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

Published online by Cambridge University Press:  10 January 2014

MAURIZIO CESERI  and
JOHN M. STOCKIE
MAURIZIO CESERI
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, via dei Taurini 19, 00185 Roma, Italy email: m.ceseri@iac.cnr.it
JOHN M. STOCKIE
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada, V5A 1S6 email: jstockie@sfu.ca
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Abstract

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.

Keywords

Free boundariesStefan problemGas dissolutionAsymptotic analysisMultiscaleMultiphysics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 4 , August 2014 , pp. 449 - 480
Copyright
Copyright © Cambridge University Press 2014 

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