The Stokes resistance of an arbitrary particle—IV Arbitrary fields of flow

Chemical Engineering Science, 1964, Vol. 19, pp. 703-727. Pergamon Press Ltd., Oxford. Printed in Great Britain.

The Stokes resistance of an arbitrary particle-IV Arbitrary

fields of flow

H. BRENNER? Department

of Chemical Engineering, New York University, Bronx 53, New York (Received 7 January 1964)

Abstract-A phenomenological scheme is formulated for calculating the quasistatic Stokes force and torque on a rigid particle of any shape immersed in a flow field which tends to an arbitrary Stokes flow at infinity. This generalizes a previous result (Part III) limited to a uniform shear flow at infinity. The phenomenological resistance coefficients are shown to be constant polyadics which are intrinsic properties of the particle, dependent only on its external shape. In particular they are independent of the density, viscosity, and state of motion of the fluid. It is demonstrated that these coefficients can be computed solely from a knowledge of the solutions of Stokes equations for translational and rotational motions of the particle, along any three non-coplanar axes, in a fluid at rest at infinity. Explicit formulae for the polyadic coefficients are given for ellipsoidal and slightly deformed spherical particles.

1. INTRODUCTION IN PARTSI and II of this sequence [l, 21 (corrigenda to Part I are given in Ref. [3]) a phenomenological scheme was developed for calculating the Stokes hydrodynamic force and torque experienced by a rigid particle of arbitrary shape undergoing translational and rotational motions in a fluid at rest at infinity. Part III [4] extended the development to the case where the fluid at infinity was being sheared. In this paper we complete the generalization by extending the analysis to the case of an arbitrary Stokes field of flow at infinity. A Poiseuillian flow is an example of such a motion. In a sense, the analysis constitutes a generalization of FAXEN’Slaws [5, 61. According to these theorems the Stokes force F’ and torque T& (about the centre 0) exerted on a rigid stationary spherical

particle of radius a immersed in an arbitrary Stokesian flow field, u = u(r), extending to infinity are (l.la) F’ = 6zpau, + npa3(?2u), T;, = 4npa3(V x u)O

(l.lb)

where p is the viscosity, $’ the Laplace operator, and the subscript 0 appended to the various functions of the unperturbed flow field u refers to evaluation at the centre of the sphere, O.$ The essence of our ultimate generalization may be seen by writing FAXEN’Slaws in GIBBS’polyadic form F’ = p[67caI.u, + na311 i (VVu),] Tb = -p4na3& : (Vu),

(1.2a) (1.2b)

where I is the dyadic idemfactor and E = -1 x I is the unit alternating isotropic triadic.5 We note

t On leave 1963-64 at the Chemical Engineering Department, University of Minnesota, Minneapolis 14, Minnesota. $ In the event the sphere is not at rest, the total force F and torque TO may be obtained by adding to equation (1.1) the additional terms F” = -6n~aUo To” = -8quuso where UO is the velocity of the sphere centre and o the angular velocity of the sphere. 5 If il, iz, i3 are any right-handed system of mutually perpendicular unit vectors we may write I = i&&k e = ijikizq82 where the summation convention on repeated indices obtains. The indices range over the values 1,2, 3. The Cartesian tensors 6/t and EI~~Z are, respectively, the Kronecker delta and the permutation symbol. With regard to multiple dot and cross products of polyads we adopt the nesting convention of CHAPMAN and COWLING [7] whereby, for example, inwhichqrc(k=1,2,3,...

. . . q1qz:q3q4 . . . = . . . (q1 .q4)(q2 .q3) . . . ) may be any vector. It is only in this one respect that we depart from the original notation

GIBBS [8].

703

of

H. BRENNER

that the fluid properties enter into (1.2) only through the viscosity coefficient, while the state of motion of the fluid at infinity enters only through the constant polyadic coefficients uc, (Vu),, (VVu),. These, in turn, are merely the leading coefficients in a Taylor series expansion of u about 0, namely u=u,+~.(vu),+r~:(VVU),+

. ..

where r. is the position vector of any point in the fluid relative to an origin at the sphere centre 0. The remaining polyadic coefficients in (1.2), i.e., the dyadic 6~1, triadic 4na3e and tetradic na31 I, are thus properties of sphere alone, which describe its intrinsic resistance. As would be expected from the fact that a sphere is an isotropic body lacking any directional properties, these constant intrinsic phenomenological resistance polyadics are themselves isotropic. This would not be true for bodies of more general shape. This interpretation of FAXEN’S laws coupled with knowledge gained from previous papers in this series [l, 2, 41 suggests that the generalization of (1.2) to rigid particles of arbitrary shape ought to be of the form

lishing the dependence of the resistance polyadics on choice of origin. Some detailed results for spherical particles are given in Section 5. It is shown in Section 6 that the resistance polyadics can be computed for any body solely from a knowledge of the solution of Stokes equations for translational and rotational motions of the body in a fluid at rest at infinity. This technique is applied in Section 7 to evaluate the resistance polyadics for an ellipsoidal particle. A slight variation of this technique is employed in Section 8 to calculate these polyadics for a “slightly deformed sphere”. Finally, in Section 9, we suggest some applications of the general techniques developed herein to other classes of problems. 2.

Consider a homogeneous viscous incompressible fluid, extending to infinity, undergoing some arbitrary Stokesian fluid motion, u = u(r). Let a rigid particle, which itself may be in motion, be immersed in the flow. Denote by (v, p) the ensuing velocity and dynamic pressure fields. These are assumed to satisfy the quasistatic Stokes equations

+2v=;vp

F’ = ,@@*uo + 3@o : (Vu), + + 4a. i (VVU), + . ..]

(2.1)

(1.3a)

v-v = 0

T; = ~[21’o~uo + “I’, : (Vu), + + 4ro ; (VVU), + . . . ]

PRELIMINARIES

u*3b)

where 0 is now any point afhxed to the particle and the entities “a, and “IO are constant n-adic intrinsic resistance coefficients which depend only upon the external shape of the body and the choice of origin 0. It will be shown subsequently that (1.3) is indeed correct. As is clear from FAXEN’S laws, the infinite series (1.3) terminates for a spherical particle after a finite number of terms. This property is unique to isotropic bodies and derives from their isotropy coupled with the fact that u, being a solution of Stokes equations, satisfies the equations V”u = 0 and V”(V x u) = 0 everywhere in the fluid. Section 2 is devoted to a discussion of the equations of motion. In Section 3 a proof of (1.3) is provided. Section 4 is devoted primarily to estab-

(2.2)

Let 0 be any point which moves with the particle and denote by U. the instantaneous velocity of this point and by o the instantaneous angular velocity of the particle. Since w is a free vector, it is not required that the axis of rotation pass through 0. As the fluid in contact with a solid body adheres to it, the boundary condition to be satisfied at its surface, sP, is v = U, + cu x r, on sP

(2.3)

where r. is the position vector of a point relative to 0. At large distances from the body the original motion is unaffected by the presence of the particle. Hence, it is required that v--tuasr+co

(2.4)

The linearity of the equations of motion and boundary conditions enable us to decompose the 704

The Stokes resistanceof an arbitrary Particle-IV. Arbitrary fields of boundary-value problem (2.1)-(2.4) separate problems as follows:

into

three

v = v’ + v” + II

(2Sa)

p = p’ + p” + 8

(2Sb)

In arriving at (2.11)-(2.12) we have utilized the fact that the unperturbed field, (u, $), being free from singularities in the interior of the space presently occupied by the particle, can cause neither a net force nor torque on the particle (cf. the Appendix to Ref. [9]). It is already known from Parts I and II [I, 21 that

where each of the individual fields (v’, p’), (v”, p”), as well as the undisturbed motion (u, j), satisfy Stokes equations. The former fields are to be chosen so as to satisfy the boundary conditions v’=

-uons,

(2.6a)

1Oatr=co

(2.6b)

4

v” =

U, + 0 x r, on sp ( Oatr=ao

flow

F” = -/L(K.U, T;; = -p(C,*U,

+ C6.o)

(2.15)

+ C&,.6.$

(2.16)

where K is the translation dyadic and Co and O. are, respectively, the coupling and rotation dyadics at 0. We need therefore examine in detail only F’ and T&. These correspond to the force and torque which the particle would experience were it held stationary in the undisturbed flow field u. Hence, we term them the stationary force and torque.

(2.7a) (2.7b)

As the vector U, + o x r, is independent of the choice of origin, all the fields in (2.5)-(2.7) are necessarily independent of the location of 0. The total hydrodynamic force and torque (about 0) arising from (v, p) are

3.

