The document summarizes modeling and computation of expanding and shrinking interface problems. It discusses how interface problems arise in physical/biological sciences involving moving boundaries. It presents the classical Hele-Shaw problem as an example of an expanding interface problem and describes its governing equations. It also discusses a modified Hele-Shaw problem with a lifting plate as an example of a shrinking interface problem. It proposes an adaptive scaling technique to speed up simulations of slowly expanding interfaces and slow down rapidly shrinking interfaces.

1. Interface problems Expanding interface problem Shrinking interface problem
Modeling and Computation of Moving Interface
Problems
Shuwang Li 1
1
Applied Math Dept., Illinois Institute of Technology, Chicago
CISC Lunchtime Matchmaking Seminar
October 18, 2017
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

2. Interface problems Expanding interface problem Shrinking interface problem
1 Interface problems
2 Expanding interface problem
Classical Hele-Shaw problem
Numerical Methods
Numerical Results
3 Shrinking interface problem
Modified Hele-Shaw problem with lifting plate
Numerical Results
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

3. Interface problems Expanding interface problem Shrinking interface problem
Interface problems in physical/biological sciences
Moving boundary/interface problems: boundary value problems defined
in a domain whose boundary is a priori unknown and evolving with time;
interface separating different domains.
Examples: multiphase flow in fluids; phase transformations in materials
including crystal growth (solid/liquid), epitaxial thin film growth
(solid/vapor), and elastic precipitate growth/shrink (solid/solid); tumor
growth, bio-membrane, and pattern formation via diffusion in
bio-systems; fluid-structure interactions;
Central question: dynamical stability of the interface
What I can help: modeling and computation (numerics); understand
dynamics and instabilities...
What you can help: new problems; experimental verifications;
interpretations...
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

4. Interface problems Expanding interface problem Shrinking interface problem
Computational challenges due to expanding/shrinking
Key questions besides solving PDEs: identify the interface and evolve
the interface (front tracking/level set/phase field). Consider a simple
interface: a sphere or a circle driven by an injection flux (growth) or
extraction flux (shrinkage)
Evolving Velocity
3D:
d(volume : 4πR3
(t)/3)
dt
= J, i.e. interface velocity:
dR(t)
dt
=
J
πR2(t)
2D:
d(area : πR2
(t))
dt
= J, i.e. interface velocity:
dR(t)
dt
=
J
πR(t)
Challenges for expanding and shrinking interfaces (open
non-equilibrium system):
slow dynamics for the growth problem (R increase); need to
increase resolution;
fast dynamics for the shrinking problem (R decrease); need to
reduce resolution.
From a computational point of view, we want to: speed up the slow
dynamics for expanding interface or slow down the fast dynamics for
shrinking interface without changing the real physics.
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

5. Interface problems Expanding interface problem Shrinking interface problem
Example: Hele-Shaw and expanding viscous fingering
Hele-Shaw problem is a classical example for studying the interface
dynamics. Application: oil recovery in petroleum engineering, natural
gas storage.
Saffman-Taylor instability (fingering pattern) occurs when less viscous
fluid is injected into existing viscous fluid.
blue: air
white: oil
Pressure jump at the interface is given by the Laplace-Young condition.
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

6. Interface problems Expanding interface problem Shrinking interface problem
Governing Equations (air-oil system)
Differential equations (exterior problem):
2
P = 0, x ∈ ΩL
V = − P · n, x ∈ Σ(t)
Pin − Pout = τκ, x ∈ Σ(t)
Σ
∂P
∂n
ds = J(t)
Interface evolution:
dx
dt
· n = V(x), x ∈ Σ(t).
Linear Stability (Mullins-Sekerka, 1963; Saffman-Taylor, 1958).
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

7. Interface problems Expanding interface problem Shrinking interface problem
Rescaling
The Idea
isolate morphological change from overall growth by mapping onto a new
time and space: (x, t) → (¯x,¯t), i.e. scale out the growth: R(t(¯t)) = ¯R(¯t).
x(α, t) = ¯R(¯t)¯x(α,¯t), ¯t =
t
0
1
ρ(t )
dt .
Integrable ¯ρ(¯t) = ρ(t(¯t)) > 0
speed up or slow down
adaptive
The normal velocity in the rescaled frame ¯V,
¯V(¯t) =
¯ρ
¯R
V(t(¯t)) −
¯x · n
¯R
d ¯R
d¯t
Set
d ¯A
d¯t
= 0,
¯Σ(¯t)
¯Vd¯s = 0 →
d ¯R
d¯t
=
π¯ρ¯J
¯A(0)¯R
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

