Orthogonal Curvilinear A Coordinate Systems - Springer

Summary of Notation B
and Brief Review of

Polyadic Algebra

The vector, dyadic, polyadic, and tensor notation used in this book
follows customary American usage, being derived from the work of Gibbs4•
Extensions of Gibbs' notation to polyadics of ranks greater than 2 (that is,
dyadics) is discussed in Drew's Handbook of Vector and Polyadic Analysis3•
For our immediate purposes, however, the formidable general symbolism
developed by Drew is unnecessary. Block! has produced a very brief and
readable textbook on the relationship between polyadics and tensors.
Milne's5 book affords an excellent example of the physical insights afforded
by polyadic symbolism in physical problems. The reader should be cautioned,
however, that Milne utilizes the "nesting convention" of Chapman and
Cowling2 with regard to multiplication of polyadics, rather than the original
notation of Gibbs4 •

In this text, physical quantities encountered are distinguished as being

scalars, vectors, and polyadics. They are distinguished, wherever feasible, *

by differences in type as follows:

*Thus, boldface Greek symbols may represent either vectors or polyadics.

524

Summary of Notation and Brief Review of Polyadic Algebra 525

s = scalar (lightface italic)
v = vector (boldface roman)
i, e = unit vectors (boldface roman)
, = polyadic (boldface sans serif)

When, without specific designation, the same letter appears in the same
context both as a vector (for example, v) and as a scalar (for example, v), the

scalar is the magnitude of the vector. When necessary for clarity Iv Iis used to

denote the magnitude of the vector v.
Vectors and polyadics are often conveniently expressed in terms of their

components in some particular system of curvilinear coordinates (qlo q20 qa),

for example, cartesian coordinates (x, y, z), spherical coordinates (r, fJ, cp),
cylindrical coordinates (p, cp, z). In this text we work only with orthogonal

curvilinear coordinates, of which the preceding systems are examples. If
(ilo ih ia) are a right-handed* triad of unit base vectors in such a system, for
example (i, j, k) in cartesian coordinates or (in ie, i~) in spherical coordinates,
then any vector v may be expressed in the form

(B-1)

where (VI' Vb Va) are the components of the vector v in the particular coor-
dinate system. This relation may also be written concisely as

(B-2)

We shall review briefly some of the more important properties of poly-
adics. The most general dyadic can be expressed in the nonion form

(B-3)

Written out explicitly this is (B-4)

+ +D = ililDll iligD 12 iliaDj3

+ igil D21 + igigDgg + i2i3 D23
+ i3 il D 3j + i3i2D32 + i3i3D33

The set of nine scalar numbers Djk (j, k = 1,2,3) are the components of

the dyadic. Though their numerical values depend on the particular system

ofcoordinates (qj, q2, q3) employed, the dyadic D itselfhas a significance which
transcends any particular choice of coordinates. If the unit vectors in Eq.
(B-4) are suppressed, one may regard the dyadic D as the 3 x 3 matrix

*The system is right-handed in the cyclic order

3

'"1,/--+ 2

if, for the scalar triple product,

526 Appendix B

=CDI2 DU) (B-5)

(D) D21 D22 D23
D31 D32 D33

The determinant of the dyadic D is the scalar

Dll DI2 DI3 (B-6)
detD = D21 D22 D23

D31 D32 D33

whose value can be shown to be an invariant which is independent of the

particular system of coordinates employed.
The transpose (or conjugate) of the dyadic D is denoted by the symbol

Dt (some authors use D, Dc, Dt) and may be defined as the dyadic obtained

by interchanging the order of the unit vectors in Eq. (B-3). Thus,

(B-7)

or, since the indices j and k are dummy indices, we have, upon interchanging

j and k, that

33 (B-8)

Dt = ~ ~ ijikDkj

j~1 k~1

This operation can be shown to have an invariant meaning. It is also equiva-

lent to the usual transposition operation with matrices, where the rows and

columns in Eq. (B-5) are interchanged.

A dyadic is said to be symmetric if it is equal to its own transpose, that is,

D = Dt (B-9)

From Eqs. (B-3) and (B-8), this is equivalent to the three scalar equations

=Djk DkJ (j, k = 1,2,3) (B-lO)

which, when written out explicitly, requires that

(B-1 I)

A symmetric dyadic thus possesses only six independent components. Any
symmetric dyadic can be written in the diagonal form

(B-12)

where (el , e2, e3) are a particular system of mutually perpendicular unit
vectors called the eigenvectors (characteristic vectors) of the symmetric dyadic
D. The three scalars D I, D2, D3 are called its eigenvalues (principal values,
characteristic values, characteristic roots). For a given symmetric dyadic D,
the problem of establishing its eigenvalues and eigenvectors is equivalent to
that involved in diagonalizing the matrix in Eq. (B-5).

