Low Reynolds number hydrodynamics: with special applications to particulate media

563 Pages·1981·12.37 MB·English

by  John Happel

Low Reynolds number hydrodynamics.
Mechanics of fluids and transport processes
editor: RJ. Moreau
Low Reynolds number
hydrodynamics
with special applications to particulate media
John Happel
Columbia University
Department of Chemical Engineering and Applied Chemistry
New York, New York USA
Howard Brenner
Department of Chemical Engineering
Cambridge, Massachusetts USA
1983 MARTINUS NIJHOFF PUBLISHERS
a member of the KLUWER ACADEMIC PUBLISHERS GROUP
THE HAGUE / BOSTON / LANCASTER
Distributors
for the United States and Canada: Kluwer Boston, Inc., 190 Old Derby Street,
Hingham, MA 02043, USA
for all other countries: Kluwer Academic Publishers Group, Distribution Center,
P.O.Box 322, 3300 AH Dordrecht, The Netherlands
Library of Congress Catalogue Card Number 72-97238
First edition 1963
Second revised edition 1973
First paperback edition 1983
ISBN-13: 978-90-247-2877-0 e-ISBN-13: 978-94-009-8352-6
DOl: 10.1007/978-94-009-8352-6
Copyright
© 1983 by Martinus Nijhoff Publishers, The Hague.
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, mechanical,
photocopying, recording, or otherwise, without the prior written permission of
the publishers,
Martinus Nijhoff Publishers, P.O. Box 566, 2501 CN The Hague,
The Netherlands.
Contents
Preface
Symbols
1. Introduction
1-1 Definition and purpose, 1. 1-2 Historical review, 8. 1-3 Application in
science and technology, 13.
2. The Behavior of Fluids in Slow Motion 23
2-1 The equations of change for a viscous fluid, 23. 2-2 Mechanical energy
dissipation in a viscous fluid, 29. 2-3 Force and couple acting on a body moving
in a viscous fluid, 30. 2-4 Exact solutions of the equations of motion for a
viscous fluid, 31. 2-5 Laminar flow in ducts, 33. 2-6 Simplifications of the
Navier-Stokes equations, especially for slow motion, 40. 2-7 Paradoxes in
the solution of the creeping motion equations, 47. 2-8 Molecular effects in
fluid dynamics, 49. 2-9 Non-newtonian flow, 51. 2-10 Unsteady creeping
flows, 52.
3. Some General Solutions and Theorems Pertaining
to the Creeping Motion Equations 58
3-1 Introduction, 58. 3-2 Spherical coordinates, 62. 3-3 Cylindrical co
ordinates, 71. 3-4 Integral representations, 79. 3-5 Generalized reciprocal
;heorem, 85. 3-6 Energy dissipation, 88.
Contents
4. Axisymmetrical Flow 96
4-1 Introduction, 96. 4-2 Stream function, 96. 4-3 Relation between
stream function and local velocity, 98. 4-4 Stream function in various co
ordinate systems, 99. 4-5 Intrinsic coordinates, 100. 4-6 Properties of the
stream function, 102. 4-7 Dynamic equation satisfied by the stream function,
103. 4-8 Uniform flow, 106. 4-9 Point source or sink, 106. 4-10 Source
and sink of equal strength, 107. 4-11 Finite line source, 108. 4-12 Point
force, 110. 4-13 Boundary conditions satisfied by the stream function, 111.
4-14 Drag on a body, 113. 4-15 Pressure, 116. 4-16 Separable coordinate
systems, 117. 4-17 Translation of a sphere, 119. 4-18 Flow past a sphere,
123. 4-19 Terminal settling velocity, 124. 4-20 Slip at the surface of a
sphere, 125. 4-21 Fluid sphere, 127. 4-22 Concentric spheres, 130. 4-23
General solution in spherical coordinates, 133. 4-24 Flow through a conical
diffuser, 138. 4-25 Flow past an approximate sphere, 141. 4-26 Oblate
spheroid, 145. 4-27 Circular disk, 149. 4-28 Flow in a venturi tube, 150.
4-29 Flow through a circular aperture, 153. 4-30 Prolate spheroid, 154.
4-31 Elongated rod, 156. 4-32 Axisymmetric flow past a spherical cap, 157.
5. The Motion of a Rigid Particle of Arbitrary Shape
in an Unbounded Fluid 159
5-1' Introduction, 159. 5-2 Translational motions, 163. 5-3 Rotational mo
tions, 169. 5-4 Combined translation and rotation, 173. 5-5 Symmetrical
particles, 183. 5-6 Nonskew bodies, 192. 5-7 Terminal settling velocity of
an arbitrary particle, 197. 5-8 Average resistance to translation, 205. 5-9
The resistance of a slightly deformed sphere, 207. 5-10 The settling of spher
ically isotropic bodies, 219. 5-11 The settling of orthotropic bodies, 220.
