Using detailed asymptotic analyses of the dynamics of the phase-field model, we show that the major sharp-interface models (Stefan, modified Stefan, Hele-Shaw, etc.) all arise as limiting cases of the phase-field equations. The scaling of the physical parameters in the microscopics leads to distinct macroscopic models with critical di8'erences.
We consider the distinguished limits of the phase field
equations and prove that the corresponding free boundary problem is
attained in each case. These include the classical Stefan
model, the surface tension model (with or without kinetics), the surface
tension model with
zero specific heat, the two phase Hele–Shaw, or quasi-static,
model. The Hele–Shaw model is
also a limit of the Cahn–Hilliard equation, which is itself
a limit of the phase field equations.
Also included in the distinguished limits is the motion by mean
curvature model that is a limit
of the Allen–Cahn equation, which can in turn be attained from the
phase field equations.
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