A dyadic D is antisymmetric if it is equal to the negative of its transpose,

that is, if
(B-13)

Summary of Notation and Brief Review of Polyadic Algebra 527

From Eqs. (B-3) and (B-S), this requires that

and Dll = D22 = Dss = 0 (B-14)
D12 = -Du, D23 = -Ds2' DSl = -DIs

An antisymmetric dyadic thus possesses only three independent components.
Any dyadic can be uniquely expressed as the sum of a symmetric and

antisymmetric dyadic as follows:

(B-1S)

the first term on the right being symmetric and the second antisymmetric.
A particularly important dyadic is the idem/actor or unit dyadic. This

may be written in the form*

where {ls _ ifj = k (B-16)
(B-17)
Jk - 0 ifj"* k

is the Kronecker delta. Hence, an equivalent form of the idemfactor is
+ +I = iii! i2i2 isia
(B-lS)

The most general triadic, say T, can be expressed in the formt

and thus has 27 independent components, T jk1• In applications one must

consider two possible transposition operations:

pre-transposition-

aas (B-20)

tT = ~ ~ ~ ikijilTjkl
j=! k=! 1=1
= ass
~ ~ ~ ijikilTkjl
j=! k=! l=!

post-transposition-

=3 as (B-2l)

Tt ~ ~ ~ ijilikTjkl
j=! k=1 l=!
=a as
~ ~ ~ ijikilTjlk
j=1 k=! 1=1

An especially useful triadic is the unit isotropic triadic (alternating triadic,
alternator),

(B-22)

*Some authors prefer U for the unit dyadic.

tSince polyadics of order greater than 2 appear only infrequently in the text, we do not
use any special type style to distinguish the different orders. If necessary, one can attach the

affix n to indicate the order of the polyadic. Thus nA is an n-adic; for example, 2A is a dyadic,
SA a triadic, etc.

528 Appendix B

where Ejk' is the permutation symbol, having the following properties: it is

+zero if any two of the three indices are equal; it has the value 1 if (j, k, /) is
an even cyclic permutation of the integers (l, 2, 3); it has the value -1 if

(j, k, /) is an odd cyclic permutation of the integers (l, 2, 3). Thus, written

out explicitly we have

€ = ili2iS - ilia i2 (B-23)

+ i2iail - i2ilia

+ iaili2 - ia i2il

The most general polyadic of rank n in three-dimensional space is the
n-adic

(B-24)

which has 3n components.
There exist several different types of "multiplication" pertaining to

vectors and polyadics. Since all such entities may be expressed in the form of

Eq. (B-24), the multiplication rules may be conveniently expressed in terms
of operations on the unit vectors in Eq. (B-24), at least for the orthogonal
systems of interest to us in this text. Attention is confined to those multi-
plicative operations which appear explicitly in this book.

For the dot multiplication of vectors, we have (for j, k = 1,2,3)

ij.ik = Sjk (B-25)

whereas for cross mUltiplication of vectors

=• • ~S Ejk,l•, (B-26)
1=1
Ij X Ik

Dot and cross multiplications may be applied to polyadics of any order

by invoking the convention that the operation denoted by the dot or cross

is to be performed on the vectors appearing immediately on either side of the

operational symbol. For example, the two possible dot products of a dyadic

with a vector are D·v = (~ ~ ijikDjk)· ~ i,v,

jk I

= ~ ~ ~ ij(ik'i,)DjkV,
jk1

= ~ ~ ~ ij Ski DjkVI
jk1

= ~ ~ ijDjkVk (a vector) (B-27)
jk

and v·D = (~ijVj)'(~ ~ ikilDkl)
j k1
= ~ ~ ~ (ij.ik)iIDk,Vj
jk ,

= ~ ~ ~ i, SjkDk,Vj
jk ,

= ~ ~ i,Dk,Vk
k,

= ~ ~ ij DkjVk (a vector) (B-28)

Summary of Notatian and Brief Review of Polyadic Algebra 529

the latter equation being obtained by replacing the dummy index 1 with j.
We observe that

v·D -=/= D·v (B-29)

unless D is symmetric, in which case Djk = Dkj• Observe, however, that it

is always true that

(B-30)

One can dot mUltiply two polyadics of any order; for example, for a triadic
, and dyadic D,

= ~ ~ ~ ~ ~ i)kin OtmTjktDmn (B-3 I)

j k tmn

= ~ ~ ~ ~ ijikinTjktDln (a triadic)
j kIn

As examples of cross products we have, with the aid of Eq. (B-26),

D x v = (~ ~ i)kDjk) x (~ itVt)t
jk

= ~ ~ ~ ij(ik X il)DjkvI
jk I

= ~ ~ ~ ~ ijim fktmDjkVI (a dyadic) (B-32)
j kIm

= ~ ~ ~ (ij X ik)ilDklVj
jkI

= ~ ~ ~ ~ imi, fjkmDklVj
j kIm

=~ ~~~ ijim flkj D km VI (a dyadic) (B-33)
j kIm

which we have obtained from the preceding equation by the substitutions

j - I, 1- m, m - j. Because of the relations

(B-34)

it is possible to express these relations in a variety of equivalent forms.

We note that

Dxv-=/=vxD (B-35)

Cross products may also be formed from higher-order polyadics; for example,

D X , = a tetradic.

