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

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves 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 the Stefan condit...

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Bibliographic Details
Published in:European journal of applied mathematics 2014-08, Vol.25 (4), p.449-480
Main Authors: CESERI, MAURIZIO, STOCKIE, JOHN M.
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Boundary value problems
Conservation
Dissolution
Fourier series
Free boundaries
Gases
Ice
Mathematical analysis
Mathematical models
Melting
Physics
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Summary:We develop a mathematical model for a three-phase free boundary problem in one dimension that involves 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 the 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 series approximations.
ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792513000430