Graduate Thesis Or Dissertation
 

Nonlinear free boundary problems arising from soil freezing in a bounded region

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  • Changes of density occur naturally in phase transition processes and introduce the bulk movement of material. It is customary in analyzing such problems to disregard this unpleasant complication and assume the densities to be equal. However, such changes are unavoidable and for one-dimensional problems the complexities introduced by this bulk movement can easily be circumvented. The key idea is posing the problem in local coordinates which are fixed in each phase. In this dissertation, we investigate freezing and thawing of soils in a bounded two-phase medium with phases whose material properties are not only distinct but their thermal dependence is also permitted. Generally speaking, when a freezing process takes place in a cooled melt situated in contact with its solid phase, an interface boundary is formed whose movement (as the freezing proceeds) results in compression of both phases. Owing to the density differences, the density of the material will increase, movements will occur in each phase, pressures and thermal stresses will build up in the respective phases, and the freezing point will decrease. Mathematically, this results in three nonlinear free boundary problems for determining: (I) the location of the interface boundary along with the temperature distribution throughout the medium, (II) the pressure and velocity distributions in the unfrozen phase, and (III) the displacement distribution and hence the thermal stresses in the frozen phase. In fact, the temperature satisfies a nonlinear parabolic differential equation on each side of the interface while the temperature is continuous across the interface and equals the transition temperature, the condition of local thermodynamic equilibrium. To consider the problems from the most general point of view, mass forces are taken into account such that the pressure and velocity distributions satisfy a nonlinear couple of hyperbolic differential equations of the first order in the unfrozen phase and the pressure is related to the density through the equation of state. The displacement satisfies a nonlinear hyperbolic differential equation of the second order in the frozen phase which is related to the thermal stresses through the generalized Hooke's law. Across the interface, the pressure is equal to the negation of the normal thermal traction on the interface. Furthermore, the movement of the interface is related to the temperatures, the velocities and the material properties at the interface through conditions of dynamical compatibility for energy and mass transfer. Based upon potential theoretic arguments, we prove existence, uniqueness and continuous dependence on the initial and boundary data of solutions to Problem I. Along with these results, explicit expressions for the densities, the specific heats and the thermal conductivities as functions of time and local coordinates in their respective phases, which fit our analysis, are also obtained. Correspondingly, the characteristic method is utilized to show existence and uniqueness of solutions to Problems II and III, and we demonstrated the continuous dependence of their solutions on the respective data. Moreover, asymptotic estimates for the critical time of breakdown in their solutions are also obtained. Some remarks on discontinuities in general are finally discussed.
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