Another form of multiplication is direct multiplication, in which no
operational symbol is employed. For example,

Dv = (~ ~ ijikDjk)(~ itvt)I
jk

= ~ ~ ~ ijiki, DjkVt (a triadic) (B-36)
jkt

It is sometimes convenient to use multiple operational symbols. The

530 Appendix B

only such operator employed extensively in this book is the double-dot

multiplication of Gibbs. In Gibbs' notation, if a, b, c, d are any vectors, then *

ab: cd = (a·c)(b·d) (B-3?)

In particular, if these be unit vectors

iji,,:izim = (ij.iz){ik'im) = 8jz 8"m (B-38)
(B-39)
For example,

D(l): D(2) = (~ ~ ij i"D)1»: (~ ~ izimDI;;)
jk Zm

= ~ ~ ~ ~ (iji,,: Um)DW Dl;;
z m
= j " ~ ~ (ij.il){ik' im) D)1) Dl;;

~ ~

j " Im

= ~ ~ ~ ~ 8jz8"mDWDl~
j " zm

= ~ ~ DW DW (a scalar)
j Ie

Similarly,

(B-40)

Another useful multiple operation is the double cross-product, which in

Gibbs' notation is defined by the relation

ab ~ cd = (a x c){b x d) (a dyadic) (B-41)

In a sequence involving more than two operations, parentheses may be

*required to define the operations unambiguously. For example,
(v(l) x v(2»'D v(1) x (v(2)'D)

whereas = =V(l) x (D'V(2»

(v(l) X D)'V(2) v(l) X D'V(2)

so that no parentheses are necessary to specify the order of the operation.

The idemfactor has the property that if '\If is any vector or polyadic, then

1·'\If = '\If. I = '\If (B-42)

Other useful properties possessed by the idemfactor are (B-43)

=I: V(J)V(2) v(l). V(2)

where v(l) and V(2) are any vectors. As particular examples we have (B-44)
(B-45)
I:Vv = V·v
and I : V V = V· V = V2

where V2 is the Laplace operator.

*Some authors2,5 define ab: cd = (a·d)(b·c)

Summary of Notation and Brief Review of Polyadic Algebra 531

The unit alternating triadic has the useful property that

E: VO ) V(2) = VO ) X V(2) (B-46)

In particular

E :VV = V x v (B-47)

We note that

E = -I xl (B-48)

If D(1) and D(2) are dyadics and VO ) and V(2) are arbitrary vectors, then if

v(l). D(t). V(2) = v(l). D(2). V(2)

the principle of equality of dyadics permits us to conclude that
D(1) = D(2)

The determinant of a dyadic may be expressed in the invariant form

det D = HD ~ D) : D (B-49)

If the determinant of a dyadic D is different from zero, then the dyadic
possesses a unique inverse D-l defined either by the relation

D·D- l = 1
(B-50)

or

The inverse possesses the properties that (B-51)
(D(1)·D(2Jt l = D(2~'DC;~ (B-52)

and

The inverse or reciprocal dyadic may be computed from the relation

D-l _- -HdDet~DD)' (B-53)

The relationship

D·v = 0

for v an arbitrary vector requires that D = O. On the other hand, if v is a

given vector the preceding relation requires that D = 0 if, and only if, det D

*- O. Conversely, if v is a given vector and det D = 0 then D need not be

zero. For example, if we let i3 be a unit vector parallel to the given vector v,

that is, v = i3 v, then the relation D· v = 0 is clearly satisfied by any dyadic of

the general form +D = ili l D l1 ili2D l2
+ +i 2i l D2l i2i2 D22
+ +i3 il D3l i3 i2D32

In such cases for which det D = 0, D is said to be an incomplete dyadic.

Conversely, D is a complete dyadic if det D *- O.

532 Appendix B

A vector v for which the vector D·v is parallel to v is an eigenvector of D.

If, for such vectors v we write

D·v = I\,V

then I\, is called the eigenvalue associated with v. The foregoing may be
written in the form

(D - 11\,)'v = 0

so that the eigenvalues are the roots of the characteristic (secular) equation

det (D - II\,) = 0 (B-54)

In the particular case where D is a symmetric dyadic, there are in general

three real roots 1\" (i = 1,2,3), not necessarily distinct, of the cubic equation

Dll - I\, DI2 DIS (B-55)
DI2
DIS D22 - I\, Dgs =0
D23 Dss - I\,

If the roots are distinct, then the corresponding three eigenvectors v" defined by

(B-56)

form a mutually perpendicular triad of vectors. The preceding equation
remains unaltered if VI is multiplied by a constant, say c,:

D,(ctvt) = I\,,(CIV;)

It is convenient to choose c, such that the vector c, Vt is normalized to unity,

that is, Ic, v, I = 1. Thus the normalized eigenvectors, e, = v';v;, satisfy
led = 1, and the equation

D·e, = I\"e, (i = 1,2,3) (B-57)

Since they are mutually perpendicular, then

ej·ek = Ojk (B-58)

By utilizing these eigenvalues and normalized eigenvectors the symmetric

dyadic D may be written in the form of Eq. (B-12), where 1\,1 = Dt•

A symmetric dyadic is said to be positive definite if for all non-zero

vectors u, the scalar u·D·u > 0

(B-59)

