New asymptotic analysis method for phase field models in moving boundary problem with surface tension

Jie  Wang; Xiaoqiang  Wang

doi:10.3934/dcdsb.2015.20.3185

Article Contents

2015, Volume 20, Issue 9: 3185-3213. Doi: 10.3934/dcdsb.2015.20.3185

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New asymptotic analysis method for phase field models in moving boundary problem with surface tension

Jie Wang^1, and
Xiaoqiang Wang^1,

1.
Department of Scientific Computing, Florida State University, Tallahassee, FL 32306-4120

Received: September 30, 2013

Revised: April 30, 2015

Published: October 2015

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Abstract

In this paper, we give an asymptotic analysis of the phase field Allen-Cahn and Cahn-Hilliard models of free surfaces with surface tension. Unlike the traditional approach that approximates the solution by the so-called matched asymptotic expansion involving outer expansion, inner expansion and matching, our new approach utilizes a uniform double asymptotic expansion to expand the whole phase field function directly. Although the main result is not new, we would like to emphasize that we derive the result under a uniform double asymptotic expansion. Thus, in this paper the detailed structure of the phase field functions in the equilibrium state is obtained, and the consistency of the phase field models with the corresponding sharp interface models is discussed, including the free surface Allen-Cahn model, Cahn-Hilliard model, and the Allen-Cahn model with volume constraint. The explicit asymptotic expansion of the phase field function reveals rich details of its structures. Moreover, it nicely explains some unusual phenomena we observed in numerical experiments. The theory introduced in this paper can be applied to guide the future modeling and simulation of other moving boundary problems by phase field models.

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Mathematics Subject Classification: Primary: 35B40, 65M12, 65M15 and 53A10.

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