Phase separation in a gravity field

We prove here well-posedness and convergence to equilibria for the solution trajectories associated to a model for solidification of a liquid content of a rigid container in a gravity field. We observe that the gravity effects, which can be neglected without considerable changes of the process on finite time intervals, have a substantial influence on the long time behavior of the evolution system. Without gravity, we find a temperature interval, in which all phase distributions with a prescribed total liquid contents are admissible equilibria, while, under the influence of gravity, the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.


Introduction
In this paper we continue the discussion started in [7], extending the model to the case in which also gravity effects are considered during the phase transition process. We derive a model for solid-liquid phase transition of a medium inside a rigid container. The main goal is to give a qualitative and quantitative description of the interaction between volume, pressure, phase, and temperature changes in the situation that the specific volume of the solid phase exceeds the specific volume of the liquid phase. We observe, in particular, that the solidification may take place at a temperature slightly above the critical temperature θ c . The overheating is due to the fact that the pressure decreases from the bottom to the top. A quantitative description of this phenomenon is given in here by means of the so-called Clausius-Clapeyron formula (cf. (2.40)).
There is an abundant classical literature on the study of phase transition processes, see e.g. the monographs [1], [3], [9] and the references therein. In particular, in [4], the authors proposed to interpret a phase transition process in terms of a balance equation for macroscopic motions, and to include the possibility of voids, while the microscopic approach has been pursued in [5] in the case of two different densities ̺ 1 and ̺ 2 for the two substances undergoing phase transitions. Let us, however, refer to the Introduction of [7] for a more detail description of the previous works in the literature on this topic.
The forces occurring as a result of solid-liquid phase transitions in small containers are very strong, much stronger than gravity forces. In a bottle of water of less than one meter height for example, they differ by at least four orders of magnitude. From the quantitative viewpoint, the gravity effects can thus be neglected without considerable changes of the process on finite time intervals. They have, however, a substantial influence on the long time behavior of the evolution system. Without gravity, we observe a temperature interval, in which all phase distributions with a prescribed total liquid contents are admissible equilibria (cf. also [7]). If even a weak gravity field is assumed to be present, then the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.
Here we proceed as follows: in Section 2, we derive a model describing the evolution of the process which is driven by an energy balance, a quasistatic momentum balance, and a phase dynamics equation. Still in Section 2, we verify the thermodynamic consistency of the model, and we study the equilibria.
The well-posedness of the corresponding system of evolution equations is proved in Section 3, while and the study of the long-time behavior of solutions and convergence to equilibria is proved in the last Section 4.

