complex variable methods and moving boundary …...complex variable methods and moving boundary...

Complex variable methods and moving boundaryproblems

Linda Cummings, NJIT

IMA, Minneapolis, March 24 2009

Linda Cummings, NJIT Complex variable methods and moving boundary problems

Overview

I Aim to give an overview of some complex variable methodsusing case studies, in particular:

I The Hele-Shaw moving boundary problemI Quick introduction of experimental setup; mathematical

model; widespread relevance;

I Complex variable methods for Hele-ShawI Conformal mappingI The Schwarz function

I Other applications of complex variable methods: Slow viscousflow (higher order!) and beyond


The Hele-Shaw moving boundary problem

I H.S. Hele-Shaw “The flow of water” (Nature, 1898)I Hele-Shaw cell: Two parallel glass/perspex plates, separated

by a narrow gap, which is partially filled with viscous fluid.I Free boundary between air and liquid can be made to move by

suction/injection of viscous fluid – moving boundary problem.


Saffman-Taylor “finger” (1958)

on 19 March 2009rspa.royalsocietypublishing.orgDownloaded from


Paterson’s radial fingers (1981)

Lots of nice subsequent work on radial geometry, e.g. Couder et al.Linda Cummings, NJIT Complex variable methods and moving boundary problems

Model equations

Fluid velocity u = (u, v ,w), and pressure p satisfy:

ρdU

dT= −PX + µ(UXX + UYY + UZZ ),

ρdV

dT= −PY + µ(VXX + VYY + VZZ ),

ρdW

dT= −PZ + µ(WXX + WYY + WZZ ),

UX + VY + WZ = 0,

with zero velocity boundary conditions at Z = 0, b. Equationssimplify since ε = b/L 1.


Reduced (dimensionless) model

I Reduced model: p(x , y)

px = uzz , py = vzz

with u = v = w = 0 on z = 0, 1

gives u =1

2pxz(z − 1) v =

1

2pyz(z − 1).

I Integrate continuity equation ux + vy + wz = 0 across cell∫ 1

0(ux + vy )dz = −[w ]10 = 0 ⇒ pxx + pyy = 0

I Pressure is a velocity potential for the averaged flow.

(u, v) :=

∫ 1

0(u, v)dz = − 1

12∇p.


Historical note

© 1898 Nature Publishing Group

I Original motivation: Flow streamlines within the cellreproduce those for 2D inviscid flow around solid obstacles,also described by velocity potential u = ∇φ.

I Hele-Shaw cell completely dominated by viscous effectsthough! Streamlines produced by a very different mechanism.


Moving boundary problem

I Averaged 2D flow problem in (x , y)-space,

v = −∇p, ∇2p = 0 in Ω(t) ⊂ R2.

I Need boundary conditions at ∂Ω(t), an initial condition Ω(0),plus any driving singularities in pressure.

I Dynamic boundary condition: jump in pressure across ∂Ω(t)balanced by surface tension, γ

p = γκ on ∂Ω(t),

Often γ 1 → zero surface tension (ZST) approximation,

p = 0 on ∂Ω(t).

I Kinematic boundary condition: At ∂Ω(t), (fluid velocity innormal direction)=(velocity of ∂Ω(t) in normal direction),

v · n = −∂p

∂n= Vn.


Widespread relevance of the model

I Free surface flow through a porous medium (e.g. flow ofgroundwater) - Darcy flow.

I One-phase Stefan problem reduces to Hele-Shaw freeboundary problem in limit as specific heat → 0.

I Electrochemical machining/forming.

I Squeeze films.


Suction/injection through a point sink/source

I Driving pressure singularity varies. Paradigm example is pointsource/sink at the origin (or ∞), strength Q,

p ∼ − Q

2πlog r as r → 0 (r = (x2 + y2)1/2).

Flow locally radial with v = −∇p = (Q/2πr)er .


Complex potential

I Pressure satisfies moving boundary problem

∇2p = 0 in Ω(t), p = 0 and∂p

∂n= −Vn on ∂Ω(t),

p ∼ − Q

2πlog r as r → 0, Ω(0) specified.

I Key observation: Writing z = x + iy , pressure is the real partof a complex potential w(z , t), complex analytic except atz = 0, and

w(z) ∼ − Q

2πlog z as z → 0, and <(w) = 0 on ∂Ω(t).