6. Interaction between Two or More Particles 235
6-1 Introduction, 235. 6-2 Two widely spaced spherically isotropic particles,
240: 6-3 Two spheres by the method of reflections and similar techniques, 249.
6-4 Exact solution for two spheres falling along their line of centers, 270. 6-5
Comparison of theories with experimental data for two spheres, 273. 6-6
More than two spheres, 276. 6-7 Two spheroids in a viscous liquid, 278. 6-8
Limitations of creeping motion equations, 281.
7. Wall Effects on the Motion of a Single Particle 286
7-1 Introduction, 286. 7-2 The translation of a particle in proximity to
container walls, 288. 7-3 Sphere moving in an axial direction in a circular
cylindrical tube, 298. 7-4 Sphere moving relative to plane walls, 322. 7-5
Spheroid moving relative to cylindrical and plane walls, 331. 7-6 k-coeffi
cients for typical boundaries, 340. 7-7 One- and two-dimensional problems,
341. 7-8 Solid of revolution rotating symmetrically in a bounded fluid, 346.
7-9 Unsteady motion of a sphere in the presence of a plane wall, 354.
Contents
8. Flow Relative to Assemblages of Particles 358
8-1 Introduction, 358. 8-2 Dilute systems-no interaction effects, 360. 8-3
Dilute systems-first-order interaction effects, 371. 8-4 Concentrated systems,
387. 8-5 Systems with complex geometry, 400. 8-6 Particulate suspensions,
410. 8-7 Packed beds, 417. 8-8 Fluidization, 422.
9. The Viscosity of Particulate Systems 431
9-1 Introduction, 431. 9-2 Dilute systems of spheres-no interaction effects,
438. 9-3 Dilute systems-first-order interaction effects, 443. 9-4 Concen
trated systems, 448. 9-5 Nonspherical and nonrigidparticles, 456. 9-6 Com
parison with data, 462. 9-7 Non-newtonian behavior, 469.
Appendix A. Orthogonal Curvilinear Coordinate Systems 474
A-I Curvilinear coordinates, 474. A-2 Orthogonal curvili,[ear coordinates,
477. A-3 Geometrical properties, 480. A-4 Differenfiation of unit vectors,
481. A-5 Vector differential invariants, 483. A-6 Relations between carte
sian and orthogonal curvilinear coordinates, 486. A-7 Dyadics in orthogonal
curvilinear coordinates, 488. A-8 Cylindrical coordinate systems, 490. A-9
Circular cylindrical coordinates, 490. A-I0 Conjugate cylindrical coordinate
systems, 494. A-ll Elliptic cylinder coordinates, 495. A-12 Bipolar cylinder
coordinates, 497. A-13 Parabolic cylinder coordinates, 500. A-14 Coordinate
systems of revolution, 501. A-15 Spherical Coordinates, 504. A-16 Conjugate
coordinate systems of revolution, 508. A-17 Prolate spheroidal coordinates,
509. A-18 Oblate spheroidal coordinates, 512. A-19 Bipolar coordinates, 516.
A-20 Toroidal coordinates, 519. A-21 Paraboloidal Coordinates, 521.
Appendix B. Summary of Notation and Brief Review
of Polyadic Algebra 524
Name Index 537
Subject Index 543
Preface
One studying the motion of fluids relative to particulate systems is soon
impressed by the dichotomy which exists between books covering theoretical
and practical aspects. Classical hydrodynamics is largely concerned with
perfect fluids which unfortunately exert no forces on the particles past which
they move. Practical approaches to subjects like fluidization, sedimentation,
and flow through porous media abound in much useful but uncorrelated
empirical information. The present book represents an attempt to bridge
this gap by providing at least the beginnings of a rational approach to fluid
particle dynamics, based on first principles.
From the pedagogic viewpoint it seems worthwhile to show that the
Navier-Stokes equations, which form the basis of all systematic texts, can be
employed for useful practical applications beyond the elementary problems
of laminar flow in pipes and Stokes law for the motion of a single particle.
Although a suspension may often be viewed as a continuum for practical
purposes, it really consists of a discrete collection of particles immersed in an
essentially continuous fluid. Consideration of the actual detailed boundary
value problems posed by this viewpoint may serve to call attention to the
limitation of idealizations which apply to the overall transport properties of
a mixture of fluid and solid particles.
It is hoped that the research worker in this and related fields will be
stimulated by noting that not only does the hydrodynamic viewpoint lead to
a clearer correlation of much existing work, but that, at every turn, there
exists a host of new problems awaiting solution. Among those which seem
especially intriguing are the effect of variation in particle size and arrange
ment on the dynamic behavior of particulate systems, and the possibilities of
extending the present treatments to higher Reynolds numbers. Engineers may
be interested in the availability of fluid dynamic models which can serve as
the framework for more extended investigations involving other transport
processes, coupled perhaps with chemical reactions.
