Executive Summary : | We are aware of and worried about the fact that the earth’s ice covers in the polar regions are receding as they melt with an increase in the earth’s average surface temperature. The widely accepted fact is that a substance changes state from solid to liquid through melting if the temperature is equal to or more than its melting point. Melting is an interfacial phenomenon that is observable as the temperature rises to the melting point leading to the coexistence of solid-liquid-gas. The liquid appears as a thin film at the surface of the solid, known as a premelted film and this phenomenon is called premelting. This film flows like a viscous liquid due to a thermomolecular pressure gradient arising due to a temperature gradient along the film. Premelting dynamics – the dynamics of the premelted film – play vital roles in many natural and engineering settings, such as the motion of glaciers, formation of snow, winter sports, frost heave, chemical uptake on atmospheric ice surfaces, cryogenic cell preservations, etc. Despite a huge range of applications, premelting dynamics remained scarcely understood. Limited studies are available in the literature based on various simple assumptions, e.g., unidirectional flow, steady-state temperature field, pure substances, etc. Inevitably all the substances are impure in nature. In cryogenic cell preservations, biological cells (e.g., tissues, heart) are preserved in saline solution in hypothermic conditions. The substance that undergoes change of state is ice/water, which contains salt in the form a solute/impurity dissolved in it. Therefore, it is essential to mathematically model premelting dynamics with impurities, which are missing in the existing models. Also, another important ingredient of most of the realistic settings is the time-varying temperature – again not considered in the majority of the earlier mathematical studies. Therefore, in this project, we aim to integrate the effects of a time-dependent temperature and impurities of the substances and provide a mathematical model that will enable us to improve our understanding of dynamics of premelted film confined in narrow gaps between its solid phase and a foreign elastic substance. The elastic substance is used as a proxy for the biological cell (or soil). Our primary (but not exclusive) objectives of this project are (a) an improved mathematical model, (b) development of numerical techniques to solve the model problem, and (c) possible implications to frost heave and/or cryogenic cell preservations. |