Table Of ContentMathematical Engineering
Alexander A. Avdeev
Bubble
Systems
Mathematical Engineering
Series editors
Claus Hillermeier, Neubiberg, Germany
Jörg Schröder, Essen, Germany
Bernhard Weigand, Stuttgart, Germany
More information about this series at http://www.springer.com/series/8445
Alexander A. Avdeev
Bubble Systems
123
Alexander A.Avdeev
Dmitrovskoe shosse
46-1-6
127238Moscow
Russia
E-mail: [email protected]
ISSN 2192-4732 ISSN 2192-4740 (electronic)
Mathematical Engineering
ISBN978-3-319-29286-1 ISBN978-3-319-29288-5 (eBook)
DOI 10.1007/978-3-319-29288-5
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Preface
The main purpose of this book is to bring together into a coherent system all our
knowledgeaboutthephysicalmechanismsgoverningthelawsofbehaviourofboth
the simplest bubble systems consisting of single bubbles and of involved systems,
which occur, for example, in bubble flows or during bubbling.
Due to the rapid progress in science, both textbooks and scarce scientific
monographs,partiallyembracingthisfieldofknowledge,tendtoageratherquickly,
often losing their actualité right after publication. I would like very much that this
book does not share such a fate. This wish has determined its structure.
First,wegiveasystematictreatmentofthefundamentalproblemsdescribingthe
dynamics of single bubbles (laws of their growth, collapse, fragmentation and
emersion in the gravitational field, pulsation and so on.). The use of modern
softwareproductsforanalyticalstudieshelpstoobtainnewquitesurprisingresults
evenfortheseclassicalproblems,whoseauthorsinduetimeremarkablycombined
deep physical sense, serious mathematical technique, and enormous amount of
computational and analytical work to obtain rigorous analytical solutions.
Moreover, despite the classical nature of the problems under consideration, for
some of them the complete analytical solutions were obtained only inthe past two
decades and at present are unknown to the broad research community.
Second, based on these fundamental solutions it is possible to put forward, in a
large number of cases, fairly rigorous analytic solutions of next-level problems,
which are traditionally farmed out to experimentalists or specialists in the field of
computational fluid dynamics. As example, we may mention the problems on
hydrodynamics of bubble flows in channels, sizes of bubbles, true void fraction,
hydraulic resistance during boiling of subcooled liquid under forced motion con-
ditions, and so on. Despite the fact that in a number of respects we cannot avoid
using the numerical methods, the particular analytic solutions, as well as the
recourse to qualitative methods of investigations residing in consistent application
of similarity methods, enable one to carry out the results of analysis to design
relations,whichsecurethecorrespondingpassagestothelimitandhavewiderange
of applications and high accuracy.
v
vi Preface
Third, from the methodological point of view, ofgreat value are theconceptual
developments in the field of bubble flows, and in particular, the theory of boiling
shock, which was created by the author in the 1980s, and the applications of the
Reynolds analogy in the study of nonequilibrium two-phase flows.
The systematic use of the theory of boiling shock has enabled one to explain,
andinmanycasestopredict,quiteanumberofexperimentaleffectsaccompanying
theprocessesofdischargeofflashingliquid,theprocessesofunsteadyflashingwith
rapid pressure relief, and to build on its basis methods for calculation of similar
phenomena.
Reynoldsanalogyforflowswith‘doubledisequilibrium’(superheatednear-wall
layer of liquid—subcooled liquid in the core), which often occur in
high-performance cooling systems, proved a fairly natural tool in the study of
processes of radial transfer of heat and momentum, providing thereby a basis for
obtainingfairlyrigorousanalyticsolutionsoftheheatexchangeandhydrodynamics
of such flows.
There are already thousands of scientific and engineering papers relating to
nonequilibrium two-phase flows. Frequently, these studies consider the same
phenomena from quite different (and in a number of cases from directly opposite)
viewpoints. In some cases (fortunately, its number is quite narrow), erroneous
approaches migrate from one paper to another to gradually become traditional.
Despitethefactthatthisbranchofsciencereacheditsfullfloweringinthe1970–
80s due to the rapid development of nuclear and aerospace engineering, the
research activity in this field has been quite intensive up to now, which can be
explained by its operational and scientific significance. Consequently, a young
scientist, who has received a good fundamental background and begins his or her
study in this field, will only after several years be able to orientate in this multi-
dimensional space, to be capable of distinguishing the important tendencies of the
theory from the dead-end tracks.
Aresearchengineerembarkingonastudyofnonequilibriumtwo-phasesystems
shouldbefamiliarwithnumerousclassicaldisciplineslikethethermodynamicsand
thetheoryofthepropertiesofmatter,themechanicsofliquid,gasdynamicsaswell
asthetheoryofheatandmasstransfer.Generaltextbooksandhandbooksonthese
scientific disciplines contain such a huge amount of information that they cannot
always be comprehended and made valuable by an amateur in thermal physics.
