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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.
This book is about two special topics in rheological fluid
mechanics: the elasticity of liquids and asymptotic theories of
constitutive models. The major emphasis of the book is on the
mathematical and physical consequences of the elasticity of
liquids; seventeen of twenty chapters are devoted to this.
Constitutive models which are instantaneously elastic can lead to
some hyperbolicity in the dynamics of flow, waves of vorticity into
rest (known as shear waves), to shock waves of vorticity or
velocity, to steady flows of transonic type or to short wave
instabilities which lead to ill-posed problems. Other kinds of
models, with small Newtonian viscosities, give rise to perturbed
instantaneous elasticity, associated with smoothing of
discontinuities as in gas dynamics. There is no doubt that liquids
will respond like elastic solids to impulses which are very rapid
compared to the time it takes for the molecular order associated
with short range forces in the liquid, to relax. After this, all
liquids look viscous with signals propagating by diffusion rather
than by waves. For small molecules this time of relaxation is
estimated as lQ-13 to 10-10 seconds depending on the fluids. Waves
associated with such liquids move with speeds of 1 QS cm/s, or even
faster. For engineering applications the instantaneous elasticity
of these fluids is of little interest; the practical dynamics is
governed by diffusion, .say, by the Navier-Stokes equations. On the
other hand, there are other liquids which are known to have much
longer times of relaxation."
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.
This substantially revised second edition teaches the bifurcation
of asymptotic solutions to evolution problems governed by nonlinear
differential equations. Written not just for mathematicians, it
appeals to the widest audience of learners, including engineers,
biologists, chemists, physicists and economists. For this reason,
it uses only well-known methods of classical analysis at foundation
level, while the applications and examples are specially chosen to
be as varied as possible.
This IMA Volume in Mathematics and its Applications PARTICULATE
FLOWS: PROCESSING AND RHEOLOGY is based on the proceedings of a
very successful one-week workshop with the same title, which was an
integral part of the 1995-1996 IMA program on "Mathematical Methods
in Materials Science." We would like to thank Donald A. Drew,
Daniel D. Joseph, and Stephen L. Passman for their excellent work
as organizers of the meeting. We also take this opportunity to
thank the National Science Foun dation (NSF), the Army Research
Office (ARO) and the Office of Naval Research (ONR), whose
financial support made the workshop possible. A vner Friedman
Robert Gulliver v PREFACE The workshop on Particulate Flows:
Processing and Rheology was held January 8-12, 1996 at the
Institute for Mathematics and its Applications on the University of
Minnesota Twin Cities campus as part of the 1995- 96 Program on
Mathematical Methods in Materials Science. There were about forty
participants, and some lively discussions, in spite of the fact
that bad weather on the east coast kept some participants from
attending, and caused scheduling changes throughout the workshop.
Heterogeneous materials can behave strangely, even in simple flow
sit uations. For example, a mixture of solid particles in a liquid
can exhibit behavior that seems solid-like or fluid-like, and
attempting to measure the "viscosity" of such a mixture leads to
contradictions and "unrepeatable" experiments. Even so, such
materials are commonly used in manufacturing and processing."
This IMA Volume in Mathematics and its Applications TWO PHASE FLOWS
AND WAVES is based on the proceedings of a workshop which was an
integral part of the 1988-89 IMA program on NONLINEAR WAVES. The
workshop focussed on the development of waves in flowing
composites. We thank the Coordinating Commit tee: James Glimm,
Daniel Joseph, Barbara Keyfitz, Andrew Majda, Alan Newell, Peter
Olver, David Sattinger and David Schaeffer for planning and
implementing the stimulating year-long program. We especially thank
the Workshop Organizers, Daniel D. Joseph and David G. Schaeffer
for their efforts in bringing together many of the major figures in
those research fields in which modelling of granular flows and
suspensions is used. Avner Friedman Willard Miller, Jr. PREFACE
This Workshop, held from January 3-10,1989 at IMA, focused on the
properties of materials which consist of many small solid particles
or grains. Let us distinguish the terms granular material and
suspension. In the former, the material consists exclusively of
solid particles interacting through direct contact with one
another, either sustained frictional contacts in the case of slow
shearing or collisions in the case of rapid shearing. In
suspensions, also called two phase flow, the grains interact with
one another primarily through the influence of a viscous fluid
which occupies the interstitial space and participates in the flow.
(As shown by the lecture of I. Vardoulakis (not included in this
volume), the distinction between these two idealized cases is not
always clear."
This book is about two special topics in rheological fluid
mechanics: the elasticity of liquids and asymptotic theories of
constitutive models. The major emphasis of the book is on the
mathematical and physical consequences of the elasticity of
liquids; seventeen of twenty chapters are devoted to this.
Constitutive models which are instantaneously elastic can lead to
some hyperbolicity in the dynamics of flow, waves of vorticity into
rest (known as shear waves), to shock waves of vorticity or
velocity, to steady flows of transonic type or to short wave
instabilities which lead to ill-posed problems. Other kinds of
models, with small Newtonian viscosities, give rise to perturbed
instantaneous elasticity, associated with smoothing of
discontinuities as in gas dynamics. There is no doubt that liquids
will respond like elastic solids to impulses which are very rapid
compared to the time it takes for the molecular order associated
with short range forces in the liquid, to relax. After this, all
liquids look viscous with signals propagating by diffusion rather
than by waves. For small molecules this time of relaxation is
estimated as lQ-13 to 10-10 seconds depending on the fluids. Waves
associated with such liquids move with speeds of 1 QS cm/s, or even
faster. For engineering applications the instantaneous elasticity
of these fluids is of little interest; the practical dynamics is
governed by diffusion, *say, by the Navier-Stokes equations. On the
other hand, there are other liquids which are known to have much
longer times of relaxation.
This volume collects papers dedicated to Jerry Ericksen on his
sixtieth birthday, December 20, 1984. They first appeared in
Volumes 82-90 (1983-1985) of the Archive for Rational Mechanics and
Analysis. At the request of the Editors the list of authors to be
invited was drawn up by C. M. Dafermos, D. D. Joseph, and F. M.
Leslie. The breadth and depth of the works here reprinted reflect
the corresponding qualities in Jerry Ericksen's research, teaching,
scholarship, and inspiration. His interests and expertness center
upon the mechanics of materials and extend to everything that may
contribute to it: pure analysis, algebra, geometry, through all
aspects of theoretical mechanics to fundamental experiment, all of
these illumi nated by an intimate and deep familiarity with the
sources, even very old ones. He is independent of school and
contemptuous of party spirit; his generosity in giving away his
ideas is renowned, but not everyone is capable of accepting what is
offered. His writings are totally free of broad claims and
attributions beyond his own study. Some are decisive, some are
prophetic, and all are forthright. His work has served as a beacon
of insight and simple honesty in an age of ever more trivial and
corrupt science. The authors of the memoirs in this volume are his
students, colleagues, admirers, and (above all) his friends."
This substantially revised second edition teaches the bifurcation of asymptotic solutions to evolution problems governed by nonlinear differential equations. Written not just for mathematicians, it appeals to the widest audience of learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at foundation level, while the applications and examples are specially chosen to be as varied as possible.
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