Time-dependent conformal map: z = f (ζ)

I Know analytic complex potential exists, with specifiedsingularity, and boundary conditions.

I Difficulty: Domain Ω(t) unknown. Introduce time-dependentconformal map z = f (ζ, t) from a known domain (e.g. theunit disc |ζ| ≤ 1) onto Ω(t).

I f exists by Riemann mapping theorem, and is uniquely definedif we specify f (0) = 0 and f ′(0) ∈ R+, for example.

!" z−plane

#

Unit disk

z=f(

(t)

Evolving fluid domain

!,

plane

$)

U


Complex potential in the ζ-plane

I Define complex potential in the ζ-plane by W (ζ) = w(f (ζ)).

I Since pressure p = <(w(z)

), where w complex analytic with

w(z) ∼ − Q

2πlog z as z → 0, and <(w) = 0 on ∂Ω(t), . . .

I . . . we must have W complex analytic on |ζ| ≤ 1, with

W (ζ) ∼ − Q

2πlog ζ as ζ → 0, and <(W ) = 0 on |ζ| = 1.

I Thus

W (ζ) = − Q

2πlog ζ.


Kinematic boundary condition n · ∇p = −Vn

I Since p = 0 defines the boundary curve, we have

n = ±∇p/|∇p|, and Vn = ∓ pt

|∇p|on ∂Ω(t).

I Kinematic condition can thus be written

∂p

∂t= |∇p|2 on ∂Ω(t).

I Recall p = <(w(z)) = <(W (ζ)), and W (ζ) = −Q/(2π) log ζ.So, on |ζ| = 1,

|∇p|2 = p2x + p2

y = |w ′(z)|2 =|W ′(ζ)|2

|f ′(ζ)|2=

Q2

4π2|f ′(ζ)|2,

and pt = <(Wζζt + Wt

)= − Q

2π<(ζtζ

),

where we determine ζt via z = f (ζ, t) ⇒ 0 = fζζt + ft .


Putting this all together: Polubarinova-Galin equation

<(ζtζ

)= − Q

2π|f ′(ζ)|2, ζt = − ft(ζ)

f ′(ζ)

⇒ <(ζf ′(ζ)ft(ζ)

)=

Q

2πon |ζ| = 1

I Conformal mapping functions f that satisfy thisPolubarinova-Galin equation give solutions to MBP.

I Time-reversible: t 7→ −t and Q 7→ −Q leaves equationunchanged. Exchanging a point source for a point sink (samestrength) exactly reverses the free boundary motion.

I Remarkable fact: Many simple exact solutions!I Polynomials; rational functions; log-rational functions all work.

I More generally:

<(ζf ′(ζ)ft(ζ)

)= −<(ζW ′(ζ)) on |ζ| = 1


Simple example: Nth order polynomial

P-G equation: <(ζf ′(ζ)fτ (ζ)

)=

Q

2πon |ζ| = 1

I f (ζ) = a1(τ)ζ + aN(τ)ζN , assume real coefficients.

I Substitute in P-G equation using fact that on |ζ| = 1, ζ = e iθ.

I Equating θ-dependence in P-G gives two ODEs for a1, aN :

d

dt

(a2

1 + Na2N

)=

Q

π,

d

dt(aN

1 aN) = 0.

I (Integrability is no accident.)

I With initial conditions a1(0) and aN(0), solution is immediate.


Evolution in time: N = 8

For the point sink case, finite-time blow-up is (almost!) inevitablevia cusp formation in the free boundary.

–1

–0.5

0

0.5

1

–1 –0.5 0.5 1


Solution blow-up with suction

I Advancing viscous fluid (injection) stable, while retreatingcase (suction) is unstable (in fact ill-posed).

I This blow-up for suction problems on finite domains may beunderstood by considering the time-reversibility.

I Can only avoid blow-up if all fluid is removed through thepoint sink (domain shrinks to nothing about this point).

I However, injection from origin into an initially empty domainleads (uniquely) to the trivial solution of an expanding circle.

I Hence only suction from the center of a circular fluid domaincan shrink smoothly to nothing. All other suction scenariosmust break down before all fluid is extracted – by singularityformation in the free boundary, or by topology change.

