AltWitterstein-limit-equations_IFB

DOI 10.4171/IFB/271

Distributional equation in the limit of phase transition for fluids

HANSWILHELMALT

Institut f¨ur Angewandte Mathematik, Universit¨at Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

E-mail: h.w.alt@t-online.de

GABRIELEWITTERSTEIN

Zentrum f¨ur Mathematik, Technische Universit¨at M¨unchen, Boltzmannstr. 3, D-85747 Garching bei M¨unchen, Germany

E-mail: gw@ma.tum.de

[Received 5 August 2010 and in revised form 13 October 2011]

We study the convergence of a diffusive interface model to a sharp interface model. The model consists of the conservation of mass and momentum, where the mass undergoes a phase transition.

The equations were considered in [W3] and in the diffuse case consist of the compressible Navier–

Stokes system coupled with an Allen–Cahn equation. In the sharp interface limit a jump in the mass density as well as in the velocity occurs. The convergence of mass and momentum is considered in the distributional sense. The convergence of the free energy to a limit is shown in a separate paper.

The procedure in this paper works also in other general situations.

2010 Mathematics Subject Classification:Primary 82B26; Secondary 35Q30.

Keywords:Compressible Navier–Stokes equations; phase transition; sharp interface model.

1. Introduction

We show by an example how phase field models converge to problems with sharp interface.

Although we are interested in flow problems connected with phase field equations, we think that the approach introduced here is quite general, and applies also to other situations. Flow problems as considered here are treated in [AR], [AF], [DG], [FG], [LW], [LT] and existence theorems were achieved in [Ab], [AR], [LS], [R]. This shows that the underlying system of differential equations is of main interest.

The phase field model (see Section3) is governed by the compressible Navier–Stokes system and the Allen–Cahn equation with a parameterδ > 0. It models the interactive flow of a mixture of two different materials. Therefore we have two mass conservations for the two masses, and one momentum conservation for the sum of the two masses. The sum of the two mass conservation laws results in conservation of the total mass, that is, the sum of the two masses. Together with the momentum equation this is the compressible Navier–Stokes system. As remaining law we do not pick the difference of the two mass conservations, but instead we use the second mass law. This turns out to be the Allen–Cahn equation, which is the second equation of system (3.1) below. The first and third equations of (3.1) are the above mentioned compressible Navier–Stokes system.

The purpose of this paper is to take the Navier–Stokes/Allen–Cahn system and to study its limit behaviour asδ → 0. We show that in the distributional sense the equations converge. For explanation of these concepts we refer to the appendix.

c

European Mathematical Society 2011

The sharp interface problem concerns two fluids occupyingΩ_t¹andΩ_t²separated locally by a free boundaryΓ_t(see Section2). The two components interact onΓ by mass transfer with a reaction rateτττ_δ. In fact, such an interface might be a thin layer, but looking at the phase transition from a certain distance, the transition zone will appear as a surfaceΓ.

Therefore, mathematically we consider the limit asδ→0 (see the proofs in Sections5and6).

In the caseδ >0 the independent variables are the two massesρ_δ¹andρ_δ², and the velocityv. With ρ=ρ_δ¹+ρ_δ²we have the following conservation laws for the two masses and the momentum:

∂tρ¹_δ+div(ρ_δ¹v)=τττδ,

∂_tρ²_δ+div(ρ_δ²v)= −τττ_δ,

∂_t(ρv)+div(ρv⊗v+Π_δ)=f_δ.







(1.1)

The quantitiesτττ_δ andΠ_δ and how they depend onδ are explained in Section3, together with the mass equations in an equivalent form forρ andφ:=(1/ρ)ρ_δ², which we treat as order parameter, since the free energy depends on∇φ.

It is shown in Sections 5 and 6 that under suitable assumptions the quantities under the derivatives converge pointwise to quantities inΩ^m,m= 1,2, andΓ. These assumptions contain the special form ofτττδ,

τττ_δ=η_δ(ρ, φ)δfδ

δφ, η_δ(ρ, φ)=η0(ρ, φ)

δ ,

f_δ= 1

δρW (φ)+δh(ρ)|∇φ|²

2 +U (ρ, φ),

(1.2)

where theδ-scaling is essential.

The basis for this convergence is the distributional formulation, which requires the measures µµµ_Ωm andµµµ_Γ defined in (2.3). We arrive at the following set of equations:

∂_t(ρ¹µµµ_Ω1)+div(ρ¹v¹µµµ_Ω1)=τττµµµ_Γ,

∂_t(ρ²µµµ_Ω2)+div(ρ²v²µµµ_Ω2)= −τττµµµ_Γ,

∂t

X

m

ρ^mv^mµµµ_Ωm

+divX

m

(ρ^mv^m⊗v^m+Π^m)µµµ_Ωm+Π^sµµµ_Γ

=X

m

f^mµµµ_Ωm,











(1.3)

where

Π^s = −γγγ (I−ν⊗ν) withν:=ν_Ω1 = −ν_Ω2, and where the surface tensionγγγ is given by

γγγ :=

Z ∞

−∞

h(R⁰)−a₂(R⁰, Φ⁰) ea(R⁰, Φ⁰)

e_h(R⁰) 2

|∂rΦ⁰|²dr

= Z ∞

−∞

h(R⁰

M⁰)−a₂(R⁰

M⁰, Φ⁰

M⁰) ea(R⁰

M⁰, Φ⁰

M⁰) e_h(R⁰

M⁰) 2

|∂_rΦ⁰

M⁰|²dr

=γγγ (M⁰), M⁰=ρ¹λ¹=ρ²λ² onΓ

(see Section6), whereλ¹andλ²are defined in5.3. For the reaction rateτττ there is no additional formula exceptτ =M⁰, so that it is defined by the distributional equation

∂_tX

m

ρ^mµµµ_Ωm

+divX

m

ρ^mv^mµµµ_Ωm

=0

via the strong equation

τττ := −ρ¹(v¹−vΓ)•ν_Ω1 = +ρ²(v²−vΓ)•ν_Ω2 onΓ .

