Crank J. Free and Moving Boundary Problems

Free.

and

moving

bou.ndary

problems

John

Crank

Professor

Emeritus

Brunel University

CLARENDON

PRESS . OXFORD

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Published in the United States

by

Oxford University Press, New York

©John

Crank, 1984

First published 1984

First issued in paperback 1987

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British Library Cataloguing in Publication Data

Crank,

J. '

Free and moving boundary problems

1. Differential equations,

Nonlinear-

Numerical solutions

2.

Boundary value

problems-Numerical solutions

L Title

515.3'55 QA371

ISBN

0-19-853357-8

ISBN

0-19-853370-5 (pbk)

Library

of

Congress Catalaging in Publication Data

Crank, John.

Free and moving boundary problems.

Bibliography: p.

Includes index.

1.

Boundary value problems.

1.

Title.

TA347.B69C731987

515.3'5 86-28484

ISBN

0-19-853357-8

ISBN

0-19-853370-5 (pbk)

Sa

and printed in

Northem Ireland

by

The Universities Press (Belfast) Ltd.

Preface

THIS

book aims

to

present a broad

but

reasonably detailed account of the

mathematical solution of free and moving

bound~

problems.

The

mov-

ing boundaries occur mostly in heat-flow problems with phase changes

and in certain diffusion processes. A free boundary does not move but its

position has

to

be

determined as

part

of

the

solution of a steady-state

problem. Seepage through porous media

is

perhaps the most common but

by

no means

the

only source of problems

of

this kind.

The

broad spectrum of active research workers includes three groups:

engineers and others with practical problems; numerical analysts produc-

ing suitable numerical' algorithms;

and

pure mathematicians who decide

that

certain problems and their solutions exist, are properly posed, and

may even

be

unique. They also examine

the

convergence and stability

properties of numerical schemes. A few people fit into all three groups.

It

is

hoped

that

this book will help

to

alleviate

the

usual difficulties of

communication_ between

the

various interested parties.

.

Both

free and moving

boundari~

have been popular subjects for

research in recent years, leading

to

an almost bewildering collection of

new mathematical methods. This seems

to

be

an opportune time to

attempt a systematic presentation of them. Authors have tended to test

their methods by solving a small number of model problems and so

studies of melting ice, simple dams, oxygen diffusion with sorption, and

electrochemical machining are referred to frequently in this book. The

earliest mathematical work concentrated on one-dimensional problems.

Modern computer developments, however, make it possible to handle

free and moving boundaries in two and three space dimensions and to

model practical systems more realistically. Such methods feature largely

in this volume, though it makes no pretence

to

be

a compendium of

industrial problems.

Parallel studies of

the

mathematical properties of

the

differeatial equa-

tions and their solutions, and of

the

numerical algorithms, have been

prolific.

It

has not been possible here

to

do more than indicate some of

the

established results and

to'

include key references in the extensive

bibliography. A separate volume

is

needed

to

do justice to this aspect of

the

subjec~.

vi Preface

It

is a pleasure to acknowledge

the

generous help I have received from

the Leverhulme Trust through

the

award of an Emeritus Fellowship. I am

grateful

to

Mrs Lesley Barker who typed the text with great accuracy and

intelligent skill from a manuscript which in parts came near

to

being

illegible.

I have been glad

to

be

able to use the BruneI University Library

and to have expert guidance from Assistant Librarian,

Mr

M.

J. Dorling.

Finally,

I should like

to

thank all who have sent

me

reprints and

unpublished copies of their work, and

the

staff of

the

Clarendon Press for

their kind help and consideration.

BruneI University

1982

J.e.

Contents

1. MOVING-BOUNDARY PROBLEMS: FORMULATION 1

1.1. Introduction 1

1.2. A simple example: melting ice 2

1.2.1. Single phase 2

1.2.2. Two phases 4

1.2.3.

