Citation
Si, Hui (2000) Numerical study of interfacial flow with surface tension in two and three dimensions. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:10052010131226155
Abstract
In the first part of this thesis, we present new formulations for computing the motion of curvature driven 3D 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 2D fluid interfaces with surface tension. Applications to 2D vortex sheets, the Kirchhoff rod model, nearly anti—parallel vortex filaments, motion by mean curvature in 3D 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 2D HeleShaw flows and 3D axisymmetric flows through porous media with suction. HeleShaw flows with suction are known to form cusp singularities in finite time with zerosurfacetension. 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 zerosurfacetension 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 zerosurfacetension cusp time, the vanishing surface tension solution is singular at the finger neck. The zerosurfacetension 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 3D 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 HeleShaw 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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Applied Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Applied And Computational Mathematics 
Thesis Availability:  Restricted to Caltech community only 
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Thesis Committee: 

Defense Date:  12 August 1999 
Record Number:  CaltechTHESIS:10052010131226155 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:10052010131226155 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6097 
Collection:  CaltechTHESIS 
Deposited By:  John Wade 
Deposited On:  05 Oct 2010 20:49 
Last Modified:  26 Dec 2012 04:31 
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