I Suction from infinity (expanding bubble) better: manynontrivial solutions that exist ∀t.


Solution blow-up

I Singularity formation in the free boundary is associated withthe arrival of a derivative-zero (or singularity) of the mappingfunction f (ζ, t) at the unit circle, from |ζ| ≥ 1.

I “Generic” solution breakdown is via a 3/2-power cusp in thefree boundary (corresponding to a simple zero of f ′(ζ) at theboundary).

I Many other non-generic blow-up scenarios possible (canalways construct desired blow-up using time-reversibility).

I Regularization of the unstable model (by surface tension,kinetic undercooling, etc.) is a whole different talk.


Method 2: Schwarz function of the free boundary

p = <(w(z)), ∇2p = 0 in Ω(t),

p = 0,∂p

∂n= −Vn on ∂Ω(t).

I Much neater way to obtain solutions to the MBP uses idea ofthe Schwarz function of ∂Ω(t).

I Schwarz function is unique function g(z , t), analytic in someneighborhood of ∂Ω(t), such that z = g(z , t) defines ∂Ω(t).

I Can be shown that for any point z ∈ ∂Ω,

∂z

∂s=

1√g ′, Vn = − i

2

gt√g ′,

where ′ = d/dz and s is arclength along ∂Ω.


Schwarz function and complex potential

∂z

∂s=

1√g ′, Vn = − i

2

gt√g ′,

I On ∂Ω(t),

dw

dz=∂w

∂s/∂z

∂s=

(∂p

∂s+ i

∂p

∂n

)√g ′ = −iVn

√g ′ = −1

2

∂g

∂t.

I Since both sides of the final equality are analytic in someneighborhood of ∂Ω(t), we may analytically continue todeduce that

dw

dz= −1

2

∂g

∂twherever both sides exist.


Schwarz function evolution equation

dw

dz= −1

2

∂g

∂twherever both sides exist. (1)

I w analytic on Ω(t) except at specified pressure singularities.I In general, though g is analytic in a neighborhood of ∂Ω(t), it

will have singularities within Ω(t).

I Idea: Eliminate singularities on RHS (1) that don’t correspondto specified pressure singularities, and match those that do.

I Yields ODEs for time-dependent coefficients in Schwarzfunction.

I Again use conformal maps as a tool.

g(z) = z = f (ζ) = f (ζ) = f (1/ζ) on |ζ| = 1.

Since first and last terms are analytic in nhd of free bdy,analytic continuation ⇒ equal everywhere, thus

g(z) ≡ f (1/ζ).



dw

dz= −1

2

∂g



will have singularities within Ω(t).I Idea: Eliminate singularities on RHS (1) that don’t correspond

to specified pressure singularities, and match those that do.I Yields ODEs for time-dependent coefficients in Schwarz

function.

I Again use conformal maps as a tool.

g(z) = z = f (ζ) = f (ζ) = f (1/ζ) on |ζ| = 1.


g(z) ≡ f (1/ζ).



dw

dz= −1

2

∂g



will have singularities within Ω(t).I Idea: Eliminate singularities on RHS (1) that don’t correspond

to specified pressure singularities, and match those that do.I Yields ODEs for time-dependent coefficients in Schwarz

function.I Again use conformal maps as a tool.

g(z) = z = f (ζ) = f (ζ) = f (1/ζ) on |ζ| = 1.


g(z) ≡ f (1/ζ).


Schwarz function approach: Implementation

dw

dz= −1

2

∂g

∂t, g(z) ≡ f (1/ζ), (2)

I Illustrate approach for mapping function (Howison)

z = f (ζ) =a(t)

ζ+

N∑

k=1

ω−k log(c(t)ωk − ζ),

a(t) > 0; ωN = 1 and c(t) > 1 (so f conformal on |ζ| ≤ 1).Maps to exterior of a “bubble”. With injection or suctionfrom infinity, w(z) ∼ (Q/2π) log z as |z | → ∞.

I Schwarz function:

f (1/ζ) = a(t)ζ +N∑

k=1

ωk log(c(t)ζω−k − 1)−N∑

k=1

ωk log ζ,

singular within unit disc at ζ = 0 and ζk = ωk/c,corresponding to z =∞ and zk = f (ωk/c) within Ω(t).


