase model can be found in J.S. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard Fluids and Topological Transitions, Proc. Roy. Soc. London A 454 (1998) 2617-2654. The three-phase model can be found in (J.S. Kim and J.S. Lowengrub, Ternary Cahn-Hilliard fluids, with J.-S. Kim, Int. Free Bound. in review).

The models are nontrivial to solve as they couple a nonlinear fourth order advection-diffusion equation (Cahn-Hilliard) with the Navier-Stokes equations. To solve the coupled system, we have developed very efficient nonlinear multigrid methods. See for example Conservative multigrid methods for Cahn-Hilliard fluids}, with J.-S. Kim, K. Kang, J. Comp. Phys. 193 (2004) 511-543. We are currently developing adaptive nonlinear multigrid methods for this problem (see below).

** B. Models of surfactants. ** We have been developing new physically based models
of interfacial flows with surfactants. The key here is to develop an Eulerian approach
that can be used together with numerical methods such as the volume-of-fluid or level-set
method. The reason for using these methods is that they are capable of capturing topology
transitions such as drop-breakup and coalescence. The presence of
surfactants has a profound effect on these processes.

Together with
Professor Ashley James we have developed a
A surfactant conserving volume-of-fluid method for interfacial flows with
insoluble surfactant, J. Comp. Phys. 201 (2004) 685-722.

The algorithm is versatile and can be applied to a number of physical and biological problems, where the local length scales are dictated by the specific problem.

The reference for this work is ( X. Zheng, A. Anderson, J.S. Lowengrub and V. Cristini, Adaptive unstructured volume remeshing algorithms: Application to level-set simulations of multiphase flows, J. Comp. Phys. in review). An example of this approach is shown below.

Comparisons with by recent experiments by Dr. Z. Mohamed-Kassim and Professor Ellen Longmire are shown.

Nonlinear multigrid method using adaptive overlapping Cartesian meshes.

Left: Coarse mesh; Middle: Refined mesh; Right: Finest mesh.

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