Table Of ContentInterdisciplinary Applied Mathematics
Valurne 3
Editors
F. John L. Kadanoff J.E. Marsden
L. Sirovich S. Wiggins
Advisors
G. Ezra M. Gutzwiller D. Holm
D.D. Joseph P.S. Krishnaprasad
J. Murray M. Schultz K. Sreenivasan
S. Winograd
Interdisciplinary Applied Mathematics
1. Gutzwiller: Chaos in Classical and Quantum Mechanics
2. Wiggins: Chaotic Transport in Dynamical Systems
3. Joseph/Renardy: FundamentalsofTwo-Fluid Dynamics:
Part 1: Mathematical Theory and Applications
4. Joseph/Renardy: Fundamentals of Two-Fluid Dynamics:
Part II: Lubricated Transport, Drops and Miseihle Liquids
Daniel D. Jo seph Yuriko Y. Renardy
Fundamentals of
Two-Fluid Dynamics
Part 1: Mathematical Theory
and Applications
With 205 illustrations, 69 in color
Springer Science+Business Media, LLC
Daniel D. Joseph Yuriko Y. Renardy
Department of Aerospace Department of Mathematics
Engineering and Mechanics Virginia Polytechnic Institute
University of Minnesota and State University
Minneapolis, MN 55455 USA Blacksburg, VA 24061 USA
Editors
F. John L. Kadanoff J.E. Marsden
Courant Institute of Department of Physics Department of
Mathematical Sciences James Franck Institute Mathematics
New York University University of Chicago University of California
New York, NY 10012 Chicago, IL 60637 Berkeley, CA 94720
USA USA USA
L. Sirovich S. Wiggins
Division of Applied Mechanics Department
Applied Mathematics Mai! Code 104-44
Brown University California Institute of Technology
Providence, Rl 02912 Pasadena, CA 91125
USA USA
Cover illustration: Cylinder coated with 12500 cs silicone oii cusps in air. (See color plate 11.6.5
(a-b).)
Mathematics Subject Classifications (1991): 76E05, 76E15, 76E30, 35B20
Library of Congress Cataloging-in-Publication Data
Joseph, Daniel D.
Fundamentals of two-fluid dynamics 1 Daniel D. Joseph, Yuriko Y.
Renardy.
p. cm. - (lnterdisciplinary applied mathematics : v. 3/4)
Includes bibliographical references and index.
Contents: part 1. Mathematical theory and applications - part
2. Lubricated transport, drops and miscible liquids.
1. Fluid dynamics. 1. Renardy, Yuriko Y. IL Title.
III. Series.
QC15J.J67 1992
620.1 '064-dc20 92-34044
Printed on acid-free paper.
© 1993 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, !ne. in 1993
Softcover reprint of the hardcover Jst edition 1993
Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the
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accordingly be used free1y by anyone.
Production managed by Henry Krell; manufacturing supervised by Vincent Scelta.
Photocomposed copy prepared from the authors' TeX file.
987654321
ISBN 978-1-4613-9295-8 ISBN 978-1-4613-9293-4 (eBook)
DOI 10.1007/978-1-4613-9293-4
Preface
Two-fluid dynamics is a challenging subject rich in physics and prac
tical applications. Many of the most interesting problems are tied to the
loss of stability which is realized in preferential positioning and shaping of
the interface, so that interfacial stability is a major player in this drama.
Typically, solutions of equations governing the dynamics of two fluids are
not uniquely determined by the boundary data and different configurations
of flow are compatible with the same data. This is one reason why stability
studies are important; we need to know which of the possible solutions are
stable to predict what might be observed. When we started our studies in
the early 1980's, it was not at all evident that stability theory could actu
ally work in the hostile environment of pervasive nonuniqueness. We were
pleasantly surprised, even astounded, by the extent to which it does work.
