Regensburg 2011

Workshop Uni Regensburg

Phase Field Models in Fluid Mechanics

Distributional equations and diffuse and sharp interfaces

Hans Wilhelm Alt

Workshop Uni Regensburg

Phase Field Models in Fluid Mechanics

Distributional equations and diffuse and sharp interfaces

Hans Wilhelm Alt

Plan of the talk:

• Distributions in spacetime

• Examples: Surfaces, lines, points

• Objectivity (Frame indifference)

• Phase field model: Asmptotic limit towards distributions

Distributions

R ⁿ

v

Γ

Γ

Surfaces: (t, x) ∈ Γ ⇐⇒ x ∈ Γ_t

Γ ⊂ R× Rⁿ (m + 1)-dimensional, T_(t,x)Γ 6= {0} ×Rⁿ Γ_t := {x ∈ Rⁿ ; (t, x) ∈ Γ} m-dimensional, 0 ≤ m ≤ n

v_Γ “surface velocity” , T_(t,x)Γ = span{(1, v_Γ(t, x))} ∪ {0} × T_xΓ_t Surface measures:

hζ , µµµ_Γi :=

Z

R

Z

Γt

ζ(t, x) dH_m(x) dL1(t)

= Z

Γ

ζ(t, x)

p1 + |v_Γ|² dH_m+1(t, x) Extreme cases m = n and m = 0:

Ω (Ω_t n-dimensional = open set) v_Ω = 0 hζ , µµµ_Ωi =

Z

R

Z

Ω_t

ζ(t, x) dH_n(x) dL1(t) = Z

Ω

ζ(t, x) dL_n+1(t, x)

Γ (Γ_t = {y_t} 0-dimensional = point) vΓ = ˙yt

hζ , µµµ_Γi = Z

R

Z

{yt}

ζ(t, x) dH0(x) dL1(t) = Z

R

ζ(t, y_t) dL₁(t)

R ⁿ

v

Γ

Γ

Theorem (Distributional and strong version)

Let t 7→ Γ_t be m-dimensional, 0 ≤ m ≤ n. The equation

∂_t(eµµµ_Γ) + div (qµµµ_Γ) = fµµµ_Γ is equivalent to

q − evΓ tangential on Γ (q − evΓ)(t, x) ∈ T_xΓ_t

∂_t^Γe − evΓ•κ_Γ + div^Γ(q − evΓ) = f on Γ ∂_t^Γe + div^Γq = f

Here

Γ_t := {x ∈ Rⁿ ; (t, x) ∈ Γ}

v_Γ(t, x) ∈T_xΓ_t^⊥ “surface velocity” of Γ_t κΓ(t, x) m-times mean curvature of Γ_t

∂_t^Γ := ∂_t + vΓ•∇

div^Γ := X

i=1,...,m

τ_k•∂_τk

hζ , −∂_t(eµµµ_Γ) − div (qµµµ_Γ) + fµµµ_Γi

= h∂_tζ , eµµµ_Γi + h ∇ζ , qµµµ_Γi + hζ , fµµµ_Γi = Z

∂_tζ · e + ∇ζ•q + ζ · f dµµµ_Γ

Examples of distributions in D

(U ), U ⊂ R × R

:

Body Ω with mass exchange at the boundary Γ = ∂Ω

∂_t(%µµµ_Ω) + div (%vµµµ_Ω) = τµµµ_Γ

(τ mass production at boundary)

⇐⇒

∂_t% + div(%v) = 0 in Ω 0 = τ + %(v − vΓ)•ν_Ω on Γ Mass balance on a membrane Γ

with mass density %^s > 0 and “particle velocity” v^s

∂t(%^sµµµ_Γ) + div (%^sv^sµµµ_Γ) = 0

⇐⇒ on Γ:

v^s − v_Γ tangential

∂_t^Γ%^s − %^s κ_Γ•v_Γ + div^Γ(%^s(v^s − v_Γ)) = 0 κΓ (n−1)-times mean curvature vector of Γ

∂_t^Γ := ∂_t + vΓ•∇ time derivative of Γ

Strong differential equality ⇐⇒ ∂_t^Γ%^s + div^Γ(%^sv^s) = 0

PDE as distribution Body Ω = U

∂_t(%µµµ_U) + div (%vµµµ_U) = 0

⇐⇒

∂t% + div(%v) = 0 in all of U

Mass in a system of M curves Γ^k_t

which meet at point P_t, everything is moving

∂_t(

M

X

k=1

%^kµµµ_Γ^k) + div (

M

X

k=1

%^kv^kµµµ_Γ^k) = a µµµ_P

⇐⇒

v^k − vΓ^k tangential

∂_t^Γk%^k + div^Γk(%^kv^k) = 0

on Γ^k, k = 1,2,3

M

X

k=1

%^k(1, v^k)•n_Γ^k q

1 + |v_Γ^k|²

= a

p1 + |v_P|² on P

n_Γ^k tangent at Γ^k, normal to P

Objectivity (Frame indifference)

