hele-shaw flows with kinetic undercoolingmech. 323, 201{236; tanveer (2000) j. fluid mech. 409,...
TRANSCRIPT
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Hele-Shaw flows with kinetic undercooling
Scott McCue, Bennett Gardiner, Michael Dallaston, Timothy Moroney
Queensland University of Technology, Brisbane, Australia
15 January 2015, Banff International Research Station
Saffman & Taylor (1958) Proc Roy Soc A 245, 312–329.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 1 / 30
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Introduction
Hele-Shaw cellviscous fingering instabilities
Design studio called Nervous System,
http://n-e-r-v-o-u-s.com/blog/. Artist Antony Hall,
http://www.antonyhall.net.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 2 / 30
http://n-e-r-v-o-u-s.com/blog/http://www.antonyhall.net
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Introduction
Flow configurationsbubbles propagating, contracting and expanding
(a) Bubble
Ω(t)
β (t)
∂Ω(t)
(b) Channel
Ω(t)
∂Ω(t)
Evolving fluid region Ω(t) in free-boundary Hele–Shaw flow(a) a bubble geometry, and (b) a channel geometry.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 3 / 30
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Introduction
Formulationimportant physics in dynamic condition
Stokes flow averaged over small gap:
q = −d2
12µ∇p, ∇·q = 0, ⇒ ∇2p = 0.
Nondimensionalise with φ representingnegative pressure:
∇2φ = 0, x ∈ Ω(t),
vn =∂φ
∂n, x ∈ ∂Ω(t)
φ ∼{
ln |x|, |x| → ∞ (bubble geometry)x, x→∞ (channel geometry) ,
φ =
0 (no regularisation)σκ (surface tension)cvn (kinetic undercooling)
, x ∈ ∂Ω(t).
No-regularisation case ill-posed in onedirection, with finite-time blow-up andcusp formation: Howison et al. (1985) Q.J. Mech. Appl. Math. 38, 343–360;Howison (1986) SIAM J. Appl. Math. 46,20–26.
Surface tension regularises blow-up, butstill unstable: Siegal et al. (1996) J. FluidMech. 323, 201–236; Tanveer (2000) J.Fluid Mech. 409, 273–308.
Kinetic undercooling received much lessattention: Howison (1992) Euro. J. Appl.Math. 3, 209–224.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 4 / 30
-
Introduction
Formulationimportant physics in dynamic condition
Stokes flow averaged over small gap:
q = −d2
12µ∇p, ∇·q = 0, ⇒ ∇2p = 0.
Nondimensionalise with φ representingnegative pressure:
∇2φ = 0, x ∈ Ω(t),
vn =∂φ
∂n, x ∈ ∂Ω(t)
φ ∼{
ln |x|, |x| → ∞ (bubble geometry)x, x→∞ (channel geometry) ,
φ =
0 (no regularisation)σκ (surface tension)cvn (kinetic undercooling)
, x ∈ ∂Ω(t).
No-regularisation case ill-posed in onedirection, with finite-time blow-up andcusp formation: Howison et al. (1985) Q.J. Mech. Appl. Math. 38, 343–360;Howison (1986) SIAM J. Appl. Math. 46,20–26.
Surface tension regularises blow-up, butstill unstable: Siegal et al. (1996) J. FluidMech. 323, 201–236; Tanveer (2000) J.Fluid Mech. 409, 273–308.
Kinetic undercooling received much lessattention: Howison (1992) Euro. J. Appl.Math. 3, 209–224.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 4 / 30
-
Introduction
Formulationimportant physics in dynamic condition
Stokes flow averaged over small gap:
q = −d2
12µ∇p, ∇·q = 0, ⇒ ∇2p = 0.
Nondimensionalise with φ representingnegative pressure:
∇2φ = 0, x ∈ Ω(t),
vn =∂φ
∂n, x ∈ ∂Ω(t)
φ ∼{
ln |x|, |x| → ∞ (bubble geometry)x, x→∞ (channel geometry) ,
φ =
0 (no regularisation)σκ (surface tension)cvn (kinetic undercooling)
, x ∈ ∂Ω(t).
No-regularisation case ill-posed in onedirection, with finite-time blow-up andcusp formation: Howison et al. (1985) Q.J. Mech. Appl. Math. 38, 343–360;Howison (1986) SIAM J. Appl. Math. 46,20–26.