Schwarz function example

dw

dz= −1

2

∂g

∂t, g(z) ≡ f (1/ζ), (3)

I Near the points ζk ,

g(f (ζ)) ∼ ωk log(ζ − ζk) + O(1)

⇒ g(z) ∼ ωk log(z − zk) + O(1).

I Thus, requiring analyticity of dw/dz at zk in (3) gives

zk = f (ωk/c) = constant, 1 ≤ k ≤ N,

N invariants of the motion.

I Matching similarly at ζ = 0 (z =∞) gives 2nd equation,d

dt(a2 − Nac) = −Q/π.


Exact solutions (Howison) versus experiments (Paterson)

By suitable choice of parameters/initial conditions can generatesolutions that mimic closely experimental observations.


Issues of solution selection, here and elsewhere

I Many examples of “selection problems” in Hele-Shaw flow,where experimental/numerical outcome well-defined, buttheoretically (ZST) one can obtain many possible solutions.

on 19 March 2009rspa.royalsocietypublishing.orgDownloaded from

finger that begins to bulge but is quickly drawn into the sink

forming a wedge. Note that the top of this finger is narrower

than the one corresponding to the larger surface tension S

!0.01 !Fig. 2". A look at the tangent angle #($ ,t) aroundthe finger tip for S!4"10#4, shown in Fig. 4, strongly

suggests the formation of a corner when the interface touches

the sink. The tangent angle # appears to develop a disconti-nuity at the finger tip ($!0), precisely when the Hele-Shawsolution breaks down. The computations of Nie and Tian,12

for another type of initial data, also suggest this breakdown

scenario. The formation of the wedge and the tip corner seem

to be generic for this type of flow.

Smaller values of surface tension reveal new features in

the interface evolution. Figure 5 shows the interface shape

for S!5"10#5. The finger clearly bulges and develops a

well-defined neck before it becomes a wedge. It is interesting

to note that this neck appears at a height close to that of the

zero-surface-tension cusp. It is conceivable that the forma-

tion of the neck and the bulging of the finger are due to the

influence of the zero-surface-tension singularity. In fact, a

look at the curvature shown at Fig. 6!a" for the time t

!0.2860, which is very close to tc!0.2842, shows alreadythe appearance of two symmetric spikes corresponding to the

location of the neck. The behavior of the interface curvature

at subsequent times is shown in Figs. 6!c" and 6!d". Note inparticular that the curvature grows almost ten times in mag-

nitude from t!0.2916 %Fig. 6!c"& to t!0.2918 %Fig. 6!d"&.The sharp and large spike at $!0 is an indication of thecorner singularity forming as the tip of the wedge touches

the sink. We use N!16 384 and 't!2"10#7 to resolve

accurately this large curvature motion. At t!0.2918, the dis-tance of the wedge tip to the sink is r!2.92"10#3 and the

tip curvature is (!#1371.11. The sink pressure #log rdominates the surface tension pressure S( .

3. The interface limiting behavior as S˜0

We investigate now the interface limiting behavior be-

fore and past tc . We present numerical evidence to show that

an asymptotic corner angle is selected in the limit as surface

tension tends to zero when the finger tip is about to reach the

sink. The computations presented here also suggest that the

vanishing surface tension solution is singular at the finger

neck.

To obtain information on the behavior of the wedge

angle in the limit as surface tension tends to zero, we com-

FIG. 4. Behavior of the tangent angle #($ ,t) around the finger tip ($!0)as the interface is about to collapse, for S!4"10#4 and A)!1. The tan-gent angle, plotted against the parametrization variable $ at the times t

!0.2920, 0.2930, and 0.2932, appears to develop a discontinuity. N

!16 384 and 't!2"10#7.

FIG. 5. Evolution of the initially circular fluid blob past tc , for S!5"10#5 and A)!1. !a" The interface plotted at t!0.2880, 0.2900, and0.29181. !b" A close-up of the interface finger at the times t!0.2840,0.2860, 0.2880, 0.2900, and 0.291 81. N!16 384 and 't!2"10#7 for the

last stage of the motion. At t!0.29181, the distance of the wedge tip to thesink is 2.92"10#3 and the tip curvature is #1371.11.

