Modelos de mudan¸ ca de fase irrevers´ ıveis Gabriela Planas Departamento de Matem´ atica Instituto de Matem´ atica, Estat´ ıstica e Computa¸ c˜ ao Cient´ ıfica Universidade Estadual de Campinas, Brazil Em colabora¸ c˜ ao com J.L. Boldrini (Unicamp) e L.H. de Miranda (UnB) G. Planas (Unicamp) Col´ oquio UFSC 1 / 36
Outline 1 Introduction to phase transitions 2 A model 3 Notation and assumptions 4 Existence of solutions 5 Influence of the convection G. Planas (Unicamp) Col´ oquio UFSC 2 / 36
Introduction to phase transitions Motivation Phase transitions are familiar occurrences, for example, the freezing of water to ice or the melting of ice to water. Perito Moreno Glacier (Argentina) G. Planas (Unicamp) Col´ oquio UFSC 3 / 36
Introduction to phase transitions Phase transitions The oldest mathematical model for phase transitions is the classical Stefan (1835-1893) problem which treats the formation of ice in the polar seas. The sharp-interface models are generally macroscopic continuum models which stipulate basic equations, such as the heat equation in each phase, and impose conditions on the interface (of zero thickness). Nowadays a large class of problems - containing a free or moving boundary - are called Stefan problems. G. Planas (Unicamp) Col´ oquio UFSC 4 / 36
Introduction to phase transitions Sharp-interface models For sharp-interface models, it is difficult to incorporate in a natural (i.e., in a physically sound) way several more complex physical phenomena which may be relevant. For instance, supercooling and superheating, finiteness of the interface thickness, surface tension effects and so on. In many real solidification/melting processes, the interfaces are actually not sharp; there may be even large transitions regions. G. Planas (Unicamp) Col´ oquio UFSC 5 / 36
Introduction to phase transitions Diffused-interface or phase-field models The main idea of the diffused-interface models is to take in consideration from the beginning that interfaces always have some thickness, maybe small, and also a structure. In the phase-field methodology each phase is distinguished by an auxiliary variable so-called phase-field . In different phases the phase-field attains different values. This methodology has emerged as a powerful tool that allows the modeling of complex growth structures occurring during phase transitions, like dendritic patterns. http://photography.nationalgeographic.com/ G. Planas (Unicamp) Col´ oquio UFSC 6 / 36
Introduction to phase transitions Phase-field models Phase field models can be interpreted and applied with two different aims, comparing to the sharp-interface models : it is a physically more detailed description of the phase transition, 1 it is easier to implement in a numerical algorithm. 2 A theoretical link between phase-field models and sharp-interface models was established by using formal asymptotic expansion. (Caginalp [Phys. Rev. A 1989], Caginalp-Xie [Phys. Rev. E 1993]) G. Planas (Unicamp) Col´ oquio UFSC 7 / 36
Introduction to phase transitions Phase-field models The classical phase-field model introduced by Caginalp [Arch. Ration. Mech. Anal. 1986] is � φ t − ǫ ∆ φ + 1 2 ( φ 3 − φ ) = θ θ t + φ t = ∆ θ. The unknown are the phase-field φ, which identifies the phases, and θ, the temperature of the material. The regions: Solid region: { φ = 1 } Mushy region: {− 1 < φ < 1 } , where the material is a mixture of solid and liquid states Liquid region: { φ = − 1 } G. Planas (Unicamp) Col´ oquio UFSC 8 / 36
Introduction to phase transitions Irreversible phase transitions We consider now a model that describes, as an example, the behaviour of material subjected to thermal hardening. For such material, increasing the temperature induces an irreversible chemical transformation of the material from phase 1 into another phase 2 (cooling down the obtained material does not permit the initial phase to be recovered). See Blanchard and Guidouche [Euro. J. Appl. Math. 1990]. As far as applications are concerned (in the case of a glue for example) phase 2 is generally harder than phase 1. Irreversibility is not a mere theoretical feature: even materials of the daily life, such as eggs, do not re-melt after solidification. G. Planas (Unicamp) Col´ oquio UFSC 9 / 36
Introduction to phase transitions Irreversible phase transitions Bonfanti, Fr´ emond and Luterotti [Adv. Math. Sci. Appl. 2000] proposed the following model: � θ t + ω t = ∆ θ, ω t + α ( ω t ) − ∆ ω + β ( ω ) ∋ θ. The unknown are the phase-field ω, which identifies the phases, and θ, the temperature of the material. Here, α and β denote two maximal monotone graphs with the following domains D ( α ) = [0 , + ∞ ) and D ( β ) = [0 , 1] . The differential inclusion means that there exist η ∈ α ( ω t ) and ξ ∈ β ( ω ) such that ω t + η − ∆ ω + ξ = θ a.e. G. Planas (Unicamp) Col´ oquio UFSC 10 / 36