Surface tension regularises blow-up, butstill unstable: Siegal et al. (1996) J. FluidMech. 323, 201–236; Tanveer (2000) J.Fluid Mech. 409, 273–308.
Kinetic undercooling received much lessattention: Howison (1992) Euro. J. Appl.Math. 3, 209–224.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 4 / 30
-
Introduction
Formulationimportant physics in dynamic condition
Stokes flow averaged over small gap:
q = −d2
12µ∇p, ∇·q = 0, ⇒ ∇2p = 0.
Nondimensionalise with φ representingnegative pressure:
∇2φ = 0, x ∈ Ω(t),
vn =∂φ
∂n, x ∈ ∂Ω(t)
φ ∼{
ln |x|, |x| → ∞ (bubble geometry)x, x→∞ (channel geometry) ,
φ =
0 (no regularisation)σκ (surface tension)cvn (kinetic undercooling)
, x ∈ ∂Ω(t).
No-regularisation case ill-posed in onedirection, with finite-time blow-up andcusp formation: Howison et al. (1985) Q.J. Mech. Appl. Math. 38, 343–360;Howison (1986) SIAM J. Appl. Math. 46,20–26.
Surface tension regularises blow-up, butstill unstable: Siegal et al. (1996) J. FluidMech. 323, 201–236; Tanveer (2000) J.Fluid Mech. 409, 273–308.
Kinetic undercooling received much lessattention: Howison (1992) Euro. J. Appl.Math. 3, 209–224.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 4 / 30
-
Introduction
Formulationimportant physics in dynamic condition
Stokes flow averaged over small gap:
q = −d2
12µ∇p, ∇·q = 0, ⇒ ∇2p = 0.
Nondimensionalise with φ representingnegative pressure:
∇2φ = 0, x ∈ Ω(t),
vn =∂φ
∂n, x ∈ ∂Ω(t)
φ ∼{
ln |x|, |x| → ∞ (bubble geometry)x, x→∞ (channel geometry) ,
φ =
0 (no regularisation)σκ (surface tension)cvn (kinetic undercooling)
, x ∈ ∂Ω(t).
No-regularisation case ill-posed in onedirection, with finite-time blow-up andcusp formation: Howison et al. (1985) Q.J. Mech. Appl. Math. 38, 343–360;Howison (1986) SIAM J. Appl. Math. 46,20–26.
Surface tension regularises blow-up, butstill unstable: Siegal et al. (1996) J. FluidMech. 323, 201–236; Tanveer (2000) J.Fluid Mech. 409, 273–308.
Kinetic undercooling received much lessattention: Howison (1992) Euro. J. Appl.Math. 3, 209–224.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 4 / 30
-
Introduction
Kinetic undercoolingwhat is it?
Kinetic undercooling condition says φ = c vn on interface.
Large Stefan number (or small specific heat) limit of one-phase Stefan problem:
1
β
∂φ
∂t= ∇2φ, φ = σκ+ cvn, vn =
∂φ
∂non solid-melt interface :
Evans & King (2000) Q. J. Mech. Appl. Math. 53, 449-473.
Change in curvature in transverse direction: Romero (1981) PhD, Caltech.
Wetting layer on plates: Park & Homsy (1984) J. Fluid Mech. 139, 291–308.
Diffusion of solvent in glassy polymers: Cohen & Erneux (1988) SIAM J. Appl. Math. 48,1451–1465.
Streamer discharges: Ebert et al (2011) Nonlinearity 24, C1–C26.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 5 / 30
-
Introduction
Kinetic undercoolingwhat is it?
Kinetic undercooling condition says φ = c vn on interface.
Large Stefan number (or small specific heat) limit of one-phase Stefan problem:
1
β
∂φ
∂t= ∇2φ, φ = σκ+ cvn, vn =
∂φ
∂non solid-melt interface :
Evans & King (2000) Q. J. Mech. Appl. Math. 53, 449-473.
Change in curvature in transverse direction: Romero (1981) PhD, Caltech.
Wetting layer on plates: Park & Homsy (1984) J. Fluid Mech. 139, 291–308.