All other quantities are explained in5.2and6.2. We mention that the standard way to describe the weak formulation of such problems makes use of test volumina (see e.g. [K], [S], [JR]). But we think that the usage of test functions is a more elegant way to formulate the system.

Besides these distributional equations there are additional boundary conditions at the interfaceΓ. They are derived in Section7and can be written as

v¹_tan=v²_tan, ρ¹=g1(ρ¹λ¹), ρ²=g2(ρ²λ²),











(1.4)

withg1(M):=R⁰_M(−∞)andg2(M):=R⁰_M(+∞).

With these boundary conditions the description of the limit problem is complete, that is, the problem is completely determined by (1.3) and (1.4).

There are also equivalent forms of these additional conditions in (1.4); one version is given by v¹_tan=v²_tan,

(v¹−v²)•ν=ωωω, ωωω=ωωω(M⁰), G(ρ¹, ρ²)=0,











(1.5)

which one can find in Section7.

The corresponding strong version of the above distributional equations is

∂_tρ^m+div(ρ^mv^m)=0 inΩ^m, m=1,2,

ρ¹(v¹−vΓ)•ν_Ω1+ρ²(v²−vΓ)•ν_Ω2 =0 onΓ ,

∂t(ρ^mv^m)+div(ρ^mv^m⊗v^m+Π^m)=f^m inΩ^m, m=1,2, divΠ^s =X

m

(ρ^m(v^m−v_Γ)•ν_Ω^mv^m+Π^mν_Ω^m) onΓ , where the last identity is the Delhaye condition [Del], which can also be written as

−γγγ κ_Γ − ∇^Γγγγ =M⁰(v²−v¹)−(Π²−Π¹)ν, M⁰=ρ¹λ¹=ρ²λ² (1.6) taking the Laplace formula into account. This can also be written as

∇^Γγγγ =((Π²−Π¹)ν)_tan, −γγγ κ_Γ •ν+M⁰(λ²−λ¹)= −ν•(Π²−Π¹)ν.

Altogether, one describes the limit problem with conditions at the interface containing quantities, hereγγγ,g1, andg2, which are described by the inner coordinate to the problem.

Therefore the phase field model enables one to derive and describe constitutive equations for the limit problem.

2. Sharp interface problem

We consider the interface problem in the distributional sense, since then it is in its natural form.

This is because the space-time divergence, that is,∂t and div, is an operation that really acts on the test function. Here the distributional formulation is also appropriate, because the interface comes by taking a limit of phase field equations (see Section3).

The problem lives in a local domainΩ⊂R×Rⁿ(for physical reasons one has to setn63, but in this papernis arbitrary), which consists of two domainsΩ¹andΩ²and a smooth interfaceΓ, that is,

Ω =Ω¹∪Γ ∪Ω². (2.1)

We assume thatΓ is timelike, so thatΩt =Ω_t¹∪Γt ∪Ω_t²for eachtwith a smooth surface

Γ_t := {x;(t, x)∈Γ}. (2.2)

HereΓ_t is the time slice ofΓ, andΩ_t¹andΩ_t²are defined in the same way. We define measures µ

µµ_Ωm(E):=Lⁿ⁺¹(E∩Ω^m), µµµ_Γ(E):= Z

R

Hⁿ⁻¹({x∈Γ_t;(t, x)∈E})dL¹(t ). (2.3) Note that

µ

µµ_Γ = 1 p1+ |v_Γ|²

Hⁿ

x

^{Γ , (2.4)}

whereHⁿ

x

^{Γ is the}n-dimensional Hausdorff measure onΓ, andv_Γ is the normal surface velocity onΓ. We consider mass and momentum equations in the distributional sense, that is, the mass equations read

∂t(ρ^mµµµ_Ωm)+div(ρ^mv^mµµµ_Ωm)=τττ^mµµµ_Γ for m=1,2,

τττ¹+τττ²=0 (τττ :=τττ¹= −τττ²), (2.5) and the momentum equation is

∂t

X

m

ρ^mv^mµµµ_Ωm

+divX

m

(ρ^mv^m⊗v^m+Π^m)µµµ_Ωm+Π^sµµµ_Γ

=X

m

f^mµµµ_Ωm+f^sµµµ_Γ. (2.6) This means that there is no mass at the interface, but there is a surface tension tensorΠ^sonΓ. These equations are equivalent to versions in the strong sense (see [Al, Section 2]). By this we mean the mass conservation

∂tρ^m+div(ρ^mv^m)=0 inΩ^m,

−τττ =ρ¹(v¹−v_Γ)•ν=ρ²(v²−v_Γ)•ν onΓ (ν:=ν_Ω1 = −ν_Ω2), (2.7) whereνΩ^m is the outer normal ofΩ^mandvΓ is the normal velocity ofΓ (it is not a “velocity”, but rather a “normal velocity”), and the momentum equation

∂t

X

m

ρ^mv^m

+divX

m

(ρ^mv^m⊗v^m+Π^m)

=0 inΩ^m, div^ΓΠ^s =f^s+X

m

ρ^m(v^m−v_Γ)•ν_Ω^mv^m+X

m

Π^mν_Ω^m onΓ .

(2.8)

The last equation on Γ is the Delhaye interface condition (see [Del]). Another version of this condition in the situation of this paper is presented in (1.6).

Besides these interfacial conditions given by the distributional equations we have to assume additional conditions on the interface to ensure the existence of a unique local solution (see for example (1.4) and (1.5)). For a phase field model this additional information is included in the δ-equation. These equations have to be extracted in the limit procedure; this is done in Section7.

