Hele-Shaw flows with kinetic undercoolingMech. 323, 201{236; Tanveer (2000) J. Fluid Mech. 409, 273{308. Kinetic undercooling received much less attention:Howison (1992) Euro. J. Appl

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

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.


http://n-e-r-v-o-u-s.com/blog/http://www.antonyhall.net

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.


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.


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.


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?


Bubble geometry

Part I: Bubble geometry

(a) Bubble

Ω(t)

β (t)

∂Ω(t)


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.


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.


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)


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.


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.


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.


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.


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.


Channel geometry

Change in scene

48◦ turnaround


Channel geometry

Part II: Channel geometry

(b) Channel

Ω(t)

∂Ω(t)


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.


movie.aviMedia File (video/avi)

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.


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).


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?


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.


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.


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).


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).


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.


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


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).


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.


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


@swmccue

IntroductionBubble geometryChannel geometrySummary

Documents

Hele-Shaw flows with kinetic undercoolingMech. 323, 201{236; Tanveer (2000) J. Fluid Mech. 409, 273{308. Kinetic undercooling received much less attention:Howison (1992) Euro. J. Appl