Modelos de mudan ca de fase irrevers veis Gabriela Planas - - PowerPoint PPT Presentation

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)

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Outline

1 Introduction to phase transitions 2 A model 3 Notation and assumptions 4 Existence of solutions 5 Influence of the convection

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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.

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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.

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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.

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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.

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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 :

1

it is a physically more detailed description of the phase transition,

2

it is easier to implement in a numerical algorithm.

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])

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Introduction to phase transitions

Phase-field models

The classical phase-field model introduced by Caginalp [Arch. Ration.

• Mech. Anal. 1986] is

φt − ǫ∆φ + 1

2(φ3 − φ) = θ

θt + φt = ∆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}

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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.

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Introduction to phase transitions

Irreversible phase transitions

Bonfanti, Fr´ emond and Luterotti [Adv. Math. Sci. Appl. 2000] proposed the following model: θt + ωt = ∆θ, ωt + α(ωt) − ∆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(ωt) and ξ ∈ β(ω) such that ωt + η − ∆ω + ξ = θ a.e.

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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 ωt ≥ 0 and thus the phase transition is irreversible.

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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.

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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 ⊂ RN, 2 ≤ N ≤ 4, be an open bounded domain with smooth boundary ∂Ω and Q = Ω × (0, T), for T > 0.

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A model

The system

We consider the following system ut − ∆u + ∇P + K(h(ω))(u + ρut) = ζθ in Qml, ∇ · u = 0 in Qml, u + ρut = 0 in Qs, θt + ωt − ∆θ − ∆pθ + u · ∇θ = g(x, t) in Q, ωt + α(ωt) − ∆ω − ∆qω + κu · ∇ω ∋ θ + f (ω) in Q, θ = ∂ω ∂ν = 0, u = 0

• n ∂Ω × (0, T),

θ(., 0) = θ0, ω(., 0) = ω0, u(., 0) = u0 in Ω.

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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: Qs = {(x, t) ∈ Q : h(ω(x, t)) = 1}, the solid region, and Qml = {(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.

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A model

The fluid equations

As we assume slow flow of the molten material, we have ut − ∆u + ∇P + K(h(ω))(u + ρut) = ζθ in Qml, u + ρut = 0 in Qs, where ζθ is the buoyancy force due to thermal differences given by the Boussinesq approximation, the Carman-Kozeny type term K(h(ω))(u + ρut), with ρ > 0 a given constant, brings a singularity in the transition layers from non-solid to solid regions, since we require that lim

s→1− K(s) = +∞.

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A model

The Carman-Kozeny term

The multiplier, namely u + ρut, 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 + ρut = 0 in Qs, 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 Qs, i.e., when the material becomes solid must immediately stop.

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A model

The temperature equation

θt + ωt − ∆θ − ∆pθ + u · ∇θ = g(x, t) in Q. ∆pθ = 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 = (k1 + k2|∇θ|p−2)∇θ, with k1 and k2 positive constants; that is, we assume the heat diffusion coefficient is given by k1 + k2|∇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.

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A model

The phase-field differential inclusion

ωt + α(ωt) − ∆ω − ∆qω + κu · ∇ω ∋ θ + f (ω) in Q. α is a maximal monotone graph in R2 with domain D(α) = [0, +∞). So, we have that ωt ≥ 0 and thus the phase transition is irreversible. ∆qω = div(|∇ω|q−2∇ω), q > 2 is the q-Laplacian. Due to the flow transport, an advection term for the phase may also be included (depending on the value of κ ≥ 0, which is given a constant). The case κ = 0 is an approximation for the case that the time scale of the solidification process is faster than the time scale of the flow. f is a Lipschitz function.

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A model

Modelling approach

Approach to the irreversibility is similar to the ones in Bonfanti et. al (2000), Bonetti (2002), Colli et al. (2007), Lauren¸ cot et al. (2002), Luterotti et al. (2002). Following the main ideas of Voller et al. (1987), Beckermann et al. (1988, 1999), Blanc et al. (1995), among others, we coupled a diffuse interface model for phase-transitions with a singular system for the fluid motion.

In this approach, the flow is modeled as if the medium in the transition layers were a type of porous media, with porosity related to the phase-field variable. This brings up a singular equation for fluid motion, which can be a singular Navier-Stokes equation or some modification of it. In the present case, since we are considering slow flows, the approach leads to a singular Stokes equation.