• The value of physical quantities depend on the observer

• The type of a physical quantity is given by a transformation rule

• The description of physical processes has to be independent of the observer Observer transformations (classical group = Newton’s physics)

t x

= y = Y (y^∗) = Y (t^∗, x^∗) =

T(t^∗, x^∗) X(t^∗, x^∗)

=

t^∗ + a

Q(t^∗)x^∗ + b(t^∗)

where a ∈ R, b : R → Rⁿ, Q : R → R^n×n orthogonal transformation, detQ = 1 DY = (Y_k⁰_l)k,l=0,...,n =

1 0 X˙ Q

=

1 0 Qx˙ ^∗ + ˙b Q

Examples of transformation rules:

% objective scalar ⇐⇒ %◦Y = %^∗

that is %(t, x) = %^∗(t^∗, x^∗) for (t, x) = Y(t^∗, x^∗)

v velocity ⇐⇒ v◦Y = ˙X + Qv^∗ (Doppler effect)

that is v(t, x) = ˙X(t^∗, x^∗) +Q(t^∗)v^∗(t^∗, x^∗) for (t, x) = Y(t^∗, x^∗)

f ^force ⇐⇒ f◦Y = %^∗( ¨X + 2 ˙Qv^∗)+ ˙QJ^∗ + r^∗X˙ + Qf^∗

(if mass conservation is ∂t% + div (%v+J) = r) They come from objectivity of balance laws

Theorem (Objectivity for certain systems of balance laws)

∂_t(e^kµΓ) +

n

X

i=1

∂_xi(q_i^kµΓ) = f^kµΓ (k=1,. . . ,N)

Physical type given by (linear) transformation rule for test functions ζ◦Y =Z^−Tζ^∗ ( ζ^∗ =Z^Tζ◦Y )

Then this system is objective, if e◦Y = Ze^∗

q_i◦Y = ˙X_iZe^∗ +

n

X

j=1

Q_ijZq_j^∗ for i = 1, . . . , n

f◦Y = Z⁰_te^∗ +

n

X

j=1

Z⁰_jq_j^∗ + Zf^∗

Examples:

mass mass-momentum mass-momentum-energy

Z = 1 Z =

1 0 X˙ Q

= DY Z :=







1 0 0

X˙ Q 0

1

2|X˙|² X˙^TQ 1







(Z = Z(Y ) is a differential operator in Y )

Proof: All µ_Γ have the same transformation behaviour Weak formulation in spacetime:

∂0 := ∂_t, q0 := e, q_i = (q_i^k)_k=1,...,N, f = (f^k)_k=1,...,N Z Xⁿ

i=0

∂_iζ•q_i + ζ•f

dµ_Γ = 0 for test functions ζ = (ζ^k)_k=1,...,N

ζ^∗ =Z^Tζ◦Y ⇒ ∂_jζ^∗ =Z⁰_j^Tζ◦Y +

n

X

i=0

Y_i⁰_jZ^T(∂_iζ)◦Y

=⇒

Z Xⁿ

j=0

∂_jζ^∗•q_j^∗ + ζ^∗•f^∗

dµΓ^∗

=

Z Xⁿ

i=0

(∂_iζ◦Y )•(

n

X

j=0

Y_i⁰_jZq_j^∗) + (ζ◦Y )•(Zf^∗ +

n

X

j=0

Z⁰_jq_j^∗)

dµΓ^∗ (det =1)

=

Z Xⁿ

i=0

∂_iζ•(

n

X

j=0

Y_i⁰_jZq^∗_j)◦Y ⁻¹ + ζ•(Zf^∗ +

n

X

j=0

Z⁰_jq_j^∗)◦Y ⁻¹ dµΓ

=

!