A necessary and sufficient condition that D be a positive definite symmetric

dyadic is that its three eigenvalues 1\,1> I\,g, I\,s each be positive scalars; for if
we write the symmetric dyadic in the form of Eq. (B-12) and note that I\,t =

D" we obtain + +u· D•u = 1\,1 u~ 1\,2 U~ I\,s u~

(B-60)

where Ut is the component of u in the direction of et.
If we restrict ourselves to situations in which the unit vectors (il> ig, is) in

Eqs. (B-3) and (B-24) are the constant cartesian unit vectors (i, j, k), the
calculus of polyadics can be made equivalent to ordinary scalar calculus. For
example, by writing

Summary of Notation and Brief Review of Polyadic Algebra 533

(B-61)

where (XI> Xg, X3) are cartesian coordinates, the divergence of a dyadic may be
written in cartesian form as follows:

(B-62)

Written out explicitly this is

+ +V ° D = i! (oODX!ll oODXgg! OoDX33!)

+ ig (oDu + ODg2 + OD32) (B-63)
OX! oXg oX3

+ i3 (OODX!!3 + ooDXgg3 + OoDX333)

In noncartesian coordinate systems, the unit vectors are not constant, but

are themselves functions of position (see Sections A-4 and A-7). Hence, in
such cases,

*-o (0 Y) 0 oY

O-::X;j- lk lk -o:X:;-j

which shows why Eq. (B-62) is not valid except when (XI> Xg, X3) and

(il> ig, i3) refer to cartesian coordinates.

Ifwe confine ourselves solely to cartesian coordinates one may, in a sense,

ignore the unit vectors and summation signs in equations such as (B-2),

(B-3), (B-16), (B-24), and (B-62), and write (B-2')
(B-3')
=v Vj (B-16')
=D Djk (B-24')
=I Ojk
=np Pk1k, •••kn

(B-62')

so that an obvious correspondence can be made to exist between polyadics
and cartesian tensors. Polyadics, however, are clearly much more general

entities than are cartesian tensors, for it is obviously not essential that the
unit vectors be cartesian unit vectors. The distinction is, of course, of signifi-
cance only in the calculus of polyadics and tensors-not in their algebras.

With regard to the integral calculus of polyadics, the only formula we
shall mention explicitly is the analog of Gauss' divergence theorem,

534 Appendix B

(B-64)

where S is a closed surface completely bounding the volume V, dS is a directed
element of surface area pointing out of the volume V, np is a polyadic of
any rank, and dV is an element of volume. Since, in general

dS.np *- np·dS

it is important to maintain the proper ordering of the directional quantities
in the integral theorem.

The significance of polyadic integrals is, perhaps, most readily grasped
by expressing them in terms of cartesian unit vectors. For example, if T is
the triadic

and we write dS = ~ ij dSj
and j

V = ~ in J--
n uXn

then Eq. (B-64) may be written
f f~ ~ ~ ilim dSkTklm = ~ ~ ~ ilim ~klm dV
k lm S k lm V UXk

which is a dyadic equation, equivalent to nine scalar relations. In differenti-
ating the unit vectors and in bringing them through the integration sign, we
have explicitly utilized the fact that they are constants, independent of posi-
tion. Thus, the preceding relation written out in component form is valid
only for cartesian systems. The relation from whence it emanated,

is, ofcourse, an invariant relation and holds true in any system ofcoordinates.

BIBLIOGRAPHY

1. Block, H. D., Introduction to Tensor Analysis. Columbus, Ohio: Merrill,
1962.

2. Chapman, S., and T. G. Cowling, The Mathematical Theory of Non-Uniform
Gases, 2nd ed. Cambridge: Cambridge Univ. Press, 1961.

3. Drew, T. B., Handbook of Vector and Polyadic Analysis. New York: Reinhold,
1961.

4. Gibbs, J. W., and E. B. Wilson, Vector Analysis (reprint). New York: Dover,
1960.

5. Milne, E. A., Vectorial Mechanics. London: Methuen, 1957.

Indices

Name Index

[Names listed here refer only to the text
itself. Additional names appear in the Bib-
liography at the end of each chapter.]

A Boardman, R. P., 414
Bohlin, T., 318, 320
Ackerberg, R. C, 141 n Bond, W. N., 153-154
Acrivos, A., 127 n, 207 n Boussinesq, 1., 8, 129
Andersson. 0., 244 Breach, D. R., 225
Aoi, T., 145 Brenner, H., 7 n, 62 n, 67 n, 85, 159, 160,
Aris, R., 7, 23, 26,129
Arrhenius, S., 465 207, 219, 226, 246, 276, 281, 288,
Ast, P. A .. 321. 391 292, 302, 330, 331, 343, 346, 349,
351,376,387,404-408,432
B Brinkman, H. C, 390-391, 413
Brodnyan, J. G., 470
Bagnold, R. A., 19 Broersma, S., 230
Bairslow, L., 48 Bueche, A. M., 461
Bakhmeteff, B. A., 419 Burgers. J. M., 82-85, 229-230, 245,
Bart, E., 273-275,315 277-278, 374, 376, 378, 385, 419,
Bartok, W., 462 441,442
Basset, A. B., 8, 125, 126 Byrne, B. 1., 299, 318
Bateman, H., 8, 129 n
Bazin, H. E., 10 c
Becker, H. A., 232
Berker, R., 7 n Carman, P. C, 10, 393, 403-404, 417,
Bird, R. B., 7, 52, 92 418
Blake, F. C, 10
Blasius, P. R. H., 32 Caswell, B., 51-52
Block. H. D.. 524 Chang, I-Dee, 7 n
Chapman, S., 49-50, 524
Charles, M. E., 36-37
537

