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John S. Lowengrub, Fellow

Topological Transitions and Singularities in Fluid and Biological Interfaces

The goal of the fluid dynamics aspect of this research is to refine and validate a physically-based method to capture pinch-off and reconnection in liquid/liquid interfacial flows. The goal of the biological aspect of this research is to determine the type of patterns formed in a model of growing, non-necrotic vascular tumors (carcinomas) and to develop experimentally validated models of tissue freezing that have application in cryopreservation and cryosurgery.

In fluid dynamics, changes in the topology of interfaces between partially miscible or nominally immiscible fluids are poorly understood. Such changes can occur when continuous jets pinch off into droplets, when sheared interfaces atomize, or when droplets of one fluid reconnect with another. These topological transitions occur in many practical applications, such as in liquid/liquid separators and mixers, where reaction and mixing rates within the systems are controlled directly by the detailed dynamics of the transition processes.

Historically, the narrow zone separating immiscible fluids has been represented as a sharp interface (i.e., a surface of discontinuity in density and viscosity). The classical Navier-Stokes equations can be solved on either side of the interface with the appropriate jump conditions prescribed across it. However, this sharp interface model breaks down and singularities form when interfaces pinch-off and reconnect. Furthermore, in these models, one has to prescribe ad-hoc cut-and-connect conditions to evolve the flow through these events.

To overcome this difficulty, these researchers have developed a diffuse interface model in which the sharp interface is replaced with a narrow, diffuse layer across which limited mixing occurs-consistent with physical chemistry. This diffusion allows pinch-off and reconnections to occur smoothly and automatically and eliminates the need for cut-and-connect conditions. In this partial miscibility model (PMM), a mass concentration field is introduced and the model consists of a generalized diffusion equation (of Cahn-Hilliard type) for the mass concentration field coupled with the Navier-Stokes equations with extra stresses that mimic surface tension.

These researchers have solved several flow regimes using the PMM. In particular, the pinchoff of liquid/liquid jets has been simulated. Several regimes have been considered. In the first, parameters were used that corresponding to an experiment performed in the laboratory of Professor Ellen Longmire from the Aerospace Engineering and Mechanics Department at the University of Minnesota. In this experiment, an axisymmetric jet of water/glycerin was pumped into a silicone oil ambient. Non-axisymmetric jets were also considered, and strong tendencies were found for the jets to axisymmetrize.

Currently, this research is moving toward more realistic simulations of topology transitions. In particular, these researchers are going beyond a Boussinesq approximation of the PMM. In the Boussinesq model, density differences are allowed only in the gravitational term. Consequently, chemical diffusion does not generate non-soleniodal velocity fields (through density variation from a diffusing concentration field). The next step is to consider a low Mach number limit of the PMM. In this limit, general density variation is allowed, but it is supposed that the chemical potential does not depend on pressure. This leads to a slightly easier system of equations than the full PMM. New numerical methods are being developed and tested to accurately and efficiently simulate these flows, and the results are being compared to actual experiments. This research is in the midst of developing an axisymmetric code that can be used to achieve high-resolution simulations of pinchoff and reconnections. Two regimes are being considered-pinchoff and reconnection of liquid/liquid jets and droplets impacting an interface. Both are being investigated experimentally in Professor Longmire's laboratory.

In the biological systems study, a very important consideration in understanding the growth of tumors is their stability. In particular, one wishes to know under what conditions tumors become spherical or tend to breakup increasing the danger of spreading their cells throughout the body. These researchers have been studying a quasi-steady limit of a model that couples a reaction-diffusion equation for nutrient with a Darcy's law for tumor motion. The resulting system is very similar to that describing motion of fluid in a Hele-Shaw cell. In this model, there is a surface tension-like term that mimics the cell-to-cell adhesiveness. To begin, the linear stability of evolving spherical solutions has been considered. Dynamics, through the tendency of the tumors to grow or shrink, can greatly affect the stability of the tumor. In particular, there is competition between the stabilizing effects of the cell-to-cell adhesiveness and the tumor growth. This leads to speculation that the simulations of the fully nonlinear evolution will reveal tumors with interesting morphology. To perform the fully nonlinear evolution, a boundary integral method is being developed to evolve the tumor boundary. This was begun by solving the equations in two dimensions. Three-dimensional simulations are considered later.

In cryobiology, or low-temperature biology, tissues may be frozen for either preservative or destructive purposes. In the former, the goal is to cryopreserve tissue systems and maintain their viability upon thawing. In the latter, the goal is to destroy undesirable tissues such as tumors by freezing them during cryosurgery. In both cryopreservation and cryosurgery, the outcome is highly dependent on the thermal and chemical history the tissues experience during the processes. These researchers are working to develop, simulate, and validate a two-compartment model of tissue freezing. In this system, one compartment models the extracellular space while the other models the intracellular space. Mass, heat, and chemical species transfer is accounted for between the extracellular and intracellular spaces. In the extracellular spaces, these fields evolve by convection-diffusion equations. The local interaction of cells with the extracellular space is governed by the Kadeem Katchalsky irreversible thermodynamic model of membrane transport. This has been done previously for pancreatic islets.

This information is available in alternative formats upon request by individuals with disabilities. Please send email to alt-format@msi.umn.edu or call 612-624-0528.

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