STATIONARY FORCE AND TORQUE

It is assumed that the undisturbed field u may be expanded in Taylor series about 0. Thus and To =

u=UO+~.(vu),+~:(vvu),+ r, x (c&*x) (2.9) . JSP +$Jvnu),+... rororO

+-3?_:(vvvu)o+

where ds is a directed element of surface area pointing into the fluid and 11:is the stress dyadic A = -1p + p[Vv + (Vv)+]

(3.1) where

(2.10)

n-times

for an incompressible Newtonian fluid. The transposition operator (1_)has its usual significance (see Parts II and III, [2, 41). Inasmuch as z depends linearly on (v, p) it follows from (2.5) that we may write

r” = rr . . . rrj= an n-adic

n-times vn =G

(2.11)

To = T;, + T;;

(2.12)

J J r,x(ds*d)

F’ =

&*a’

(2.13)

SP

T;, =

(2.14)

SP

with similar expressions for the double-primed fields. Here, A’ is the stress dyadic arising from (v’, p’).

= an n-adic operator

and

F = F’ + F” where

. ..

q

More generally, the symbol inserted between two polyadics whose ranks are each equal to or greater than m denotes m successive dot multiplications in the order prescribed by the nesting convention of CHAPMAN and COWLING [7]. The symbol is suggested by DREW [IO] though he himself eschews the nesting convention, preferring instead

q

705

H.

BRENNER

the original convention of GIBBS [8] relating to the order of polydot multiplication. Because we shall encounter such scalar operators as the Laplacian, biharmonic and others of higher order, it is convenient to denote these scalar operators by the symbols

ential equations and n = 0, 1,2,3, . . .

v.(.+2vo) n+2VO= n+2

to distinguish them from the corresponding polyadic operators. The constant polyadics (V”& are not wholly arbitrary since u must satisfy the equations V-u = 0, \i2V x u = 0 and V”u = 0 at all points of the fluid, in particular at 0. This does not affect the form of the final results (i.e., equation (1.3)) but only the number of independent scalar components of the resistance polyadics. In view of the general identity = Ir$YZij(V’“u),

(a vector) (3.2)

we find from (2.6a) and (3.1) that v’ satisfies following boundary condition on sp: -I.u,-;Ir,:(vu),-...

= 0

(3.6)

-$Irions,

= 0

(3.9)

V4(n+2Vo) = 0

(3.10)

R’ = -1p’

+ /L[Vv’ + (Vv’)+]

+ 4II,

: (Vu), + 5II, i (VVU), +

v’+Oasr+co

(3.3b)

In consequence of linearity, the solution of Stokes equations satisfying these boundary conditions may be expressed in the form v’=~V~uo+3Vo:(VU)O+4VO~(VVU)O+...+ + “+2Vo+(V”u)o

+ .. .

(3.4a)

F’ = n[20*uo + ... ]

+ v(n+2vo> + + +[v(“+2vo)]

(3.13)

+ 3@o : (Vu), + 4@o : (VVu), +

+ . . . + “+2~,(~+v?l), (3.4b) Tb = p[2ro'Uo

in which ‘“V. = “V,(r) and mPo spectively, m-adic “velocity” and At any given point r in the fluid upon the shape of the particle and These polyadic fields satisfy the

(3.12)

in which n+3110(r) is an (n + 3)-adic “stress” field. At a given point, r, it depends only on the shape of the particle and the location of 0. When (3.12) is substituted into (2.13) and (2.14) the following invariant expressions result for the vector force and torque on a stationary particle:

+ 2P. : (Vu), + “PO ; (VVU), +

+ . . . + n+ lPo+~(V”u),

+ . ..]

where n+31-Io = -I(“f’Po)

that

(3.11)

in the form

+ . . . + “+3110~~(V”u)o (3.3a)

(3.8)

so that the “pressure” field is harmonic and the “velocity” field biharmonic. These relations provide the key to obtaining solutions of the polyadic equations of motion. In consequence of (3.4), the stress dyadic

n’ = p[3 II-u,

- . . . on sp

(3.7)

Vo+Oasr+co

V2(n+iPo)

the

-

- -$ Ir;jjn+1J(V”u),

p’ = @P*u,

(3.5)

It is clear from these relations that (“+‘Vo, “+‘Po) depend on the choice of origin only through the dependence of r. on 0. It is easily established from (3.5)-(3.6) that

may be written

while (2.6b) requires

For

V2, V4 and, in general

m VZm = v2v2 . . . v2v2

v’=

conditions:

+2(n+2Vo) = VC’iP,)

m-times

r$YLJm(V”u)o

boundary

= “PO(r) are, re“pressure” fields. they depend only the location of 0. following differ-

+ . ..]

(3.14)

+3 r. : (VU), -k4r. : (vv~>~ f

+ . . . + n+2ro~(v”u)0

+ ... ]

(3.15)

in which

(3.16) J sp 706

The Stokes resistance of an arbitrary particle-IV.

and n+zro =

ro x (&“+?I,) s SP

(3.17)

are constant (n + 2)-adics dependent only on the shape of the particle and the location of 0. These intrinsic phenomenological coefficients characterize the hydrodynamic resistance of the body. From (2.11)-(2.12), (2.15)-(2.16) and (3.14)(3.15) the total hydrodynamic force and torque on a non-stationary body are, respectively, F = -p(K*Uo

+ C&n) +

2 n+ZQD,/+[(V”u), n=O

+ p T o=

(3.18)

-p(C~‘UO+n,~o)+

+ p F “+Zrow+(V”“,o n=O

Arbitrary fields of flow

where, to make the arguments clear, subscripts have been attached to 2@even though the latter is already known to be independent of position. Now, in the Taylor series expansion (3.1), u may also be expanded about P instead of 0. Equating the result to (3.1) the following is obtained: Ilo

+

?.(Vu),+ 5:

(VVU), + . . .

=u,+z.(Vu),+%:(VVu),+... .

*

(4.6)

The variation of ‘(I$+ 3@p, 4
(3.19)

GD,*u, = +Dp*up ug = up = u I

4.

~OPERTIES OF THE FORCE AND TORQUEPOLYADICS

and thus (%I$ - 2@,)‘U = 0

The first few polyadic resistance coefficients in (3.14)-(3.15) are either identical or closely related to comparable coefficients discussed in earlier papers [l, 2, 41. In particular, ‘Q,=K

(4.1)

2ro = co +.Do= -(&

(4.2)

+ $Cf,.E)

(4.3)

3ro = -(To + J&,‘&)

(4.4)

which correctly shows ‘CD to be independent of choice of origin. Next, consider the case where u is such that Vu is constant throughout the fluid. Equations (4.5) and (4.6) now reduce to ~~o~uo+~~o:vu=~Q)p~up+~~)p:vu 1

2@o. {@P - r,)*Vu) = (3@p - 3@o) : Vu Since rap = r, - rp is the position vector relative to 0, the above may be written as

of P

- 2@ro, : vu = pDp - %Jo) : vu

But Vu is an arbitrary dyadic. Thus, it is required that 3@‘p= 3@o - 2@oro, (4.8)

: (Vu), + (VVU), + . . .

(4.7)

Utilizing (4.7) and eliminating u. - up from the above, we are ultimately led to the expression

+D, ’ llg + 3qJ : (Vu), + +D, ; (VVU), + . . . + +I$:

2(X$= +Do

Ilo + r,*Vu = up + r,*vu

where & and Q are, respectively, the shear-force and shear-force triadics at 0 [4]. The dependence of the force and torque polyadics on the location of 0 may be established by generalizing techniques developed in earlier papers for the lower-order coefficients. Consider the expression for F’ in (3.14) and note that F’ must be independent of the choice of origin. Thus, if P be any point other than 0, equation (3.14) may be written with the affix P appearing in place of 0. Equating the result to (3.14) we obtain = +Dp’Up + %,

As u is an arbitrary vector this requires that

(4.5)

which expresses the variation of 3@ with the location of the origin.

707

H.

BRENNER

Similarly, determination of the dependence of 4@‘pon origin requires that we consider the case where VVu is constant throughout the fluid. The calculation entails use of the identities (4.7) and (4.8). It is apparent that we may proceed, step-bystep, to establish the dependence of any “a, on position, and that the resulting equation may be expressed in terms of the polyadics “00, “-1@0, 2 n-2 and rap, rOp, . . . , I-“&?. We shall a 0, **. , ‘@, not attempt to give the general result. Determination of the dependence of the torque polyadics ‘rp on origin may be established along similar lines, apart from a minor complication resulting from the fact that the torque itself varies with origin according to the relation Tb = Tb - rep x F’

(4.9)

The most general n-adic, say “Ao, contains 3” independent scalar components; that is, if i,, i,, i, be a triad of mutually perpendicular unit vectors, “A0 may be expressed in the general form “A, =

i

i

ml=1

mz=l

. . . m$li,,,,i,, . . . i,,,,,A$$,z. . . ,,,, ”

(4.10) The scalar components A$,,,,__ may, if desired, be regarded as the components of the comparable &h-rank Cartesian tensor. Note that “r. is not a true tensor but is, rather, a pseudo tensor (axial tensor). Even for a particle of arbitrary shape, “@, and To will not generally possess 3” independent components. For example, the translation dyadic ‘@ is known to be symmetric (i.e., ‘@ = ‘a+) and hence possesses six rather than nine independent components. But even in the absence of such unique dynamical symmetry properties the number of independent components is necessarily reduced owing to (i) the multiple symmetry of the VV . . . VV operator, and (ii) the existence of the three relations V-u = 0, v2V x u = 0 and 9”~ = 0 satisfied by the Stokesian flow field u. Superimposed on these restrictions may be geometrical symmetry conditions arising from the macroscopic symmetry of the body, if any. General methods are available [2, 41 for treating such geometric symmetry reductions.

