Numerical study of interfacial flow with surface tension in two and three dimensions

Citation

Si, Hui (2000) Numerical study of interfacial flow with surface tension in two and three dimensions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8n4d-y915. https://resolver.caltech.edu/CaltechTHESIS:10052010-131226155

Abstract

In the first part of this thesis, we present new formulations for computing the motion of curvature driven 3-D filament and surface. The new numerical methods have no high order time step stability constraints that are usually associated with curvature regularization. This result generalizes the previous work in [23] for 2-D fluid interfaces with surface tension. Applications to 2-D vortex sheets, the Kirchhoff rod model, nearly anti—parallel vortex filaments, motion by mean curvature in 3-D and simplified water wave model are presented to demonstrate the robustness of the methods. In the second part of this thesis, we investigate numerically the effects of surface tension on the evolution of 2-D Hele-Shaw flows and 3-D axisymmetric flows through porous media with suction. Hele-Shaw flows with suction are known to form cusp singularities in finite time with zero-surface-tension. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. Our computations reveal an asymptotic shape of the wedge as surface tension tends to zero. Moreover, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. For 3-D axisymmetric flow, similar behavior is observed. The surface develops a narrow finger which evolves into a cone as it approaches the sink. The finger diameter is smaller than the finger width for Hele-Shaw flow and the surface moves faster. The azimuthal component of the mean curvature enhances the definition of the finger neck while smoothing the interface there.

Thesis Files