Z Xⁿ

i=0

∂_iζ•q_i + ζ•f dµ_Γ

⇐=

q_i◦Y =

n

X

j=0

Y_i⁰_jZq_j^∗ for i = 0, . . . , n f◦Y =

n

X

j=0

Z⁰_jq_j^∗ + Zf^∗

Examples:

Mass-momentum system

∂_t(%µ_U) + div(%vµ_U) = 0

∂_t(%vµ_U) + div((%v v^T+ Π)µ_U) = fµ_U The transformation matrix is

Z =

1 0 X˙ Q

= DY Objectivity says

% %v^T

%v %v v^T+ Π

◦Y =

1 0 X˙ Q

%^∗ %^∗v^∗T

%^∗v^∗ %^∗v^∗v^∗T+ Π^∗

1 X˙^T

0 Q

0 f

◦Y =

0 0 X¨ Q˙

%^∗

%^∗v^∗

+

n

X

j=1

0 0 X˙ ⁰_j 0

%^∗v_j^∗

· · ·

+

1 0 X˙ Q

0 f^∗

⇐⇒

% is an objective scalar, i.e. %◦Y = %^∗ v is a velocity, i.e. v◦Y = ˙X + Qv^∗

Π is an objective tensor, i.e. Π◦Y = QΠ^∗Q^T f is a force, i.e. f◦Y = %^∗( ¨X + 2 ˙Qv^∗) +Qf^∗

Choice of Π as an objective tensor:

For fluids (Velocity v is independent variable)

Π = pId − S

p = %f⁰_% − f free energy: f = fb(%, v) = %

2|v|² + fb0(%) S = λ1divv Id + λ2

(Dv)^S − 1

ndivv Id

(Dv)^S objective tensor

λ1 = bλ1(%), λ2 = bλ2(%) objective scalars For solids (Velocity v is dependent variable)

Π = −

n

X

i=1

λ_ie_ie_i^T

e_i = b^ei(x) objective vectors λ_i = λb_i(x) objective scalars

Reference coordinates x = ξ(t, x) components ξ_i are objective scalars (t, x) 7→ (t, x) = τ(t, x) := (t, φ(t, x))

v(t, x) = ∂_tφ(t, x) for x = φ(t, x) or x = ξ(t, x) It follows: % = %(x) , ∂_t(%V ) − divP = f

where P = J · (−Π◦τ)F^−T first Piola-Kirchhoff stress tensor J = det Dφ, V = ∂tφ, F = Dφ, % = J · %◦τ, f = J · %fτ

Mass-momentum system for an interface problem Let Γ := ∂Ω

∂_t(%µµµ_Ω) + div(%vµµµ_Ω) = r^sµµµ_Γ

∂_t(%vµµµ_Ω) + div((%v v^T+ Π)µµµ_Ω + Π^sµµµ_Γ) = fµµµ_Ω + f^sµµµ_Γ The transformation matrix is still

Z =

1 0 X˙ Q

= DY Objectivity says on Γ

0 0 0 Π^s

◦Y =

1 0 X˙ Q

0 0

0 Π^s∗

1 X˙^T

0 Q

r^s f^s

◦Y =

0 0 X¨ Q˙

0 0

+

n

X

j=1

0 0 X˙ ⁰_j 0

0

· · ·

+

1 0 X˙ Q

r^s∗

f^s∗

⇐⇒

r^s is an objective scalar, i.e. r^s◦Y = r^s∗

Π^s is an objective tensor, i.e. Π^s◦Y = QΠ^s∗Q^T f^s is surface force, i.e. f^s◦Y = r^s∗X^˙ + Qf^s∗

Lemma A possible choice is f^{s =} r^sv + f₀^{s with} f₀^s◦Y = Qf₀^s∗

Surface tension

Drop given by distributional conservation laws for mass and momentum in Ω and on Γ := ∂Ω

System gives

transformation rules

∂_t(%µµµ_Ω) + div(%vµµµ_Ω) = 0

∂_t(%vµµµ_Ω) + div

(%v v^T+ Π)µµµ_Ω + Π^sµµµ_Γ

= fµµµ_Ω

⇐⇒

mass and momentum equation in Ω

on Γ: (v − vΓ)•ν_Ω = 0, Π^sν = 0 for normal ν, div^ΓΠ^s = ΠνΩ=: f^s

Therefore it contains the

Surface tension law on Γ div(Π^sµµµ_Γ) = f^sµµµ_Γ

⇐⇒

Z

(Dζ•Π^s + ζ•f^s) dµµµ_Γ = 0 for ζ ∈ D(U;Rⁿ)

⇐⇒ Dζ = (∂_jζ_i)i,j=1,...,n

div^ΓΠ^s = f^s,

Π^sν = 0 for ν = ±ν_Ω Objectivity

vΓ is “surface velocity”: vΓ◦Y = ˙X•(Qν^∗)Qν^∗ + QvΓ^∗

Π^s objective tensor ν_Ω objective vector

Π^sνΩ = 0 is objective f^s= ΠνΩ objective vector

Surface tension as function of normal

It is Π^sν_Ω = 0, and if Π^s symmetric tensor, n = 3 Result: If Π^s = Πb^s(ν) then

