A bottle in a freezer

TL;DR

This paper develops a mathematical model for the solidification process of liquids in elastic bottles during freezing, explaining high stress occurrences and deriving related formulas.

Contribution

It introduces a coupled PDE model for bottle solidification, proving existence, uniqueness, and analyzing long-term behavior of solutions.

Findings

High stresses explained by the model

Derived formula for undercooling coefficient

Proved well-posedness and long-term stability

Abstract

We propose here a model for solidification of a liquid contents of an elastic bottle in a freezer. The main goal is to explain the occurrence of high stresses inside the bottle. As a by-product, we derive a formula for the undercooling coefficient in terms of the elasticity constants, latent heat, and the phase expansion coefficient. We investigate the well-posedness of the three-dimensional model: we prove the existence and uniqueness of a solution for the corresponding initial-boundary value problem which couples a PDE with an integrodifferential equation and an ordinary differential inclusion ruling the evolution of the phase parameter. Finally, we prove some results on the long time behavior of solutions.