The model
As reference state, we consider a liquid substance contained in a bounded connected bottle Ω ⊂ R 3 with boundary of class C 1,1 . The state variables are the absolute temperature θ > 0, the displacement u ∈ R 3 , and the phase variable χ ∈ [0, 1]. The value χ = 0 means solid, χ = 1 means liquid, χ ∈ (0, 1) is a mixture of the two.
We make the following modeling hypotheses.
(A1) The displacements are small. Therefore, we state the problem in Lagrangian coordinates, in which the mass conservation is equivalent to the condition of a constant mass density ̺ 0 > 0.
(A2) The substance is compressible, and the speed of sound does not depend on the phase.
(A3) The evolution is slow, and we neglect shear viscosity and inertia effects.
In agreement with (A1), we define the strain ε as an element of the space T 3×3 sym of symmetric tensors by the formula ε = ∇ s u := 1 2 (∇u + (∇u) T ). (2.1) Let δ ∈ T 3×3 sym denote the Kronecker tensor. By (A4), the elasticity matrix A has the form Aε = λ(ε : δ) δ , (2.2) where " : " is the canonical scalar product in T 3×3 sym , and λ > 0 is the Lamé constant (or bulk elasticity modulus), which we assume to be independent of χ by virtue of (A2). Note that λ is related to the speed of sound v 0 by the formula v 0 = λ/̺ 0 . We want to model the situation where the specific volume V solid of the solid phase is larger than the specific volume V liquid of the liquid phase. Considering the liquid phase as the reference state, we introduce the dimensionless phase expansion coefficient α = (V solid − V liquid )/V liquid > 0, and we define the phase expansion strainε bỹ We fix positive constants c 0 (specific heat), L 0 (latent heat), θ c (freezing point at standard atmospheric pressure), β (thermal expansion coefficient), and consider the specific free energy f in the form where I is the indicator function of the interval [0, 1]. The stress tensor σ is decomposed into the sum σ v + σ e of the viscous component σ v and elastic component σ e .
The state functions σ v ,σ e , s (specific entropy), and e (specific internal energy) are given by the formulas where ν > 0 is the volume viscosity coefficient. The scalar quantity is the pressure, and the stress has the form σ = −p δ. The process is governed by the balance equations − div σ = f vol (mechanical equilibrium) (2.10) ̺ 0 e t + div q = σ : ε t (energy balance) (2.11) where γ 0 is the phase relaxation coefficient, ∂ χ is the partial subdifferential with respect to χ, f vol is a given volume force density (the gravity force) with standard gravity g and vector δ 3 = (0, 0, 1), and q is the heat flux vector that we assume in the form q = −κ∇θ (2.14) with a constant heat conductivity κ > 0. The equilibrium equation (2.10) can be rewritten in the form ∇p = −̺ 0 g , hence with a function P of time only, which is to be determined. On ∂Ω, we assume boundary conditions in the form with a given positive measurable function h (heat transfer coefficient), and a constant θ Γ > 0 (external temperature). Identity (2.16) means that the boundary is rigid. Other possibilities (elastic or elastoplastic boundary response) have been considered in another context ( [7,8]).
By Gauss' Theorem, we have Ω div u(x, t) dx = 0 (2.18) We have ε : δ = div u. Using (2.9), we write the mechanical equilibrium equation (2.15) as Integrating over Ω and using (2.18) we obtain We see that in liquid (χ = 1) and at temperature θ = θ c , the pressure p(x, t) vanishes on the "midsurface" of Ω given by the equation Hence, p(x, t) can be interpreted as the difference between the absolute pressure and the standard pressure. This difference is higher below and lower above the midsurface.
Eq. (2.20) enables us to eliminate P (t) and rewrite (2.19) in the form As a consequence of (2.4), the energy balance and the phase relaxation equation in (2.11)-(2.12) have the form where ∂ denotes the subdifferential. For simplicity, we now set The system now completely decouples. For the unknown absolute temperature θ , local relative volume increment U = div u, and liquid fraction χ, we have the evolution system (note that mathematically, ∂I(χ) is the same as L∂I(χ)) with boundary condition (2.17), (2.14), that is, We then find u as a solution to the equation div u = U in Ω, u = 0 on ∂Ω. It is indeed not unique, and due to our hypotheses (A3), (A4), we lose any control on possible volume preserving turbulences. This, however, has no influence on the system (2.25)-(2.27), which is the subject of our interest here.
Let us describe the set of all possible stationary states. It follows from (2.25) and (2.28) that the only temperature equilibrium is θ = θ Γ . The equilibrium values χ ∞ and U ∞ satisfy the system almost everywhere in Ω, that is, We claim that unlike in the case without gravity, (2.31) determines the equilibria uniquely. The set Ω is connected. We can therefore define (a, b) ⊂ R as the maximal interval such that Ω ∩ ( We introduce two dimensionless constants For water, we have for instance d ≈ 0.055, G 0 ≈ 2 · 10 4 /ℓ. Eq. (2.31) then reads The quantity is independent of x, so that the left hand side of (2.33) is positive for x 3 < m + G 0 ℓZ and negative for x 3 > m + G 0 ℓZ . By definition of the subdifferential, we necessarily have Let Ω(r) denote the set {x ∈ Ω : x 3 > r} for r ∈ R. Eq. (2.35) states that the set Ω(m + G 0 ℓZ) corresponds to the solid domain. We have |Ω(r)| = 0 for r ≥ b, |Ω(r)| = |Ω| for r ≤ a, and We easily identify Z as the only solution to the equation since F is nonincreasing. We see that one of the following three cases necessarily occurs: in Ω and we have pure solid with temperatures in Ω and we have pure liquid with temperatures in Ω \ Ω(m + G 0 ℓZ), and We observe that solidification may take place at temperatures slightly above θ c . For water in a container of ℓ = 50 cm height, the relative size of the "overheated ice temperature domain" is smaller than d/G 0 ≈ 1.4·10 −6 , hence it is far beyond the standard measurement accuracy. The overheating is due to the fact the pressure decreases from the bottom to the top, as pointed out after formula (2.20). A quantitative characterization of this phenomenon is given by the so-called Clausius-Clapeyron equation, which relates the freezing temperature with the pressure. It can be derived here as follows. The equilibrium relative pressure p ∞ depends only on x 3 , and is given, by virtue of (2.15) and (2.20), by the formula The phase interface at temperature θ Γ is located at level x 3 if the right hand side of (2.39) vanishes. Setting L β = L 0 − αβθ c /̺ 0 , we thus obtain the Clausius-Clapeyron condition for phase transition in the form of [10, Equation (288)], namely In terms of the new variables θ, U, χ, the energy e and entropy s can be written as and the energy and entropy balance equations now read The entropy balance (2.44) says that the entropy production on the right hand side is nonnegative in agreement with the second principle of thermodynamics. The system is not closed, and the energy supply through the boundary is given by the right hand side of (2.43).