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A model

Modelling approach - Some remarks

Not only does convection influence the solidification pattern, but the evolving microstructure can also produce unexpected and complicated phenomena. It is a delicate issue to decide how to take into consideration, both in physically sound and in mathematically feasible manner, the effects of fluid motion in a solidification process. The inclusion of convection brings several new technical difficulties to an already hard problem. The major difficulty of our problem is to balance these two approaches, i.e., to properly couple these models.

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A model

Some related references

Blanc et. al (1995): stationary problem for binary alloys with convection by using the solid fraction to distinguish the phases Boldrini-P. (2005): 2D phase-field model with convection for alloys with a singular Navier-Stokes equation Boldrini-P. (2005): 3D phase-field model with convection for alloys with a modified Navier-Stokes equations Boldrini-de Miranda-P. (2012): 2D irreversible problem with a singular Navier-Stokes equation and κ = 0 Remark These works employ the usual Carman-Kozeny term (ρ = 0).

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Notation and assumptions

Notation

We use standard notation. W s,p(Ω), 1 ≤ p ≤ ∞ and 0 ≤ s ≤ ∞, denote Sobolev spaces. Spaces of divergence-free vector fields: V = {v ∈ (C ∞

0 (Ω))N : ∇ · v = 0}

and H = V

(L2)N

, V = V

(W 1,2)N

, Since the difference will be clear from the context, we will use the same notation for vectors in RN and scalars.

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Notation and assumptions

Assumptions

(H1) Ω ⊂ RN, N = 2, 3 or 4, is an bounded domain with ∂Ω of class C 2; (H2) α ⊂ R2 is a maximal monotone operator such that α(0) ∋ 0; its domain is given by D(α) = [0, +∞); (H3) f : R → R, a Lipschitz continuous function with f (0) = 0; (H4) g : Q → R, belonging to L2(0, T; L2); (H5) K : [0, 1) → R, K ≥ 0, K(0) = 0, K ∈ C 1([0, 1)), K ′(x) ≥ 0 and lim

x→1− K(x) = +∞;

(H6) h : R → R is a C 1(R)-increasing function such that h(z) = 0 when z ≤ 0 and h(z) = 1 when z ≥ 1 (and thus 0 ≤ h(z) ≤ 1, ∀z ∈ R); (H7) ρ > 0 and κ ≥ 0. In order to simplify the notation we fix ρ = 1; (H8) Let q > N and p > max

• 4,

2q q − 2

• ; we take the initial data such

that u0 ∈ V , θ0 ∈ W 1,p and ω0 ∈ W 2,q.

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Existence of solutions

Definition of solution - 1

A quadruple (u, θ, ω, η) is a solution when u ∈ C([0, T]; H) ∩ L∞(0, T; V ) with ut ∈ L2(0, T; H), θ ∈ C([0, T]; Lp) ∩ L∞(0, T; W 1,p ) with θt ∈ L2(0, T; L2), ω ∈ C([0, T]; C(Ω)) ∩ L∞(0, T; W 1,q) with ωt ∈ L2(0, T; W 1,2) ∩ L∞(0, T; L2), η ∈ L2(0, T; Ls), for some 1 < s ≤ 2,

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Existence of solutions

Definition of solution - 2

satisfying T

• Ω

ut · φ + ∇u · ∇φ + K(h(ω))(u + ut) · φ = T

• Ω

ζθ · φ, ∀ φ ∈ L2(0, T; V ) with compact support in Qml, θt + ωt − ∆θ − ∆pθ + u · ∇θ = g a.e. in Q, η + ωt − ∆ω − ∆qω + κu · ∇ω = θ + f (ω) a.e. in Q, and also η ∈ α(ωt) ⊂ L2(0, T; Ls), u + ut = 0 a.e. in Qs = {(x, t) ∈ Q : h(ω(x, t)) = 1}, u(., 0) = u0, θ(., 0) = θ0, ω(., 0) = ω0 a.e in Ω.

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Existence of solutions

Theorem of the existence of solutions

Theorem Suppose that 1 < s < 2 if N = 2, and 2N N + 2 ≤ s ≤ min

• N

N − 1, 2q q + 2

• if N = 3, 4.

Under hypotheses (H1)-(H8), there exists a solution in the sense of the previous definition. Remark The restriction on s is related to the non-monotone perturbation κu · ∇ω. In the case that κ = 0, we recover the case s = 2.