Intimately related to the observations in the preceding paragraph is the fact that the polyadic resistance coefficients are not, in general, uniquely defined. For example owing to the condition of incompressibility we have that I : (Vu), = (V-u), = 0 Thus, it is clear that we are free to add to 3@o and 3ro in equations (3.14) and (3.15) any triadics of the general form q1, where q may be any vector. This lack of uniqueness does not, of course, apply to the forces and torques derived from the resistance polyadics. And from a physical point of view only the experimentally observable quantities (i.e., force and torque) in the theory need be unique. This lack of uniqueness is therefore irrelevant (see the discussion of the “principle of indeterminacy” in Part III 141). 5. SPHERICALPARTICLES The existence of the fields (n+‘Vo, “+lPo) defined in (3.5)-(3.8) is most simply illustrated for a spherical particle. Solutions are readily obtained by employing (3.9)-(3.10). In particular, “+?Vo can be expressed as the sum of a series of (n + 2)-adic harmonic (“‘“A(r)) and biharmonic functions (r ’ “‘“A(r)). Thus, for n = 0, 1 and 2 we ultimately obtain for a sphere of radius a 2V = -+a(r’

- a2)H2 - aIH,

(5.1)

‘P = -$aH, 3Vo = -a3(r2

I - a2)H3 - a31H,

(5.2)

2P. = -ya3H2 4Vo = -+a’(r”

I - a2)H4 -

- &a”(r’

- a’)H,I

- +a51H2 (5.3)

- +a31 IH0 3Po = -$a5H3

- ta3H,I

1

in which r is measured from the sphere centre 0 and H, is a normalized n-adic “solid spherical harmonic” of degree -(n + 1) defined by

708

(5.4)

The Stokes resistance of an arbitrary particle-IV.

The first few of these polyadic harmonics are Ho=;

(5.5)

These solutions derive from the boundary tion (3.7) by observing that?

That H, does indeed have the harmonic properties attributed to it may be seen as follows: Firstly, because of the n-adicity of the polyadic operator V”, H, is obviously an n-adic. Secondly, since V’(l/r) = 0 it is clear that V’H,, = 0 and so H,, satisfies Laplace’s equation. Finally, since the V-operator is homogeneous in r-l it is apparent that V”(l/r) and, hence, H, is homogeneous in r-(“+l). Thus, H, is indeed what we have stated it to be. By utilizing the general identities which follow, it is easily verified that (5.1~(5.3) satisfy (3.5)-(3.8). These identities are

r”

0 r

I

VH, = -(n + l)H,,+, (5.6)

V-H, = 0

i

V*(r2H,) = 2H,_, V*(IIHO) = -H,I

I

I

It is a matter of some analytical difficulty to attempt to solve (3.5)-(3.8) for arbitrary n, even for the simple case of a sphere. At best we are able to state that for such a body the leading terms in the “expansions” of “+2Vo and “+‘Po are Vo= -

2”-l(n + l)! (2n)!

a2”+l(r2 - a’)H,+,

-

p

n

=r.+‘H,-(-t’~+lVd n.

r

(5.9)

is a normalized n-adic “surface spherical harmonic” of degree n. A proof of the expansion (5.8) is given in Appendix I. ALTERNATIVE SCHEME FOR CALCULATING THE RESISTANCEPOLYADICS

Computation of the force and torque polyadics from their fundamental definitions, (3.16)-(3.17), is obviously a matter of great analytical difficulty, for it requires a detailed solution of the basic differential equations and boundary conditions, (3.5)-(3.8), for the body in question. And, as is clear from Section 5, this is a formidable problem even for as simple a body as a sphere. Fortunately there is an alternative and vastly simpler scheme for computing these resistance polyadics. The method requires detailed knowledge of the solutions of Stokes equations only for the cases where the body in question translates and rotates in a fluid at rest at infinity. Let v,,. be the solution of Stokes equations satisfying the translational boundary conditions

2”n! - a’“+‘IH,, + (2n)!

no other (n + 2)-adics containing + solid spherical harmonics of order greater than n I (5.7a)

(5.

where

i

V2(r2H,) = -2(2n - l)H,

2”(n !)’ =(2n)!Pn+

condi-

n-adic surface spherical harmonics of ’ degrees n - 2, n - 4, n - 6, etc. I

6. V’H, = 0

n+1+

no other (n + I)-adics containing + solid spherical harmonics of order t greater than n - 1 I (5.7b)

,etc.

n-I-2

= - 2% + 1)!(2n + 3) a2n+lH (2n) ! (n + 2)

“flpo

1

HI=;

Arbitrary fields of flow

vt, = U on sp

(6.la)

vt,. + 0 as r -b a3

(6.lb)

t The correspondingexpansion of the scalar pa in ordinary Legendre polynomials, P,(p), is [l 11 IL” -_ we2 W)!

709

“(“’

Legendre polynomials of orders + PA(~), Pn4G), etc. I

I

H. BRENNER

for a particle of any shape, and let Gbe any solution of Stokes equations satisfying arbitrary conditions on sP and vanishing at infinity. The constant vector U is arbitrary. There is then a reciprocal theorem [l] of the form J+G.“,,.

+&.%..V

(4.2)

The validity of this relation depends in part on the vanishing of vtr. and 8 at infinity. Substituting (6.la) into the latter equation yields (Is/+)

*U = [sJ.(ds.n,..)

then satisfies Stokes equations and vanishes at infinity. From this definition, the force F arising from 8 is clearly equal to the difference between the force F arising from v and the force arising from u. But, as discussed, u possesses no singularities within the interior of the space presently occupied by the particle and cannot therefore produce a force on the particle (cf. Ref. [9]).t Consequently, 6=F

Equations (6.6)-(6.8) therefore yield the relation

(6.3)

dsII;,;(v

F=p

s SP

ds.il

Furthermore, in the form

as shown in [2], ICY,. may be expressed (6.5)

RI,. = ML. * u

where II,,,(r) is a triadic “stress” field dependent only on the shape of the particle. When (6.4) and (6.5) are introduced into (6.3) one obtains G.u=fi

at the surface of the body and the arbitrarily prescribed Stokes field u at infinity. Apart from potential difficulties in evaluating the resultant integrals, it is clear that computation of this force requires knowledge only of the solution of Stokes equations for translational motion of the body along any three non-coplanar axes, for this information suffices to determine II,,.. The analog of (6.9) for the torque To may be obtained by replacing (6.1) by the rotational Stokes satisfying the boundary conditions field 0~~~~.

is SP or, since U is an arbitrary vector

F=j.l But, by identity

s

Ov,,t. = Q x r, on sP

(6.10a)

o~,,~.+ 0 as r --) co

(6.1Ob)

where o is an arbitrary constant vector. The affix 0 on the velocity field merely indicates its dependence on choice of origin. The counterpart of (6.9) ultimately obtained is

V.(ds.TI,,j %J

V*(ds*II,,.) = (ds.J&)+.V

(6.9)

directly in terms of the arbitrarily prescribed field v

(6.4)

s SP

- u)

The above relation gives the force F on the body

The force experienced by the particle, sP, due to the field 0 is F=

(6.8)

= (dsl$)~V

To =

whereupon

P

r

ds. J&,,..(v

- u)

(6.11)

J SP

Equation (6.6) applies only to fields 8 which vanish at infinity. To remove this restriction, let v by any solution of Stokes equations satisfying arbitrary boundary conditions on the particle and reducing to any arbitrary Stokes field u(r) at infinity. The field v=v-u (6.7)

in which JIrot_ is the intrinsic triadic “stress” field defined by (6.12) 0%. = P o%t. *@J To illustrate simply the utility and significance of (6.9) and (6.11), consider a spherical particle of radius a and choose 0 to be the sphere centre. t By this we mean that when the stress vector arising from u is integrated over sp it yields the null vector.

710

The Stokes resistance of an arbitrary particle-IV.

fields of flow

For bodies of more general shape equations (6.9) and (6.11) must be treated rather differently. Let v and u in the latter be defined as in (2.3) and (2.4) so that F and T, have their previous significance. From (2.3) and (3.1) we have

Then [12] 3P (ds*%.),=, = - Yj-oUds=

Arbitrary

-$_Jds

and (dS~orc,,J,=, = -3/H” x f ds = 3/l a.1 ‘W ds r

(

r>

v-u=U,+oxr,-

where ds = u2 sin 0 & dcj is an element of surface area on the sphere (s,) and r is to be measured from the centre of the sphere. Comparison with (6.5) and (6.12) shows that

+$gv”“),

g

n=On.

on sP (6.13)

Equations (6.9) and (6.11) therefore adopt the form

-gps

(ds*rI,,.),_= and

-

(ds~JI,,~),=, = 3~; ds and But It = I and (e.r)+ = --E-r. Furthermore, if q be any vector, 1.q = q and (a*r)*q = -r x q. Substitution in (6.9) and (6.11) then yields F = -

To=,[(s.~.~~~~,.~.U,-

s s

2

-

(v - u) ds

ds*J&

x r,

1

1

*co -

s.