3. Phase field problem

In [W2] the flow of a mixture of two different materials has been considered, which is governed by the compressible Navier–Stokes system and the Allen–Cahn equation. It can be written as

∂tρ+div(ρv)=0, ρ(∂_tφ+v• ∇φ)= −τττ_δ,

∂_t(ρv)+div(ρv⊗v+Π_δ)=f_δ,







(3.1)

where the termf_δ is an external force and is there for an arbitrary observer (see e.g. [Al]). The reaction rateτττδfulfills (3.8) below, andΠδis given by (3.5). The unknown functionsρ(t, x) >0, φ (t, x)∈R, andv(t, x)∈Rⁿdenote the total mass density, the mass fraction, and the velocity. We write

ρ:=ρ_δ¹+ρ_δ², φ_δ¹:=ρ_δ¹/ρ, φ²_δ :=ρ_δ²/ρ, φ:=φ²_δ,

ρ_δ¹=(1−φ)ρ, ρ_δ²=φρ. (3.2)

This means that the mass densities (ρ_δ¹, ρ_δ²) are equivalent variables to (ρ, φ). The reaction rate τττ_δ is a function of (ρ, φ,∇ρ,∇φ,D²φ), and the total tension tensor Π_δ is a function of (ρ, φ, v,∇φ,Dv). The dependence on∇φcan be understood as a dependence on a certain linear combination of the gradients ofρ_δ¹andρ_δ², since

∇φ= 1

ρ(φ_δ¹∇ρ_δ²−φ_δ²∇ρ_δ¹).

The system can be written as the mass conservation for each mass and the momentum conservation for the total mass:

∂tρ¹_δ+div(ρ_δ¹v)=τττδ,

∂tρ²_δ+div(ρ_δ²v)= −τττδ,

∂t(ρv)+div(ρv⊗v+Πδ)=f_δ.







(3.3)

So far the system has rarely been treated mathematically. In this connection we refer to [HS], [W1].

Concerning the systems we state the following

3.1. THEOREM With definition (3.2) the system (3.1) is equivalent to (3.3).

Proof. This has been shown in the appendix of [W2]. The conservation of momentum is the same in both systems. The sum of the first two equations of (3.3) gives the conservation of the total mass.

The first equation of (3.3) reads∂t(ρφ)+div(ρφv)= −τττδand conservation of the total mass turns

it intoρ(∂tφ+v• ∇φ)= −τττδ.

2

The quantitiesτττδandΠδare defined in the general case by the internal free energy density

fδ≡fδ(ρ, φ,∇φ) (3.4)

and besides(Dv)^Sby the Lam´e coefficientsa₁(ρ, φ)anda₂(ρ, φ). The tensorΠ_δhas the form

Π_δ=P−S, (3.5)

wherePis determined by the internal free energyfδ, containing the pressure pf_δ,

P≡P(ρ, φ,∇φ):=p_fδI+ ∇φ⊗f_δ⁰_∇φ, (3.6) and the stress tensor

S≡S(ρ, φ, (Dv)^S). (3.7)

The mass transition rateτττ_δis given by

τττδ≡τττδ(ρ, φ,∇ρ,∇φ,D²φ):=ηδ(ρ, φ)δf_δ

δφ. (3.8)

The variation of a functiongdepending on(ρ, φ,∇φ)with respect toφ, and the pressure p_g, are defined by

δg

δφ :=g⁰_φ−div(g⁰_∇φ), p_g :=ρg⁰_ρ−g. (3.9) Hereg⁰_ρandg⁰_∇φdenote the derivatives ofgwith respect to the variableρand∇φ(no new notation is introduced for these variables). The total free energyf includes the dynamical part and is given by

f ≡f (ρ, φ, v,∇φ)=f_δ(ρ, φ,∇φ)+ρ

2|v|². (3.10)

For the total free energyf the following energy identity has been shown in [W2].

3.2. THEOREM The total free energyf defined in (3.10) together with the free energy flux ψ:=f v+Π_δ^Tv− ˙φf⁰_∇φ

satisfies

∂tf +divψ−v•fδ= −1 ρτττδ

δf_δ

δφ −Dv:S60. (3.11)

Here for the inequality the assumptions in3.3are required. For every functiongthe total derivative is defined by

g˙:=(∂_t+v• ∇)g =∂_tg+v• ∇g.

Proof. Let(ρ, φ, v)be a solution of system (3.1) (or (3.3)). For the dynamical part one computes

∂_t 1

2ρ|v|²

+div 1

2ρ|v|²v+Π_δ^Tv

=v•f_δ+Dv:Π_δ.

Then for a general total free energy fluxψwithψ=f v+ψ₀andf as in (3.10) one obtains

∂_tf+divψ=∂_t

f_δ+1 2ρ|v|²

+div

f_δv+1

2ρ|v|²v+ψ0

=∂_tf_δ+div(f_δv+ψ₀−Π_δ^Tv)+v•f_δ+Dv:Π_δ

= ˙fδ+div(ψ0−Π_δ^Tv)+v•f_δ+Dv:(fδI+Πδ).

Now, for the free energy one gets

f˙δ=f_δ⁰_ρρ˙+f_δ⁰_φφ˙+f_δ⁰_∇φ•(∇φ)˙

=f_δ⁰_ρρ˙+f_δ⁰_φφ˙+f_δ⁰_∇φ• ∇ ˙φ−Dv:(∇φ⊗f_δ⁰_∇φ)

=f_δ⁰_ρρ˙+(f_δ⁰_φ−div(f_δ⁰_∇φ))φ˙+div(φf˙ _δ⁰_∇φ)−Dv:(∇φ⊗f_δ⁰_∇φ).

