A diffuse interface model of a multiphase
fluid with surfactant
Oliver Dunbar
MASDOC, Warwick Mathematics Institute
Joint Work with Bjorn Stinner
MASDOC - CCA Conference,
Thursday 17/04/15
Table of Contents
Introduction
Sharp interface model
Diffuse interface model
Simulation
Future work
Introduction
What is diffuse interface modelling?
A process which is used to increase regularity of models that describe
phenomena involving discontinuities along free boundaries. Obtain a nonzero
interface width ε, in such a way to recover information in the original model
when ε → 0.
Why may one consider this technique?
I
This method allows for a large range of phenomena to be observed
compared to a ‘sharp’ model.
I
Numerically, easier to construct smooth interfaces (via continuous finite
elements)
I
Most common applications in fluid mechanics (mixing/separation,
multi-fluids), chemistry (polymer blending/reaction fronts), and image
processing.
Our problem
What are we modelling?
Our model will be of a multi (three or more) phase fluid flow in a fixed region,
for example a mix of different incompressible fluids (e.g oil, water and vinegar)
or a fluid which tends to form different phases. In particular we wish to see the
effects of adding surfactants to this model.
What is a surfactant?
A surfactant, or surface active agent, is a chemical quantity which adsorbs to
the boundary layers of the fluid due to a -philic/-phobic chemical structure and
leading to it being more soluble in one fluid than another. The fluid molecules
will then weakly bind to it - resulting in a loss in surface tension.
Sharp interface model
Notation
Consider a fixed body Ω ⊂ Rd in dimension d = 2, 3:
I Partition into M connected disjoint time dependent subdomains
(i)
Ω = ∪M
i=1 Ω (t).
I Interface between subdomains (not necessarily connected) hypersurface:
(i)
(j)
Γ(i,j) (t) := Ω (t) ∩ Ω (t).
Assumption: We restrict to at most three hypersurfaces intersecting at a
(i)
(j)
(i)
single point (red dots) Ω (t) ∩ Ω (t) ∩ Ω (t) = T (i,j,k) (t)
Sharp interface model
The Mass and momentum equations
in Ω(i) (t)
∇·v =0
(i)
(i)
(i)
(i)
∂t (ρ v) + ∇ · (ρ v ⊗ v) = ∇ · T
[v]ji
= 0,
uΓ(i,j) = v · ν
(i,j)
(1a)
in Ω (t)
(1b)
(i,j)
(1c)
on Γ
(t)
fluid velocity v , with normal velocity uΓ(i,j) on Γ(i,j) , and ρ(i) density of fluid i.
T(i) is a corresponding momentum flux.
Explanation:
I
(1a) Mass conservation, (incompressibility)
I
(1b) Momentum conservation (stress tensor ‘yet to be determined’)
I
(1c) Free boundary conditions along interface - velocity is continuous and
the interface is transported by the fluid flow.
Sharp interface model
The surfactant equations
∂t• c (i) = − ∇ · J(i)
c
∂t• c (i,j)
+c
(i,j)
∇Γ(i,j) · v = − ∇Γ(i,j) ·
J(i,j)
c
+q
(i,j)
Jc(i,j) · µ(i,j,k) + J(j,k)
· µ(j,k,i) + J(k,i)
· µ(k,i,j) = 0
c
c
(i)
in Ω(i) (t)
(2a)
(i,j)
(2b)
on Γ
(t)
on T (i,j,k) (t)
(3)
(i,j)
c bulk surfactant concentration, c
interfacial surfactant concentration,
(i)
(i,j)
Jc , Jc are associated fluxes for surfactant and the q(i,j) is the adsorption from
bulk to interface. µ(i,j,k) is the conormal of interface Γ(i,j) pointing into fluid k.
Material derivative: ∂t• (w ) = ∂t (w ) + v · ∇w
Surface Gradient: ∇Γ(i,j) w = ∇w − (∇w · ν (i,j) )ν (i,j)
Explanation:
I
(2a) Mass balance equation for surfactant in the bulk fluid
I
(2b) Mass balance equation for surfactant on the interface
I
(20) gives a force balance at the triple junction of the interfaces. (no mass
can be concentrated/lost at the junction).
Sharp interface model
Energy and constitutive assumptions
For M phases, we postulate the following free energy:
X Z ρ(i) 2
E (c (·) , c (·,·) , v) :=
|v| + Gi (c (i) ) +
2
Ω(i)
i=1...,M
X
i,j=1,...,M
i<j
Z
γij (c (i,j) )
Γ(i,j)
Gi and γij are the bulk and surface chemical free energy densities respectively
assumed with Gi00 > 0, γij00 > 0.
Assumption: Instantaneous adsorption to the interface. Generates a local
equilibrium at the interfaces for the bulk/surface free energy potential.