538 Name Index

Charnes, A., 48-49 Faxen, O. H., 12, 48, 258, 259, 272, 322,
Cheng, P. Y., 413, 462, 467 323, 325, 328, 330, 344, 379, 385
Chester, W., 281 n
Christiansen, E. B., 188,219-220 Fayon, A. M., 316, 423
Christopherson, D. G., 92 Feodorofl', N. V., 419
Citron, S. J., 354 Feshbach, H., 117 n, 402
Cole, J. D., 45 Fidleris, V., 318, 320
Collins, E. R., 9, 16 Fitch, E. B., 416
Collins, W. D., 157 Ford, T. F., 464
Copley, A. L., 17, 18 n Frisch, H. L., 432, 459, 461, 463
Coull, J., 232 Fujikawa, H., 49, 283, 386
Coulter, N. A., 469 Fulton, J. F., 469
Cowling, T. G., 524
Cox, R. G., 281 n G
Craig, F. F., 315
Cunningham, E., 11, 130, 132 n, 387, Gans, R., 160, 232
Gardner, G. C., 420
411 Geckler, R. D., 465
Ghildyal, C. D., 350
o Gibbs, J. W., 7 n, 86, 201, 524, 530
Gilliland, E. R., 421
Dahl, H., 259-260, 272 Glansdorfl', P., 92
Dahler, J., 25 n Goldsmith, H. L., 296, 371
DallaValle, J. M., 13,417 Goldstein, S., 141 n
Darcy, H. P. G., 8-10 Grace, H. P., 421
Dean, W. R., 60 Graetz, L., 38
Debye, P., 461 Green, A. E., 51
Dowson, D., 92 Green, H. L., 416
Drew, T. B., 7,524 Greenhill, A. G., 38
Dryden, H. L., 8, 129 n Gupta, S. C., 149
Dupuit, A. J. E. 1., 10 Guth, E., 12-13,443-447

E H

Eagleson, P. S., 416 Haberman, W. L., 72, 130, 134, 318, 320,
Einstein, A., 12, 207, 441-448, 452, 455, 321,349,351,392

461-470 Hadamard, J. S., 127
Eisenschitz, R., 459, 460 Hall, W. A., 404
Emersleben, 0., 48 Happel, J., 133, 273, 276, 283, 299, 302,
Epstein, N., 363, 364, 418
Epstein, P., 396 316, 318, 321, 363, 364, 391, 415,
Epstein, P. S., 50-51, 126 417,418,423,424,442,443,450,
Ericksen, J. L., 51 455, 463
Eveson, G. F., 273, 275 Harris, C. C., 412
Hasimoto, H., 48-49, 377-379, 385, 386,
F 396
Hatch, L. P., 403
Fair, G. M., 403 Hawksley, P. G. W., 466
Famularo, J., 309, 321, 381, 383, 384, Hayes, D. F., 92
Hiemenz, K., 32
385
Heiss, J. F., 232

Name Index 539

Helmholtz, H. L. F., 91-93 Ladenburg, R., 11,72,331
Hermans, I. I., 432 Ladyzhenskaya, O. A., 7 n, 61 n
Hill, R., 92 Lagerstrom, P. A., 7 n, 45
Hirsch, E. H., 51 Lamb, H., J I. 48, 62, 63, 83, 106, 183,
Hocking, L. M., 269-270
Hooper, M. S., 36 192, 220-221, 252, 301, 302, 321,
Howland, R. C. I., 353 342,343-344,349,388,439
Hutto, F. B., 421-422 Landau, L. D., 26, 160, 166, 353
Lane, W. R., 415, 416
lllingworth, C. R., 7 n Langlois, W. E., 7 n
fnce, S., 8 Lanneau, K. P., 425
frmay, S., 404 Larmor, J., 183, 192
Lee, H. M., 130, 132 n
leffery, G. B., 12, 115, 215, 236, 269-
272, 274, 330, 348-349, 441, 456- Lee, R. c., 442
460
Leibenson. L. S., 399
lenson, V. G., 40, 46 Leslie, F. M., 52
Leva, M., 422-423
lohansen, F. c., 154 Levich, V. G., 129
Lichtenstein. L., 61
Johnson, E., 465-466 Lifshitz, E. M., 26, 160, 166. 353
lones, A. M., 230,232 Lightfoot, E. N., 7
Jones, G., 468 Lin. P. N., 374, 375
Loeffler, A. L., Ir., 396, 398, 399
Lorentz, H. A., 11, 85. 87, 315, 329