Introduction to phase transitions Irreversible phase transitions Example: α = ∂ I + and β = ∂ I where I + is the indicator function of the interval [0 , + ∞ ) and I of the interval [0 , 1]. We have that ω t ≥ 0 and thus the phase transition is irreversible . G. Planas (Unicamp) Col´ oquio UFSC 11 / 36
Introduction to phase transitions Irreversible phase transitions with fluid flow Many of the mathematical questions related to this model that arose, like existence of solutions for a more complete model with high-order nonlinearities or simplified versions, were answered by several authors. Among other authors we may cite Aso, Bonetti, Colli, Fr´ emond, Kenmochi, Lauren¸ cot, Luterotti, Schimperna, Stefanelli. These previous works do not consider the possibility of fluid flow inside the non-solid region. Melt convection has important effects since, in some sense, the heat can be transported by the fluid flow. Consequently, it adds new length and time scales to the problem and results in morphologies that are potentially different from those generated by purely diffusive heat. G. Planas (Unicamp) Col´ oquio UFSC 12 / 36
A model A model We consider a model for the evolution of the process of irreversible solidification of certain materials taking into account the effects of fluid flow in the molten regions. The quantities which describe the progressive transformation of the material from liquid to solid state are: the phase-field variable ω, the temperature of the material θ, the velocity of the material u , and the hydrostatic pressure P . Let Ω ⊂ R N , 2 ≤ N ≤ 4 , be an open bounded domain with smooth boundary ∂ Ω and Q = Ω × (0 , T ), for T > 0. G. Planas (Unicamp) Col´ oquio UFSC 13 / 36
A model The system We consider the following system u t − ∆ u + ∇ P + K ( h ( ω ))( u + ρ u t ) = ζθ in Q ml , ∇ · u = 0 in Q ml , u + ρ u t = 0 in Q s , θ t + ω t − ∆ θ − ∆ p θ + u · ∇ θ = g ( x , t ) in Q , ω t + α ( ω t ) − ∆ ω − ∆ q ω + κ u · ∇ ω ∋ θ + f ( ω ) in Q , θ = ∂ω on ∂ Ω × (0 , T ) , ∂ν = 0 , u = 0 θ ( ., 0) = θ 0 , ω ( ., 0) = ω 0 , u ( ., 0) = u 0 in Ω . G. Planas (Unicamp) Col´ oquio UFSC 14 / 36
A model The regions h ( . ) is a given function depending on the material being considered and relating the solid fraction and the phase-field variable: h ( ω ( x , t )) gives the solid fraction at ( x , t ). The unknown space-time phase regions are: Q s = { ( x , t ) ∈ Q : h ( ω ( x , t )) = 1 } , the solid region , and Q ml = { ( x , t ) ∈ Q : 0 ≤ h ( ω ( x , t )) < 1 } , the non-solid region . We assume that h is a smooth real increasing function such that h ( z ) = 0 when z ≤ 0 and h ( z ) = 1 when z ≥ 1. G. Planas (Unicamp) Col´ oquio UFSC 15 / 36
A model The fluid equations As we assume slow flow of the molten material, we have u t − ∆ u + ∇ P + K ( h ( ω ))( u + ρ u t ) = ζθ in Q ml , u + ρ u t = 0 in Q s , where ζθ is the buoyancy force due to thermal differences given by the Boussinesq approximation, the Carman-Kozeny type term K ( h ( ω ))( u + ρ u t ), with ρ > 0 a given constant, brings a singularity in the transition layers from non-solid to solid regions, since we require that s → 1 − K ( s ) = + ∞ . lim G. Planas (Unicamp) Col´ oquio UFSC 16 / 36
A model The Carman-Kozeny term The multiplier, namely u + ρ u t , acts as a relaxation factor, allowing a smooth decay of the velocity once the material becomes solid since it forces the velocity to satisfy u + ρ u t = 0 in Q s , implying the exponential decay in time. The usual form of the Carman-Kozeny term corresponds to ρ = 0 and forces the velocity to satisfy u = 0 in Q s , i.e., when the material becomes solid must immediately stop. G. Planas (Unicamp) Col´ oquio UFSC 17 / 36
A model The temperature equation θ t + ω t − ∆ θ − ∆ p θ + u · ∇ θ = g ( x , t ) in Q . ∆ p θ = div( |∇ θ | p − 2 ∇ θ ) , p > 2 is the p -Laplacian. g is an external force. For the heat flux, we assume that is of form q = ( k 1 + k 2 |∇ θ | p − 2 ) ∇ θ , with k 1 and k 2 positive constants; that is, we assume the heat diffusion coefficient is given by k 1 + k 2 |∇ θ | p − 2 and thus increases as the temperature gradient increases. Due to the flow transport, an advection term for the temperature is also included. G. Planas (Unicamp) Col´ oquio UFSC 18 / 36
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