Aseriousproblemforastudentisthechoiceofsubdisciplinesthatarevitalinhisor
her studies.
In this process, contacts with elder colleagues and supervisors are invaluable.
Maturingofayoungscientistisbelievedtobeadditionallyfacilitatedbythisbook,
which includes the principal achievements in the construction of the theory and
illustrates application of the principal tools for its further development.
Inwritingthisbook,theauthorwasmainlyfocusedonthedemandsofgraduate
and postgraduate students specializing in two-phase flows physics. This book will
bea usefulcomplementtoexistingtextbooks on heat exchange andgasdynamics,
with the aim toward in-depth study of these disciplines and providing help in
comprehending the fundamentals of the theory of bubble systems.
Preface vii
The true needs of practicing professionals and postgraduate students are not far
apart. Both require, above all, an understanding of the fundamentals of running
processes, which is the main purpose of this book.
All solutions, without exception (both analytic and numerical), obtained in this
book expand to fairly simple design formulas, many of which are put forward for
the first time. Hence, this book also serves as a detailed handbook that can be
conveniently used in analytical, numerical and design practice.
In conclusion, the author expresses his sincere gratitude to his numerous col-
leagues, coworkers and friends, whose discussions and valuable comments were
instrumental in bringing the book to its final form.
TheauthorisdeeplyindebtedtoProf.BernhardWeigand(StuttgartUniversity),
whose courteous attention and support encouraged him in this endeavour.
Agreatroleintheformingoftheauthor’sentirescientificoutlookwasplayedin
due time by his supervisor Prof. Dmitry Labuntsov (1929–1992). The fact that the
readertakesthisbookinhisorherhandsmeansthatLabuntsov’sideascontinueto
develop and his blessed memory will live on.
Moscow Alexander A. Avdeev
December 2015
Contents
1 Introduction. General Principles of Description of Two-Phase
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Two-Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Methods of Mathematical Description
of Two-Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Detailed Description. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Unit Cell Models. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Single Continuum Models. . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Models of Multispeed Medium . . . . . . . . . . . . . . . . . 7
1.3 Incorporation of Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Coupling Conditions. . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Intensity of Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Quasi-Equilibrium Scheme . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Phase Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 The Structure of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Dynamics of Bubbles in an Infinite Volume of Liquid . . . . . . . . . . 35
2.1 The Problems of Force Controlled Bubble Evolutions. . . . . . . . 35
2.2 The Rayleigh-Lamb Equation . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Collapse of a Vapour Bubble. . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Change of the Bubble Radius in Time . . . . . . . . . . . . 42
2.3.2 The Pressure Field in the Liquid . . . . . . . . . . . . . . . . 47
2.3.3 The Influence of Capillary Effects and Viscosity
Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Dynamic Growth of Vapour Bubble. . . . . . . . . . . . . . . . . . . . 54
2.4.1 The Effect of Inertial Forces . . . . . . . . . . . . . . . . . . . 55
2.4.2 The Effect of the Surface Tension Forces . . . . . . . . . . 58
2.4.3 The Effect of Viscosity Forces. . . . . . . . . . . . . . . . . . 61
ix
x Contents
2.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Pulsations of Bubbles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 The Problem Under Consideration . . . . . . . . . . . . . . . . . . . . . 69
3.2 The Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 The Statement of the Problem . . . . . . . . . . . . . . . . . . 71
3.2.2 The Mathematical Description . . . . . . . . . . . . . . . . . . 73
3.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.1 Linearization of the Equations . . . . . . . . . . . . . . . . . . 75
3.3.2 Bubble Fundamental Frequency. . . . . . . . . . . . . . . . . 77
3.3.3 Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . 78
3.4 Consideration of the Results of the Analysis . . . . . . . . . . . . . . 81
3.4.1 Set of Similarity Numbers. . . . . . . . . . . . . . . . . . . . . 81
3.4.2 Oscillation of the Internal Parameters . . . . . . . . . . . . . 82
3.4.3 The Possibility of the Polytropic Approximation . . . . . 84
3.4.4 Oscillations of the External Pressure
and Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Pulsation at Stepwise Variation of Pressure
(the Adiabatic Approximation). . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.1 Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.2 Energy Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5.3 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Thermally Controlled Bubble Growth. . . . . . . . . . . . . . . . . . . . . . 99
4.1 TheMathematicalFormulationoftheThermalBubbleGrowth
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Dimensional Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.1 Early Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.2 Searching for Solution of the Boundary Value
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.3 Self-similarity Solutions . . . . . . . . . . . . . . . . . . . . . . 115
4.4 The General Analytical Solution . . . . . . . . . . . . . . . . . . . . . . 116
4.4.1 Solution in the Integral Form. . . . . . . . . . . . . . . . . . . 116
4.4.2 Analytical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.3 Analysis of the Results. . . . . . . . . . . . . . . . . . . . . . . 123
4.5 The Asymptotic Laws of Bubble Growth . . . . . . . . . . . . . . . . 128
4.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
