**University Lecture Series**

Volume: 3;
1992;
72 pp;
Softcover

MSC: Primary 30; 31; 58; 76;

Print ISBN: 978-0-8218-7002-0

Product Code: ULECT/3

List Price: $20.00

AMS Member Price: $16.00

MAA Member Price: $18.00

**Electronic ISBN: 978-0-8218-3218-9
Product Code: ULECT/3.E**

List Price: $20.00

AMS Member Price: $16.00

MAA Member Price: $18.00

# Why the Boundary of a Round Drop Becomes a Curve of Order Four

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*A. N. Varchenko; P. I. Etingof*

This book concerns the problem of evolution of a round oil spot surrounded by water when oil is extracted from a well inside the spot. It turns out that the boundary of the spot remains an algebraic curve of degree four in the course of evolution. This curve is the image of an ellipse under a reflection with respect to a circle. Since the 1940s, work on this problem has led to generalizations of the reflection property and methods for constructing explicit solutions. More recently, the results have been extended to multiply connected domains. This text discusses this topic and other recent work in the theory of fluid flows with a moving boundary. Problems are included at the end of each chapter, and there is a list of open questions at the end of the book.

#### Readership

Advanced undergraduates, graduate students, and others interested in integrable systems and fluid mechanics.

#### Reviews & Endorsements

The book is well written.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Why the Boundary of a Round Drop Becomes a Curve of Order Four

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Preface vii8 free
- 1. Mathematical model 110 free
- 2. First integrals of boundary motion 514
- 2.1. Richardson's integrability theorem 514
- 2.2. Reconstruction of a domain from the values of its moments, and the inverse problem of two-dimensional potential theory 615
- 2.3. Results on the uniqueness of a domain with given moments (potential) 716
- 2.4. The result of injection does not depend on the order of work of the sources and sinks 918
- Problems 1019

- 3. Algebraic solutions 1120
- 4. Contraction of a gas bubble 2130
- 4.1. Formulation of the problem 2130
- 4.2. The inclusion property 2231
- 4.3. Contraction of a convex domain 2332
- 4.4. Contraction points 2433
- 4.5. An analogue of Richardson's theorem 2534
- 4.6. Dynamics of the gravity potential 2736
- 4.7. The gravity potential as the solution of a boundary value problem 2736
- 4.8. Proof of the main theorem 2837
- 4.9. Self-similar solutions 2938
- 4.10. Asymptotics of contraction 3039
- 4.11. Several sources 3140
- Problems 3241

- 5. Evolution of a multiply connected domain 3544
- 5.1. Statement of the problem 3544
- 5.2. Integrals of motion 3544
- 5.3. Algebraic solutions 3645
- 5.4. Riemann's theorem 3746
- 5.5. Singularity correspondence 3847
- 5.6. Proof of the theorem on algebraic solutions 3948
- 5.7. Construction of solutions 3948
- 5.8. Reconstruction of an annular domain from its moments 4049
- Problems 4251

- 6. Evolution with topological transformations 4352
- 7. Contraction problem on surfaces 5564
- Answers and clues to the problems 6170
- A few open questions 6978
- References 7180
- Back Cover Back Cover182