Mathematical Sciences: Well-Posed Numerical Calculations of Free-Surface Flows

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9308075 Baker The investigator studies the motion of free surfaces in fluid flows by investigating the development of singularities in a related analytic problem. In many free-surface flows, the effects of viscosity and surface tension appear to be so small that they are neglected. However, attempts to solve the motion of the free surface by numerical techniques, boundary integral techniques for example, soon run into difficulties due to the formation of curvature singularities. Only when one of the fluids is effectively a vacuum do numerical calculations indicate that the motion of the free-surface is well-posed. Obviously, the effects of surface tension and viscosity, no matter how small the coefficients are, become very important close to the time of singularity development. Precisely how these regularizing agents prevent singularity formation, and what the nature of the subsequent motion is, are the main thrusts of this proposal. To achieve this objective, the investigator uses a simple mathematical view that describes singularity formation through the analytic extension of the boundary integrals into the complex arclength variable (or other suitable parametrization variable). In this complex plane, singularities are born, depending on the details of the initial conditions. These singularities then track through the complex plane, and may reach the real axis in finite time. At that point they become physically relevant. Preliminary studies of the influence of surface tension show that the singularities slow down and fail to reach the real axis in finite time, but they get extremely close. The close proximity of these singularities to the real axis plagues standard numerical methods, because of the high resolution needed to capture the extremely distorted free-surface. Instead, the investigator uses a method that represents any singularities in the complex plane explicitly, so that the rest of the behavior of the free-surface is analytic. In particular, this analytic part can be represented by Fourier series very accurately. The first task is to use this approach on a simpler classical free-surface flow problem, the fingering instability in a Hele-Shaw cell. The second is to explore the method on the Kelvin-Helmholtz instability of an interface between two incompressible fluids of equal density and in the presence of surface tension. Surfaces between immiscible liquids and gases occur abundantly in nature and technology. Some examples include the rise of bubbles in chemically reacting units, falling rain-drops, spray jets, and oil and water flowing in the ground. Mathematical models that attempt to describe the motion of surfaces typically exhibit great sensitivity to the details of the motion. Consequently, computer simulations can suffer from inaccuracies unless extraordinary care is taken. The investigator develops a conceptually new mathematical approach that will allow severely deformed geometries to be tracked reliably. In particular, the approach will be able to track the long term behavior of water penetrating oil in a Hele-Shaw Cell, a mathematically similar model for the motion of oil and water during secondary stage recovery from an oil-field. Once completely understood on this important problem, the approach will be adapted to the study of rising bubbles and dendritic formation during crystal growth. ***