The

Stefan condition 4

1.3. Generalizations of the classical Stefan problem 5

1.3.1. Non-linear heat parameters 5

1.3.2. Linearized forms 8

1.3.3. Non-dimensional forms 9

1.3.4. Density change

and

convection 10

1.3.5. Ablation and

the

inverse Stefan problem 11

1.3.6. Multi-phase problems 12

1.3.7. Simultaneous diffusion and heat

flow:

alloys

14

1.3.8. Muskat's problem 16

1.3.9. Two and three space dimensions 16

1.3.10: Implicit boundary conditions: oxygen diffusion

problem 19

1.3.11. Concentrated thermal capacities 23

2.

FREE-BOUNDARY

PROBLEMS: FORMULATION 30

2.1. Introduction 30

2.2. Simple rectangular

dam

31

2.2.1. Classical formulation: physical derivation 31

2.2.2. Stream function 34

2.2.3. Formulation on fixed domain: Baiocchi transfor-

mation

2.2.4. Variational inequality formulation

2.2.5.

In

terms of stream functions

2.3.' Other two-dimensional dams

2.3.1. Rectangular

dam

with toe drain

2.3.2. Rectangt4ar dam with sheetpile

34

38

40

41

44

viii Contents

2.3.3. Rectangular dam with toe drain and sheetpile

,2.3.4.

Dam

with inlet face slanted

2.3.5.

Dam

with inlet face slanted and a toe drain

2.3.6.

Dam

with sloping base

2.3.7. Arbitrary-shaped

dam

2.3.8. Arbitrary-shaped

dam

with

toe

drain; split-domain

method

2.4. Three-dimensional dams

2.5. Coastal aquifer

2.6. Canals and ditches

2.7. Axisymmetric

flow

2.8. Heterogeneous porous media

2.8.1. Stratified dams

2.9. Evaporation

or

infiltration

2.10. Unsaturated

flow:

capillary fringe

2.11. A generalized dam problem: a new formulation

2.12. Degenerate free-boundary problems

2.12.1. Hele-Shaw

flow

analogy

2.12.2. Hele-Shaw injection

2.12.3. Electrochemical machiniltg

2.12.4. Evolution dam problem

2.13. Lubrication cavitation in a journal bearing

2.14. A more general look at the Baiocchi transformation

2.15. Connections between certain free-boundary problems

3. ANALYTICAL SOLUTIONS

3.1. Exact similarity solutions

3.2. Neumann's solution; generalizations; volume changes

3.3. Similarity solutions in cylindrical and spherical coordi-

nates

3.4. Similarity solutions

and

moving heat sources

3.5. Integral-equation formulations

3.5.1. Integral transforms

3.5.2. Green's functions

3.5.3. Embedding technique

3.5.4. Heat-balance method: Goodman

3.5.5. Heat-balance method with spatial subdivision

3.5.6. Multi-dimensional problems

3.6. Approximate solutions

3.6.1. Steady-state approximation and improvements

3.6.2. Spacial subdivision

46

47

49

50

52

54

57

60

63

66

70

71

74

75

79

81

82

85

90

93

96

100

101

110

111

114

117

122

128

133

137

139

, 139

142

3.6.3.

The

inverse Stefan problem

3.6.4. Fictitious diffusivity approximations

3.6.5. Perturbation methods; asymptotic

stability

4. FRONT-TRACKING METIIODS

4.1. Numerical techniques

4.2. Fixed finite-difference grid

4.3. Modified grids

4.3.1. Variable time step

4.3.2. Variable space grid

4.3.3. Finite-elements: adaptive meshes

4.4. Method of lines

Contents

ix

144

147

solutions;

150

163

168

170

173

181

5.

FRONT-FIXING METIIODS 187

5.1. One-dimensional problems 187

5.2. Body-fitted curvilinear coordinates 192

5.3. Examples of use of body-fitted coordinates 195

5.4. Isotherm migration method 199

5.4.1. IMM in one space dimension 199

5.4.2. IMM in multi-dimensional problems 201

6.

FIXED-DOMAIN METIIODS 217

6.1. Introduction 217

6.2. Enthalpy method 217

6.2.1. Weak solutions 220

6.2.2.