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Existence of solutions

Scheme of the proof - 1

1 We construct an approximated problem by

modifying the fluid equation in order to make it to hold in the whole domain Q by replacing the Carman-Kozeny term with a family of smooth and bounded functions Kτ, τ > 0 which approaches K, Kτ(x) = Kext(h(x) − τ), x ∈ R, where Kext is the extension of K defined on (−∞, 1) by letting Kext(x) = 0 when x < 0. changing the monotone graph α in the phase-field inclusion by suitable approximations ατ, and discretization in time.

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Existence of solutions

Scheme of the proof - 2

Definition of ατ

Let γτ be the standard Yosida’s regularization of γ = α−1. Notice that Fτ : R → R, where Fτ(x) = γτ(x) + τ|x|s−2x, 1 < s ≤ 2, is maximal monotone, differentiable a.e. and bijective. We set ατ = (γτ + τ|I|s−2I)−1, which is also bijective and differentiable a.e.. ατ satisfies:

ατ(0) = 0 and ατ(.) is maximal monotone, ατ(x)x ≥ 0, ατ is locally Lipschitz continuous.

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Existence of solutions

Scheme of the proof - 3

2 We prove the existence of solutions for the approximated problem by

a fixed point argument.

3 We obtain a priori estimates which do not depend on τ. 4 We pass to limit when the parameter τ tends to zero by compactness

and monotonicity arguments.

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Influence of the convection

Improved regularity

N s,p(Ω) the Nikolskii space, with s = 1 + σ, 0 < σ < 1 and p ≥ 2, which is a Banach space with respect to the following norm uN s,p =

• |u|p

N s,p + up Lp

1/p , where |u|p

N s,p = N

• i=1

sup

h=0

• Ω|h|
• ∂xiu(x + h) − ∂xiu(x)
• p

|h|σp being Ωδ = {x ∈ Ω : d(x, ∂Ω) > δ}. The spaces N s,p and W s,p are closely related in the following sense N s+ǫ,p ֒ → W s,p ֒ → N s,p, ∀ǫ > 0.

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Influence of the convection

Improved regularity

Theorem Under hypotheses (H1)-(H8), suppose additionally that κ = 0 and ∂Ω ∈ C 3. Then there exists a solution satisfying η ∈ α(ωt) ⊂ L2(0, T; L2) and ω ∈ L∞(0, T; W 2,2) ∩ L∞(0, T; N 1+2/q;q). Remark Notice that besides improving the space Ls to L2, we also obtain extra fractional regularity for ω due to the following continuous embedding N 1+2/q,q ֒ → W 1,q.

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Final remarks

Final remarks

To obtain the improved regularity, we apply certain tools on fractional regularity for p-Laplacian operators, which will require ∂Ω ∈ C 3 (Boldrini-de Miranda-P. (2013)). It is unknown whether this regularity can be obtained for the full model when κ = 0. In progress: improved regularity results for the p-Laplacian operators (de Miranda-P. (2017)). We do not know whether uniqueness of solutions hold for the present

• problem. The main difficulty in proving such a result appears when
• ne tries to compare the flow velocities associated to two possible

solutions with same initial conditions; since we have a free-boundary value problem, the corresponding fluid equations hold in different subsets of Q, and we cannot proceed as usual to get a equation for the difference of the velocities.

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Final remarks

Final remarks

Some interesting problems: Navier-Stokes equations instead of the Stokes equations Different Carman-Kozeny terms leading to different equations in the solid region Maximal monotone graph β with domain D(β) = [0, 1] instead of the q-Laplacian

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Final remarks

Some references

J.L. Boldrini, L.H. de Miranda, G. Planas, On Singular Navier-Stokes Equations and Irreversible Phase Transitions, Communications on Pure and Applied Analysis 11(5), 2055-2078, 2012. J.L. Boldrini, L.H. de Miranda, G. Planas, Existence and fractional regularity

• f solutions for a doubly nonlinear differential inclusion, Journal of Evolution

Equations 13(3) 535-560, 2013. J.L. Boldrini, L.H. de Miranda, G. Planas, A mathematical analysis of fluid motion in irreversible phase-transitions, Zeitschrift fur angewandte Mathematik und Physik - ZAMP, 66(3), 785-817, 2015. L.H. de Miranda, G. Planas, Parabolic p-Laplacian revisited: global regularity and fractional smoothness, submitted

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Final remarks

Thank you for your attention

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