Mathematical Modelling of Heat and Mass Transfer Processes

330 Pages·1995·6.731 MB·English

by  V. G. Danilov, V. P. Maslov, K. A. Volosov (auth.)

Mathematical Modelling of Heat and Mass Transfer Processes
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathel1lQtics and Computer Science, Amsterdam, The Netherlands
Volume 348
Mathematical Modelling
of Heat and
Mass Transfer Processes
by
v.
G. Danilov
v.
P. Maslov
and
K. A. Volosov
Moscow Institute of Electronics and Mathematics,
Moscow, Russia
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4183-6 ISBN 978-94-011-0409-8 (eBook)
DOI 10.1007/978-94-011-0409-8
This is a completely revised and updated translation
of the original Russian work of the same title,
Moscow, Nauka © 1987. Translation by M. A. Shishkova.
Printed on acid-free paper
AlI Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover Ist edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form Of by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vll
From the Preface to the Russian Edition. . . . . . . . . . . . . . . . . . IX
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter I. Properties of Exact Solutions of
Nondegenerate and Degenerate Ordinary Differential
Equations
1.1. Standard equations. . .. . .. .. .. .. . . . .. . . .. ... ... . . . ... .. . 19
1.2. Examples............................................... 30
Chapter II. Direct Methods for Constructing Exact
Solutions of Semilinear Parabolic Equations
2.1. Preliminary notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2. Representation of self-similar solutions in terms of rational
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3. Construction of exact one-phase and two-phase solutions 47
2.4. Formulas for solutions of semilinear parabolic equations
with common cubic nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5. Relation between the number of phases in the solution and
the degree of nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.6. Asymptotics of wave creation for the KPP-Fisher equation 70
Chapter III. Singularities of Nonsmooth Solutions to
Quasilinear Parabolic and Hyperbolic Equations
3.1. Main definitions. . . . . . . . . . . . .. . .. . . .. .. . .. .. .... . . . . . .. . 74
3.2. Asymptotic solutions bounded as c -+ 0 ................ 75
3.3. Asymptotic solutions unbounded as c -+ 0 .............. 78
3.4. The structure of singularities of solutions to quasilinear
parabolic equations near the boundary of the solution
support................................................. 81
3.5. The structure of singularities of nonsmooth self-similar
solutions to quasilinear hyperbolic equations.... . . . . . . .. 107
Chapter IV. Wave Asymptotic Solutions of Degenerate
Semilinear Parabolic and Hyperbolic Equations
4.1. Self-stabilizing asymptotic solutions .................... 127
4.2. Construction of nonsmooth asymptotic solutions.
Derivation of basic equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
vi CONTENTS
4.3. Global localized solutions and regularization of ill-posed
problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.4. Asymptotic behavior of localized solutions to equations
with variable coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.5. Heat wave propagation in nonlinear media. Asymptotic
solutions to hyperbolic heat (diffusion) equation ........ 180
4.6. Localized solutions in the multidimensional case ........ 193
Chapter V. Finite Asymptotic Solutions of Degenerate
Equations
5.1. An example of constructing an asymptotic solution ..... 201
5.2. Asymptotic solutions in the one-dimensional case. . . . . . . . 209
5.3. Asymptotic finite solutions of degenerate quasilinear
parabolic equations with small diffusion ................ 219
5.4. Relation between approximate solutions of quasilinear
parabolic and parabolic equations ...................... 229
Chapter VI. Models for Mass Transfer Processes
6.1. Nonstationary models of mass transfer... .... .. ... .. .... 235
6.2. Asymptotic solution to the kinetics equation of
nonequilibrium molecular processes with external
diffusion effects ........................................ 240
6.3. The simplest one-dimensional model. . . . . . . . . . . . . . . . . . . . 248
Chapter VII. The Flow around a Plate
7.1. Introduction............................................ 254
7.2. Uniformly suitable asymptotic solution to the problem
about the flow of low-viscous liquid around a semi-infinite
thin plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
7.3. Asymptotic behavior of the laminar flow around a plate
with small periodic irregularities. . . . . . . . . . . . . . . . . . . . . . . . 269
7.4. Critical amplitude and vortices in the flow around a plate
with small periodic irregularities .... . . . . . . . . . . . . . . . . . . . . 287
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Appendix. Justification of Asymptotic Solutions
(S. A. Vakulenko)
1. One-dimensional scalar case ............................ 301
2. Complete description of the solution behavior in
a neighborhood of the manifold M ..................... 309
3. Zeldovich waves ........................................ 310
Index 315
PREFACE
In the present book the reader will find a review of methods for constructing a
certain class of asymptotic solutions, which we call self-stabilizing solutions. This
class includes solitons, kinks, traveling waves, etc. It can be said that either the
solutions from this class or their derivatives are localized in the neighborhood of a
certain curve or surface.
For the present edition, the book published in Moscow by the Nauka publishing
house in 1987, was almost completely revised, essentially up-dated, and shows our
present understanding of the problems considered.
The new results, obtained by the authors after the Russian edition was published,
are referred to in footnotes. As before, the book can be divided into two parts: the