T,2!!

a

ds*JI~,,,.r~

r x (v - u) ds

.%

(6.15)

FAXEN’Slaws [5, 61 may be derived from these as follows: set v = 0 in the above so that F and To are replaced by F’ and Th, respectively. Now, from equation (AIL23) of Appendix II (with it = 0 and 1, respectively), we have for any vector function u(r) possessing continuous derivatives of all orders at the origin,

s

“ds = 4na2 u. + ; %

s

rxuds= S,

Comparison of these expressions with (3.18) and (3.19) shows that? (6.16) CA =

(V’“)o + $ (V”“)o -t

ds*II&. x r. J sp

ds*Q$.r$

+$(Pu),+ ...I

and

But, since u is a Stokes flow, V4u = 0 and V’(V x u) = 0 everywhere. FAXEN’S laws (see (1.1a-b)) now follow immediately.

. ..) (6.18)

x “)o + & {i@(V x “)}o +

+&PYvx”))0+...]

(n = 1,2,3,

(6.17)

and Co = 21’o = -

T[(V

~(V”u)o

ds-ol-&e s SP

(6.19)

t The “irregular” terms, K, CO and no appear here in a slightly different form than were originally given in Part II [2]. That these alternative forms are consistent with the present forms may be seen by taking the transposes of (6.16), (6.17), (6.19) and (6.20) and utilizing the following relations: K = Kt, SLo = !&t, (ds.lTt)t = &II and (&lTt x ro)t = -ro x (&IT), the last two being identities and the first two consequences of the known symmetry of the translation and rotation dyadics.

711

H. BRENNER no =

ds. ,J&.

s %J

dwolIfo&

“+21-‘0= -A

x r.

soid at the point (x1, x2, xs) lying on its surface. In addition, ds is a scalar element of surface area on the ellipsoid. Consequently, for translational motion with vector velocity U in any direction,

(6.20)

(n=1,2,3

,...)

s SP

(6.21)

Note that (6.17) and 6.19) provide alternative means of calculating Co. Equations (6.18) and (6.21) provide the means whereby the higher-order resistance polyadics may be computed solely from a knowledge of the solutions of Stokes equations for translational and rotational motions in a fluid at rest at infinity along any three non-coplanar axes. 7.

ELLIPSOIDALPARTICLES

Explicit expressions are derived in this section for the force and torque polyadics for the ellipsoidal particle x: 6+“:=1 (7.1) Tif al a; a: From the work of OBERBECK(see LAMB [13]) on the translation of an ellipsoid parallel to a principal axis (say Xi) with scalar velocity Vi, a straightforward, but lengthy, calculation of the stresses at the surface s, of the ellipsoid shows that for this special case 4P

(ds . mtr.)s, = - ei 7 ala2a3 X +

1

where (e,, e2, e3) are unit vectors (x1, x2, ~3) and x =

s m

aj =

in which

Uip ds

Ui Mi

parallel to

s

m dl

(7.2)

om

dil

o (a; + W(~>

(j = 1,253)

A(2) = [(a: + A)(a; + A)(a; + /I)]“2

K-Up ds

(7.6)

in which e2e2 e3e3 K = 167~ elel + x + a&x, + x + a$~, [ X + a:%

1

(7.7)

is the translation dyadic for the ellipsoid (see Section 4 of Part III [4]). Comparison of (7.6) with (6.5) yields = - 4na h a Kp ds 1 2 3

(da-%.),.

(7.8)

Since K is symmetric, substitution of (7.8) into (6.18) gives 1 “+2@* = L K r”p ds (7.9) n! 47cala2a3

s S.

where r is to be measured from the centre, 0, of the ellipsoid. The n-adic surface integral appearing in (7.9) may be evaluated by the following procedure. HOBSON[14, p. 4871 gives the following relation: If Yk(xl, x2, x3) is any scalar solid spherical harmonic of order k and I,/&, x2, x3) is any continuous scalar function possessing derivatives of all orders at the origin, then

fs3k,2, ~j$(% x3)pds X2,

2kk!

(7.3)

D2

= 4’a1a2a3 (2k + l)! [I ’ + 2(2k + 3) + D4

(7.4)

+2*4(2k+

Furthermore,

3)(2k+

a X

P=

p 4na,a,a,

(ds.x,.>,. = -

(7.5)

is the perpendicular distance from the centre of the ellipsoid to the plane which is tangent to the ellip-

yk

al

ax,’

5) + *”

a ‘2

axp)

a3

Lax3

ml,

x2,

x3>

1

(7.YO) where the subscript 0 implies evaluation of the function in square brackets at the centre of the

712

The Stokes resistanceof an arbitrary particle-1V. Arbitrary fields of tlow ellipsoid, X1 = X, = X3 = 0. differential onerator

Also,

0’

is the

where Q is the constant symmetric dyadic

Consider the case k = 0. It is obvious that (7.10) is not limited to scalar fields, $, but is equally applicable to polyadic fields, say \Ir, of any degree of polyadicity, satisfying appropriate conditions of continuity and differentiability. Thus,

+

e3e3

(7.17)

u& + aga2

This yields = 4nna“, a Q.E.PP ds 1 2 3

(ds. JL)S,

(7.18)

where r is again measured from the centre 0 of the ellipsoid. Equation (6.21) therefore takes the form

X(I),+;

(D2Q)o+;

(D4Q)o+ ...

)

(7.12)

n+2ro = -

$ 4~a “, a 1

In particular, upon setting $(r) = f” we obtain

s

se

D”r” for m even 471a,a2a3 (m + l)! i

II+2

r 0=

3ro=

(7.14)

(7.15)

A similar treatment may be given of the torque polyadics for an ellipsoid. From JEFFERY’S[I 51 solution for the rotation of an ellipsoid in a uniform shear field we find, after suppressing that portion of his solution dealing with the shear contribution (and correcting some minor typographical errors), that = -

4aa

“,a

1

2

r-E*@

ds

n!(n + 2)!

Q-a*D”+lr”+l

_.E

3 [(

e,e,e3ut

for n odd

+ (e,e,e,ai

1

- e1e3e2ui)

+ (e2e3e1u: - e2e1e3u$)

2@ ==I(

Vs. o%ds.

(7.20)

3

For example, when n = 1 we obtain

Except for the tedious differentiations required in (7.14), the force polyadics for the ellipsoid may be regarded as completely determined. The first few of the non-zero coefficients are

4@o = +K(e,e,a~ + e2e2ai + e3e3a:), etc.

-

1

/O for n odd 1 KD”r” for n even n!(n + l)!

(7.19)

s.

0 for n even

(7.13)

No subscript 0 need be affixed to the m-adic D”P since it is obviously a constant polyadic and thus independent of position. Substitution of (7.13) into (7.9) yields

n+2(Do =

r”+lp ds

Q-E-/ 3

In view of (7.13), the final expressions for the torque polyadics are

0 for m odd r”‘p ds =

2

- e3e2e1u:>

uza2 + a:a3 1

&x3 + u:Ur 1 a&

+ u&

+

+

1

(7.21)

Equation (4.4) permits us to compare this expression with a comparable formula for the shear-force triadic, ro, for an ellipsoid, tabulated in Part III [4]. The unique symmetry of the ellipsoid permits the results of our analysis to be expressed in a more easily manageable form, not involving polyadic resistance coefficients. When the ellipsoid is stationary, equations (6.9) and (6.11) take the form F’= -p

ds-l-I!,.*u f S.

and

(7.16)

3

713

T; = -p

ds..I& s S.

‘II

H. BRENNER

Inserting (7.8) into the former latter they become

’

F’s

and (7.18) into the

s

K.

474a2a3 3P 4na,a,a,

Tb = -

up ds

(7.22)

rup ds s s.

(7.23)

scalar differentiations. obtain Tb=pQ*

uo + ;

Q-E :

(D2u)o + ;

+$

(D4u), +

(Pu),

+ ...

Since

1

[r

a;

the vector differential

(7.29)

0’ and vector operator

D2=E-1:VV

Uj-&U 1

310

operator + e,az $

+ e,ag $ 2

(7.25)

(7.31)

Equations (7.24) and (7.27) for the ellipsoid are the direct counterpart of FAXEN’S laws for the sphere, (1.1 a and b). FAXEN’S results follow easily from (7.24) and (7.27). If we set

equations

(7.7) and (7.17) become K = 16za,

Q = 14rca

while (7.11) and (7.25) degenerate

Equations

3

F’ =

jOU 1

... \

1 +$P+++ [i

it commutes

to

0 = a2V

(7.24) and (7.27) therefore

&pa

Tb = 4xpa

0

(7.26) E : is constant

(7.30)

l-J = E-i.V

the stationary torque about the centre of the ellipsoid can then be expressed in the form

As the operator

0

and

D2 = a’V’ ,

Tb = -pQ.a:

(7.1) may be

a, = a2 = a3 = a

l+g+&+...