γij0 (c (i,j) ) = Gi0 (c (i) ) =: q
⇐⇒ c (i) = (Gi0 )−1 (q), c (i,j) = (γij )−1 (q)
Surface Tension:
σ̃ij (q) := σij (c (i,j) (q)) = γij (c (i,j) (q)) − c (i,j) (q)q
Sharp interface model
Energy dissipation
Thermodynamic consistency requires E to have non positive dissipation, we
show some sufficient conditions for it to satisfy
Z
d
E ≤
(working terms) · νΩ
dt
∂(Ω)
Constitutive assumptions:
T(i) := pI + 2η (i) D(v)
Jc(i)
J(i,j)
c
[T(·) ]ij ν (i,j)
:=
−Mc(i) ∇Gi0 (c (i) (q))
:=
−Mc(i,j) ∇Γ(i,j) γij0 (c (i,j) (q))
= σ̃ij (q)κ
(i,j) (i,j)
0 = σ̃ij (q)µ
ν
(i,j,k)
(4a)
(i)
(4b)
(i,j)
(4c)
(i,j)
(4d)
(i,j,k)
(4e)
in Ω
on Γ
+ ∇Γ(i,j) σ̃ij (q)
+ σ̃jk (q)µ
in Ω(i)
(j,k,i)
+ σ̃ki (q)µ
on Γ
(k,i,j)
on T
Explanation:
I
(4a) Navier stokes stress tensor (p pressure, η (i) viscosity)
I
M c , Mc
I
(4d) Balance on the interface between hydrodynamic and surfactant forces
I
(4e) gives rise to contact angle condition between phases.
(i)
(i,j)
chemical mobilities.(4b)-(4c) yield surfactant diffusion
Remark: These conditions, with the equilibrium relation, close the system.
Diffuse interface model
The phase field variables
We now wish to build up the diffuse model, or phase field model.
Gibb’s Simplex
M
n
o
X
ΣM := u = (u (1) , . . . , u (M) ) ∈ RM :
u (i) = 1, where 0 ≤ u (i) ≤ 1
i=1
Then the phase field variables (or order parameters) are functions
ϕ = (ϕ(1) , . . . , ϕ(M) ) : Ω → ΣM
(i)
which for the multiphase flow model, we shall define
P ϕ to be the volume
fraction associated to fluid i, of course we have i ϕ(i) = 1
We are able to transfer useful quantities
P of(i)the(i)system, for example the total
mass density of the fluid is ρ(ϕ) = M
i=1 ρ ϕ , similarly for viscosity etc.
P
In particular we set the volume averaged velocity: v = i ϕ(i) ṽ
(ṽ was previous mean velocity)
Diffuse interface model
Hydrodynamic equations
With v now a volume averaged velocity, summing the continuity equation in
each fluid, we obtain ∇ · v = 0, allows conservation of linear momentum and
we construct
∇·v =0
(5a)
∂t (ρv) + ∇ · (ρv ⊗ v) = ∇ · T
(5b)
∂t (ϕ(i) ) + ∇ · (ϕ(i) v) = −∇ · J(i)
ϕ
(5c) =⇒ ∂t (ρ) + ∇ · (ρv) = −∇ · J
(5c)
with J :=
M
X
ρ(i) J(i)
ϕ
(5d)
i=1
Explanation:
I
(5a),(5b) Mass and momentum conservation for the volume averaged fluid
velocity
I
(5c) a phase field balance that implies a total mass balance law for the
phase field (5d)
Diffuse interface model
Ansatz for surfactants
The surfactant equations from the sharp model may be written in terms of
characteristic functions of the fluid bulk χΩ(i) and fluid interfaces χΓ(i,j) .
We may then approximate the characteristic functions,by ξi (ϕ) ≈ χΩ(i) and
introducing the interfacial width ε > 0 , δij (ϕ, ∇ϕ) ≈ χΓ(i,j) .
∂t
X
ξi c (i) (q) +
i
X
δij c (i,j) (q)
i<j
+∇ ·
X
i
ξi c (i) (q) +
X
δij c (i,j) (q) v
i<j
+∇·
X
i
ξi J(i)
c +
X
i<j
δij J(i,j)
=0
c
Diffuse interface model
Ginzburg-Landau Energy
The Energy functional we postulate for the phase field model is of a
Ginzburg-Landau type.
δ(q, ϕ, ∇ϕ) := GL free energy density
}|
{
Z z
1
ρ
εa(q, ϕ, ∇ϕ) + w (q, ϕ) + |v|2 + f (q, ϕ)
Eε (ϕ, ∇ϕ, v, q) :=
ε
2
Ω
X
ξi (ϕ(i) )Gi (c (i) (q))
I
Bulk free energy density f (q, ϕ) :=
I
a : R × ΣM × (T ΣM )d → R an admissible free energy gradient term
X
a(q, ϕ, ∇ϕ) :=
γij (c (i,j) (q))aij (ϕ, ∇ϕ)
i=1,...,M
i,j=1,...,M
i<j
I
w : R × ΣM → R an admissible multi-well potential
X
w (q, ϕ) :=
γij (c (i,j) (q))wij (ϕ)
i,j=1,...,M
i<j
Diffuse interface model
Ginzburg-Landau Energy
Eε (ϕ, ∇ϕ, v, q) :=
Z 1
ρ
εa(q, ϕ, ∇ϕ) + w (q, ϕ) + |v|2 + f (q, ϕ)
ε
2
Ω
Why this energy?