8. Interface problems Expanding interface problem Shrinking interface problem
Rescaling contd.
Take ¯ρe =
¯A(0)¯R2
(¯t)
π¯J
, then ¯R(¯t) = exp(¯t), i.e. exponential growth.
Take ¯ρl = b
log(a)(a+b¯t)
¯A(0)
π¯J
¯R, then ¯R(¯t) = log(a+b¯t)
log(a)
, i.e. logarithmic growth.
If you want to use the boundary integral method, the rescaled integral
form for ¯ρe case
¯µ(¯x) −
1
π ¯Σ(¯t)
¯µ(¯x )[
∂ ln |¯x − ¯x |
∂n(¯x )
+ ¯R(¯t)]d¯s(¯x ) = 2τ ¯κ
+ 2¯R(¯t)¯J(ln(¯R(¯t)) + ln |¯x|).
The normal velocity in scaled frame ¯V is given by,
¯V(¯x) =
¯A
2π2¯J
(
1
¯R ¯Σ
¯µ¯s
(¯x − ¯x)⊥
· ¯n(¯s)
|¯x − ¯x|2
d¯s + 2π¯J
¯x · ¯n
|¯x|2
) − ¯x · ¯n,
where ¯x⊥
= (¯y, −¯x). We evolve the interface in the scaled frame
d ¯x(¯t, s)
d¯t
· ¯n = ¯V(¯t, s).
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

9. Interface problems Expanding interface problem Shrinking interface problem
Critics and Suggestions (exponential fast growth)
Good: ∆¯t is fixed, while equivalent ∆t in the original frame is increasing.
Bad: fast growth requires small time step for numerical stability (a waste
of CPU time at the early growth stage when the interface size is small.)
Ugly (Adaptive): Choose the time scale to be (1) a log function when R
is small; (2) an exponential function when R is large.
0 10 20 30 40 50 60
10
−4
10
−3
10
−2
10
−1
10
0
R
∆t
ρe
ρ
s
original
Figure: shows the corresponding time step in the original real frame.
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

10. Interface problems Expanding interface problem Shrinking interface problem
Adaptive scaling function
Logarithmic scaling at small R(t)
ρl =
b
log(a)(a + b¯t)
¯A(0)
π¯J
¯R,
¯R =
log(a + b¯t)
log(a)
t =
¯A(0)
2 log2
(a)π¯J
(log2
(a + b¯t)
− log2
(a)),
¯Vl =
b¯A(0)
log(a)(a + b¯t)π¯J ¯R
(
1
2π ¯R ¯Σ
¯µ¯s
(¯x − ¯x)⊥ · ¯n(¯x)
|¯x − ¯x|2
d¯s
+¯J
¯x · ¯n
|¯x|2
)
−
b
log(a)(a + b¯t)¯R
¯x · ¯n.
Switch back to exponential at large R(t)
Switch at ¯t = ¯t0 with ¯R = ¯R0
ρe = c ¯R2
,
¯R = ¯R0 exp(c(¯t − ¯t0)),
t =
R2
0
¯A(0)
2π¯J
(exp(2c(¯t − ¯t0)) − 1)
+
¯A(0)
2π¯J
(¯R2
0 − 1), ¯t ≥ ¯t0,
¯Ve = c(
1
2π ¯R ¯Σ
¯µ¯s
(¯x − ¯x)⊥ · ¯n(¯x)
|¯x − ¯x|2
d¯s
+¯J
¯x · ¯n
|¯x|2
) −
π¯Jc
¯A(0)
¯x · ¯n.
where c =
b¯A(0)
log(a)(a + b¯t0)πJ ¯R0
and
¯R0 =
log(a + b¯t0)
log(a)
is the space scaling factor at
¯t0.
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