1

(7.28)

a:

of the ellipsoid

while the scalar operator adopt the forms

ds = 1 ejaj %pds j=l s se aj

Cl = ela: &

e3e3

I

E:rr=l

x. 4n: _1 np ds = - ala2a3 X 3 s. ‘j X

f e2e2

a:

C?jXj

Now Xj is a solid spherical harmonic of order 1. Therefore, setting k = 1 in (7.10) and putting Y1 = xjand\Ir = uwefindforj= 1,2,3

s

elel

Thus, the equation written as

3 rup

_

(7.24)

the dyadic surface integral in (7.23) may be evaluated by writing it in the form

s .%

1

for the torque about the centre of a stationary ellipsoid. The results of this Section may be written in a completely invariant form by defining the constant dyadic

3 C j=l

$(D60xu),+...

(7.27)

E

r =

xu),+

+%(D40xu),+

[

By defining

xu),+~(D20

x u we

se

These integrals may be evaluated by HOBSON’S theorem. Thus, setting $ = u in (7.12) and substituting the result into (7.22), one obtains for the force on a stationary ellipsoid F’ = PK.

(0

Since E : mu = -0

u. + g

(~2~)o

a2(V x u)~ + ‘$

+ $

become

(v4u)o

+

...

a4(q2V x u)~ +

1 ...1

But V”u = 0 and V2V x u = 0 everywhere, whence we recover FAXEN’S laws from the above.

with the 714

The Stokes resistance of an arbitrary particle-IV.

SLIGHTLY DEFORMED

spherical co-ordinates having their origin at the centre 0 of the undeformed sphere, and fk(6, 4) is a surface spherical harmonic of degree k which is of O(1) with respect to E. The lower order polyadic resistance coefficients for the body (8.3), available from previous papers [4, 121, or deriving from (4.1)-(4.4), are? 20 = K = 6rca[I(l + afo) - s&V2(r2f2)] + 0(s2)

SPHERE

Theoretical applications of greatest current interest relate to the behaviour of particles immersed in shear and Poiseuillian flow fields. For flows of the latter type u = 2Y[l - (R/R&

fields of flow

(r, O,C#I)are

An essential difference clearly exists between the form of the results for the sphere and the ellipsoid. In particular, the expressions for the force and torque on a sphere terminate after a finite number of terms, whereas the corresponding results for the ellipsoid do not. 8.

Arbitrary

2ro = Co = -.s6rca2s-V(rjl)

+ 0(s2)

Qo = 8na3[I(1 + 3sfo) - &V2(r2fi)]

+ O(s’)

+o = - s3na2[rV(rfi> + {IV(rfi)>+ -

(8.1)

- z”i_v3(r3fJ)] + 0(s2)

where Y is the mean velocity of flow vector (i.e., a vector whose magnitude is equal to the mean velocity and which points in the direction of net flow) through a tube of radius R,, and R is the distance from the cylinder axis. The term

ro = s27ca3[s*V2(r2f,) + {s*V2(r2fi)}+] + 0(s2)

VVu= -$&Y-y)

3@c = s6na2[IV(rfi) - &V3(r3f3)] + 0(s2)

3ro= -4na3[(1

+ +V2(rTf2)*E}] + O(E2) (8.4)

(8.2)

If C is the centroid of the slightly deformed sphere and R is its centre of reaction it is also known [12] that rcc = rcR + 0(&2)= saV(rfi) + O(E2) (8.5)

is already constant for such flows and so V”u = 0 for n > 2. In this case the values of the resistance polyadics “+2@o and “+‘I’o for IZ> 2 are irrelevant. The ellipsoid, of course, furnishes some indication of the general behaviour of anisotropic bodies in Poiseuillian flows, but its orthotropic symmetry prevents it from serving as a model of the behaviour which particles of completely asymmetric shape may display. In previous papers [4, 121 it has been shown that a “slightly deformed” spherical particle furnishes a useful model of these more general bodies, and that it leads to results of surprising accuracy even for radical departures from the spherical shape. Accordingly, in this section we shall calculate in detail the resistance coefficients 4@, and 41’0 for such a body. Because of the generality of the methods of computation we shall also be able to draw some limited conclusions about the properties of “f2@, and “+‘I’o for arbitrary n. A slightly deformed sphere is the body whose surface is described by the equation r = a

where

1

]E]4 1 is a small arbitrary

Thus, to the order in E indicated, the resistance polyadics in (8.4) at either C or R may be obtained by simply suppressing thef,-terms. To this same order in E, it is immaterial whether r and& (k # 1) in (8.3)-(8.5) are measured with respect to origins at 0, C or R. As has been shown in Part III [4, equations (6.7) 1 and (7.2)], the force F and torque To experienced by a particle of arbitrary shape, on whose surface

(s,) is satisfied some arbitrary boundary condition P, immersed in a fluid at rest at infinity, are (8.6a)

To = -

3pa2

‘, x ($),.=, dl2 s s*

(8.6b)

t In obtaining the 31’~ relation from (4.4) we have utilized the identity {e*VV(rzfi)}t = EeVV(r2fZ)+ VV(r2f2)+E

(8.3) parameter;

+ 3afO)~f s{e*V2(r2f2) +

whose validity depends on the symmetry of the VV-operator and the fact that 02(r2fi) = 0.

715

H. BRENNER

where 0 may be any point and r and r are measured from 0. Here, s1 is a sphere of unit radius having its origin at 0; dQ = sin 8 dtl d+ is an element of surface area on sl, a is the radius of any sphere concentric with sr, and (+)r)r=(I denotes the value of Q on the sphere Y = a. Though the integration is over the surface of a sphere we emphasize that the formulae (8.6) apply to a body of arbitrary shape. For the present applications we choose Q=v’=

2 “+2Voln+(V”u)o n=O

Substitution in (8.10~(8.11) yields “+%IB~= n+z@$‘) - $a& (“+‘Vg)),=. da + 0(e2) s s1 (8.14a) n+210 = n+zIg) _ 3a2E

x (“+2Vg))r=II dC2 + O(e2) (8.14b)

c8a7)

where v’ is the Stokes field defined by (2.6), its expansion being given in (3.4a). This choice requires that &F’=p

5 “+Q$Z$~“), n=O

z X s SI

where “+‘@,dO)and n+21$‘) are clearly the resistance polyadics for the undeformed sphere of radius a. In particular, it follows from FAXEN’Slaws that we may write 6rcaI for n = 0 “+‘@$) =

(8.8)

7ra311for n = 2

(8.15a)

0 for all other iz and

and z “+21,~(V”u), n=O

TO=T;=p

n+zI$B =

(8.9)

0 for all other n

where we have utilized the expansions of F’ and Tb given in (3.14) and (3.15). Now substitute these results into (8.6). As the resulting equations must be satisfied for all possible choices of the constant coefficients, (V”&, this requires that *+2Q)o =

-+a

s

(nf2V&.

da

SI

and

-47ca3E for n = 1

Consequently, if aj, is the Kronecker delta “+‘QO = 6,,6~aI + 6,,na311 - $aeJsl (n+2Vg))I=II dR + O(E’) (8.16a)

(8.10)

*+210 = -_6,147ta3e - 3a2E

r s SI

; x

x (n+2V&r))I=,,dR + ‘+‘I0 = -3a2

‘, x (n+2V0)r=. dCl

(8.11)

f s1 The fields ““VO are defined by equations (3.5)(3.8). Now choose the surface sP in (3.7) to be the deformed sphere (8.3); that is, “+‘VO is to satisfy the boundary condition +on

n.

Sd

E(“+‘vg))

+

(8.12)

OfE’)

(8.16b)

In order to calculate (n+2V(,$))r=0,i.e., the firstorder perturbation field at the sphere surface, we observe from (8.12) and (8.13) that

Now, by Taylor series expansions about the sphere surface we obtain

where s,, is the surface (8.3). Next, expand the field “+‘Vo in powers of E as follows: “+‘VO = “+‘Vg) +

o(E2)

“+2V$‘) + E(“+~V~)) + 0(s2) = - -$ Ir” on sd (8.17)

1

“+2v*=

(8.15b)

“+‘Vg) = (n+2Vg))l,, + (r - a) + O(r - a>2 and

(8.13) 716

“+‘Vg) = (n+2Vc)),=0 + O(r - a)

The Stokes resistance of an arbitrary particle-IV.

Thus, (8.17) may be written as

(n+2Vp)r=a+ (r -

where we have utilized the identity q1 x q, = -E : q1q2. These expressions permit us to obtain the resistance polyadics “+‘a0 and n+210 for the slightly deformed sphere, correctly to the first-order in the deformation parameter E, solely from a knowledge of the detailed “velocity” field ‘+‘Vb’) for the original undeformed sphere. For the particular case n = 2, “VP) is given in (5.3). To differentiate the latter with respect to r we note that

a)y+;y)‘],;a +

+ E(“+~V~))~=~+ O(r - a)’ + .5O(r- a) + +

0(E2)= -

$1 Fnrnonsd *

0

But

Arbitrary fields of flow

r - a = &a 2 fk on sd k=O

II, = r-(“+‘)P,

whence

(8.21)

in which

(-I>” r”+lVnl

p _ n

+

a

~+;~“‘,_ $fk)

= -~I(yu+

the latter being an n-adic “surface spherical harmonic” of degree n, and hence independent of r.

