A three-phase free boundary problem with melting ice and dissolving gas

TL;DR

This paper presents a mathematical model for a complex three-phase free boundary problem involving melting ice, dissolving gas, and water, incorporating novel integral conditions and asymptotic solutions validated by numerical simulations.

Contribution

It introduces a new three-phase free boundary model with an integral mass conservation condition and develops series solutions for interface dynamics and gas concentration.

Findings

Series solutions accurately approximate interface positions.

Numerical simulations validate the asymptotic and series solutions.

The model captures complex interactions between melting, dissolution, and phase interfaces.

Abstract

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves the interactions between gas, water and ice. The dynamics are driven by melting of the ice layer, while the pressurized gas also dissolves within the meltwater. The model incorporates a Stefan condition at the water-ice interface along with Henry's law for dissolution of gas at the gas-water interface. We employ a quasi-steady approximation for the phase temperatures and then derive a series solution for the interface positions. A non-standard feature of the model is an integral free boundary condition that arises from mass conservation owing to changes in gas density at the gas-water interface, which makes the problem non-self-adjoint. We derive a two-scale asymptotic series solution for the dissolved gas concentration, which because of the non-self-adjointness gives rise to a Fourier series expansion in eigenfunctions that do not satisfy the usual orthogonality conditions. Numerical simulations of the original governing equations are used to validate the series approximations.