Diffusion of solvent in glassy polymers: Cohen & Erneux (1988) SIAM J. Appl. Math. 48,1451–1465.
Streamer discharges: Ebert et al (2011) Nonlinearity 24, C1–C26.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 5 / 30
-
Introduction
Kinetic undercoolingwhat is it?
Kinetic undercooling condition says φ = c vn on interface.
Large Stefan number (or small specific heat) limit of one-phase Stefan problem:
1
β
∂φ
∂t= ∇2φ, φ = σκ+ cvn, vn =
∂φ
∂non solid-melt interface :
Evans & King (2000) Q. J. Mech. Appl. Math. 53, 449-473.
Change in curvature in transverse direction: Romero (1981) PhD, Caltech.
Wetting layer on plates: Park & Homsy (1984) J. Fluid Mech. 139, 291–308.
Diffusion of solvent in glassy polymers: Cohen & Erneux (1988) SIAM J. Appl. Math. 48,1451–1465.
Streamer discharges: Ebert et al (2011) Nonlinearity 24, C1–C26.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 5 / 30
-
Introduction
Kinetic undercoolingwhat is it?
Kinetic undercooling condition says φ = c vn on interface.
Large Stefan number (or small specific heat) limit of one-phase Stefan problem:
1
β
∂φ
∂t= ∇2φ, φ = σκ+ cvn, vn =
∂φ
∂non solid-melt interface :
Evans & King (2000) Q. J. Mech. Appl. Math. 53, 449-473.
Change in curvature in transverse direction: Romero (1981) PhD, Caltech.
Wetting layer on plates: Park & Homsy (1984) J. Fluid Mech. 139, 291–308.
Diffusion of solvent in glassy polymers: Cohen & Erneux (1988) SIAM J. Appl. Math. 48,1451–1465.
Streamer discharges: Ebert et al (2011) Nonlinearity 24, C1–C26.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 5 / 30
-
Introduction
Kinetic undercoolingstability of a straight interface
Say x = f(y, t), thenf = f0(0)− t+ δeµt cosny,
where
µ =µ(0)nt
1 + cn.
All modes unstable for injection case (unlike surface tension case, in which higher modesare stablised).
How does kinetic undercooling regularise problem?
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 6 / 30
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Bubble geometry
Part I: Bubble geometry
(a) Bubble
Ω(t)
β (t)
∂Ω(t)
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 7 / 30
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Bubble geometry
Linear stabilityperturbing a circular bubble
Interface r = s(θ, t).
Small perturbation with δ � 1:
s ∼ s0 + δγn(s0) cosnθ.
Stability determined by Gn(s0) = γn(s0)/s0;turns out
Gn(s0) = Gn(1)(s0 + n)n−1
s0(1 + n)n−1.
r = s0(t)
r
θ
r = s0(t) + �γn(t) cosnθ
1 2 3 4 5 60
2
4
6
8
10
G2
G3
G4
G5
s0
Note for fixed s0:
Gn(s0)/Gn(1) ∼ 1/s0
as n→∞.Unstable both directions.
But well-posed.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 8 / 30
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Bubble geometry
Conformal mappingtwo ideas that we all know about
1 Since φ satisfies ∇2φ = 0 in R2,
W (z, t) = φ(x, y, t) + iψ(x, y, t)
must be analytic function of z = x+ iy in flow domain.
2 Riemann’s mapping theorem says there exist mapping z = f(ζ, t) from disc |ζ| ≤ 1to flow domain R2 \ Ω(t).
Here f must be univalent and analytic in disc except at a point (which representsz =∞).
Choose this point to be ζ = 0, so that
f(ζ, t) ∼ a(t)ζ
as ζ → 0,
where a(t) is real.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 9 / 30
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Bubble geometry
Conformal mappingmap from unit disc
Physical z-plane
φ = c vn
vn =∂φ
∂n
φ ∼ log |z|
∇2φ = 0
Auxiliary ζ-plane
1
z = f(ζ, t)
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 10 / 30
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Bubble geometry
Differential geometry of curveswith complex variables
Say ζ = |ζ|eiν , so boundary is |ζ| = 1.
Recall tangent vector t̂ =dx
ds, where s is arclength.
