Non parabolic interface motion for the 1-D Stefan problem: Dirichlet boundary conditions

Abstract

Over a finite 1-D specimen containing two phases of a pure substance, it has been shown that the liquid-solid interface motion exhibits parabolic behavior at small time intervals. We study the interface behavior over a finite domain with homogeneous Dirichlet boundary conditions for large time intervals, where the interface motion is not parabolic due to finite size effects. Given the physical nature of the boundary conditions, we are able to predict exactly the interface position at large time values. These predictions, which to the best of our knowledge, are not found in the literature, were confirmed by using the heat balance integral method of Goodman and a non-classical finite difference scheme. Using heat transport theory, it is shown as well, that the temperature profile within the specimen is exactly linear and independent of the initial profile in the asymptotic time limit. The physics of heat transport provides a powerful tool that is used to fine tune the numerical methods. We also found that in order to capture the physical behavior of the interface, it was necessary to develop a new non-classical finite difference scheme that approaches asymptotically to the predicted interface position. We offer some numerical examples where the predicted effects are illustrated, and finally we test our predictions with the heat balance integral method and the non-classical finite difference scheme by studying the liquid-solid phase transition in aluminum.

Non-parabolic interface motion for the one-dimensional Stefan problem: Neumann boundary conditions

“…The non-parabolicity of ξ for small time intervals in the example shown in fig. 1 is a consequence of the boundary conditions imposed on the specimen, and this is a particular behavior that can not be observed in a Dirichlet boundary value problem [13].…”

“…It is important to note that, just by using Neumann boundary conditions, we obtain eqs. (17) and (18), which are of completely different nature than the interface position in the asymptotic limit, for a Dirichlet boundary value problem [13]. Even more, for a Dirichlet boundary value problem, the asymptotic interface position only depends on the thermal conductivities of each phase [13], on the other hand, for the Neumann boundary value problem, the interface position for long time intervals depends on the densities, specific heat capacities and latent heat of fusion.…”

“…(17) and (18), which are of completely different nature than the interface position in the asymptotic limit, for a Dirichlet boundary value problem [13]. Even more, for a Dirichlet boundary value problem, the asymptotic interface position only depends on the thermal conductivities of each phase [13], on the other hand, for the Neumann boundary value problem, the interface position for long time intervals depends on the densities, specific heat capacities and latent heat of fusion. We will illustrate a few examples where the non-parabolicity of the interface dynamics is evident, but we will also show by using the NC-FDS and HBIM, that the interface position strongly depends on the shape of the initial temperature profile, as predicted by eqs.…”

“…For the Dirichlet boundary conditions problem [13], the time behavior of the interface is far from being parabolic at large values of time, and one of the physical implications of these type of boundary conditions, is that temperature profiles within each phase of the specimen and the interface position at large time values, are completely independent of the initial temperature profile. By using Neumann boundary conditions, the physical picture changes drastically, since in this case, the solution of the problem depends completely on the initial temperature profile, due to conservation of energy.…”

“…By using Neumann boundary conditions, the physical picture changes drastically, since in this case, the solution of the problem depends completely on the initial temperature profile, due to conservation of energy. The goal of this work is to show that the interface motion is not parabolic as in the Dirichlet boundary value problem [13], and show that in this case, the solution depends entirely on the initial temperature profile. Even more, since energy conservation must be satisfied at every time step of the simulation, we developed a generalization of the non-classical finite difference scheme (NC-FDS).…”