M

K MacKay, G. D. M., 330
McNown, J. S., 42, 317, 374
Kaplun, F., 45 MacRobert, T. M., 135 n
Kawaguchi, M., 376, 385 Manley, R. SI. J., 459
Kaye, B. H., 413-414 Mason, S. G., 296, 330, 371, 459, 462
Keller, 1. B., 316,371 Mathews, H. W., 277
Kelvin, Lord, 183, 191-192 Maude, A. D., 330n, 412, 471
Kirkwood, 1. G., 461 Merrill, E. W., 17, 18
Kneale, S. G., 98 Milne. E. A., 524
Milne-Thomson, L. M., 27, 106,390
Knight, R. c., 353 Miyagi, T., 386
Mooney, M., 446, 463, 464, 470
Knudsen, 1. G., 230, 232 Mori, Y., 466
Knudsen, M ... 50 Morse, P. M., 117 n, 402
Kozeny, I., 10,403 Murnaghan, F. P., 8, 129 n
Krakowski, M., 48-49
Kuenen, P. H., 18 N
Kuhn, H., 459, 460, 470
Kuhn, W., 459, 460, 470 Navier, L. M. H., 8, 27
Kunitz, M., 465-466 Noda, H., 413
Kuwabara, S., 283, 386, 390 Noll, W., 51
Kynch, G. I., 268-269, 274, 276-277,

376, 416, 447, 453, 454, 456, 464,
466

540 Name Index

o Saito, N., 446, 471
Sampson, R. A, 98, 133, 135, 141, 145,
Oberbeck, H. A., 11, 145,220,230,332
Odquist, F. K. G., 61 149, 153,208
Oldroyd, J. G., 51, 52 Saunders, F. L., 464-465
Oliver, D. R., 371 Savic, P., 134
Oman, A. 0., 417, 419 Sayre, R. M., 72, 130, 134, 318, 321
Oseen, C. W., 7, 12 Schachman, H. K., 413, 462, 467
Othmer, D. F., 16,417,422,423,424 Scheidegger, A. E., 7, 403-404, 410,416,
Ototake. J., 466
417
p Scheraga, H. A, 470
Schmitt, K., 50
Pappenheimer, J. R., 469 Schwarz, W. H., 51-52
Payne, L. E., 115, 145, 156-157, 208 Scriven, L, E., 25 n, 127
Pearson, J. R. A., 42, 45-46, 48, 51, 225, Segre, G., 296, 316, 370, 371, 463
Shapiro, A H., 3
281,283 Shustov, S. M., 368, 370
Pell, W. H .. 115, 145, 156-157,208 Silberberg, A, 296, 316, 370, 371, 463
Pettyjohn, E. A., 188,219-220,225 Simha, R., 12-13, 71, 301, 432, 442-449,
Pfeffer, R., 273, 276, 283, 415
Philippoff, W., 462,463 455,456,459-464
Piercy, N. A. Y., 36 Slack, G. W., 27"1
Poiseuille, J. L. M., 8,469 Slattery, J. C., 52
Power, G., 92 Slezkin, N. A., 140 n, 368, 370
Prandtl, L., 41 Slichter, C. S., 10
Prigogine, I., 92 Smoluchowski, M., 11, 49, 93, 236, 238,
Proudman, 1., 42, 45-46, 48, 51, 225,
243, 249, 258-259, 276, 281, 321,
281, 283 373,374,415
Sonshine, R. M., 349, 351
R Sparrow, E. M., 396, 398, 399
Squires, L., 225
Ray, M., 149 Squires, W., Jr., 225
Rayleigh, Lord, 8 Stainsby, G., 17, 18 n
Redbeq~er, P. J., 36-37 Steinour, H. H., 416
Reynolds, 0., 58-59, 92, 401 Stewart, W. E., 7, 92
Richardson, J. F., 391, 413 Stimson, M., 115, 236, 269-272, 274, 330
Riseman, J., 461 Stokes, G. G., 8, 11, 98, 119
Rivlin, R. S., 51 Streeter, Y. L., 402
Robinson, J., 466 Sullivan, R. R., 399
Roscoe, R., 149, 196 Suzuki, M., 330
Rothfus, R. R., 342 Sweeney, K. H., 465
Rouse, H., 8 Swindells, J. F., 468
Rubinow, S. 1.,316,371
Rutgers, R., 417, 463 T
Rybczynski, W., 127
Takaisi, Y., 283, 345-346
s Talmadge, W. P., 416
Tamada, K., 49, 386
Sadron, c., 432 Taneda, S., 283
Tanner, R. I., 331 n
Saffman, P. G., 371,409 Taylor, G. I., 29, 459, 462
Taylor, T. D., 127 n, 207 n
Tchen, C., 230

Name Index 541

Theodore, L., 369 Wells, R. E., Jr., 17, 18
Tiller, F. M., 421 Westberg, R., 31, 345
Tilley, A. K., 468 White, C. M., 230, 343
Twenhofel, W. H., 18 Whitehead, A. N., 43-45
Whitmore, R. L., 318, 320, 412, 462,
u
471
Uchida, S., 376 Williams, W. E., 130, 133
Wilson, B. W., 416
v Wilson, E. B., 7 n
Winkler, F., 15
Yand, Y., 446-447, 466 Winny, H. F., 36
Yillat, H., 7 n
y