An

explicit finite-difference scheme 223

6.2.3. Alternative forms of the enthalpy function 226

6.2.4. Other numerical schemes and multi-dimensional 228

problems

6.2.5. Accurate determination of phase-change boundary 235

6.2.6. Body heating 239

6.2.7. Other conservation forms and weak solutions 242

6.2.8. Conditions not amenable to enthalpy formulation 252

6.3. Truncation method: alternating phase

6.4. Variational inequalities

6.4.1. Introductory example

6.4.2. Variational form

6.4.3. Minimization formulation

6.4.4. Review of equivalent forms

6.4.5. One-phase Stefan problem

253

257

260

263

265

267

269

x Contents

6.4.6. Two-phase Stefan problem

6.4.7. Mathematical results

7 . ANALYTICAL SOLUTION

OF

SEEPAGE

PROBLEMS

7.1. Introduction

7.2. Approximate methods

7.2.1. Dupuit's approximation

7.2.2. Boussinesq's equations

7.2.3. Other approximate methods

7.3. Hodograph method: conformal transformations

7.3.1. Hodograph plane

7.3.2. Some useful transformations

7.3.3. Trapezoidal drainage ditch

275

281

283

285

287

288

293

297

7.4. Polubarinova-Kochina's solution for the simple

dam

300

8.

NUMERICAL

SOLUTION

OF

FREE-BOUNDARY

PROBLEMS 313

8.1. Numerical solutions 313

8.2. Trial free-boundary methods 313

8.2.1. Computation of approximate trial solutions

'314

8.2.2. Moving

the

free boundary

319

8.2.3. Garabedian's modified boundary conditions

329

8.3. Boundary singularities 332

8.4. Methods using variable interchange

336

8.4.1. Adaptive variational formulation 337

8.4.2. Transformation of equation and flow region 343

8.5. Variational inequality and complementarity methods 355

8.5.1. Finite-difference and finite-element forms of the

Cryer algorithm 359

8.5.2. Axisymmetric problems

372

8.6. A general algorithm for partially unsaturated flow 376

8.7. Integral equation methods

384

8.7.1. Two-dimensional annular electrochemical machin-

ing problem 385

REFERENCES

394

AUTHOR

INDEX

409

SUBJECT

INDEX

414

1.

Moving-boundary problems:-

formulation

1.1. Introduction

PROBLEMS

in which

the

solution of a differential equation has to satisfy

certain conditions on

the

boundary of a prescribed domain are referred to

as boundary-value problems.

In

many important cases, however, the

boundary of

the

domain

is

not known in advance but has

to

be

deter-

mined

as

part of

the

solution.

The

term 'free-boundary problem'

is

commonly used when

the

boundary

is

stationary and a steady-state

problem exists. Moving boundaries, on the other hand, are associated

with time-dependent problems and

the

position of the boundary has

to

be

determined

as

a function of time and space.

The

terms 'free' and 'moving'

are used in this separate way in this book. Some authors, however, prefer

to include both types of problem under the single term 'free-boundary

problem' and a very few authors have used 'moving boundary' in a

composite sense.

In

all cases, two conditions are needed on

the

free

or

moving boundary,

one

to determine

the

boundary itself and

the

other to complete

the

definition of

the

solution of

the

differential equation. Suitable conditions

on

the

fixed boundaries and, where appropriate, an initial condition are

also prescribed

as

usual.

In

this book, a free-boundary problem requires

the

solution of an

elliptic partial differential equation. For a moving-boundary problem the

equation

is

of parabolic type.

The

practical applications are mainly but

not exclusively concerned with fluid

flow

in porous media and with

diffusion and

heat

flow

incorporating phase transformations

or

chemical

reactions. Many other important free and moving boundary problems

arising, for example, as shock waves in gas dynamics,

as

cracks in solid

mechanics,

or

as optimal stopping problems in decision theory have

perforce had

to

be

excluded, in order to keep

the

book within reasonable

bounds.

Moving-boundary problems are often called Stefan problems, with

reference to

the

early work of J. Stefan who, around 1890, was interested

in

the

melting of

the

polar ice cap. Successive authors have referred,

Crank J. Free and Moving Boundary Problems

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