Existence and uniqueness of solutions
We construct the solution of (2.26)-(2.27) by the Banach contraction argument. The method of proof is independent of the actual values of the material constants, and we choose for simplicity System (2.25)-(2.27) with boundary condition (2.17) then reads In this section we prove the following existence and uniqueness result.
Remark 3.2 For existence and uniqueness alone, we might allow the external temperature θ Γ to depend on x and t, and assume only that it belongs to the space W 1,2 loc (0, ∞; L 2 (∂Ω)) ∩ L ∞ loc (∂Ω × (0, ∞)). For the global bounds, the assumption that θ Γ be constant plays a substantial role.
The proof of Theorem 3.1 will be carried out in the following subsections. Notice first that the term U 2 t − θU t − (U + χ + 1)χ t on the right hand side of (3.2) can be rewritten alternatively, using (3.4) and (3.3), as We now fix some constant R > 0 and construct the solution for the truncated system first in a bounded domain Ω T := Ω × (0, T ) for any given T > 0, where Q R is the cutoff function Q R (z) = min{z + , R}. We then derive upper and lower bounds for θ independent of R and T , so that the local solution of (3.6)-(3.8) is also a global solution of (3.2)-(3.4) if R is sufficiently large.

A gradient flow
with a constant C ψ ≥ 0 to ensure that ψ(v) ≥ 0. Every solution U of (2.26) necessarily satisfies the condition Ω U dx = 0, hence (2.27) can be written in the form in L 2 (Ω) × L 2 (Ω). We have used the obvious identity We state here the following Lemma, whose proof can be found in [7,Lemma 4.3]. We apply the above result to the case H = L 2 (Ω)×L 2 (Ω), and v , f , ψ as above and we see that Eqs. (3.7)-(3.8) with θ replaced byθ can be equivalently written as a gradient flow (3.9)-(3.13). For its solutions, we prove the following result.
In what follows, we denote by C 1 , C 2 , . . . any constant independent of x, t and R.
Proof. We rewrite (3.9)-(3.13) as two scalar gradient flows U t + ∂ψ 1 (U) = a , (3.16) The bounds (3.14) are obvious. To prove (3.15), we consider two different inputs. As above, we denote the differences {} 1 − {} 2 by {} d for all symbols {}. By [6, Theorem 1.12], we have for all t > 0 and a.e. x ∈ Ω that We multiply the difference of (3.16) by U d , the difference of (3.17) by χ d , and sum them up to obtain that (3.19) We first integrate (3.19) over Ω. Using the symbol | · | 2 for the norm in L 2 (Ω), we get for a.e. t > 0 that a.e., and integrating over t, we find that Using again (3.19), we find for a.e. (x, t) ∈ Ω ∞ the inequality 1 2 Hence, we get (3.23) which in turn implies that a.e. (3.24) This enables us to estimate the right hand side of (3.18) and obtain the bound for a.e. x ∈ Ω and all t ≥ 0. This completes the proof.

Proof of Theorem 3.1
The unique solution (θ, U, χ) to (3.6)-(3.8), (2.46)-(2.48) exists globally in the whole domain Ω ∞ . We now derive uniform bounds independent of t and R. Take first for instance any R > 2θ * . We know that the solution component θ of (3.6)-(3.8) remains smaller than R in a nondegenerate interval (0, T ) with T > θ * /(C 6 (1 + R) 2 ). Let (0, T 0 ) be the maximal interval in which θ is bounded by R. Then, in (0, T 0 ), the solution given by Lemma 3.5 is also a solution of the original problem (3.2)-(3.4). Moreover, due to estimate (2.50), we know that θ admits a bound in L ∞ (0, T 0 ; L 1 (Ω)) independent of R. In order to prove that T 0 = +∞ if R is sufficiently large, we need the following variant of the Moser iteration lemma, whose proof can be found in [7,Prop. 4.6].
(v) System (3.38)-(3.40) admits a solution v ∈ W 1,2 loc (0, ∞; (W 1,2 ) ′ (Ω)) ∩ L 2 loc (0, ∞; W 1,2 (Ω)) ∩ L ∞ loc (Ω × (0, ∞)) satisfying the estimate We now finish the proof of Theorem 3.1 by showing that T 0 introduced at the beginning of this subsection is +∞ if R is sufficiently large. Using (3.3), we obtain that |U(x, t)| ≤ C 8 1 + By (2.50), the function U is in L ∞ (0, ∞; L 2 (Ω)) and the bound does not depend on R. Eq. (3.2), with θ added to both the left and the right hand side, thus satisfies the hypotheses of Proposition 3.6 for N = 3 and q = 2. This enables us to conclude that θ(x, t) is uniformly bounded from above by a constant, independently of R, so that θ never reaches the value R if R is sufficiently large, which we wanted to prove. By (3.43)-(3.45), also U , U t , and χ t are uniformly bounded by a constant.