In complex variables, this is
t̂ =dz
ds=
zν|zν |
,
where z = f(ζ, t), ζ = eiν gives
zν = ieiνfζ(e
iν , t)
= iζfζ on |ζ| = 1.
That is, the unit tangent “vector” is
t̂ =iζfζ|ζfζ |
on |ζ| = 1.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 11 / 30
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Bubble geometry
Differential geometry of curveskinematic condition
Normal vector is perpendicular to tangent, so
n̂ = −it̂ = ζfζ|ζfζ |on |ζ| = 1.
Velocity in normal direction
vn = v · n̂ = R{v ¯̂n
}= R
{ft ¯ζfζ|ζfζ |
}.
Normal derivative
∂φ
∂n= ∇φ · n = R
{ ¯∂W∂z
¯̂n
}= R
{ζwζ|ζfζ |
}.
So kinematic condition vn = ∂φ/∂n becomes
R{ft ¯ζfζ
}= R{ζwζ} on |ζ| = 1.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 12 / 30
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Bubble geometry
Complex variable formulationPolubarinova-Galin equation
Solution for complex potential in mapped plane is
w ∼ log ζ + cV,
where V is analytic function in disc and whose real part on |ζ| = 1 coincides withnormal velocity of the boundary:
R{V} = vn = R{ftζfζ|ζfζ |
}.
Kinematic condition gives Polubarinova-Galin equation
R{ftζfζ
}= 1 + cR{ζVζ} on |ζ| = 1
Howison (1992) Euro. J. Appl. Math. 3, 209–224.
Determine f that satisfies this equation and initial condition.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 13 / 30
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Bubble geometry
Simple numerical resultsfour-fold symmetric bubbles
Mapping will have series expansion
z(ζ, t) =a−1(t)
ζ+∞∑n=0
an(t)ζn.
Truncate series and enforce PG equation at equally spaces points on circle |ζ| = 1.
−1 0 1
−1
0
1
x
y(a) Contracting boundary
−5 0 5−5
0
5
x
y(b) Expanding boundary
0 π/2−10
−8
−6
−4
−2
ν
κ
(c) Contracting boundarycurvature
0 π/2−2
−1
0
ν
κ
(d) Expanding boundarycurvature
Numerical results for contracting andexpanding bubbles, showing (a,b) theboundary. Both cases exhibit cornerformation.
Curvature κ in (c,d) curvature as afunction of ν = arg ζ over the firstquadrant 0 < ν < π/2.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 14 / 30
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Bubble geometry
Elliptic initial conditioncorner persists
−0.1 0 0.1
−0.05
0
0.05
x
y(a) Contracting ellipse
0.022 0.044 0.0670
π/2
π
t̃
θ(b) Corner angle
(a) The numerical solution (circles) for a contracting bubble with kinetic undercooling,compared to the exact solution from (solid lines) to the simple geometric rule vn = 1.
(b) The corner angle θ of the exact solution, starting at π (no corner), and ending at zero.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 15 / 30
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Channel geometry
Change in scene
48◦ turnaround
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 16 / 30
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Channel geometry
Change in scene
48◦ turnaround
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 16 / 30
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Channel geometry
Part II: Channel geometry
(b) Channel
Ω(t)
∂Ω(t)
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 17 / 30
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Channel geometry
Saffman-Taylor fingers with surface tensiona well-studied problem
Experiments in channel geometry demonstrate a single finger emerging, of width λ = 1/2.
Time-dependent solutions in channel geometry with surface tension (Dylan Lusmore):
Isolating single travelling finger, exact solutions with γ = 0 have arbitrary λ.
Naive perturbation expansion in powers of γ still have arbitrary λ: McClean & Saffman(1981) J. Fluid Mech. 102, 455–469.
Numerical solutions show discrete set of λ: Vanden-Broeck (1983) Phys. Fluids 26,2033–2034.
Terms exponentially small in γ responsible for selection of λ = 1/2:Combescot et al. (1986) Phys. Rev. Lett. 56, 2036–2039.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 18 / 30
movie.aviMedia File (video/avi)
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Channel geometry
Discrete families of Saffman-Taylor fingersdiscrete families of solutions
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0.5
0.6
0.7
0.8
0.9
γ
λ
m = 0
m = 1
m = 2
m = 3m = 4
..