Therefore, using definition (3.9), one finally obtains

∂_tf +divψ=div(ψ₀−Π_δ^Tv+ ˙φf_δ⁰_∇φ)+v•f_δ+f_δ⁰_ρρ˙+δfδ

δφ φ˙ +Dv:(f_δI− ∇φ⊗f_δ⁰_∇φ+Π_δ).

Insertingρ˙andφ˙from the mass equations, that is,ρ˙= −ρdivvandρφ˙ = −τττ_δ, one is led to

∂_tf +divψ=div(ψ₀−Π_δ^Tv+ ˙φf_δ⁰_∇φ)+v•f_δ− 1 ρτττ_δδfδ

δφ +Dv: (f_δ−ρf_δ⁰_ρ)I− ∇φ⊗f_δ⁰_∇φ+Π_δ

.

Now, if the free energy fluxψ0is chosen as in the assertion, and if the tensorΠδ is defined as in (3.5) and (3.6), one ends up with the identity (3.11). The inequality comes from3.3below.

2

Let us now introduce the special representations forSandfδ. The stress tensorSin (3.7) is linear in(Dv)^Sand given by the classical formula

S:=a₁(ρ, φ)divvI+a₂(ρ, φ)

(Dv)^S−1 ndivvI

(3.12) with Lam´e coefficientsa1 anda2depending on the mass quantities, that is, ρandφ. For the free energy densityf_δwe consider the following representation:

f_δ(ρ, φ,∇φ):= 1

δρW (φ)+δh(ρ)|∇φ|²

2 +U (ρ, φ), U⁰_φ(ρ,0)=0, U⁰_φ(ρ,1)=0,

Whas two local minima at 0 and 1.

(3.13)

There is no assumption on the values ofW (0)andW (1). The first variation with respect toφis δfδ

δφ = 1

δρW⁰_φ(φ)−δdiv(h(ρ)∇φ)+Ψ⁰_φ(ρ, φ). (3.14) The functionWdepending on(ρ, φ)stands for a “double-well potential” and has two local minima inφatφ=0 andφ=1. The data fulfil the following three assumptions.

3.3. LEMMA The entropy condition is satisfied if

η_δ>0 and a₁>0, a₂>0.

We assume the stronger conditionsη_δ > 0,a₂ > 0. Hence we consider Newtonian flows in this paper.

4. Asymptotic expansion

We consider the case that the phase change happens in a small region around a smooth timelike surface Γ, that is, we assume that Γ is at least a C²-interface. In a neighbourhood of Γ one introduces coordinates

(t, x)=(t, y+δrν(t, y)) with (t, y)∈Γ , r ∈R, (4.1) whereν=ν_Ω1 = −ν_Ω2 onΓ. One considers a neighbourhood ofΓ given by

Γδ:= {(t, y+sν(t, y)); (t, y)∈Γ ,|s|6εδ}, (4.2) whereεδis chosen so that the solution outside the setΓδconverges towards the outer solution in the open setsΩ¹andΩ², that is, the total domain is decomposed into the three sets

Ω=Ω_δ¹∪Γ_δ∪Ω_δ²=Ω¹∪Γ ∪Ω²,

whereΩ_δ^mconverges toΩ^m locally in Hausdorff distance, and alsoΓ_δ converges toΓ locally in Hausdorff distance (see Appendix, (A3)).

After the setΓ_δis stretched one has to consider in the(t, y, r)-coordinates with(t, y)∈Γ and r ∈ Rthe inner expansion, which in our case satisfies the equations (4.5)–(4.7) below. Therefore εδ→0 asδ→0 andrδ:=(1/δ)εδ→ ∞asδ→0. Also we define

Γ_t := {x;(t, x)∈Γ}. (4.3)

In a phase field model the functionφstands for an order parameter depending onδ >0, here for the equations (3.1). Depending on this function we define for smallε > 0 and allt an interfacial region

V_δ,ε(t ):= {x; ε < φ (t, x) <1−ε}, V_δ,ε:= {(t, x);x ∈V_δ,ε(t )}. (4.4) In the limitδ &0 for eachε >0 the setV_δ,εapproaches the interfaceΓ. This is becauseφ →0 pointwise inΩ¹andφ→1 pointwise inΩ², thereforeV_δ,ε⊂Γ_δfor small enoughδ, ifεis fixed.

Let x ∈ Γδ. We denote by Pt(x) the projection ofx ontoΓt. The function s(t, x) denotes the signed distance fromx toΓt, withs < 0 inΩ¹ands > 0 inΩ². SinceΓδ,t lies in a small neighbourhood aroundΓt, we assign to each point(t, x)∈Γδa unique pair(y(t, x), r(t, x))by

y(t, x):=P_t(x), r(t, x):= 1 δs(t, x), s(t, x):=

(−dist(x, Γ_t) for(t, x)∈Ω¹, +dist(x, Γ_t) for(t, x)∈Ω².

With these definitions we calculate some first and second derivatives of the transformation:

∇s(t, x)=ν(t, P_t(x)),

∂_t(P_t(x))=v_Γ(t, P_t(x))+O(δ),

∂_ts(t, x)= −v_Γ(t, P_t(x))•ν(t, P_t(x)), D²s(t, x)=D^Γν(t, P_t(x))+O(δ),

trace(D²s(t, x))=∆s(t, x)= −κΓ(t, Pt(x))•ν(t, Pt(x))+O(δ),

where the differential operators D^Γ and∂_t^Γ for a vector functionware defined by D^Γw(t, y):=

n−1

X

k=1

(∂_τkw(t, y))⊗τ_k, ∂_t^Γw(t, y):=(∂_t+v_Γ(t, y)• ∇)w(t, y),

whereτk,k =1, . . . , n−1, is an orthonormal system of the tangent space toΓt aty. The vector ν_Ω^m(t, y)is the unit outer normal vector toΓ_t atywith respect toΩ_t^m, andν =ν_Ω1. The function v_Γ(t, y)is the normal velocity vector ofΓ_t andκ_Γ(t, y)is a normal vector denotingn-times the mean curvature ofΓ_t aty.