Π^s = −γ(Id − ν ν^T) γ ∈ R surface tension

Result: If Π is constant, then Π = pId

p ∈ R pressure

General: Π^s (and Π) can depend also on objective scalars, like % or p or θ Laplace formula: If γ : Γ → R and Π^s = −γ(Id − ν ν^T) then

div^ΓΠ^s = div^Γ(−γ(Id − ν ν^T)) = −γκ_Γ − ∇^Γγ

Hence the surface law of the momentum equation is γκΓ + ∇^Γγ + ΠνΩ = 0 Liquid drop If Π^s = −γ(Id − ν ν^T) and Π = pId − S then

mass and momentum equation in Ω (v − v_Γ)•ν = 0

∂_τγ = τ•SνΩ for tangential vectors τ γκ_Γ•ν_Ω + p = ν•Sν







on Γ

Further constitutive constraints Dv•S ≥ 0, f^s = γ = const

come from free energy inequality:

Free energy inequality

∂tF + div Φ − G0 ≤ 0

that is

0 ≥ ∂_t( fµµµ_Ω + f^sµµµ_Γ

| {z } F

) + div ( (f v + Π^Tv)µµµ_Ω + (f^sv + Π^sTv)µµµ_Γ

| {z }

Φ

) − v•f µµµ_Ω

| {z } G0

=: g µµµ_Ω + g^sµµµ_Γ

that is

g ≤ 0 on Ω and g^s ≤ 0 on Γ that is on Ω:

0 ≥ g = ∂_tf + div (f v + Π^Tv) − v•f f = f(%, v) = %

2|v|² + f0(%)

= ∂_tf0 + div (f0v) + Dv•Π Π = pId − S

= Dv•((f0 − %f0⁰%)Id + Π) = −Dv•S p = %f0⁰% − f0

and on Γ: (f^sv + Π^sTv)•ν_Ω = f^svΓ•ν_Ω, (v − vΓ)•ν_Ω = 0 0 ≥ g^S = ∂_t^Γf^s + div^Γ(f^sv + Π^sTv) − (f(v − v_Γ) + Π^Tv)•ν_Ω

= (∂_t^Γ + v•∇^Γ)f^s + (D^Γv)•(f^sId + Π^s) + v•( div^ΓΠ^s − Πν_Ω

| {z }

= 0

)

= (∂_t + v•∇)f^s + (D^Γv)•(f^sId + Π^s) D^Γv = X

τ

(∂_τv)⊗τ satisfied, if Dv•S ≥ 0, f^s = γ = const

Liquid drop It is γ = const, and it follows

∂_t% + div (%v) = 0

∂_t(%v) + div (%v v^T+ pId − S) = f p = %f0⁰% − f0, Dv•S ≥ 0







on Ω

(v − v_Γ)•ν = 0

τ•Sν = 0 for tangential vectors τ γκ_Γ•ν_Ω + p = ν•Sν

f^s = γ











on Γ

Remark If v = 0, vΓ = 0, % = const, f = 0 define (Γ = ∂Ω) EΓ =

Z

Γ

γdH_n−1 − Z

Ω

pdL_n

= Z

Γ

f^sdH_n−1 + Z

Ω

f dL_n

and solution is “stationary point” of EΓ

In general the solution is related to the free energy inequality

∂_t(fµµµ_Ω + f^sµµµ_Γ) + div (...) − v•fµµµ_Ω ≤ 0

Phase field limit

An Allen-Cahn model for compressible fluids Witterstein model

∂_t% + div (%v) = 0 f_δ = f_δ(%, ϕ,∇ϕ)

%(∂_tϕ + v•∇ϕ) = −r_δ r_δ ^:= c_δ(%, ϕ)δδδf_δ δδδϕ

∂_t(%v) + div (%v v^T+ Π_δ) = f_δ ^Π_δ ⁼ P_δ − S with

P_δ := (%f_δ⁰_% − f_δ)Id + ∇ϕ(f_δ⁰_∇ϕ)^T S := λ1(%, ϕ)divv Id + λ2(%, ϕ)

(Dv)^S − 1

ndivv Id

f_δ(%, ϕ,∇ϕ) := 1

δ%W(ϕ) + δ h(%) ∇ϕ

2

2 + U(%, ϕ)

W has two local minima at 0 and 1, U⁰_ϕ(%,0) = 0, U⁰_ϕ(%,1) = 0 δδδf_δ

δδδϕ = 1

δ% W⁰_ϕ(ϕ) − δdiv (h(%)∇ϕ) + U⁰_ϕ(%, ϕ)