.m = 8
Discrete set of finger widths for each γ.
On upper branches, α1/2((1 + α)1/2 − 1) = γ2(n+ 7/4)2,where α = (2λ− 1)/(1− λ)2Chapman (1999) Euro. J. Appl. Math. 10, 513-534.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 19 / 30
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Channel geometry
Discrete families of Saffman-Taylor fingerssome with exotic shapes
-0.1 -0.05
0.4
0.8
x
y
m = 1
m = 2m = 3m = 4m = 5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0.5
0.6
0.7
0.8
0.9
γ
λ
m = 1
m = 2
m = 3
m = 4m = 5
..
.m = 9
γ2,1
γ3,2
γ4,3γ4,1
γ9,8γ9,6γ9,4
γ9,2
Discrete set of finger widths for each γ.
For sufficiently large γ, nonconvex finger shapes (here γ = 2.03).
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 20 / 30
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Channel geometry
Channel geometry with kinetic undercoolingtravelling fronts and fingers
Large kinetic undercooling (c > 1) gives travelling front with corner formation: Chapman& King (2003) J. Eng. Maths 46, 1–32.
Smaller kinetic undercooling (c < 1) gives single finger.
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
−0.5
0
0.5
1
t→ ∞
t = 0
t = tcnr
x
y(a) c = 2: Corner formation
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
−0.5
0
0.5
1
t→ ∞
t = 0
x
y(b) c = 0.18: Finger formation
Selection of λ = 1/2 by kinetic undercooling: Chapman & King (2003).
But numerical results give continuum of solutions. Why?
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 21 / 30
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Channel geometry
Interacting streamersfinger-shaped electrical discharges
Streamers caused by applying strong electric field to weakly ionized gas, leading toionization reaction via collisions of highly energetic electrons with neutral molecules.
Thin charge layer and associated ionization front forms finger shape.
Periodic array evolve to travelling finger profiles:Luque, Brau & Ebert (2008) Phys. Rev. E 78, 016206.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 22 / 30
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Channel geometry
Saffman-Taylor fingers with kinetic undercoolingmathematical formulation
Frame of reference so velocity vanish at nose, complex potentialw(z) = φ(x, y) + iψ(x, y), where ψ is streamfunction.
Conformal map ζ = ξ + iη = e−πw from strip to upper half χ-plane. Interface isξ ∈ [0, 1], where ξ = 0 is tail and ξ = 1 is nose.
Redefine kinetic undercooling � = c π/2(1− λ).
Dynamic condition and Hilbert transform:
q = 2�qξ cos θdθ
dξ+ cos θ,
log q = −ξ
π−∫ 10
θ(ξ′)
ξ′(ξ′ − ξ)dξ′,
θ(0) = 0, q(0) = 1, θ(1) = −π
2, q(1) = 0.
Exact solution for � = 0 is q0 = ((1− ξ)/(1 + αξ))1/2, θ0 = cos−1 q, whereα = (2λ− 1)/(1− λ)2.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 23 / 30
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Channel geometry
Saffman-Taylor fingers with kinetic undercoolingthe selection problem
Analytically continue governing equations into ζ-plane.
Look for asymptotic solution:
θ ∼∞∑n=0
�nθn, q ∼∞∑n=0
�nqn as �→ 0.
Can solve explicitly for θ0, q0, θ1, q1, but λ arbitrary.
Factorial/power divergence, truncate optimally by considering limit n→∞.
Exponentially small remainder switched on across Stokes line that emerges fromsingularity and intersects real axis.
Two contributions must exactly cancel, giving solvability condition
α1/2((1 + α)1/2 − 1) = �(2n+ 7/5),
where n = 0, 1, 2, . . . (discrete branches for fixed �): Chapman & King (2003).
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 24 / 30
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Channel geometry
Saffman-Taylor fingers with kinetic undercoolingcontinuum of solutions
λmin
0 0.5 10
0.2
0.4
0.6
ε
λ(b) λ vs ε
Numerical scheme of Mclean & Saffman, Vanden-Broeck: continuum of solutions, nodiscrete set like Chapman & King (2003).
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 25 / 30
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Channel geometry
Saffman-Taylor fingers with kinetic undercoolingand surface tension too
Numerical scheme of Mclean & Saffman, Vanden-Broeck, cannot distinguish betweenanalytic fingers and corner-free but not analytic.