There are many simple solutions, called basic flows, which are never stable,
but we may always compute growth rates and determine the wavelength
and frequency of the unstable mode which grows the fastest. This proce
dure appears to work well even in deeply nonlinear regimes where linear
theory is not strictly valid, just as Lord Rayleigh showed long ago in his
calculation of the size of drops resulting from capillary-induced pinch-off of
an inviscid jet. In two-fluid problems, there are many sources for instability
and the active ones may be determined to a degree by analysis of different
terms which arise in the energy budget of the most dangeraus disturbance.
Though we have presented many results from nonlinear analysis of two
fluid problems, this side of the subject is not yet well-developed. We are
also certain that the direct Simulations oftwo-fluid problems which have
commenced only in the years just passed have a potentially huge domain
for increased understanding.
Applications of two-fluid dynamics range from manufacturing to lu
bricated transport. Different mechanisms which are unique to the flow of
two fluids can be exploited for this purpose. Density-matching can be used
to depress the effect of gravity, or of centripetal acceleration in rotating
systems, allowing one to manipulate the places occupied and the shapes of
vi Preface
the interfaces between fluids. Viscosity segregation can be used to promote
mixing and demixing, to promote say the displacement of one fluid by an
other as in the problern of oil recovery, or to segregate one molten plastic
from another by encapsulation. The lubrication of one fluid by another is a
particularly important branch oftwo-fluid dynamics; if one fluid has a sur
passingly large viscosity, it may be lubricated by a less viscous fluid. The
most beautiful of the lubricated flows are the rollers discussed in chapter II
of the first part, Mathematical Theory and Applications. The most useful of
the lubricated flows are the water-lubricated pipelines, which are discussed
in chapters V through VIII of the second part, Lubricated Transport, Drops
and Miscible Liquids. It is our hope that this book will lead to deeper
understanding of the principles and applications oftwo-fluid dynamics.
The topics treated in this book are displayed in the table of contents.
It has been divided into two self-contained parts. Most of the techniques of
analysis used in the study oftwo-fluid problems are developed in the first
part, Mathematical Theory and Applications. The analyses are fully worked
there, with enough details to teach students. Four of the six chapters of
the second part, Lubricated Transport, Drops and Miscible Liquids, treat
problems which arise in the study of water-lubricated pipelining. We have
done a much more complete comparison of theory and observations than
was previously available with the serious intent of advancing this energy
efficient technology. Chapter IX is about immiscible vortex rings and is a
report and explanation of experimental results. Chapter X develops a new
theory of binary mixtures of miscible incompressible liquids in which the
density changes and the stresses induced by diffusion are considered. As
always, it is certain that a number of excellent studiesoftwo-fluid dynamics
which deserve mention have not been mentioned.
Our research for this project could not have been done without the
help of certain persons: Mike Arney, Runyan Bai, Nick Baumann, Gor
don Beavers, Kangping Chen, Howard Hu, Paul Mohr, John Nelson, Ky
Nguyen, Luigi Preziosi and Michael Renardy. We are especially indebted
to Chen, Hu and Preziosi for their excellent analytical and numerical stud
ies of lubricated pipelining and to Bai for the design and execution of very
elegant experiments. We thank Michael Renardy for reading through the
manuscript.
The work of Joseph was supported mainly by the Department of En
ergy, Office of Basic Energy Seiences and also by the fluid mechanics branch
of the National Science Foundation, the mathematics division of the Army
Research Office, by the Army High Performance Computing Center, and the
Minnesota Supercomputer Institute. Joseph's research on water-lubricated
pipelining was funded initially under a special small NSF grant for inno
vative research involving the lubricated transport of coal-oil dispersions.
Joseph is grateful to Steve Traugott for this initial grant which was later
picked up by Oscar Manley at the DOE.
Renardy's research was funded by the National Science Foundation
Preface vü
under Grant No. DMS-8902166. This project was begun during the Winter
Quarter of 1989 at the Institute for Mathematics and Its Applications at
the University of Minnesota.
Yuriko dedicates this book with love to her father Sadayuki Yamamuro
("Papa, arigato"), and to her mother Akiko ( "osewani narimashita''). Dan
dedicates this book to Adam, Bai, Chris, Claude, Dave, Geraldo, Harry,
Howard, John, Kangping, Luigi, Mike, Paul, Pushpendra and Terrence.