+ 0(E2)

+nep]

A simple calculation therefore yields

+0(&z)

a(4vp)

Equating terms of equal order in E yields the two relations (“+2vgyr_

= -

$ If)”

(8.22)

r

n!

[

dr

1

= a(@IPo + IP, - QP21 - P4)

r=ll

Moreover, it is easily shown that

(8.18) 1; 0

and

2=411Po+31P2

Consequently, r

al

r” +(n-l)!I;01

4cD o = na311 + $a3E C

a"-l

Cfk

(8.19)

J

k=O

k

+ SIP2 - iP,I

The fomler result is, of course, trivial. Substitution of (8.19) into (8.16) gives

(3IIPo + s,

- P4)fk dR + O(E’) (8.23a)

and

4ro = -3a4aa : 2

“+‘a0 = 6,06rcaI + ~,,,Yuz~II+

k=O

r (@IPo + s s1

r

P&k dS1 + O(E’) (8.23b)

+ SIP, - +P,1-

As proved in Appendix II, the following orthogonality relation obtains :

0 for n # k

and

k!(;i+

n+21* = Equation

(8.23a) therefore

4@o = 7CU3[(l +

a”-l

y ++l)!

I ; ” fk da + O(E’) (8.20b) 01 717

-

1) Vk(9fk) for n = k (8*24)

3Ef,)II

+

adopts E{Iv2(r2f2)

&V2(r2f2)I - &V4(r4f4)>]

the form -

+ O(E’)

(8.25)

H. BRENNER

Evaluation of the integrals appearing is rather more involved. Write 410 = - 3a4e(+B,I

in (8.23b) 4711 V”+ l(r”+ ‘f,+ 1)

+ +B, - +B31 - B4) + O(.s*) (8.26)

= z

V”+‘(r”+‘fn_,)

_(2n + 1)(2n + 3) - (2n - 1)(2n + 1) +

where

B,

@S,,;fd+

=E:

+ &

(8.27) where

B,=e:

f IP2 fk dd

f k=O

B,=E:

s

SI r

s

s-1 r

B,=E:

s

slT

2 k=O

The first two integrals ing to the identity

(8.30)

may be simplified

E: for q any vector. replaced by

(8.29)

5 P, fk dQ

QI =

Thus,

2

where qj is any becomes

E’q

B,=E*

$ k=O

s

s

(8.31)

f P2fk dQ

(8.32)

s1 r

that

a.V(rfJ

(8.33)

---I :

l)!

F-1

12 -

+

E : Tr V*(rrfi)

= - $

s*V(rfi)

B4 = GE : Tr V4(rr3f3)

(8.35)

(8.36)

(8.37)

5+1

Substitution of (8.33) and (8.35)-(8.37) into (8.26) furnishes the formula for 410 to the first-order in E. The resulting expression is greatly simplified by referring to an origin at C or R rather than 0, for all terms containing fi may then be deleted. Therefore

1 VR-l(rlZ-lfn-l)

the last term in (8.34) may also be written as ( Tr V+~)~V~-l(rn-l~~-,) n+1+1

where we have noted that Vn-l(rn-lfn-l) (n - I)-adic. The “multiplier”

(8.32)

and

t Because of the identity (see Appendix AII.6)

- l)!

- &V3(r3fl)

equation

3-l

The remaining integrals are all of the same general structure and may be evaluated via the following identity proved in Appendix II:?

1

Hence,

Evaluation of B, and B, may be similarly accomplished. The calculation is simplified by observing that the vector operator E : VV is identically a zero operator because of the symmetry of the VVoperator. Consequently B, = $

B, = f

(2n -l)(n

vector.

B, = 27t~*[&V~(r~f~)

r

Equation (8.31) may be evaluated by noting r/r = P, and utilizing (8.24). Hence

(n

. . . = qmq1q2...qm-1qm+l...

3-l

rfkdR %

(8.34)

Wl+l

+ j$ Tr V2(rrfi)] k=O

=

1

the symbol

Tr qlq2 ...qm-lqmqm+l

by resort-

(8.27) and (8.28) may be

B,=E-

P-%-l

V”(rr”-lf,_l)

Tr is a transposition operator m-1 which operates on a polyadic whose rank is equal to or greater than m in such a way that the m-th vector (numbering from the left-hand side) in each of the polyads of which the polyadic is comprised is to be moved to the position of the first vector in each of the polyads, i.e.,

(8.28)

f Pzfk dR

5 k=O

Tr nfl+l

is a constant

Tr Vnrn 7&+1+1

4rc = 41YR+ O(E2) = - n14 U4& 4E.V3(r3f3) - E : Tr V4(rr3f3)

is then a constant Zn-adic operator.

5+1

718

1

+ O(E*) (8.38)

The Stokes resistance of an arbitrary particle-IV.

Also, since (8.25) contains that

nof,-term,

+ s{IV’(rZfi)

+

- &V’(r’fi)I

- &V4(r4f4>}]

9.

+ O(E*> (8.39)

Equations (8.38) and (8.39) give the main results of this section. In view of (8.5) it is immaterial to the first-order in E whether r and fk (k # 1) in the above are measured relative to origins at 0, C or R. Determination of “+‘mO and n+210 for arbitrary n is obviously a matter of great difficulty. However, on the basis of our partial knowledge of the fields “+2V$‘) given in (5.7a), equations (8.14), (8.24) and (8.34) permit us to establish the following limited results : The force polyadics for the slightly deformed sphere (8.4Oa) r = aC1 + s(fO +_fn+2>1 are, for II = 0, 1, 2, 3, . . . “f2@0 = 6,,6na(l

+ .$,,)I + 6,,rca3(1 + 3Ef,)II 6xa”+’ 2 n

-s(2n)!(n

+ 2)(2n + 5) ’

x V”+2(r”+2fn+2) + O(E~) The torque sphere

polyadics

r = 0

for the slightly

+ a(f0

(8.4Ob) deformed

+f,+dl

(8.41a)

are, for iz = 0, 1, 2, 3, . . . n+2ro = -6,,4na3(1

fields of flow

(8.41b) have the same values at C and R as they do at 0, at least to the first-order in E. These results agree with those tabulated in (8.4), (8.38)(8.39).1

it is apparent

4ac = 4@R + O(e’) = rca3[(1 + 3af,)II

Arbitrary

+ 3&fO)&-

_ E 6na”+‘(n + 1)(2n + 1)2”+3 x

DISCUSSION

The techniques outlined in this and preceding papers in the series contain the germ of an apparently novel approach to the solution of certain classes of linear (and perhaps non-linear) problems. Basically, the idea is this: if some of the constant parameters characterizing the general problem are directional in nature, e.g., velocity, spin, shear, etc., it may be possible to find transformations of the dependent variablest which remove all of the directional (and scalar) parameters from the governing differential equations, boundary and initial conditions. The penalty one pays is an increase in the “order” (i.e., degree of directionality or polyadicity) of the problem. On the credit side, however, is the inescapable fact that one gains certain a priori information which would otherwise be lacking; that is, even without attempting a detailed solution of the resulting higher-order problem, one’s knowledge of the general properties of the solution is enhanced. Consider, for example, the symmetry questions we have resolved in Parts II and III without knowing the solutions of Stokes equations for any of these bodies. In a sense the idea is a natural extension of dimensional analysis to the case where the dimensional parameters are directional rather than scalar quantities. To the extent that the dimensional analysis of a system of differential equations, initial and boundary conditions is regarded as furnishing a priori information as to the functional structure of

(2n + 4)! x

--El

[

(n + 2)c*Vn+1(r”+1fn+l) Tr n+3+1

Vnf2(rr”+‘j”+,)

t In order to demonstrate the following identities:

-

1

this for the

n+2r~

we

require

E : Tr V2(rrf) = -e.V(rf) 3-1

+ 0(s2)

and E : Tr V3(rr2fi) = -[c.V2(r2f2) + {wV2(r2fi)}f]

4’1

(8.41b)

The point 0 refers, of course, to the centre of the undeformed sphere. Except for ‘IO, which contains an _&-term, the resistance polyadics in (84Ob) and

plus the identity given in the footnote on p. 715. i It is conceivable in certain Droblems that one might find it hecessary to transform so&e or all of the independent variables as well.

