Study of a system for the isothermal separation of components in a binary alloy with change of phase

Daniel Kessler (1) , Jean-Francois Scheid (2, 3) , Giulio Schimperna (4) , Ulisse Stefanelli (5)

1 Department of Mathematics [College Park]
2 CORIDA - Robust control of infinite dimensional systems and applications
3 IECN - Institut Élie Cartan de Nancy
4 UNIPV - Dipartimento di Matematica “Felice Casorati” Department of Mathematics [Univ Pavia]
5 Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R.

An isothermal model describing the separation of the components of a binary metallic alloy is considered. A process of phase transition is also assumed to occur in the solder; hence, the state of the material is described by two order parameters, i.e. the concentration c of the first component and the phase field ϕ. A physical derivation is provided starting from energy balance considerations. The resulting system of PDEs consists of a rather regular second-order parabolic equation for ϕ coupled with a fourth-order relation of Cahn-Hilliard type for c with constraint and solution-dependent mobility. Global existence of solutions is proved and several regularity properties are discussed under more restrictive assumptions on the physical parameters. Continuous dependence on data is shown in a special case. An asymptotic analysis of the model is also performed, yielding at the limit step a coupling of the original phase field equation with a Stefan-like system for c.

Mots clés: Faedo–Galerkin scheme; variational formulation; maximum principle; constraint; fourth-order parabolic system; phase transition; binary alloy

Domaines: Equations aux dérivées partielles [math.AP]

Daniel Kessler, Jean-Francois Scheid, Giulio Schimperna, Ulisse Stefanelli. Study of a system for the isothermal separation of components in a binary alloy with change of phase. IMA Journal of Applied Mathematics, 2004, 69 (3), pp.233--257. ⟨10.1093/imamat/69.3.233⟩. ⟨hal-00141794⟩