Now we write the functions in the new coordinates(t, y, r)as

ρ(t, x)=R(t, y(t, x), r(t, x))=R(t, y(t, x),¹_δs(t, x)), φ (t, x)=Φ(t, y(t, x), r(t, x))=Φ(t, y(t, x),¹_δs(t, x)),

v(t, x)=V (t, y(t, x), r(t, x))=V (t, y(t, x),¹_δs(t, x)).

In [W2] the equations (3.1), that is, the mass conservation, the momentum conservation, and the Allen–Cahn equation, in these new inner variables are shown to be equivalent to

1

δ∂r(RΛ)=∂_t^ΓR+div^Γ(RV )+O(δ), (4.5)

η₀(R, Φ)

δ RW⁰(Φ)−∂_r(h(R)∂_rΦ)

=RΛ∂_rΦ−η₀(R, Φ) Ψ⁰_φ(R, Φ)+κ_Γ •ν h(R)∂_rΦ

+O(δ), (4.6) 1

δ

∂_r

e_h(R)|∂rΦ|² 2

ν−∂_r(a∂_rV •ν)ν−∂_r 1

2a₂∂_rV

=∂r(RΛV )−∂r(pΨ)ν− ∇^Γ

ph

|∂_rΦ|² 2

−∂r(h(R)∂rΦ∇^ΓΦ) +κ_Γ •ν h(R)|∂_rΦ|²ν+ ∇^Γ(a∂¯ _rV •ν)+∂_r(a¯div^Γ V )ν +div^Γ

1

2a₂ν⊗∂_rV

+∂_r 1

2a₂∇^ΓV

ν−κ_Γ •ν1

2a₂∂_rV +O(δ), (4.7) where

Λ:=(v_Γ −V )•ν, η_δ(ρ, φ)= 1

δη₀(ρ, φ), a:=a₁+n−2

2n a₂, a¯ :=a₁−1 na₂.

(4.8)

The termO(δ) in (4.5)–(4.7) indicates that there are additional terms in the equation, which are estimated byδ. In [W2] it is further shown that if one takes theinner expansioninδ,

R(t, y, r)=R⁰(t, y, r)+δR¹(t, y, r)+O(δ²), Φ(t, y, r)=Φ⁰(t, y, r)+δΦ¹(t, y, r)+O(δ²), V (t, y, r)=V⁰(t, y, r)+δV¹(t, y, r)+O(δ²), whereR⁰, R¹, Φ⁰, Φ¹, V⁰, V¹are bounded functions,















(4.9)

one derives from the equations in (4.5)–(4.7) the corresponding equations for (R⁰, Φ⁰, V⁰)and for (R¹, Φ¹, V¹), which are linear equations in (R¹, Φ¹, V¹) with coefficients depending on (R⁰, Φ⁰, V⁰)(see [W3, Section 8]). Higher order equations are not required for the purpose of this paper.

We assume anouter expansion, for(t, x)∈Ω^m,m=1,2, ρ(t, x)=ρ_m⁰(t, x)+δρ_m¹(t, x)+O(δ²),

φ (t, x)=φ_m⁰(t, x)+δφ_m¹(t, x)+O(δ²), withφ₁⁰=0, φ₂⁰=1, v(t, x)=v_m⁰(t, x)+δv_m¹(t, x)+O(δ²),

f_δ =f_m+O(δ),

whereρ_m⁰, ρ¹_m, φ_m¹, v⁰_m, v_m¹,f_mare bounded functions.



















(4.10)

Besides this, for the values of the expansion of (R, Φ, V ) there are boundary conditions at r = ±∞. These conditions come from the fact that in the regionδr = s ≈ ±εδ we have, for example for the values ofv, the identity

V (t, y, r)=v(t, y+δrν(t, y)). (4.11) This identity implies, with the inner expansion (4.9) and the outer expansion (4.10) atx = y + δrν(t, y), that

V⁰(t, y, r)+δV¹(t, y, r)+O(δ²)=v⁰₂(t, y+δrν(t, y))+δv₂¹(t, y+δrν(t, y))+O(δ²) forr >0, and analogously forr <0. Now setr=r_δ,

V⁰(t, y, rδ)+O(ε_δ)=v₂⁰(t, y+εδν(t, y))+O(δ), rδ → +∞, εδ:=δrδ→0 asδ→0, and obtain, asδ→0,

V⁰(t, y,+∞)=v₂⁰(t, y), V⁰(t, y,−∞)=v₁⁰(t, y)

(the second identity follows in the same way). Similarly taking the derivative with respect tor in (4.11) one gets

∂rV (t, y, r)=δ(ν(t, y)• ∇)v(t, y+δrν(t, y)), (4.12) from which one deduces, by the same procedure as above, that∂rV⁰(t, y,±∞)=0 and

∂_rV¹(t, y,−∞)=(ν(t, y)• ∇)v₁⁰(t, y), ∂_rV¹(t, y,+∞)=(ν(t, y)• ∇)v⁰₂(t, y).

In the following sections we writev^m:=v⁰_mform=1,2. The same holds forRandΦ.

5. Mass conservation

The mass conservation forρ_δ²=ρφin (3.3) is

∂_t(ρφ)+div(ρφv)= −τττ_δ, τττδ=ηδ(ρ, φ)δf_δ

δφ, ηδ(ρ, φ)=η₀(ρ, φ)

δ , η0(ρ, φ) >0.