Yang, J. T., 470
Yih, C. S., 98

w z

Wakiya, S., viii, 266-267, 278-281, 318, Zaki, W. N., 391
327,330,331,336,337,338,354 Zakin, J. L., 468
Zenz, F. A., 16,417,422,423,424,426
Ward, S. G., 462 Zierep, J., 369, 370
Watson, G. N., 303, 304
Watson, K. M., 417, 419

Subject Index

A Axisymmetrical flow (cont.):
separable coordinate systems, 117-119

Accommodation coefficient, 50 sink, 106-107

Angles, Eulerian, 205-207 source of equal strength, 107-108

Archimedes' law, 31 slip at surface of a sphere, 125-126

Avogadro's number, 415 stream function, 96-98

Axisymmetrical flow, 60, 96-157 boundary conditions satisfied by,

circular disk, 149 111-113

concentric spheres, 130-133 dynamic equation satisfied by, 103-

definition sketches for, 97 106

drag on a body, 113-116 local velocity and, 98-99

elongated rod, 156 properties of, 102-103
finite line source, 108-110 in various coordinate systems, 99-
fluid sphere, 127-129
general solution in spherical coordi- 100
terminal settling velocity, 124-125
nates, 133-138
intrinsic coordinates, 100-102 through a circular aperture, 153-154
oblate spheroid, 145-149
past an approximate sphere, 141-145 through a conical diffuser, 138-141
past a sphere, 123-124 translation of a sphere, 119-123
past a spherical cap, 156-157
uniform, 106
unit vectors for, 99
in a Venturi tube, 150-153

point force, 110-111

point source, 106-107 B
sink of equal strength, 107-108

pressure, 116 Bessel functions, 72

prolate spheroid, 154-156 Bessel's modified equation, 73

543

544 Subiect Index

Biharmonic equation, 60, 79-80 Circular cylindrical coordinates, 490-
Bingham bodies, 51
Biology, 17-18 494
Bipolar coordinates, 516-519
Bipolar cylinder coordinates, 497-499 Circular disks
Bodies joined to form a "screw-propeller," 179

anisotropic, 199 oblate spheroids as, 149
boundary layer, 41 oblique fall of, 204
center of reaction of, 160 settling asymmetrically near an inclined
drag on, 113-116
force and couple action on, 30-31 plane wall, 295
helicoidal symmetry, 189-191 side view of an "impeller" formed
helicoidally isotropic, 191-192
isotropic helicoid, 191-192 from, 182
nonskew, 192-196 Civil engineering, 16-17
orthotropic, 187
resistance of a slightly deformed Complex geometry, systems with, 400-

spherical, 207-219 410
of revolution, 188-189 Concentrated systems, 387-399, 448-
settling of orthotropic, 220-232
settling of spherically isotropic, 219- 456
Concentric spheres, 130-133
220 Conical diffuser, flow through a, 138-
skew-symmetry, 189
spherically isotropic, 187-188,240 141
wake, 41 Conjugate coordinate systems of revolu-
See also Particles
Boundaries tion, 508
between concentric spheres, 66 Conjugate cylindrical coordinate systems,
closed,61
coefficients for typical, 340-341 494-495
conditions satisfied by the stream Conjugate system of revolution, 105
Continuity equation, 23-24, 32, 59-60,
function, 111-113
effect on settling, 380 64
multiple, 61-62 complete solutions of, 77-78
open, 61 Coordinate systems
value problems involving circular intrinsic, 100-102

cylinders, 77-78 of revolution, 501-504
Brownian motion of particles, 6, 207 separable, 117-119
stream function in, 99-100
c Couette flow, 32, 444
Couette viscometer formula, 33
Carman-Kozeny equation, 393, 395, 401, Coupling tensor, 174-177
417-422 Creeping flows, unsteady, 52-54
Creeping motion equations, 41-47
Cartesian coordinates, 79, 486~487 in bounded systems, 59
Cartesian tensors, 85 complete solution of, 77-78
Cauchy linear momentum equation, 52 cylindrical coordinates, 71-78
Cauchy-Riemann equations, 59, 494 energy dissipation, 88-93
Center of hydrodynamic reaction, 174 general solutions and theorems, 58-94
Chemical engineering, 13-16
Circular apertures, flow-through, 153-154 generalized reciprocal theorem, 85-88
integral representations, 79-85
limitations of, 281-283
paradoxes in the solution of, 47-49
quasi-static, 53, 54

spherical coordinates, 62-71
Creeping motion equations (cont.):

three-dimensional, 60

two-dimensional, 59-60
variational principles for, 91-93

Subject Index 545

Cubic suspension equation, 384 Drag (cont.):
Curvilinear coordinates, 474-477 coefficient for a sphere, 45-47
Cylinders coefficients for, on a translating parti-
cle in the presence of rigid boun-
comparison of theories for flow rela- daries, 341
tive to circular, 399 form (profile), 122-123
on a single sedimenting sphere situated
cylindrical rod located along the axis axially in a circular cylinder, 317
of another cylinder and moving skin, 122-123
perpendicular to the axis, 343 on a sphere, 46, 157