2476 Phys. Fluids, Vol. 11, No. 9, September 1999 Ceniceros, Hou, and Si

Copyright ©2001. All Rights Reserved.

pare the interfaces for a set of decreasing values of surface

tension. Since the velocity of the interface depends on sur-

face tension, we compare the interfaces when their finger tips

reach the same level above the sink rather than at a fixed

time. As surface tension is reduced, the finger tip reaches the

given level faster. Figure 7 provides some indication of the

asymptotic trend of the fingers as surface tension is succes-

sively halved from S!8"10#4 to S!5"10#5. The fixed

level is y!0.01 so that the finger tips are very close to the

sink. As surface tension is decreased, the fingers develop a

neck at about y!0.27. However, away from the neck, the

finger width changes very little. More precisely, as surface

tension is reduced, the change in the finger width decreases.

Table I gives the difference !(S ,S/2) between the width of

the finger corresponding to a surface tension S and that cor-

responding to S/2 at three different levels. It is observed that

!(S ,S/2) decreases as surface tension is reduced. The fin-

gers are converging to an asymptotic shape. Table II sug-

gests that an asymptotic angle is selected for the wedge as it

touches the sink. The difference between consecutive angles

FIG. 6. Interface curvature "(# ,t) versus # around the finger tip (#!0) at

different times past tc for S!5"10#5 and A$!1. %a& t!0.2860. %b& t

!0.2880. %c& t!0.2916. %d& t!0.291 81. N!16 384 and !t!2"10#7

were used to resolve the largest curvature %d&.

FIG. 7. Comparison of the interface finger for a sequence of surface ten-

sions with A$!1. From the outer curve inwards, the fingers correspond to

the surface tension values S!8"10#4, 4"10#4, 2"10#4, 1"10#4, and

5"10#5. Each interface is plotted when the tip of the finger reaches the

fixed level y!0.01 at x!0. N!16384 and !t!2"10#7.

FIG. 8. Limiting behavior of the interface before tc!0.2842 for A$!1.

This figure shows a close-up picture of the interface around x!0 for a set of

surface tension values, decreasing from top to bottom, and plotted at time

t!0.2840. The zero-surface-tension solution is also shown. N!4096 and

!t!5"10#5.

TABLE I. Change in the finger widths as surface tension is decreased for

A$!1. The first column shows the height level at which the fingers are

compared. Columns 2–5 give the difference !(S ,S/2) between the width of

the finger corresponding to a surface tension S and that corresponding to

S/2.

y

!%0.0008,

0.0004&!%0.0004,

0.0002&!%0.0002,

0.0001&!%0.0001,

0.00005&

0.1 1.63"10#3 1.17"10#3 8.96"10#4 6.62"10#4

0.06 1.24"10#3 9.14"10#4 6.98"10#3 5.39"10#4

0.02 5.99"10#4 4.57"10#4 3.45"10#4 2.47"10#4

TABLE II. The angle of the wedge %in radiants& for a decreasing set of

surface tensions. The variation %third column& is the difference between

consecutive angles, corresponding to surface tensions S and 2S .

S Wedge angle Variation

8"10#4 0.674 59 ¯4"10#4 0.657 19 0.0174

2"10#4 0.643 99 0.0132

1"10#4 0.636 60 0.0074

5"10#5 0.633 59 0.0030

2477Phys. Fluids, Vol. 11, No. 9, September 1999 Numerical study of Hele-Shaw flow with suction

Copyright ©2001. All Rights Reserved.

(S & T, P; Ceniceros, Hou & Si)Linda Cummings, NJIT Complex variable methods and moving boundary problems

Nonzero surface tension (NZST) generalization

I The difficulty is apparent when we consider NZSTgeneralization of Schwarz function equation:

dw

dz= −1

2

∂g

∂t+

iγ

2

∂

∂z

(g ′′

(g ′)3/2

)

I ZST solutions typically have zeros of g ′(z) within Ω(t), whichare not permissible when γ > 0.