In the distributional formulation this reads Z

Ω

(∂tζ (ρφ)+ ∇ζ •(ρφv)−ζτττδ)dLⁿ⁺¹=0 (5.1) forζ ∈C₀^∞(Ω;R). The formulation of the mass conservation forρ_δ¹=ρ(1−φ)looks similar (see the first equation of (3.3)).

We consider two classes of test functions in (5.1). The first choice gives as a result the ordinary differential equations one has to solve in the inner expansion. This result is then used in the second choice of the test functions. These test functions are chosen as functions of the global variables.

Therefore one gets the equations of the outer expansion, and in addition a distributional equation across the interface. We show the following results whenδ → 0, where the first result yields the 1/δ-term at the boundary.

5.1. THEOREM Assume (4.9) and (4.10). Then for(t, y)∈Γ we have in local coordinates R⁰W⁰(Φ⁰)−∂_r(h(R⁰)∂_rΦ⁰)=0 for allr∈R.

This theorem is the version of the usual theorem on the zeroth orderΦ⁰of the phase field. It is necessary to show the following result.

5.2. THEOREM Assume (4.9) and (4.10). Then asδ → 0 the solution converges pointwise in the sense of distributions (see Appendix, (A1)) as follows:

ρφLⁿ⁺¹→ρ²µµµ_Ω2, ρφvLⁿ⁺¹→ρ²v²µµµ_Ω2, τττδLⁿ⁺¹→τττµµµ_Γ, where

τττ :=

Z +∞

−∞

η₀(R⁰, Φ⁰) R¹W⁰(Φ⁰)+R⁰W⁰⁰(Φ⁰)Φ¹−∂_r(h⁰(R⁰)R¹∂_rΦ⁰)

−∂r(h(R⁰)∂rΦ¹)+κΓ •ν h(R⁰)∂rΦ⁰+Ψ⁰_φ(R⁰, Φ⁰) dr.

Therefore the limit equation is

∂t(ρ²µµµ_Ω2)+div(ρ²v²µµµ_Ω2)= −τττµµµ_Γ.

Similarly one obtains the limit for the massρ¹inΩ¹(see the end of this section).

The value ofτττ is uniquely determined, as shown in Section8, but we have not been able to derive a constitutive equation for it, except

τττ =M⁰ Z +∞

−∞

∂rΦ⁰dr=M⁰,

which comes from inserting the definition of(R¹, Φ¹). Therefore this seems to define an arbitrary quantityτττ given by the distributional equations only. This is in analogy to the arbitrary pressure value in the incompressible limit of the Navier–Stokes equations.

For the proofs we use local and global test functions.

Local test functionζ

With the choice of a local test functionζ =ξ with aC₀^∞-functionξ around the free boundary we derive the well known first equation of the inner expansion (see (5.1)). This yields the 1/δ-term in (5.1). Explicitly we choose

ζ (t, x)=ξ(t, y, r), x =y+δrν(t, y), (5.2) where(t, y)∈Γ,r∈R, andν=ν_Ω2. The support ofr7→ξ(t, y, r)is contained in a fixed interval [−r_ξ, r_ξ], so that [−r_ξ, r_ξ]⊂[−r_δ, r_δ] for smallδ >0. We compute the derivatives:

∂tζ =∂_t^Γξ−1

δvΓ •ν∂rξ+O(δ),

∇ζ = ∇^Γξ +1

δ∂_rξ ν+O(δ),

(5.3)

and we get, ifδis small, Z

Ω

∂tζ·(ρφ)+ ∇ζ •(ρφv)−ζτττδ dxdt

= Z

R

Z +εδ

−εδ

Z

Γt

∂_t^Γξ·ρφ+ ∇^Γξ•(ρφv)−ξτττδ+1

δ∂rξ ·ρφ (v−vΓ)•ν +O(δ)χ_suppξ

(1+O(s))dHⁿ⁻¹(y)

dsdt

= Z

R

Z +rδ

−r_δ

Z

Γ_t

δ(∂_t^Γξ·ρφ+ ∇^Γξ•(ρφv))−ξ η0(ρ, φ)δf_δ

δφ +∂rξ·ρφ (v−vΓ)•ν +O(δ²)χ_suppξ

(1+O(δ))dHⁿ⁻¹(y)

drdt

= Z

R

Z +rδ

−rδ

Z

Γt

ξ η₀(ρ, φ)

−1

δρW⁰(φ)+δdiv(h(ρ)∇φ)+Ψ⁰_φ(ρ, φ)

+O(1)χ_suppξ

(1+O(δ))dHⁿ⁻¹(y)

drdt

= 1 δ

Z

R

Z +r_δ

−rδ

Z

Γt

ξ η0(R⁰, Φ⁰)(−R⁰W⁰(Φ⁰)+∂_r(h(R⁰)∂_rΦ⁰)) +O(δ)χsuppξ

(1+O(δ))dHⁿ⁻¹(y)

drdt

= 1 δ

Z

R

Z +rδ

−r_δ

Z

Γ_t

ξ η0(R⁰, Φ⁰)(−R⁰W⁰(Φ⁰)+∂r(h(R⁰)∂rΦ⁰))dHⁿ⁻¹(y)drdt+O(1).

Then it follows that the 1/δ-term vanishes asδ&0. Sinceξ is arbitrary, usingη0>0 one gets the identity in Theorem5.1.