719

H. BRENNER

the solution, so must the directional analysis suggested be regarded as furnishing supplementary a priori information (not obtainable by dimensional arguments). The higher-order system of equations resulting from the directional analysis (e.g., equations (3.5)(3.8) of the present paper) represents a greater number of scalar equations than does the original system. Though their structure appears rather forbidding, this is more apparent than real and stems largely from our relative inexperience with such entities. One has only to observe that equation (5.3), for the motion of a sphere in a “quadratic” field, represents eighty-one scalar equations. Yet its appearance is scarcely more complex than, say, Stokes solution for a translating sphere which, being a vector equation, represents but three scalar equations. It appears that polyadic harmonic functions of the type introduced here may be utilized to solve linear polyadic partial differential equations in almost precisely the same way as scalar harmonic functions are used to solve scalar linear partial differential equations. The only distinguishing (and complicating) feature of the former is the problem of specifying the order of a sequence of non-commutative operations. No comparable problem arises for scalars. Our point of view may be somewhat forcefully summarized as follows : a directional quantity-be it a velocity, a shear or the like-is an invariant and primitive entity unto itself. Its decomposition into scalar components, as is the usual practice in solving equations, is conceptually wrong and can, in the long run, serve only to retard progress. To paraphrase an expression previously paraphrased by SCRIVEN(see Ref. [16]), a polyadic is a polyadic

APPENDIX I

The expansion of (r/r)” in a series of polyadic surface spherical harmonics We define a k-adic surface spherical harmonic of degree j, kSj, to be a function which is independent of r and possessing the property that ri kSj satisfies Laplace’s equation. As is readily shown, kSj satisfies the partial differential equation

{r*?* + i(j + l)}“Sj = 0

(AI.l)

Of special interest is the normalized k-adic surface spherical harmonic of degree k, P, defined in equation (5.9). We shall prove here that the expansion of the n-adic (r/r)” in a series of “Sj’s has the general form

fn t-1

2”(n !)*

+Y3,_, +“S,_,l.

=-P,

r

(2n)!

+%,_e+ ... (AI.2)

the series terminating in either “S1 or “So according as n is odd or even. It is unnecessary for our present purposes to explicitly know the “Sj for j # It. As specific examples of the expansion (AI.2) we have? r/r = P, (r/r)* = 3P2 + 41P. (r/r)” = 3P3 + -RIP, + (IPI>++ P,I], etc.

To prove (AI.2) we utilize the following theorem [14, p. 1271: if +,(x, y, z) is any scalar function of position, homogeneous in r” (n > 0), then

x

is a polyadic.

l[

1

r4V4

r"+"

2(2n-1)+2*4(2n-l)(2n-3)-.”

’

x 4+,(x,Ys z) Acknowledgement-This paper was written while the author was on sabbatical leave in the Chemical Engineering Department at the University of Minnesota. Thanks are due to Professor N. R. AMUNDSON and his staff for their cordiality during my stay. Above all, I am grateful to Professor L. E. SCRIWN with whom I shared many pleasant moments discussing various aspects of this work. I am also indebted to the Pulp and Paper Research Institute of Canada for their continued interest in this work.

where r* = x2 + y* + z*. This expression clearly remains valid even when $, is a polyadic function, t Compare these with their counterparts (scalar) Legendre functions [ll] p =

720

involving ordinary

Pi(P)

P3

=

u%(p)

+&PO

~3

=

W3b)

+

W1b4,

etc.

The Stokes resistance of an arbitrary particle-IV.

say Jr,, (not necessarily an n-a&c). As a special case? we may write

1

r”V”

1-2(2n-1)+2+4(2n-1)(2n-3)-‘**

r2Q2

If we set h(r) from (5.9) r”p,

(AI.3)

= r” this makes q”(V) = V”whence,

+ r2nY,_2 + r4nY,_4

+ ..,

(AI.8)

“Yj is an n-adic solid spherical harmonic of order j (i.e., an n-adic of the form “Yj = ri “Sj) given by (AI.9) “Y, = O,(F)

2(2n - 1)

r”V” +2*4(2n-1)(2n-3)-‘*’

1 r”

(AI.4)

which provides an alternative scheme for computing P” Now, HOBSON[14, p. 1471also gives the following theorem : if t,k,,(x,y, z) is any scalar function, homogeneous in r”, the expansion of $,, in a series of solid spherical harmonics is necessarily of the form $, = Y, + r2Y._2 + r4Y,_4 + . . .

nY,_2 =

1 0,(V2r”), etc. 2(2n - 1)

(AI.lO)

Comparison of (AI.9) with (AI.4) shows that “y

2”(nD2

=

”

”

(2n)! r pn

Hence, (AU) becomes r”

(AI.5)

- -2% !I2r"P, + r”nSn_2

+ r”nS,_4 + . . .

_

(2n!)

where Y,,, Y,- 2, Y,,-4, . . . are scalar solid spherical harmonics of the respective orders indicated by the indices, the last being Yr or Y0 according as n is odd or even. Explicitly, these harmonics are [14, p. 1481

Division by r” then yields the desired theorem, (AI.2). APPENDIX II

Proof of equations (8.24) and (8.34)

r, = @,($,)

Proof of the basic integration theorems, (8.24) and (8.34), requires that we first prove the auxiliary theorems tabulated below.

Y”_, = 2(2n1_ 1) @“(V2$“) Y”_4 =

r” =“Y, where

=-$f$ [I - r2’2+ n.

r”Q”

2(2n - 1) + 24(2n - 1)(2n - 3) - .** (AI.7)

Again, these relations may be applied when @, is a polyadic, say \Ir,. In particular if we choose \Ir, = r”, equation (AI.5) takes the form

’

x +,(r)

fields of flow

where 0, is the scalar operator O,=l-

r2v2

Arbitrary

1 O,(V’$,), etc. 2*4(2n - 3)(2n - 5)

s

4n2kk!

(dr)kf

dfl

=

(2k

+

1)!

Vk(lkfk)

(AII.l)

S1

t It is important to note here that the symbol &(r) does not merely mean that +,, is a function of position, say &(x, y, z), but rather that the polyadic function can be expressed in terms of the single entity r. For example, the triadic function +1(x, y, 2) = 1(1x + jy + It.4 is homogeneous in r1 but cannot be expressed in terms of the single variuble r = ix + jy + kz. On the other hand, were we to replace the kx term by kr the resulting function could be written as I(ix + jy + kz) = Ir = +1(r)

4n2kk !

(r/r)kPk

da

=

(2k

+

l)!

Vk(#Pk)

(AII.2)

s S1

s

pkfrn dfi = SI

/Ofork#m 47~ Vk(rkfk)for k = m k!(2k + 1)

(AII.3)

[Oforkfm (AII.4) Vk(r“Pk)for k = m

The concept embodies the same idea as encountered in complex variable theory.

721

H. BRENNER

s

Pk(r/r)k dR = (r/r)kPk dR s1 s SI rkfk = j$ rkkvk(rkfk)

(AII.6)

s

\Irk(r>yk(xY

j$rkllflvk(ek)

rkPk=

This relation is clearly applicable even if $k is taken to be a polyadic (not necessarily a k-adic), say +k. Consequently, as a special case of this generalization we may write (see footnote on p. 721)

(A11.5)

(AII.7)

y, z> ds

2kk ! 4n (2k + l)!

=

x Vk(rkPk)mVk(?fk)

= $$

Vk(ryk)

Upon

(AILS)

+kb)

p

k

J-l)k -

1

,.k+lvk

k!

(r/r)kfk

s

y,

z)ds

X &($

= 4n

$3

c2k

+

$),k(&

1j!

r2k+2

y, z,

x

-9 ax

-Yay

=“k(-&,$;)

Applying

s

‘bk(T

xkcx,

jS,f, s1

z>yk(x,

x

Y, Z) (-411.12)

In particular

2kk ! 4x (2k + l)! the validity

of (AII.1).

dR = 0 if k # m

Y, z)

(AIMS)

Pkfm dR = 0 for k # m s s1

s

(I’/r)kfk dR = $$

ux,

vk@kfk)

this makes

Sl

2kk ! r2kt2 Y, z) ds = 471(2k + 1) 1

$k($$;)

=

so that attention need be directed only to the case k = m. If we multiply (AI.2) by fk dR and integrate over a sphere of unit radius, the orthogonality condition (AII. 15) shows that

this to (AII. 11) yields Y,

dQ

(scalar) surface harmonic of degree k, it is evident that .fk in (AII.l) may be replaced by Pk, for the order of the vector operations is unaffected. Hence, (AII.2) is correct. In order to prove (AII.3) we note firstly that, as an immediate consequence of the orthogonality properties of scalar surface harmonics

s

ak(&y,z)

az

r2 dil

is a whereSm,m,...m,

(AII.11)

a a\

/a

rkfk

=

thereby demonstrating Furthermore, since

the integration being over the surface of a sphere of radius r. But [14, p. 1261, if&(x, Y, z) and rrk(x, Y, z) are any two scalar functions each homogeneous in rk, then Xk

=

ds

(AII.lO)

r

2kk ! y, z)ll/k(x,

= rk

Sl

We shall take as fundamental the following theorem [14, p. 1571: let t,Gk(x,Y, z) by any scalar function homogeneous in rk and let Yk(z, Y, z) be a scalar solid spherical harmonic of order k. Then yk(x,

cA11.14)

obtain

s

The notation here is as follows: fk is a scalar surface spherical harmonic of degree k (so that rkfk is a scalar solid spherical harmonic of degree k), and Pk is the normalized k-adic surface spherical harmonic of degree k defined by

y, z,

y, z,

(AII.9) we

x

setting yk(xY

Pkmvk(r’tf,) = $$fk

+kmh?

r2k+2

*

j- Pkfk dn s1

The value of the left-hand integral is given (AILl), whence the validity of (AII.3) follows. It is clear that we are free to replace the scalar f, in (AII.3) by P, whereupon equation (AII.4) then follows directly.

x

(~11.13)

722

The Stokes resistance of an arbitrary particle-IV.

