Abstract
A new phase field model is introduced, which can be viewed as a nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a generalized Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).
Highlights
Phase field models are widely used in various physical contexts in which a material exhibits two distinct phases
We are interested in a phase field model designed for solid-liquid mixtures at rest, which consists of an Allen-Cahn type equation for the order parameter coupled with a modified heat equation taking into account both the latent heat and the increase of entropy due to the non-equilibrium situation inside phase-transition regions
This model turns out to be a refined version - in a nontrivial way - of what is known as the Caginalp model [3], and it can be viewed as a special case of another one designed by Ruyer [13] for moving liquid-vapor mixtures
Summary
Phase field models are widely used in various physical contexts in which a material exhibits two distinct phases. We are interested in a phase field model designed for solid-liquid mixtures at rest, which consists of an Allen-Cahn type equation for the order parameter coupled with a modified heat equation taking into account both the latent heat and the increase of entropy due to the non-equilibrium situation inside phase-transition regions. This model turns out to be a refined version - in a nontrivial way - of what is known as the Caginalp model [3], and it can be viewed as a special case of another one designed by Ruyer [13] for moving liquid-vapor mixtures.
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