cylindrical rod moving axially inside a Dumbbell, rotation of a, 195
stationary circular, 341-343 Dynamic equation, 103-106
Dynamic pressure, 28
definition sketch for axial movement
of, 342 E

experimental data for rods in circular, Earth sciences, 18-19
343-344 Eigenvalues, 526
Eigenvectors, 526
flow parallel to, 395 Einstein summation convention, 79
flow perpendicular to, 395 Einstein's law for suspension viscosity,
flow through random orientation of,
12, 441
395 Ellipsoids
moving between two parallel plane
translation of, 220-227
walls, 344-346 values of equivalent radius for, 223
resistance of, 227-231 Elliptic cylinder coordinates, 495-497
rotation of, 353-354 Elliptic equations, 61
settling factor for, 231-232 Elongated rods, prolate spheroids as, 156
suspension of spheres in, 379 Emersleben equation, 396, 398
Cylindrical coordinates, 71-78, 490 Energy dissipation, 88-93
boundary value problems including instantaneous mechanical rate of, 177-

circular cylinders, 77-78 178
particular solution of inhomogeneous three sources of, 361-362
in a viscous fluid, 29-30
equations, 72-76 Engineering
solution of homogeneous equations, chemical, 13-16
civil, 16-17
76-77 mining, 17
Epstein Zeta function, 396
D Equations
Bessel's modified, 73
Darcy unit of permeability, 9 biharmonic, 60, 79-80
Darcy's law, 8-9, 389-404 Carman-Kozeny, 393, 395, 401, 417,
Diffuse elastic reflection, 50
Dilute systems 422
Cauchy linear momentum, 52
first-order interaction effects, 371-386, Cauchy-Riemann, 59, 494
443-448 of change for a viscous fluid, 23-29
of continuity, 23-24, 32, 59-60, 64
no interaction effects, 360-371, 438-
443 complete solution of, 77-78
creeping motion, 41-47
Dimensionless variables and parameters,
53

Dirac delta function, 378
Dirichlet's formula, 220
Drag

acting on particles in a dilute suspen-
sion, 382

on a body, 113-116

546 Subiect Index

Equations (cont.): Euler equation, 28, 369
creeping motion (cont.): Eulerian angles, 205-207
Euler's theorem, 63
in bounded systems, 59
complete solution of, 77-78 F
cylindrical coordinates, 71-78
energy dissipation, 88-93 Falling-ball viscometer, the, 17
general solutions and theorems, 58- Faxen's equations, 71
Faxen's law, 67, 226, 227
94 Finite line source, 108-110
generalized reciprocal theorem, 85-
coordinates for, 109
88 streamline for flow from uniform, 110
Fluid dynamic models, viii
integral representations, 79-85 Fluidization, 422-426
limitations of, 281-283 Fluid-bed technique, 14-16
paradoxes in the solution of, 47-49 Fluid-particle systems, viii
quasi-static, 53-54
spherical coordinates, 62-71 Fluids
three-dimensional, 60 behavior in slow motion, 23-55
two-dimensional, 59-60 Couette flow, 32, 444
variational principles for, 91-93 external forces per unit volume, 24
cubic suspension, 384 gaseous bubble rising slowly through,
dynamic, 103-106 129
elliptic, 61 Hagen-Poiseuille flow, 34
Emersleben, 396, 398 laminar circular motion of, 32
Euler, 28, 369 laminar flow in ducts, 33-39
Faxen's,71 molecular effects in fluid dynamics, 49-
Gegenbauer's, 134-135 51
homogeneous, 76-77 newtonian, 26
inhomogeneous, 72-76 equation of motion for, 27
Lagrange's, 178 mechanical energy dissipation in, 29
Laplace's, 35, 58, 62, 72, 78, 211, 322 stress vector acting across a sphere,
Legendre's, 134-135 66
non-newtonian flow, 51-52
of linear momentum, 24-25 paradoxes in the solution of the creep-
mean normal pressure, 26 ing motion equations, 47-49
plane Poiseuille flow, 34
of motion for a newtonian fluid, 27 Poiseuille flow, 34, 43
of motion for a viscous fluid, 31-33 rate of creation of momentum per unit
Navier-Stokes, 27-28, 104, 393 volume, 24
simplifications of the Navier-Stokes
inertial terms in, 32 equations, 40-47
nondimensional form, 54 solid of revolution rotating symmetri-
nonlinear nature of, 31 cally in a bounded, 346-354
omitting inertial terms from, viii sphere, 127-219
Oseen's differential equations and, streamlines for a droplet, 129
unsteady creeping flows, 52-55
45-56 viscous
practical applications of, vii determination of, 17
reduced to a single scalar equation, equations of change for a, 23-29

33
relaxation methods, 40
simplifications of, 40-47
Oseen's, 44-46, 49, 81

Prandtl's, 45
random suspension, 384
rhombohedral suspension, 384

translational, of motion, 163-169