Mathematical Models and Numerical Simulations of phase change in Lagrangian and Eulerian descriptions

Abstract

This thesis presents development of mathematical models for liquid-solid and solid-liquid phase change phenomena 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, thermal conductivity, density and latent heat are continuous and differentiable functions of temperature. The width of the interface region can be as large or as small as desired by a specific application. The derivations assume the matter to be homogeneous and isotropic. In case of Lagrangian description we assume zero velocity field i.e. no flow with free boundaries i.e. stress free medium. Under these assumptions the mathematical model reduces to the first law of thermodynamics i.e. energy equation. The derivation is based on specific total energy and the heat vector. The constitutive theory for heat vector is assumed to be Fourier heat conduction law. The specific total energy incorporates the physics of phase change in the transition region between the solid and the liquid phases. This results in a time dependent non-linear diffusion equation in temperature. The physics of initiation of the phase change as well as formation and propagation of the transition region (front) is intrinsic in the mathematical model and hence no other means of front tracking are required. For the purposes of numerical simulation, the mathematical model can also be recast as a system of first order partial differential equations. In case of Eulerian description, the mathematical model consists of the continuity equation, momentum equations, energy equation, constitutive theories for stress tensor and heat vector in the liquid phase, solid phase and as well in the transition region. In the liquid phase we assume the matter to be Newtonian fluid, hence the details of the mathematical model are straight forward. In the solid region we assume the solid to be hypoelastic, hence the rate constitutive theory is valid for the stress tensor. We also assume Fourier heat conduction law for the solid phase. In the transition region containing a mixture of solid and liquid phases, use of mixture theory is most appropriate for conservation laws as well as the constitutive theory. Such mathematical models are beyond the scope of the work considered in this thesis. Instead, we present a simple model that is based on representative volume fractions in the transition region. Eulerian descriptions are necessitated when phase change occurs in a flowing medium. Regardless of whether the mathematical models utilize Lagrangian or Eulerian description, the resulting mathematical models consist of a system of non-linear partial differential equation in space and time, i.e. they constitute initial value problems. Numerical solutions of these mathematical models are obtained using space-time least squares finite element process based on minimization of residual functional. This approach results in space-time variationally consistent integral forms that yield symmetric algebraic systems with positive definite coefficient matrices that ensure unconditionally stable computations during the entire evolution. The local approximations for the dependent variables in the mathematical model are considered in h,p,k framework which permits higher degree as well as higher order space-time approximations in space and time. Numerical values of the evolution are computed using a space-time strip or a space-time slab corresponding to an increment of time with time marching. Numerical studies in R^1 and R^2 are presented to demonstrate simulation of the initiation of the phase change as well as its subsequent propagation during evolution. These studies cannot be performed using sharp interface and phase field models. The smooth interface approach considered in the present work has many significant benefits: (i) Continuous and differentiable transition region permits desired physics and avoids singular fronts that are nonphysical. (ii) The mathematical model in Lagrangian description result in a single non-linear PDE from the first law of thermodynamics which provides the ability to initiate as well as locate the phase transition front during evolution without using special front tracking methods. (iii) In Eulerian description the Navier Stokes equations and the constitutive theories for stress tensor and heat vector result in a system of non-linear PDEs with the same features for phase change initiation and propagation as in the case of Lagrangian description. (iv) The mathematical models and the computational approach presented here permits initiation of the phase transition interface and its propagation without employing any special means. This is not possible in sharp interface and phase field mathematical models. (v) The computational methodology employed in this work ensures unconditionally stable computations in which very high accuracy of evolution is possible for each time step during evolution.