The numerical solution of well posed Stefan problems in a two dimensional region are considered using a boundary integral technique. The numerical method is an extension of that used in previous work in that the boundary integral formulation takes account of heat flow both ahead of and behind the phase change front. This allows more realistic problems to be considered. Furthermore, it is found that when parameter values appropriate for water are used, the previously applied routine, based on Newton's method, for determining the location of the phase change front, is unstable. This is overcome by using a dissection based method for these parameter values. This numerical formulation is found to have a number advantages over finite difference and finite element techniques. For example, complex boundaries can be easily considered and the discretisation of the Stefan condition is not required. Numerical solutions of the Stefan problem are found for different parameter values and, more specifically, the freezing of water is considered. Employing a model of crystal formation, the numerical method is applied to predict the size of the crystals in the crystalline micro-structure that is formed when a material freezes. The predictions of this model are compared against experimental results and it is found that they are in good qualitative agreement. To obtain a more accurate model of the freezing of a liquid, the numerical method is extended to include the fluid motion in the closed region ahead of the phase change front. A numerical procedure is outlined for dealing with genuine two-phase problems, using a different approach in each phase. The fluid flow problem in the liquid phase of the material is solved using a time dependent finite difference method on a non-uniform mesh, whereas the previously derived boundary integral method is used to determine the temperature distribution in the solid phase. The numerical scheme in the liquid phase is found to be second order in space and time.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:651205 |
| Date | January 1998 |
| Creators | Galloway, Stuart J. |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/16985 |
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