Erich Miersemann - Mathematical Theory of Liquid Interfaces: Liquid Layers, Capillary Interfaces, Floating Drops and Particles

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Mathematical Theory of
Liquid Interfaces

Liquid Layers, Capillary Interfaces,
Floating Drops and Particles

Mathematical Theory of Liquid Interfaces

Liquid Layers, Capillary Interfaces, Floating Drops and ParticlesErich MiersemannUniversitt Leipzig, GermanyPublished byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office:27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office:57 Shelton Street, Covent Garden, London WC2H 9HELibrary of Congress Control Number: 2020006577British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.MATHEMATICAL THEORY OF LIQUID INTERFACESLiquid Layers, Capillary Interfaces, Floating Drops and ParticlesCopyright 2020 by World Scientific Publishing Co. Pte. Ltd.All rights reserved. In this case permission to photocopy is not required from the publisher.ISBN 978-981-121-565-0 (hardcover)ISBN 978-981-121-566-7 (ebook for institutions)ISBN 978-981-121-567-4 (ebook for individuals)For any available supplementary material, please visithttps://www.worldscientific.com/worldscibooks/10.1142/11699#t=supplDesk Editor: Soh Jing WenPrinted in Singapore

Contents

Preface

In the last decades many problems, suggested mainly through space technology and micro-mechanics where liquids are involved, are studied. In particular, liquids under low or zero gravity, liquids of small volume or liquids in small containers are of interest. The results are qualitatively different from results which one finds for related linear problems. The reason for that is the strong non-linearity of the governing equations. A striking example is the behavior of a liquid in a narrow wedge, see [Concus and Finn (1970)], or a new numerical procedure which is used to establish a new method to measure the surface tension of a liquid, see [Miersemann (2012)]. We study static problems for liquid layers, capillary interfaces, floating drops and floating particles. In all these cases interfaces are involved which separate an incompressible inviscid fluid and its vapor. The leading term in the associated equilibrium equation for the interface is the mean curvature. In the case of liquid layers no contact angle occurs. Thus the admissible variations which yield necessary or sufficient conditions for equilibrium surfaces are much easier than the variations which we will use for the floating drop or floating particle problem. In the floating particle problem we have to take into account contact angles between the capillary surface and the solid materials as well as rigid motions of the body.The chapter on liquid layers is based on a cooperation with Peter Schiller and Hans-Jrg Mgel from the Institute of Physical Chemistry at the TU Bergakademie Freiberg.Concerning capillary interfaces we recommend the inspiring book Equilibrium Capillary Surfaces of Robert Finn [Finn (1986)]. Part I (Differential geometry of surfaces in three-dimensional Euclidean space) of the book [Dierkes, Hildebrandt, Kster and Wohlrab (1992)] was an essential help for me to calculate some variational formulas.The floating drop problem was studied in [Neumann (1894)]. Franz Neumann derived, in particular, the angle conditions which satisfy the involved surfaces along the edge of the drop, see also [Slobozhanin (1986)].For the convenience of the reader we tried to keep some chapters independently from each other with the consequence that some formulas used in previous chapters follow from more general formulas derived in later chapters.Some of the exercises at the end of the chapters are just exercises and others are open problems.This book is partly based on previous lecture notes and on the reports [Miersemann (2007, 2019)]. The content of this book is not encyclopedic at all and it is dominated by my own interests and results.

Leipzig

Erich Miersemann

Chapter 1

Introduction

It is well known that a liquid can rise at the interior and at the exterior wall of a capillary tube, see

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