methods for constructing asymptotic solutions ( Chapters I-V) and the application of
these methods to some concrete problems (Chapters VI-VII). In Appendix a method
for justification some asymptotic solutions is discussed briefly.
The final formulas for the asymptotic solutions are given in the form of theorems.
These theorems are unusual in form, since they present the results of calculations.
The authors hope that the book will be useful to specialists both in differential
equations and in the mathematical modeling of physical and chemical processes.
The authors express their gratitude to Professor M. Hazewinkel for his attention
to this work and his support.
The authors are especially grateful to M. A. Shishkova for editing, translating,
and typesetting the continuously varying manuscript.
FROM THE PREFACE TO THE RUSSIAN EDITION
Dissipative structures introduced by I. Prigogine are the central objects of a new,
rapidly developing field of science called synergetics. Actually, solutions of some
standard problems, localized in space or, in a sense, "almost" localized, are consid
ered. Such solutions propagate with time, and their structure varies slowly. They
describe different phenomena such as flame spread, mould growth, crystal growth,
motion of a drop along the inclined plane, a grove evolution, etc. Therefore, from
the viewpoint of natural science, such solutions are no less interesting than solitons.
The present monograph deals with such solutions, which we call synergets.
In this book the evolution of dissipative structures is studied by using constructive
methods, i.e., by constructing asymptotic solutions of semilinear parabolic equations.
The problem reduces to solving certain ordinary differential equations. As a rule,
this is simpler than the investigation of the initial partial differential equation.
An important advantage of asymptotic methods is the possibility to examine
equations with variable coefficients of rather general form by analytical methods
and thus to investigate the stability of dissipative structures with respect to the
properties of inhomogeneous media.
The authors wish to thank C. P. Kurdyumov, V. P. Myasnikov, O. S. Ryzhov, and
V. V. Pukhnachev for useful contacts, comments, and advice. They also express their
gratitude to S. Yu. Dobrokhotov, V. A. Tsupin, G. A. Omel'yanov, P. N. Zhevandrov,
and the scientific editor of this book V. E. Nazaikinskii, with whom they have
discussed the topics of this book at different times.
INTRODUCTION
The classical models of mathematical physics are based on the concept of a contin
uous (qualitatively) homogeneous medium. However, even the simplest observations
show that objects and phenomena around us often have the form of localized struc
tures, which appear, move, and interact. Localized structures (i.e., groups or families
of objects) are naturally formed among the objects of the surrounding world. They
are everywhere, from star systems to biological populations.
Though different in nature, macroscopic structures have much in common. The
most general common property is the existence of a boundary that isolates the
structure. There are two types of such structures, with sharp boundary and with
fuzzy boundary.
A group of plants, for example, an oak grove gives an example of a structure with
sharp boundary (see Figure 1). Here one can see three different regions: the space
around the grove, the grove itself, where the density oftrees is approximately stable,
and a narrow (in contrast to the grove) boundary region (the border of the grove),
where new trees grow intensively. There are no trees beyond the border of the grove.
border
FIGURE 1
The process of combustion gives another example.
In chemistry the process of combustion is considered as parallel chemical reactions
proceeding at different rates. In the case of two interacting gases, these reactions
can be conditionally described as follows [33, 34]
A + B2 -+ AB + B, B+A2 -+AB+A.
2 INTRODUCTION
The slower reacting substance is usually called the leading center. The rate of
reaction, in which this substance is involved, determines the rate of combustion.
Ya. B. Zeldovich considered the case when the product of decay (dissociation), which
forms the leading center, is in excess in the products of reaction. In this case, as
Ya. B. Zeldovich showed, it is necessary to take into account the diffusion of leading
centers into the flame region, which leads to a self-consistent motion of the flame
front.
T
T
FIGURE 2. I - reagents, II - flame region, III - reaction products
Figure 2 shows the distributions of temperature T and reagent concentrations
(curve 1 shows the concentration of substance A, curve 2 shows the concentration of
substance B, i.e., of the leading center). Since heat is released in chemical reactions
of combustion, in the region of reaction products the temperature is maximal. In
the region of reagents the temperature is lower and the probability that a chemical
reaction starts at this temperature is small.
There is a natural question: what is in common between different mathematical
models describing the evolution of these structures? The answer is: the evolution of
both structures is described by quasilinear parabolic equations, which mathemati
cally express the balance relations characterizing these structures.
The equation describing the evolution of a grove has the form*:
~~ =
- DdivU gradU - U(A - BU) O.
= =
Here U is the density of plants, A A(r), B B(r) are given time-dependences
describing the distribution of resources responsible for vital activity of the population
and for laws of birth and death, D is the transport coefficient.
In dimensionless coordinates, this equation takes the form
cO;: - c2 divugrad u - u(a - bu) = O.
*The data characterizing this model were obtained by S. M. Semenov, who told us about them.