Suppose that solutions with kinetic undercooling and surface tension must be analytic:Tanveer & Xie (2003) Comm. Pure Appl. Math. 56, 353–402.
Take limit γ → 0.
−0.2−0.4−0.6−0.8−1
−0.5
0
0.5
x
y
0 0.2 0.4 0.6 0.8 10.56
0.58
0.6
γ
λ
Here � = 0.1, γ = 0.03, 0.5, 1.
Extrapolate to estimate analytic finger with γ = 0.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 26 / 30
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Channel geometry
Saffman-Taylor fingers with kinetic undercoolingrepeat exercise
For example, here is � = 0, 0.1, 0.2.
0 0.2 0.4 0.6 0.8 1
0.5
0.58
0.66
0.74
γ
λ
0 0.2 0.4 0.6 0.8 1
0.5
0.58
0.66
0.74
γ
λ
0 0.2 0.4 0.6 0.8 1
0.5
0.58
0.66
0.74
γ
λ
Putting it together:
0.5 1 1.5 2
0.25
0.5
0.75
�
λ
λmin(�)
0.9 1
0.96
1
0.2 0.4 0.6 0.8 1
0.5
0.6
0.7
0.8
0.9
1
c
λ
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 27 / 30
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Channel geometry
Saffman-Taylor fingers with kinetic undercoolinglarge kinetic undercooling
−0.2−0.4−0.6−0.8−1
−1
−0.5
0
0.5
1
x
y
Here γ = 0.03, � = 0, 1, 10, 3500.
Dashed line is semi-circle with unit radius: Chapman & King (2003).
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 28 / 30
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Summary
Summary of ideasif time permits
Hele-Shaw flows amenable to complex variable techniques.
Bubbles with kinetic undercooling unstable (but well-posed) in both directions, withcorners developing and persisting.
Exponentially small terms in hyperasymptotic expansion responsible for selecting discretefamilies of fingers, for both surface tension and kinetic undercooling.
Numerical solution of fingers with kinetic undercooling bit subtle: we add surface tensionand take away.
Exotic shapes of Saffman-Taylor fingers with surface tension on higher branches.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 29 / 30
-
Summary
Summary of ideasif time permits
Hele-Shaw flows amenable to complex variable techniques.
Bubbles with kinetic undercooling unstable (but well-posed) in both directions, withcorners developing and persisting.
Exponentially small terms in hyperasymptotic expansion responsible for selecting discretefamilies of fingers, for both surface tension and kinetic undercooling.
Numerical solution of fingers with kinetic undercooling bit subtle: we add surface tensionand take away.
Exotic shapes of Saffman-Taylor fingers with surface tension on higher branches.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 29 / 30
-
Summary
Summary of ideasif time permits
Hele-Shaw flows amenable to complex variable techniques.
Bubbles with kinetic undercooling unstable (but well-posed) in both directions, withcorners developing and persisting.
Exponentially small terms in hyperasymptotic expansion responsible for selecting discretefamilies of fingers, for both surface tension and kinetic undercooling.
Numerical solution of fingers with kinetic undercooling bit subtle: we add surface tensionand take away.
Exotic shapes of Saffman-Taylor fingers with surface tension on higher branches.
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 29 / 30
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Summary
Thanks to the Australian Research Council Discovery Project Scheme DP140100933,Banff International Research Station
See:
M.C. Dallaston & S.W. McCue (2013) Bubble extinction in Hele-Shaw flow with surfacetension and kinetic undercooling regularization, Nonlinearity 26, 1639–1665.
M.C. Dallaston & S.W. McCue (2014) Corner and finger formation in Hele-Shaw flowwith kinetic undercooling regularisation, European Journal of Applied Mathematics 25,707–727.
B.P.J. Gardiner, S.W. McCue, M.C. Dallaston & T.J. Moroney (2015) Saffman-Taylorfingers with kinetic undercooling, Physical Review E, under review.
Contact: Scott McCue, Queensland University of Technology, Brisbane, Australia
Twitter: @swmccue
Scott McCue (QUT) Hele-Shaw flows with kinetic undercooling 15 January 2015 30 / 30
@swmccue
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