11. Interface problems Expanding interface problem Shrinking interface problem
Adaptive scaling function contd.
Combined Scaling Function
ρs =
b
log(a)
¯A(0)
π¯J
¯R2
(
1
a¯R ¯R
logarithmic part
+
1
a¯R0 ¯R0
exponential part
)
¯Vs =
b¯A(0)
log(a)π¯J
(
1
a¯R ¯R
+
1
aR0 R0
)
(
1
2π ¯R ¯Σ
¯µ¯s
(¯x − ¯x)⊥
· ¯n(¯x)
|¯x − ¯x|2
d¯s
+¯J
¯x · ¯n
|¯x|2
−
π¯J
¯A(0)
¯x · ¯n)
= ¯Vl + ¯Ve.
0 10 20 30 40 50 60
10
0
10
1
10
2
10
3
10
4
R
ρ
ρ
l
ρ
e
ρ
s
R
0
=11
[a]
0 10 20 30 40 50 60
10
1
10
2
10
3
10
4
10
5
10
6
R
CPUtime
ρ
e
ρ
s
[b]
Figure: [a] Relation between the time
scaling function ρ and the radius R. [b]
shows the CPU time to different
scaling factor ρ.Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

12. Interface problems Expanding interface problem Shrinking interface problem
Efficiency of adaptive scaling function
0 50 100 150
10
−1
10
0
10
1
10
2
10
3
10
4
R
currenttimeT
−10 0 10
−10
0
10
1994, T=45, R=9.41
−50 0 50
−50
0
50
2006, T=500, R=31.62
−100 0 100
−100
0
100
2007, T=2300, R=65.52
−200 0 200
−200
0
200
2016, T=7514, R=122
(40 mins)
(7.9 hours)
(1.09 days)
1994 Hou et al.
50 days
2007 Li et al.
21 days 5.8 days
2016 Zhao et al.
2006 Fast et al.
50 days
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

13. Interface problems Expanding interface problem Shrinking interface problem
Example: Hele-Shaw and shrinking viscous fingering
Shrinking interface: (1) Hele-Shaw cell with suction has singularity at the
sink; (2) Hele-Shaw cell with time dependent gap has no singularity.
The interior region Ω is oil with viscosity µ.The exterior region is air. ∂Ω
represents the interface. The time dependent gap is b(t). The normal n
is pointing inward.
Air pushes oil from the exterior instead of from the interior for the
expanding fingering problem. This is an interior problem.
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

14. Interface problems Expanding interface problem Shrinking interface problem
Governing equations
Equations in the viscous fluid domain
u = −
b2
(t)
12µ
P in Ω, (1)
· u = −
˙b(t)
b(t)
in Ω, (2)
[P]t = σκ on ∂Ω, (3)
V = −
b2
(t)
12µ
∂P
∂n
on ∂Ω. (4)
Eq. (1) follows from the Darcy’s law, where u is the velocity, P is the
pressure, b(t) is the time dependent gap, and µ is the viscosity of oil.
Eq. (2) specifies the incompressible fluid with conserved volume. ˙b(t) is
the time derivative of b(t), which is the lifting speed.
Eq. (3) is the Laplace-Young condition given by the product of surface
tension σ and the curvature κ of the interface.
Eq. (4) expresses the normal velocity V and n is the unit inward normal.
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

15. Interface problems Expanding interface problem Shrinking interface problem
Comparison with experiment
Experiment: τ = 9.6 × 10−6
and b(t) = 1 + t from Nase et al. physical
of fluid (2011). Use random initial shape in simulations.
0 5 10 15
10
0
10
1
10
2
t
numberoffingers
experiment
simulation
ψ= −0.13
[a] [b]
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT

16. Interface problems Expanding interface problem Shrinking interface problem
Fission with exponential gap
(x2
+ y2
)2
= 20x2
3
+ 20y2
5
, b(t) = exp(t) ⇒ ¯b(¯t) = 1 + 0.5¯t; N = 16, 384,
∆¯t = 1E − 4 and τ = 2E − 5.
Show a movie...
Acknowledgement: NSF-DMS
Thank you for your attention...
Modeling and Computation of Moving Interface Problems Shuwang Li, IIT