Preface
The treatment developed here is based almost entirely on the linearized
form of the equations of motion which results from omitting the inertial
terms from the Navier-Stokes equations, giving the so-called creeping motion
or Stokes equations. This is tantamount to assuming that the particle Reynolds
numbers are very small. Many systems which involve bulk flow relative to
external boundaries at high Reynolds numbers are still characterized by low
Reynolds numbers as regards the movement of particles relative to fluid.
Also, inertial effects are less important for systems consisting of a number of
particles in a bounded fluid medium than they are for the motion of a single
particle in an unbounded fluid.
The subject matter is largely confined to a development of the macro
scopic properties of fluid-particle systems from first principles. General
hydrodynamic and mathematical concepts are not treated in detail beyond
what is required for further development. Most of the experimental data
reported are confined to critical experiments aimed at demonstrating the
applicability of the theoretical results to actual physical systems. Following
the first few introductory chapters, subsequent material is organized on the
basis of the class of boundary-value problems involved, similar to the ap
proach used by C. W. Oseen in his classical "Hydrodynamik." Starting with
the motion of a single particle in an unbounded medium, the problem of
the motion of several particles interacting with each other, of particles moving
in the presence of bounding walls and finally of combinations of these factors
are considered successively. Final chapters deal with the movement of fluids
relative to particulate assemblages and with the viscosity of suspensions of
particles. The latter treatment is more provisional and includes comparison
of theory with empirical equations and data.
Much of the material presented is based on original investigations of the
authors and their students, especially Jack Famularo. We were also fortunate
in having the advice of o. H. Faxen, who provided us with many early papers
by himself and C. W. Oseen, and who also carefully read the entire manu
script. S. Wakiya was also kind enough to review the manuscript and to
provide additional useful suggestions. Support in the form of grants and
fellowships from the Petroleum Research Fund of the American Chemical
Society, The National Science Foundation, The Institute of Paper Chemistry,
The Pulp and Paper Research Institute of Canada, and The Texas Company
is gratefully acknowledged. The Courant Institute of Mathematical Sciences
of New York University graciously provided considerable computer time.
The authors are convinced that the foundation of a scientifically sound
development of the fluid dynamics of particulate media at low Reynolds
numbers is now available, which should serve as a sound basis for future
investigations. They hope that this book will serve to illustrate the versatility
of this field of study.
JOHN HAPPEL
HOWARD BRENNER
SYMBOLS
The following is a list of the most frequently occurring sym
bols used in the book. Symbols not defined here are defined
at their first place of use. A few of these symbols are oc
casionally used in other contexts.
a Sphere radius Kjk Translation dyadic in multiparticle
system
A Cross-sectional area
I Length; distance to boundary; distance
b Distance to cylinder axis
between particles
B Center of buoyancy
m Hydraulic radius
c Particle dimension
mf ; mp Mass of displaced fluid; mass of
Cn Drag coefficient particle
C ; (C); Cij Coupling dyadic; matrix; tensor "'1 Center of mass
on
Cjk Coupling dyadic in multiparticle system Distance ,measured normal to surface
d Diameter n Unit normal vector
D/ Dt Material (convected) time derivative Nile Reynolds number
== o
(eh e2, e,l) e Triad of right-handed Arbitrary point fixed in particle
j
orthonormal eigenvectors P Local pressure
£ Total energy dissipation rate
PI/ Solid spherical harmonic
£2 Stokes stream function operator
PI/(cosf)); P::'(cosf}) Legendre function;
fk«(J, <1» Surface spherical harmonic associated Legendre function
F Force !1P or!1p Pressure drop
F, F Vector force P Intrinsic vector "pressure" field
j
(fF) Force matrix !1P Vector pressure drop
g Local acceleration of gravity vector [ijJ Intrinsic triadic "stress" field
(h], h." kj) c" h Metrical coefficients (ql> q2' q3) = qj Curvilinear coordinates
j
Yf", I" . .f;, Gegenbauer functions Q Volumetric 'flow rate or volume
i =,..j-=1 Unit imaginary number (r, f), <1» Spherical polar coordinates
(i], i2, i;l) ecCe ij Unit vectors r Position vector
i, j, k, or i.e, iy, iz Cartesian unit vectors r() Position vector relative to origin at 0
/", K" Modified Bessel functions rop Vector from 0 to P
R Center of reaction
I Dyadic idemfactor
R. Radius of circular cylinder
k Wall-effect constant; Kozeny constant
R Position vector
k Wall-effect dyadic
RR Vector defined in Eq. (5-7.14)
K Translational resistance coefficient for os
Distance measured along a surface
isotropic particle; Darcy permeability
constant s Unit tangent vector in intrinsic coordi-
K Average translational resistance nates
coefficient S Surface area
K], K2, K.l Eigenvalues of translation dyadic S p Particle surface
dS Directed element of surface area
(.]f') Resistance matrix
t Time
K; (K) ; Kij Translation resistance dyadic;
matrix; tensor t Unit tangent vector