Author and Subject Cumulative Index Including Table of Contents Volume 1-34

199 Pages·1998·59.782 MB·English

by  Erik van der Giessen and Theodore Y. Wu (Eds.)

MECHANICS
/MIXTURES
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Series on Advances in Mathematics for Applied Sciences - Vol. 35
MECHANICS
"/MIXTURES
K. R. Rajagopal
Department of Mechanical Engineering
University of Pittsburgh, USA
L. Tao
Viscoustech Inc., Pittsburgh, USA
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Rajagopal, K. R. (Kumbakonam Ramamani)
Mechanics of mixtures / K.R. Rajagopal, L. Tao.
p. cm. — (Series on advances in mathematics for applied sciences ; vol. 35)
Includes bibliographical references and index.
ISBN 9810215851
1. Fluid dynamics. 2. Continuum mechanics. I. Tao, L.
II. Title. III. Series.
QA911.R27 1995
532'.05-dc20 95-16400
CIP
Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd.
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Printed in Singapore by UtoPrint
This book is dedicated to my parents Hemalata and K. R. Ramamani, with love
and affection,
K. R. Rajagopal,
and to my late uncle Ruantao Zhang,
L. Tao.
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Preface
An author usually faces the onerous task of justifying the need for his or her treatise,
as there are few subjects left where not enough has been written. Fortunately for
us, the continuum theory of mixtures is one of the few areas where it is unnecessary
to plead the case for a book. While there are many mathematical treatments on
the subject matter of this book, their approach is quite different from that which is
presented here. A few review articles on mixture theory have come close to providing
a comprehensive overview of the theoretical foundations of the topic; however they
do not discuss, in any detail, initial and boundary value problems within the context
of the theory. It is to this aspect of the subject that our book is primarily devoted.
Our bias towards applications can be felt throughout the book, though the last two
chapters are devoted to developing new approaches to modelling a mixture of two
fluids and a fluid infused with solid particles with a special class of applications in
mind.
We have tried to inject new ideas or look at old problems within the context of
a new perspective, wherever possible in the book. For instance, the exact boundary
conditions that are to be prescribed for boundary value problems involving mixtures
in general are far from settled. Boundary conditions based on thermodynamic argu­
ments have been used previously to analyze steady state problems. These boundary
conditions are valid when the boundary of the mixture is saturated. However, there
are situations when such a condition does not obtain on the boundary. Thus, we have
revisited some steady state boundary value problems and solved them within the con­
text of a new set of boundary conditions that are based on a very different philosophy
from that adopted in earlier attempts at resolving these problems. Similarly, in the
treatment of a mixture of a fluid infused with solid particles in this book, we present
an approach that is markedly different from my earlier treatment of such problems
with Johnson and Massoudi. We envisage a different class of problems where we feel
our present approach is applicable. The presentation here concerning a. mixture of
two fluids also sets forth an approach that might be extended to a turbulent motion
of the mixture of two fluids. Here, we recognize that the turbulent motion of a single
constituent is as yet not well understood. However, given the fact that mixtures do
undergo turbulent motion, it is necessary to address this problem even if the attempt
is crude at best.
This book is not meant to be a text on the theory of mixtures, as it does not
VII
Vlll Preface
develop the basic frame-work for the study of mixtures in the kind of detail and
completeness that a beginning graduate student needs. The book presumes that
the reader is quite conversant with continuum mechanics. No exercises or examples
are provided, and in many instances, results from papers published recently in the
literature are used, without recording them in any detail. Issues that we consider
to be unresolved are discussed with the hope that it might stimulate the reader to
think about them. A great many assumptions, approximations and simplifications,
possibly oversimplifications are made in order to put the theory to use, presenting
the opportunity for one with more ability to address the same problems in a more
rational and rigorous manner.
My interest in the continuum theory of mixtures stems from my interactions with
Alan Wineman at the University of Michigan at Ann Arbor. The numerous discus­
sions that we had have influenced my thinking about the mechanics of mixtures, and
is reflected in all my work on mixtures. I would be remiss if I did not acknowledge
the contributions of my teachers and collaborators, too numerous to mention individ­
ually, from whom I have learnt a great deal of continuum mechanics over the years.
Of course, it goes without saying, that any errors or misunderstandings of the subject
are solely attributable to me.
K. R. Rajagopal
Pittsburgh
January 1995
Contents
1 INTRODUCTION 1
2 PRELIMINARIES 7
2.1 Kinematics 7
2.2 Partial Tractions and Partial Stresses . .. 10
2.3 Balance of Mass 10
2.4 Balance of Linear Momentum . . 11
2.5 Balance of Angular Momentum . .. 14
2.6 Conservation of Energy . .. . .. 15
2.7 Second Law of Thermodynamics . .. . .. 16
2.8 Volume Additivity Constraint . . . .. . . .. . .. 18
3 DIFFUSION OF A FLUID THROUGH A SOLID UNDERGOING
LARGE DEFORMATIONS: CONSTITUTIVE RESPONSE FUNC­
TIONS 21
4 STEADY STATE PROBLEMS 29
4.1 Boundary Conditions 30
4.2 Diffusion of a Fluid Through a Rubber Slab 37
4.3 Diffusion of a Fluid Through a Rubber Annulus . . 46
5 DIFFUSING SINGULAR SURFACE 57
5.1 Analysis of a Time-dependent Diffusing Singular Surface 58
5.2 Unsteady Diffusion of a Fluid Through a Nonlinearly Elastic Solid Slab 75
5.3 Unsteady Diffusion of a Fluid Through a Nonhnearly Elastic Solid
Cylindrical Annulus . .. . . .. 87
6 WAVE PROPAGATION IN SOLIDS INFUSED WITH FLUIDS 99
6.1 Biot's Equations 100
6.2 Counterpart of Biot's Equations 105
6.3 Wave Propagation in Anisotropic Swollen Mixtures 110
ix
x MECHANICS OF MIXTURES
7 MIXTURE OF TWO NEWTONIAN FLUIDS 123
7.1 Basic Balance Equations . . .. 124
7.2 Constitutive Variables 125
7.3 Constitutive Modelling 128
7.4 Two Special Motions 131
8 MIXTURE OF A FLUID AND SOLID PARTICLES 139
8.1 Turbulence Modelling in Newtonian Fluids . .. 142
8.2 Balance Equations for the Mixture 149
8.3 On Constitutive Modelling 154
8.4 Specific Model 158
A SOME RESULTS FROM DIFFERENTIAL GEOMETRY 163
B STATUS OF DARCY'S LAW WITHIN THE CONTEXT OF MIX­
TURE THEORY 167
Bibliography 171
Index 191