Global test functionζ

We now choose test functions as functions of (t, x). Since we claim that the terms converge in the sense of distributions, we have to choose independent test functionsα ∈ C₀^∞(Ω;R)andβ ∈

C₀^∞(Ω;Rⁿ). We obtain Z

Ω

(αρφ+β•(ρφv)−ζτττ_δ)dxdt

= Z

Ω_δ²

(αρφ+β•(ρφv))dxdt+ Z

Γδ

(αρφ+β•(ρφv)−ζτττ_δ)dxdt+O(1). (5.4) Here we have used the particular form ofτττ_δand thatφ≈1 onΩ_δ²,φ ≈0 onΩ_δ¹forδ &0. This implies thatτττ_δ =O(1)inΩ\Γ_δ. Sinceρandφare bounded and pointwise convergent with respect to the Lebesgue measure, we obtain further

Z

Ω_δ²

(αρφ+β•(ρφv))dxdt → Z

Ω²

(αρ²+β•(ρ²v²))dxdt, Z

Γδ

(αρφ+β•(ρφv))dxdt=O(ε_δ)→0, forδ&0. And theτττ_δ-term converges to

Z

Γδ

ζτττ_δdxdt= Z

Γδ

ζ1

δη₀(ρ, φ)δf_δ δφ dxdt

= Z

Γδ

ζ1

δη0(ρ, φ) 1

δρW⁰(φ)−δdiv(h(ρ)∇φ)+Ψ⁰_φ(ρ, φ)

dxdt

= Z

R

Z +rδ

−r_δ

Z

Γ_t

ζ (η₀(R⁰+δR¹, Φ⁰+δΦ¹)+O(δ²))

· 1

δ R⁰W⁰(Φ⁰)−∂_r(h(R⁰)∂_rΦ⁰)

+ R¹W⁰(Φ⁰)+R⁰W⁰⁰(Φ⁰)Φ¹−∂r(h⁰(R⁰)R¹∂rΦ⁰)

−∂r(h(R⁰)∂rΦ¹)+κΓ •ν h(R⁰)∂rΦ⁰+Ψ⁰_φ(R⁰, Φ⁰) +O(δ)

·(1+O(ε_δ))dHⁿ⁻¹(y)drdt.

Thus, due to identity (5.1), we see that the 1/δ-term vanishes and that the expression forδ & 0 converges to

Z

R

Z

Γt

ζ Z +∞

−∞

η0(R⁰, Φ⁰) R¹W⁰(Φ⁰)+R⁰W⁰⁰(Φ⁰)Φ¹−∂r(h⁰(R⁰)R¹∂rΦ⁰)

−∂_r(h(R⁰)∂_rΦ¹)+κ_Γ •ν h(R⁰)∂_rΦ⁰+Ψ⁰_φ

drdHⁿ⁻¹(y)dt, which is the result of Theorem5.2.

The two mass equations

The strong version of the equation in5.2is

∂_tρ²+div(ρ²v²)=0 inΩ², τττ =ρ²(v²−vΓ)•ν_Ω2 onΓ .

(5.5)

Forρ¹we obtain an analog with+τττ on the right hand side,

∂_t(ρ¹µµµ_Ω1)+div(ρ¹v¹µµµ_Ω1)=τττµµµ_Γ, (5.6) or equivalently

∂_tρ¹+div(ρ¹v¹)=0 inΩ¹, τττ+ρ¹(v¹−vΓ)•ν_Ω1 =0 onΓ .

(5.7) From (5.5) and (5.7) we conclude that

2

X

m=1

ρ^m(v^m−v_Γ)•ν_Ω^m =0 (5.8)

onΓ or

ρ¹(v¹−v_Γ)•ν=ρ²(v²−v_Γ)•ν (ν=ν_Ω1 = −ν_Ω2).

This identity belongs to the conservation of the total mass, which in the distributional sense is the sum of the conservation of the individual masses and reads

∂_tX

m

ρ^mµµµ_Ωm

+divX

m

ρ^mv^mµµµ_Ωm

=0. (5.9)

The equation for the total mass of the phase field problem, which in the weak sense is Z

Ω

(∂tζ ·ρ+ ∇ζ •(ρv))dLⁿ⁺¹=0, (5.10) has one consequence in local coordinates, which occurs in a differentδ-term than in the Allen–Cahn equation.

5.3. THEOREM Assume (4.9) and (4.10). Then for(t, y)∈Γ we have in local coordinates

∂r(R⁰Λ⁰)=0 for allr∈R.

The boundary conditions forΛ⁰:=(v_Γ −V⁰)•νare (without writing the arguments(t, y)) Λ⁰(−∞)=λ¹:=(v_Γ −v¹)•ν, Λ⁰(+∞)=λ²:=(v_Γ −v²)•ν.

For the proof we use local test functionsζ = ξ with aC₀^∞-functionξ around the free boundary.

One infers from (5.10) that 0=

Z

Ω

∂_t^Γξ −1

δv_Γ •ν∂_rξ

ρ+

∇^Γξ+1 δ∂_rξ ν

•(ρv)

dLⁿ⁺¹

= 1 δ

Z

R

Z +εδ

−ε_δ

Z

Γ_t

(−vΓ •ν∂rξ ·R⁰+∂rξ ν•(R⁰V⁰)+O(δ))(1+O(s))dHⁿ⁻¹(y)

dsdt

= Z

R

Z +rδ

−r_δ

Z

Γ_t

(∂_rξ·R⁰·(V⁰−v_Γ)•ν+O(δ))(1+O(δ))dHⁿ⁻¹(y)

drdt.

Lettingδ→0 one gets 0=

Z

R

Z +∞

−∞

Z

Γt

∂_rξ·R⁰·(V⁰−v_Γ)•νdHⁿ⁻¹(y)drdt and it follows that∂r(R⁰(V⁰−vΓ)•ν)=0.