I Asymptotics beyond all orders required in neighborhood ofsuch zeros to analyze selection of appropriate ZST solution.(Combescot et al; Shraiman; Hong & Langer; Chapman;Tanveer; C & King)


Other notable results

I Baiocchi transform (Elliott & Janovsky; weak formulation)

I Conserved quantitles (Richardson)

I Quadrature domains preserved under the dynamics(Gustafsson & others)

I Other powerful solution techniques using theory of Fuchsiandifferential equations (Polubarinova-Kochina; Craster &co-workers)

556 L. J. Cummings

2 4 6 8 10

-10

-7.5

-5

-2.5

2.5

5

7.5

10

Figure 3. Plot showing the free boundary at times t = 0.5, 1, !/2 with wedge angle ! = 0.3!. Thefluid flow is from left to right, and the initial free boundary is along the positive y-axis.

that " be real. In terms of the wedge angle !! and a = 1 ! ! we have

L1 = !iei!![(2a/3)(2a + 1)(a + 1)]! 23 , L2 = ei!![(2a/3)(2a + 1)(a + 1)]! 2

3 ;

M1 = ! sin !![(2a/3)(2a + 1)(a + 1)]! 23 , M2 = ! cos !![(2a/3)(2a + 1)(a + 1)]! 2

3 ;

R0 =

!9(2a + 1)

32a2(a + 1)2

" 13

.

Hence the solution to the problem. In terms of the original variables a parametric

representation of the free boundary is given, for 0 ! # ! 1, by

z = Z(#)

# t

0f($) d$ =

L1

# t

0f($) d$

(1 ! #)(1 ! i cot !!)(B1(#) + iB2(#)),

=ie!ia!(i

"# +

"1 ! #)2a

# t

0f($) d$

(1 ! #)[(2a/3)(2a + 1)(a + 1)]23

$1

2+ 2a(1 ! #) ! a

"1 ! #

i"# +

"1 ! #

%.

The function f(t) is arbitrary, being the dimensionless timescale on which the pressure

I . . .


Beyond Hele-Shaw: 2D Stokes flow with free boundaries

I Relevant for many real-world applications (industrial sinteringprocesses; glass manufacturing,. . . )

I 2D slow flow, ∇p = µ∇2u and ∇ · u = 0, described bystreamfunction ψ and Airy stress function A, both biharmonic,such that u = (ψy ,−ψx) and

σ11 = −p + 2∂2ψ/∂x∂y = −2∂2A/∂y2

σ12 = σ21 = ∂2ψ/∂y2 − ∂2ψ/∂x2 = 2∂2A/∂x∂yσ22 = −p + 2∂2ψ/∂x∂y = −2∂2A/∂x2

I Stress conditions and kinematic condition

σn = −γκn− pn, u · n = Vn, on ∂Ω(t).

I Initial conditions + driving singularities complete the model.


2D Stokes flow: complex variable formulation

I Biharmonic functions have a complex analytic representation(Goursat). A, ψ biharmonic conjugates, Azz − iψzz = 0, and

A + iψ = −(zφ(z) + χ(z)), z = x + iy ,

for φ, χ analytic on Ω(t). These functions fully specify flow.

I Stress conditions may be written in terms of the Airy stressfunction in integrable form, leading to

∂

∂s

[φ(z) + zφ′(z) + χ′(z)− iγ

2µ

∂z

∂s+ α + iβ +

pz

2

]= 0 on ∂Ω(t).

I With simply-connected domain integrate and setα = β = p = 0 w.l.o.g.


Simply-connected case: Procedure

I Time-dependent conformal map z = f (ζ, t) again maps Ω(t)to unit disk, and Φ(ζ) = φ(f (ζ)), X (ζ) = χ(f (ζ)).

I Again reformulate boundary conditions on |ζ| = 1.

I KBC u · n = Vn becomes

2Φ(ζ)− iγ

2

∂z

∂s=∂f

∂t+ f ′(ζ)

dζ

dton |ζ| = 1

∂z

∂s= iζ

f ′(ζ)

|f ′(ζ)|,

dζ

dt= iζθ on ζ = e iθ

I Divide through by ζf ′(ζ) and take real part to obtain

<[

1

ζf ′(ζ)

(2Φ(ζ)− ∂f

∂t

)]= − γ

2|f ′(ζ)|on |ζ| = 1.

I Use freedom to fix values of Φ and f at ζ = 0 to definefunction P(ζ) = [ · ], analytic on |ζ| ≤ 1. Dirichlet problem for<(P) can be solved explicitly, given f .