February 1992 Minneapolis, Minnesota
Blacksburg, Virginia
Contents
Contents of Part I: Mathematical Theory and Applications
Prefaee V
Color Insert follows page 240
Chapter I. Introduetion 1
1.1 Examples 2
1.1 (a ) Fingering 2
1.1 (b) Lubricated Pipelining 3
1.1 (c ) Segregation and Lubrication of Solids in Liquids 5
1.1 (d) Lubricated Pipelining of Solid Particulates 7
1.1 (e ) Manufacturing 11
1.1 (f) Lubricated Extensional Flows: A Rheological Application 13
1.1 (g) Microgravity Through Density Matehing 14
1.1 {h) Geophysical Applications 15
1.1 (i) Transient Flow of Two Immiscible Liquids in a
Rotating Container 16
1.2 Formulation of Equations 18
1.2 (a) Transport ldentities 19
1.2 (b) Balance of Momentum 22
1.2 (c) Balance of Energy 23
1.2 {d) Boundary Conditions 25
1.2 (e) Summary 26
1.3 Nonuniqueness of Steady Solutions 27
1.3 (a) Bubbles 27
1.3 {b) Parallel Shear Flows 28
1.3 (c) Two-Fluid Convection 31
1.3 {d) Rotating Couette Flow 31
1.3 (e ) Nonuniqueness and Stability 31
1.3 (f) Nonuniqueness and Variational Principles 32
x Contents
Chapter II. Rotating Flows of Two Liquids 44
11.1 Rigid Motions of Two Liquids Rotating in a Cylindrical Container 45
11.1 (a) Steady Rigid Rotation of Two Fluids 45
11.1 (b) Disturbance Equations 49
11.1 (c ) Energy Equation for Rigid Motions of Two Fluids 49
11.1 (d) The Interface Potential 52
11.1 (e ) Integrability of the Energy 58
11.1 (f) Minimum of the Potential 60
11.1 (g) Spatially Periodic Connected Interfaces 62
11.2 The Minimum Problem for Rigid Rotation of Two Fluids 63
11.2 (a) The Cylindrical Interface 63
11.2 (b) Mathematical Formulation of the Minimum Problem 67
11.2 (c ) Analysis of the Minimum Problem 68
11.2 (d) Periodic Solutions, Dropsand Bubbles 76
11.2 (e ) All the Salutions with J < 4 Touch the Cylinder 76
11.3 Experiments on Rigid Rotation of Two Fluids in a
Cylindrical Container 78
11.3 (a) Experiments with Heavy Fluid Outside- the Spinning
Rod Tensiometer 78
11.3 (b) Experiments with Heavy Fluid Inside - Coating Flows 85
11.4 Experiments with Liquids on Immersed and Partially lmmersed
Rotating Rods. Rollers, Sheet Coatings and Emulsions 93
11.4 (a ) Rollers 94
11.4 (b) Sheet Coatings 101
11.4 (c ) Fingering lnstabilities and the Formation of Emulsions 103
11.4 (d) Centrifugal lnstabilities 110
11.5 Taylor-Couette Flow of Two Immiscible Liquids 111
11.5 (a) Experimentsand Parameters 115
11.5 (b) Circular Couette Flows 117
11.5 (c) RollerE 119
11.5 (d) Emulsions, Tall Taylor Cells, Cell Nucleation 119
11.5 (e ) Phase Inversion 127
11.5 (f) Phase Separation 127
11.5 (g) Phase Inversion and Phase Separation 130
11.5 (h) Chaotic Trajectories of Oil Bubbles in an Unstable
Water Cell 133
11.6 Two-Dimensional Cusped Interfaces 140
11.6 (a) lntroduction 141
11.6 (b) Experiments 144
11.6 (c ) Theory 150
11.6 (d) Numerical Results 157
11.6 (e) Conclusions 168