6. Momentum conservation

The momentum conservation forvin system (3.3) is

∂t(ρv)+div(ρv⊗v+Πδ)=fδ,

wheref_δstands for an external term. In the distributional formulation this reads Z

Ω

∂_tζ •(ρv)+ ∇ζ :(ρv⊗v+Π_δ)+ζ•f_δ

dLⁿ⁺¹=0, (6.1)

where we consider vector-valued test functionsζ ∈ C₀^∞(Ω;Rⁿ). We show the following results when δ → 0. The first result yields again, as for the mass conservation, the 1/δ-term at the boundaryΓ.

6.1. THEOREM Assume (4.9) and (4.10). Then for(t, y)∈Γ we have in local coordinates eh(R⁰)|∂_rΦ⁰|²

2 =

ea(R⁰, Φ⁰)∂rV⁰•ν for allr∈R, where

eh:=ρh⁰_ρ+h and ea:=a1+n−1

n a2=a+1

2a2. (6.2)

This theorem is necessary to show the following result.

6.2. THEOREM Assume (4.9) and (4.10). Then asδ → 0 the solution converges pointwise in the sense of distributions as follows:

ρvLⁿ⁺¹→X

m

ρ^mv^mµµµ_Ωm, (ρv⊗v+Πδ)Lⁿ⁺¹→X

m

(ρ^mv^m⊗v^m+Π^m)µµµ_Ωm+Π^sµµµ_Γ, f_δLⁿ⁺¹→X

m

f^mµµµ_Ωm. Here, withc¹=0 inΩ¹andc²=1 inΩ²,

Π^m:=p_Ψ(ρ^m, c^m)I−S(ρ^m, c^m, (∇v^m)^S) inΩ^m, Π^s := −γγγ (I−ν⊗ν) onΓ ,

γγγ := Z ∞

−∞

h(R⁰)|∂rΦ⁰|²−a2(R⁰, Φ⁰)∂rV⁰•ν dr

= Z ∞

−∞

h(R⁰)−a₂(R⁰, Φ⁰) ea(R⁰, Φ⁰)

e_h(R⁰) 2

|∂rΦ⁰|²dr

= Z ∞

−∞

h(R⁰

M⁰)−a2(R⁰

M⁰, Φ⁰

M⁰) ea(R⁰

M⁰, Φ⁰

M⁰) eh(R⁰

M⁰) 2

|∂_rΦ⁰

M⁰|²dr

=γγγ (M⁰), M⁰=ρ¹λ¹=ρ²λ² onΓ .

(6.3)

Therefore the limit equation is

∂_tX

m

ρ^mv^mµµµ_Ωm

+divX

m

(ρ^mv^m⊗v^m+Π^m)µµµ_Ωm+Π^sµµµ_Γ

=X

m

f^mµµµ_Ωm.

The equation (2.6) is satisfied withf^s =0. The strong formulation of this weak equation reads

∂_t(ρ^mv^m)+div(ρ^mv^m⊗v^m+Π^m)=f^m inΩ^m, m=1,2, div^ΓΠ^s =X

m

ρ^m(v^m−v_Γ)•ν_Ω^mv^m+Π^mν_Ω^m

onΓ ,

and in addition the conditionΠ^sν =0 onΓ is satisfied by the above matrixΠ^s. The functionγγγ is the surface tension, which is here given in terms of the local coordinates. We mention thatγγγ has no sign. The second representation ofγγγ follows from6.1. The third representation is in advance of (7.4) in Section7. It implies thatγγγ is a function ofM⁰alone.

For the proof of the theorems we consider again two classes of test functions, where the different meaning of these test functions is the same as for the mass.

Local test functionζ

Consider test functionsζ =ξ in the special case of local coordinates as in (4.1), that is,ζ (t, x)= ξ(t, y, r)with compactly supported function(t, y, r) 7→ξ(t, y, r), that is,ζ has compact support in a small neighbourhood ofΓ shrinking towardsΓ whenδ→0. We conclude, using (5.3), that

0= Z

R

Z

Ωt

(∂tζ•(ρv)+Dζ :(ρv⊗v+Πδ)+ζ•fδ)dxdt

= Z

R

Z +εδ

−ε_δ

Z

Γ_t

(∂tζ•(ρv)+Dζ :(ρv⊗v+Πδ)+ζ•f_δ)(1+O(δ))dHⁿ⁻¹(y)dsdt

= Z

R

δ Z +rδ

−r_δ

Z

Γ_t

∂_t^Γξ−1

δv_Γ •ν∂_rξ

•(ρv)+

D^Γξ+1 δ∂_rξ⊗ν

:(ρv⊗v+Π_δ)+ξ•f_δ

·(1+O(δ))dHⁿ⁻¹(y)drdt.

In a small neighbourhood ofΓ we compute Π_δ=P−S=p_ΨI+δ

2p_h|∇φ|²I+δh∇φ⊗∇φ−

a₁−a₂ n

divvI−a₂(∇v)^S

= 1 δ

1

2p_h|∂_rΦ⁰|²I+h|∂_rΦ⁰|²ν⊗ν−

a₁−a2

n

ν•∂_rV⁰I−1

2a₂(ν⊗∂_rV⁰+∂_rV⁰⊗ν)

+O(1),

where the coefficientsh,a1, anda2have to be taken at the values(R⁰, Φ⁰). Since the main term is of order 1/δwe get for the above integral the value

Z

R

Z +r_δ

−rδ

Z

Γt

(∂_rξ⊗ν):Π_δdHⁿ⁻¹(y)drdt+O(1)

= Z

R

Z +rδ

−rδ

Z

Γt

∂rξ •(Πδν)dHⁿ⁻¹(y)drdt+O(1).

Now

Π_δν= 1 δ

1

2e_h|∂_rΦ⁰|²ν−

a1+(n−2)a₂ 2n

∂_rV⁰•ν ν−1 2a2∂_rV⁰

+O(1),

eh=ρh⁰_ρ+h=ph+2h (phdefined in (3.9)),