John S. Lowengrub, Fellow

Topological Transitions and Singularities in Fluid, Crystal, and Biological Interfaces

This research group is studying topological transitions and singularities of interfaces in fluid flows and biological systems. They use a dual approach of sharp and diffuse interface methods and are developing accurate and efficient algorithms. In fluid systems, they are investigating ternary fluid flows in which two of the components are immiscible but the third component is miscible with at least one of the other components. They are also continuing to perform high resolution simulations of droplet coalescence and breakup. In crystals, the group is studying the evolution of three-dimensional crystals with isotropic and anisotropic surface tension. In biological systems, the group is studying the morphology of non-necrotic vascular tumors.

In their fluid flow work, the researchers developed modelling pinchoff and reconnection methods for use in a Hele-Shaw cell. The group analyzed two diffuse interface models, called the HSCH model and BHSCH model, to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell. In the HSCH model, the binary fluid may be compressible due to diffusion. The BHSCH model uses a Boussinesq approximation and the fluid is incompressible. Having found that the breakup of an unstably stratified fluid layer is smoothly captured by both models, the researchers concluded further that the HSCH model seems to be more diffusive than the BHSCH model and predicts an earlier pinchoff time, which causes subtle differences between the two in the pinchoff region. Away from the pinchoff region, both models yield nearly identical results and were demonstrated in convergence with the classical sharp interface model as the interface thickness vanishes.

These researchers also focused on producing boundary integral methods for multicomponent fluids and multiphase materials. As part of this research, the group participated in the development of methods that accurately and efficiently include surface tension on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities in inviscid fluids; the generation of capillary waves on the free surface; and problems in Hele-Shaw flows involving pattern formation through the Saffman-Taylor instability, pattern selection, and singularity formation.

In crystals, the group is studying the important question of the three-dimensional morphology of growing solid crystals in a liquid melt as well as the growth of precipitates in solid-solid phase transformations. This involves solving a diffusionally evolving free boundary problem. Linear analysis suggests that under certain conditions, the morphology of growing crystals may be controlled. The researchers confirmed this with their numerical experiments. They have developed a fully developed, three-dimensional boundary integral method to solve the diffusional evolution of interfaces. They began with isotropic surface tension, and have begun to extend their results to anisotropic interface kinetics.

In their research on tumor growth, the group is working to determine the conditions under which tumors become spherical or tend to break up, increasing the danger of spreading their cells throughout the body. They are using a model that couples a reaction-diffusion equation for nutrient with a Darcy’s law for tumor motion. The model contains a surface-tensionlike term that mimics the cell-to-cell adhesiveness. The resulting system is very similar to that describing motion of fluid in a Hele-Shaw cell. In the evolution process, competition arises between the growth/shrinkage of the tumor due to food supply and the stabilizing effect of the cell-to-cell adhesive forces. The group has performed linear analysis and developed boundary integral methods to solve the problem in two dimensions.

Research Group

Anthony M. Anderson, Undergraduate Student Researcher
Vittorio Cristini, Supercomputing Institute Research Scholar
Jacob J. Hageman, Graduate Student Researcher
Russell Hooper, Graduate Student Researcher
Leonard Imas, Research Associate
Jun-Seok Kim, Graduate Student Researcher
Trygve Kristiansen, Graduate Student Researcher
Hyeong-Gi Lee, Visiting Researcher
Perry H. Leo, Faculty Collaborator
Shuwang Li, Graduate Student Researcher
Xiaofan Li, Faculty Collaborator
Ellen K. Longmire, Faculty Collaborator
Milton Lopes, Visiting Researcher
Qing Nie, Department of Mathematics, University of California at Irvine, Irvine, California
Michael Renardy, Mathematics Department, Virginia Tech, Blacksburg, Virginia
Galyna Vasko, Graduate Student Researcher
Nicolas Vera, Graduate Student Researcher
Nicolas Versluis, Research Associate
Xiaoming Zheng, Research Associate
Hua Zhou, Research Associate