Analysis of a two-phase field model for the solidification of an alloy

Abstract

In this paper we present some theoretical results for a system of nonlinear partial differential equations that provide a phase field model for the solidification/melting of a metallic alloy. It is assumed that two different kinds of crystallization are possible.Consequently, the unknowns are the temperature τ and the phase field functions u and v. The time derivatives u t and v t appear in the equation for τ (the heat equation). On the other hand, the equations for u and v contain nonlinear terms where we find τ .

Local existence of solutions of a three phase‐field model for solidification

“…In this article the model is deduced from physical principles and supported by some numerical simulations. We remark that, from the mathematical point of view, the present model can be seen as a generalization of the solidification models treated by Hoffman and Jiang in [3] and is related to Boldrini et al in [20,21], where the interaction potentials were similar to the ones considered here, but the diffusion mechanism was much simpler than the present nonlinear ones.…”

“…For technical reasons, the result is presently limited to the one-dimensional setting; we stress that, from the point of view of its mathematical analysis, the problem studied here is much more difficult than the corresponding one analyzed in Boldrini et al [20], where it was possible to obtain a global existence theorem, even in higher dimension. The difference is that in the present case we have very strong nonlinearities involving the higher-order derivatives; this makes the problem very difficult to handle from the technical point of view, and we have to be careful in the many estimates that are required for the arguments.…”

Some optimal control problems for a two-phase field model of solidification

“…These results are proved in [3]. Theorem 2.1 Let us assume that hypotheses (7) hold, f ∈ L q (Q) with q > 5/2 and τ 0 , u 0 , v 0 ∈ W 2 2 ( ) with ∂τ 0 /∂n| ∂ = ∂u 0 /∂n| ∂ = ∂v 0 /∂n| ∂ = 0.…”

“…Then, arguing as in [3], the following can be proved: Theorem 6.1 Let ⊂ R 3 be a bounded C 2 domain. Let us assume that k 1 , k 2 , a 1 and a 2 are positive constants, m 1 , m 2 ∈ L ∞ (Q) and u 0 , v 0 ∈ W 2 2 ( ) satisfy ∂u 0 /∂n| ∂ = ∂v 0 /∂n| ∂ = 0 and 0 ≤ u 0 , v 0 ≤ K .…”

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

“…Consequently, a wide variety of industrial applications are covered. For detailed discussions on the phase-field transition system we refer to [12][13][14][15][16][17]20,22,26,27,[30][31][32]34]. In [33] the reader can found more details relative to a more extensive class of problems on the type those treated in this paper (reaction-diffusion equation), as well as different types for the nonlinear term F (ϕ).…”