Free energy inequality f = f(%, ϕ, v,∇ϕ) = ₂^%|v|² + f_δ(%, ϕ,∇ϕ)

∂_tf + div (f v + Π^T_δ v − ϕf˙ ⁰_∇ϕ) − v•f_δ = −1

%r_δ^δδδfδ

δδδϕ − Dv•S ≤ 0

System written for the two masses %¹ and %²

%¹_δ = (1− ϕ)% , %²_δ = ϕ% , ^or % = %¹_δ + %²_δ , ϕ = %²_δ

%¹ + %² becomes

∂_t(%¹_δµ_U) + div (%¹_δv_δµ_U) = r_δµ_U

∂t(%²_δµ_U) + div (%²_δv_δµ_U) = −r_δµ_U

∂_t(%vµ_U) + div (%v v^T+ Π_δ)µ_U

= fµ_U Theorem In the limit δ → 0 this becomes

∂_t(%¹µµµ_Ω¹) + div (%¹v¹µµµ_Ω¹) = rµµµ_Γ

∂t(%²µµµ_Ω²) + div (%²v²µµµ_Ω²) = −rµµµ_Γ

∂_t X

m

%^mv^mµµµ_Ω^m

+ div

X

m

(%^mv^mv^mT+ Π^m)µµµ_Ω^m + Π^sµµµ_Γ

= X

m

f^mµµµ_Ω^m

The convergence is in the sense of distributions, in particular (%v v^T+ Π_δ)µµµ_U −→ X

m

(%^mv^mv^mT+ Π^m)µµµ_Ω^m + Π^sµµµ_Γ

Π^s = −γ(Id − ν ν^T), γ given by an integral over “local coordinates”

where U = Ω¹ ∪Γ∪ Ω²

For η ∈ D(U;Rⁿ × Rⁿ), U ⊂ R × Rⁿ, U = Ω¹_δ ∪Γ_δ ∪ Ω²_δ Z

U

η•(%v v^T+ Π_δ) dL_n+1 = X

m

Z

Ω^m_δ

. . . dL_n+1 + Z

Γδ

. . . dL_n+1 In a small neighbourhood of Γ

Π_δ = p_U Id + δ

2p_h|∇ϕ|²Id + δh∇ϕ(∇ϕ)^T− (λ1 − λ2

n )divvId − λ2(∇v)^S

= 1 δ

₁

2p_h|∂_rΦ⁰|²Id + h|∂_rΦ⁰|²ν ν^T

−(λ1 − λ2

n )ν•∂_rV ⁰ Id − 1

2λ2(ν(∂_rV ⁰)^T+ ∂_rV ⁰ν^T)

+

O

δΠ_δν → 1

2p_h|∂_rΦ⁰|²ν +h|∂_rΦ⁰|²ν − (λ1 − λ2

n )ν•∂_rV ⁰ν − λ2ν•∂_rV ⁰ν

= (1

2p_h + h)|∂_rΦ⁰|²ν − (λ1 + (n − 1)λ2

n )∂_rV ⁰ = 0 (δΠ_δ)_tan := δΠ_δ − (δΠ_δν)ν = δΠ_δ(Id − ν ν^T)

→ ₁

2p_h|∂_rΦ⁰|² − (λ1 − λ₂

n )ν•∂_rV ⁰

(Id − ν ν^T)

=

₁

2p_h(R⁰)|∂_rΦ⁰|² − λ1(R⁰,Φ⁰) − λ2(R⁰,Φ⁰) n

ν•∂_rV ⁰

(Id − ν ν^T)

γ = Z

R

− 1

2p_h(R⁰)|∂_rΦ⁰|² + λ1(R⁰,Φ⁰) − λ2(R⁰,Φ⁰) n

ν•∂_rV ⁰

dr

References

H.W. Alt: “Entropy principle and interfaces. Fluids and Solids”.

AMSA 2009

G. Witterstein: “Sharp interface limit of phase change flows”.

AMSA 2011

H.W. Alt, G. Witterstein: “Distributional equation in the limit of phase transition”.

Submitted to IFB 2011

———————————————————————————————————- D. Bedaux: “Nonequilibrium Thermodynamics and Statistical Physics of Surfaces”.

Advance in Chemical Physics. Volume LXIV. John Wiley & Sons 1986 W. Kosi´nski: “Field Singularities and Wave Analysis in Continuum Mechanics”.

Halsted Press (John Wiley & Sons) 1986 (Polish original published in 1981) J. C. Slattery: “Interfacial Transport Phenomena”.

Springer 1990