Simply-connected case: Procedure

I Time-dependent conformal map z = f (ζ, t) again maps Ω(t)to unit disk, and Φ(ζ) = φ(f (ζ)), X (ζ) = χ(f (ζ)).

I Again reformulate boundary conditions on |ζ| = 1.I KBC u · n = Vn becomes

2Φ(ζ)− iγ

2

∂z

∂s=∂f

∂t+ f ′(ζ)

dζ

dton |ζ| = 1

∂z

∂s= iζ

f ′(ζ)

|f ′(ζ)|,

dζ

dt= iζθ on ζ = e iθ

I Divide through by ζf ′(ζ) and take real part to obtain

<[

1

ζf ′(ζ)

(2Φ(ζ)− ∂f

∂t

)]= − γ

2|f ′(ζ)|on |ζ| = 1.

I Use freedom to fix values of Φ and f at ζ = 0 to definefunction P(ζ) = [ · ], analytic on |ζ| ≤ 1. Dirichlet problem for<(P) can be solved explicitly, given f .


Stress boundary condition

I Reformulating SBC on unit circle leads to an equation thatcan be analytically continued off |ζ| = 1:

∂

∂t

(f ′(ζ)f (1/ζ)

)+

∂

∂ζ

(ζP(ζ)f ′(ζ)f (1/ζ)

)= −2X ′(ζ),

where P is analytic on |ζ| ≤ 1 and known (previous slide).

I Solution method here relies on analyticity of X (ζ) on Ω(t).Proposing a conformal map f with time-dependentcoefficients, f (1/ζ) has singularities in |ζ| ≤ 1 (Liouville).Elimination of singularities on LHS of this governing equationgives ODEs for the time-dependent coefficients in the map.

I Many exact solutions can be written down (any rationalfunction, e.g.), even when γ 6= 0.

I Unlike the Hele-Shaw problem, the limit γ → 0 is regular.


Pioneers in these methods

I Garabedian (1959, steady flows with rigid or free boundaries)

I Hopper (1984; 1990s, exact solutions to the MPB driven bysurface tension)

I Richardson (1992 and many subsequent papers)

I More recently: Crowdy, Tanveer, Siegel, Howell, Howison,King, LJC,. . .


Example: Sintering solutions

I Hopper, Richardson, Crowdy, and others; relevant for opticalfiber manufacture.

Stokes flows with free boundaries 327

F!"#$% 7. Five disks as in Figures 5 and 6, but the outer ones are of radius !(13!5!5)"8" 1.739while the central one is still of radius 1; as in Figure 6, only the first quadrant is shown. The freeboundary passes through a cusped configuration when t" t

!" 1.22. With t"A tan &, the outlines

are drawn for seven equal increments in & from 0 to !"2, and A is chosen so that the fourth incrementyields t" t

!.

F!"#$% 8. The coalescence of five circular disks, all of radius 1. Their centres are at z" 0, 2 ei!/", #2,#2#2 ei!/# and 2 e!i!/#. With t" tan &, the outlines are drawn for seven equal increments in & from0 to !"2.

The present example should be compared with those of Howison & Richardson (1995)

and Tanveer & Vasconcelos (1994, 1995), where the inclusion of a non-zero surface tension

in some Stokes flow problems is shown to prevent the formation of cusps that do appear

when the surface tension is taken to be zero. In particular, the nature of the cusps here is

quite di!erent. To use the customary descriptions, it seems that, while surface tension may

keep at bay the 3"2-power cusps, it does not prevent the appearance of 5"2-power cusps.

Note, too, that the near-cusps occurring in the earlier works arise as a consequence of an

imposed suction mechanism, while the genuine cusps here evolve in a motion driven solely

by surface tension. While the near-cusps persist once formed (but disappear if the suction

is removed), the free boundary here passes through the cusped state.