The Ginzburg-Landau energy density can approximate the delta function in such
a way that when ε → 0 the G-L energy will Γ− converge to the sharp energy.
How to choose a, w?
To recover the correct surface energy (for given Γ(i,j) ) as ε → 0 requires
conditions on a, w . [Garcke,Nester,Stoth], [Boyer et al.] have shown that the
goal is to reduce the system to the two phase case away from triple junctions.
(Vague) assumptions
I
Gradient term a: positivity, quadratic in ∇ϕ, on straight paths running
perpendicularly through an interface - locally a is constant.
I
Multiwell potential term w : positivity, multiwell potential, on straight
paths running perpendicularly through an interface - locally w becomes a
consistent double well potential.
Diffuse interface model
Full phase field model
By once again finding constitutive assumptions for an energy dissipation
inequality we state the full phase field model:
∇·v =0
∂t (ρv) + ∇ · (ρv ⊗ v) = ∇ ·
X
X
σij δij I −
∇ϕ(k) ⊗ aij ,∇ϕ(k)
i<j
k
− v ⊗ J − pI + 2ηD(v)
For k = 1, . . . , M
M
X
∂t (ϕ(k) ) + ∇ · (ϕ(k) v) = ∇ ·
L(k,l) (ϕ, q)∇µ(l)
l=1
µ
(k)
=
X
−ε∇ · (σ̃ij aij ,∇ϕ(k) ) + εσ̃ij aij ,ϕ(k)
i<j
+
∂t
X
i
ξi c (i) (q) +
X
1
σ̃ij wij,ϕ(k) + ξk0 (Gk − c (k) q)
ε
X
X
δij c (i,j) (q) + ∇ ·
ξi c (i) (q) +
δij c (i,j) (q) v
i<j
i
=∇·
X
i
ξi Mc(i) ∇q +
i<j
X
i<j
δij Mc(i,j) ∇q
Diffuse interface model
Formal asymptotic methods
I have stated previously that one may perform recovery for the sharp model as
ε → 0. This is acheived by a Formal method called matched asymptotics.
Summary of method:
You assume that there exist appropriate asymptotic expansions of quantities in
the bulk and in the interface layer. Then you wish to match these to different
orders to form the appropriate bulk and interface equations.
For simple models this approach has been used to construct analytically
rigorous results. The models was also successfully employed in [Garcke, Lam,
Stinner 2014], in the two phase case. There has also been work for triple
junctions for the Allen-Cahn equation [Garcke, Nestler, Stoth 1997].
Discretisation and Simulation
A particular model
[Boyer Lapurta 2006]
X
3
|∇ϕ(k) |2
aij (∇ϕ) = |∇ϕ(i) |2 + |∇ϕ(j) |2 −
8
k6=i,j
(i) 2
(i) 2
(j) 2
wij (ϕ) =6((ϕ ) (1 − ϕ ) + (ϕ ) (1 − ϕ(j) )2 −
X
(ϕ(k) )2 (1 − ϕ(k) )2
k6=i,j
(i) 2
(j) 2
+ Λ(ϕ ) (ϕ )
Y
(k) 2
(ϕ
)
k6=i,j
This gives
3
µ(l) = −ε ∇ · (Σ(l) ∇ϕ(l) ) + l.o.t
4
where Σ(l) = σli + σlj − σij . Wetting effects occur if Σ(l) < 0
Also defined:
I
Free energies, Gi , γij and an isotherm.
I
L(k,l) constant and O(10−2 ), densities of fluids equal, viscosities equal.
I
surfactant mobilities Mc , Mc
I
BCs: no flux for phase field, “Natural” angle condition for interfaces,
Dirichlet/No flux for surfactants
(i)
(i,j)
constant and O(1)
Discretisation and Simulation
Simulation - Surfactants supplied on left. Highly viscous fluid
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Discretisation and Simulation
Simulation - Marangoni force with surfactants(low fluid viscosity)
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Discretisation and Simulation
Simulation - Wetting effects
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Jc(i,j) · µ(i,j,k) + J(j,k)
· µ(j,k,i) + J(k,i)
· µ(k,i,j) = 0
c
c
will not be satisfied in the sharp interface model.
on T (i,j,k) (t)
Future work
Conclusions and future work
Conclusion:
We have presented a model of multiphase fluid with surfactant, then built a
multi phase field construction, and described briefly how one may recover
important quantities or relations. We have thus shown that diffuse models can
be used to approximate very complex models, in many applications. Not
discussed in this presentation was the more numerical benefits of these models,
the ease of construction, their robustness and they can be shown to even
capture physical effects that are not yet covered by asymptotics analysis (e.g
wetting effects).
Future work:
I
Asymptotic matching analysis around triple junctions
I
More complex surfactant regimes (e.g different existing states in the bulk
such as monomers/micelles)
I
Numerical analysis and implementation of different discrete coupling
schemes
I
Comparisons with real physical experiments (For example liquid lenses).
Thank you for listening!