Resumen de Beyond the classical Stefan problem
Francesc Font Martínez
In this thesis we develop and analyse mathematical models describing phase change phenomena linked with novel technological applications. The models are based on modifications to standard phase change theory. The mathematical tools used to analyse such models include asymptotic analysis, similarity solutions, the Heat Balance Integral Method and standard numerical techniques such as finite differences. In chapters 2 and 3 we study the melting of nanoparticles. The Gibbs-Thomson relation, accounting for melting point depression, is coupled to the heat equations for the solid and liquid and the associated Stefan condition. A perturbation approach, valid for large Stefan numbers, is used to reduce the governing system of partial differential equations to a less complex one involving two ordinary differential equations. Comparison between the reduced system and the numerical solution shows good agreement. Our results reproduce interesting behaviour observed experimentally such as the abrupt melting of nanoparticles. Standard analyses of the Stefan problem impose constant physical properties, such as density or specific heat. We formulate the Stefan problem to allow for variation at the phase change and show that this can lead to significantly different melting times when compared to the standard formulation. In chapter 4 we study a mathematical model describing the solidification of supercooled liquids. For Stefan numbers larger than unity the classical Neumann solution provides an analytical expression to describe the solidification. For Stefan numbers equal or smaller than unity, the Neumann solution is no longer valid. Instead, a linear relationship between the phase change temperature and the front velocity is often used. This allows solutions for all values of the Stefan number. However, the linear relation is an approximation to a more complex, nonlinear relationship, valid for small amounts of supercooling. We look for solutions using the nonlinear relation and demonstrate the inaccuracy of the linear relation for large supercooling. Further, we show how the classical Neumann solution significantly over-predicts the solidification rate for values of the Stefan number near unity. The Stefan problem is often reduced to a 'one-phase' problem (where one of the phases is neglected) in order to simplify the analysis. When the phase change temperature is variable it has been claimed that the standard reduction loses energy. In chapters 5 and 6, we examine the one-phase reduction of the Stefan problem when the phase change temperature is time-dependent. In chapter 5 we derive a one-phase reduction of the supercooled Stefan problem, and test its performance against the solution of the two-phase model. Our model conserves energy and is based on consistent physical assumptions, unlike one-phase reductions from previous studies. In chapter 6 we study the problem from a general perspective, and identify the main erroneous assumptions of previous studies leading to one-phase reductions that do not conserve energy or, alternatively, are based on non-physical assumptions. We also provide a general one-phase model of the Stefan problem with a generic variable phase change temperature, valid for spherical, cylindrical and planar geometries.
