Mathematical Modeling and Numerical Simulation of Liquid-Solid and Solid-Liquid Phase Change

Abstract

This thesis presents numerical simulations of liquid-solid and solid-liquid phase change processes using mathematical models in Lagrangian and Eulerian descriptions. The mathematical models are derived by assuming a smooth interface (or transition region) between the solid and liquid phases in which the specific heat, density, thermal conductivity, and latent heat of fusion are continuous and differentiable functions of temperature. In the derivations of the mathematical models we assume the matter to be homogeneous, isotropic, and incompressible in all phases. The change in volume due to change in density during phase transition is neglected in all mathematical models considered in this thesis. In one class of mathematical models we assume the velocity field to be zero i.e. no flow assumption, and free boundaries i.e. zero stress field. Under these assumptions the mathematical models reduce to first law of thermodynamics i.e. the energy equation, a nonlinear diffusion equation in temperature if we assume Fourier heat conduction law relating temperature gradient to the heat vector. These mathematical models are invariant of the type of description i.e. Lagrangian or Eulerian. In the second group it is shown that when the stress field and the velocity field are assumed nonzero in all three phases, then the resulting mathematical model from the conservation and balance laws in Lagrangian description for solid phase, Eulerian description for liquid phase, and mixed descriptions in the transition region are inadequate in describing the interaction between the media. Validity and usefulness of these models from the point of view of continuum mechanics as well as computational mathematics are considered and discussed. The third group of mathematical models are derived using conservation and balance laws with the assumption that stress field and velocity field are nonzero in the fluid region but are assumed zero in the solid region. In the transition zone the stress field and the velocity field transition from nonzero at the liquid state to zero at the solid state based on temperature in the transition zone. These models are consistent based on principles of continuum mechanics, hence provide correct interaction between the media and are shown to work well in the numerical simulations of phase transition applications with flow. Numerical solutions of the nonlinear diffusion equation in R1 and R2 resulting from the first group of models (zero stress and zero velocity field in all phases) and the nonlinear partial differential equations resulting from the third group of mathematical models are obtained using space-time hpk finite element processes based on spacetime residual functional in which the space-time integral forms are space-time variationally consistent, hence the resulting computations remain unconditionally stable during the entire evolution regardless of the choices of h, p, and k and the dimensionless parameters in the mathematical model. Numerical studies are presented in R1 and R2 for liquid-solid and solid-liquid phase transitions using the first group of models and the computed solutions in R1 are compared with the theoretical solution from the sharp-interface method. Numerical studies are presented using the third group of mathematical models for liquid-solid phase transition to demonstrate the phase transition simulation ability of this group of mathematical models in the presence of flow.