Figure 8 shows an example involving five initial disks, all of radius 1, that possesses no

symmetry. Following the advice in §2, the origin during the calculations has been chosen


Viscous fiber evolution under stretching

I Can also allow for slow axial variation along fibers to generatesolutions for “drawn” fibers. Leading-order flow extensional(along axis); this solution feeds into 2D cross-flow problem.Complex variable methods then apply: evolving cross-sectionmay again be described by time-dependent conformal map, inwhich distance along axis appears as a parameter (Cummings& Howell).380 L. J. Cummings and P. D. Howell

0

0.5

1.0

1.5–0.4

0–0.2

0.2

–0.2

0

0.2

0.4

(a)

0

0.5

1.0

1.5–0.4

0–0.2

0.2

–0.2

0

0.2

0.4

(b)

Figure 5. Evolution of fibres with surface tension and axial inertia. The value of the surfacetension parameter !! is 0.02", Re = 1.0, and St cos # = 0.

0

0.5

1.0

1.5–0.4

0–0.2

0.2

–0.2

0

0.2

0.4

(a)

0

0.5

1.0

1.5–0.4

0–0.2

0.2

–0.2

0

0.2

0.4

(b)

0

0.5

1.0

1.5–0.4

0–0.2

0.2

–0.2

0

0.2

0.4

(c)

0

0.5

1.0

1.5–0.4

0–0.2

0.2

–0.2

0

0.2

0.4

(d)

Figure 6. Evolution of fibres with surface tension, axial inertia, and axial gravity. Gravity acts to theleft along the fibre axis in (a) and (b), and to the right in (c) and (d). The surface tension parameteris !! = 0.02", and Re = 1.0 for both cases. In (a, b) St cos # = "1.5, and in (c, d) St cos # = +1.5.

8. Transformation to Lagrangian variablesWe have seen in § 7 that for many interesting initial shapes, the system of par-

tial di!erential equations governing the evolution of the fibre can readily be solvednumerically (certainly more readily than the three-dimensional Navier–Stokes equa-tions!). However, without running very many such simulations one cannot deducemuch general qualitative information from purely numerical results. Moreover wehave already noted that the complex-variable formulation of the evolution problem(4.9), although elegant, is not particularly helpful when seeking insight into the be-


Hele-Shaw cf. Stokes flow

I Interesting to compare mathematical structure of 2 problems(Cummings, Howell & King).

ZST Hele-Shaw∂g(z)

∂t=

dw(z)

dz

ZST Stokes∂

∂t(f ′(ζ)f (1/ζ)) = −2X ′(ζ)

I Singularities of Schwarz function in Hele-Shaw are fixed inz-plane; while for Stokes they are fixed in ζ-plane.

I Infinite set of conserved quantities for each:

ZST Hele-Shaw Mk =∫∫

Ω zk dxdy

ZST Stokes Mk =∫∫

Ω ζk dxdy


Multiply-connected Stokes flows are tricky

I Some applications involve viscous domains with “holes”.

I “Nice” formulation of SBCs relies on setting arbitrary functionof integration (and xs pressure) to zero (w.l.o.g.).

I Not true in general with more than one free boundary!

Analytic continuation



into interior !> firstglobal equation

global equationinto interior !> second

into interior !> global equation !(")

!(")

I Analytic continuation of SBCs off two free boundaries leads totwo globally-holding equations, which must be consistent.

I Consistency arises naturally when arbitrary functions ofintegration are zero for all free boundary components (Crowdyet al.; Richardson). General case is an ongoing study.


Summary

I Complex variable methods extremely useful for classes ofMBPs governed by Laplacian or biharmonic equations, wheredependent variable can be expressed in terms of complexanalytic function(s). (”Streamers” in ionization; voidelectromigration, and other applications.)

I Key step: Transform to known (fixed) domain, e.g. unit disc,and recast “moving boundary” aspect of the problem as atime-dependent conformal map, with coefficients to bedetermined.

I Boundary conditions give constraints on conformal map. Trickis to find a class of mapping functions that will yield solutions.

I Much ongoing research still, for example, into selection issuesand singularity formation in Hele-Shaw, relation with theory ofquadrature domains, viscous sintering in multiply-connecteddomains, viscous fiber-drawing, effect of body forces, . . . .


Acknowledgements

IMAP Saffman & G I TaylorL PatersonH S Hele-ShawP Ya Polubarinova-KochinaL A GalinS D HowisonS RichardsonH D Ceniceros, T Hou & H SiM J Shelley & W S DaiV Entov & co-workersP R GarabedianR W HopperD G Crowdy & S TanveerJ R KingJ R Ockendon & co-workers


complex variable methods and moving boundary …...complex variable methods and moving boundary...

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