To prove (AII.S), set IZ= k in (AI.2), premultiply by Pk &I and integrate over sl. By the general orthogonality relation

s s

Pk(r/rjkdSZ=

s1

Had we postmultiplied have obtained

z[

is

(r/r)“Pk dt2

1 a r sin 8 a4

where i,, ie, i, are unit vectors in spherical co-ordinates, and observe that i, = r/r,

ai,/& = 0,

i;i,

= 1, i;i,

k-times k-times

= 0, i;i,

= 0

n+l : p, = -P r 2n+l

+ . ..) .r. (i,:

{rkmvk(rrfk))

n+l+

“+lS”_l

(AII.16)

i.e.

2 p, - n+l r

2n + 1

p,+l

(A11.17)

We propose to show that “+lS,,_l is an (n + l)adic surface spherical harmonic of degree n - 1, as the notation suggests. Inasmuch as this quantity + .Ij(rf,j is obviously an (n + 1)-adic, homogeneous in r”, it suffices to prove that r-““+lS,_l satisfies Laplace’s

kkfk)

+2(r-”

=r

Vk(rkfk)

From (AII.7), the k-adic in braces on the left-hand side is k!rkP,, while from (AII.6) the scalar in braces on the right-hand side is k!rkf,. Thereby, equation (A11.9) is proved. There remains now only the problem of providing a proof of (8.34). Define the (n + lj-adic ““S,_, by the relation?

rk

ak = Ik ark

= &

{rk@k(r?k)}j?+k(rkfk) =g

“+ls,_l

$)m&i

qJ”(rX)

The value of the integral in braces is given in (AII.2), from which the validity of (AII.8) now follows. To prove (AII.9), premultiply both sides of (AII.8) by the operator fim. This yields

then

= ti(C

vk(rkfk)

Sl

The right-hand sides of each of the above two equations are identical, thereby confirming (AILS). Equation (AII.6) results from repeated applications of EULER’S theorem relating to homogeneous polynomials; that is, if we write V=i,~+i,~$+i+__

=2&

vk(r-kfkk)

SI

PkPk dQ *

i

In view of (AII.5) this is equivalent to

(AI.2) by Pk dQ we would

(r/r)kPk dCI = ss s1

dQ Ei

Pk(r/r)k

s*

scalar or-

PkPkda *

fields of flow

We begin the proof of (AII.8) by combining (AII.3) and (AII.6). Since vk(rkfk) is a constant k-adic it may be removed from beneath the integral sign. Thus

jS,‘S, da = 0 for k # m s s1 which follows from the corresponding thogonality relation, we then obtain

Arbitrary

equation. II+1

From

(AII.17)

S,_ 1) = $‘(rH,) - f-$

we have t2(r2Hn+ d

fk

where H = r-(n+l)P ” n

= rkk!fk

whence (AII.6) is demonstrated. Equation (AII.7) follows from the fact that we are free to replace fk in (AII.6) by any polyadic which depends on 8 and 4 but not r. 723

7 This is suggested by the scalar recursion formula [ll] P~nW = &$

pla+m

+ *

valid for ordinary Legendre polynomials.

Pn--1&)

H. BRENNER

is the

n-adic solid

-(n + 1) defined

spherical harmonic of order in (5.4) (see also (5.9)). As is

But, from (AII.6),

readily shown

L-1 = &$)‘-‘l”-‘JvYrY_L)

V*(rH,) = 2VH, = -2(n + l)H,+l The polyadic V”- I(?- ‘f,_ 1), being constant, may be removed from beneath the integral sign, whence

and (see (5.6)) V2(r2H,+l) = -2(2n + l)H,+,

1

“+lw = (n _ l)! .+y+,

Consequently

da

V*(r-” “+lS”_& = 0

On the basis of (AII.16), it is convenient to write : Pnfk dQ = “+lZ + “+lW sI

(AII.18)

x

yr”- ‘f,- 1)

x pp-

which proves the contention.

I

The value of the integral appearing above may be obtained from (AIIS) and (AII.2); hence 4x2%

“+lW = on

r

+

Tr V”(r”P”)p[

x

nfl+l

0!

x v”-l(r”-%-l)

where

p,+ lfk dQ

(~11.19)

and n+lW=

F k=O

are constant

“+ %,_ Jk dR s

(AII.20)

s,

(n + l)-adics (dependent

only on the

Now, as can be easily demonstrated . r”P” = ii r2n+1V(r-“P,_i) so

that (2n + l)! .+:I

index n), and “flSn_l is as defined in (AII.17). It follows immediately from (AII.3) that n+lz = n,(2n +4&2n + 3) V”+‘(r”+‘f,+l)

47~2”

“+lW = -

V”{r

x In_1Iv”-‘(P-

(AII.21)

With regard to (AI1.20), it is clear from orthogonality that only the single term k = n - 1 contributes to the infinite sum over k. Hence, using the definition in (AIL 17)

;P&dR-s s1

The last integral vanishes more, we may write

(r/r)P, =

Furthermore,

dQ

Tr

Further-

since V”+‘(r”+Ifn_r)

n+lw =

P,r/r

4n

Tr Vn(rr”n-lfn_ I) n!(2n - 1) n+l+1

where Tr is the transposition operator defined in the latter part of Section 8. Since the operations of integration and transposition commute, we obtain Tr n+l+l

s SI

P. ;L-l

= V”+‘(r”+‘f,_,)

we finally obtain

n+1+1

“+lW=

- (2n + l)rr”-‘f”...,

n+l+l

by orthogonality.

Tr

= V(r”+lj”_l)

Consequently, P”+,f”-,

= 2!_2:(,-_2)~),j”-1

by identity

r2”+1V(r-“j”_l)

n+l 2n + 1 s sI

Y”_ 1)

But, from (AII.9) P”_l~l(V”-‘(r”-lj”_l)

“+lW =

*“+ ‘V(r-“P,_ 1)} X

da

- &

V”+l(S+l&_l)

(AII.22) 1

Substitution of (AII.22) and (AII.21) into (AII. 18) yields equation (8.34) and thereby completes the required 724

proof.

The Stokes resistance. of an arbitrary particle-IV.

There are a host of other potentially useful relations obtainable by appropriate generalization of the scalar relations given in HOBSON'S book [14]. The only one we shall mention explicitly is the theorem [14, p. 1611

s

X,(x,Y, Mx,

Y,

z>ds r’P ‘+2(2n+3)+.

r”?” +2*4(2n+3)(2n+5)

x

+

**’I ’

K(&,$3;)W Y, z)],

(AII.23)

the integration being over the surface of a sphere of radius r. Here, Y,, is any scalar solid spherical harmonic of order n, and $(x, y, z) is any scalar function which can be represented by an absolutely convergent power series within some region y > r. As usual, the subscript 0 means evaluation at the sphere centre, x = y = z = 0. The generalization of this formula to the case in which Y,, and + are polyadics of any order is clearly valid. Of special interest is the case n = 0. This furnishes a proof of a theorem used in Ref. [12, equation (6.25)] for which no proof was originally supplied.

Arbitrary

fk A surface spherical

harmonic of degree k F, F’, F”, i Vector hydrodynamic forces exoerienced by a particle in wnseq&ce of the Stokesian motions v, v’, v”, i, respectively I-L An n-udic solid spherical harmonic of degree -(n + 1) detied bv (5.4) h, iz, i3 Righihandeh triad of mutually perpendicular unit vectors Unit vector in the xj-direction i5 0’ = 1,2, 3) . 1% Unit vector in the xmk-direction (mk = 1,2,3) i,, ie, ie Unit vectors in spherical w-ordinates, (r, 0, +) Unit vectors parallel to (x, y, 2) i, i, k I Dyadic idemfactor Translation resistance dyadic Denotes m successive dot multiplications PtPI, P”, i Local Stokesian pressure fields wrresponding to the local velocity fields v, v’, v”, u, respectively P?Z Legendre polynomial of order n P* n-a&c surface spherical harmonic of degree n dei?ned by (5.9) mPo m-adic “pressure” field, dependent on choice of 0, defined bv (3.4b) _ . ’ 4, q1. q2. ... Arbitrary vect&s Constant dyadic defined by (7.17) Q r Distance or a spherical co-ordinate Position vector r Position vector relative to an origin r0 at 0 Position vector of a point P relative rap to an origin at 0 n-times i

An n-adic defined as rz R Cylindrical w-ordinate Ro Radius of a circular cylinder so Surface of a sphere of radius a Surface of the deformed sphere (8.3) Sd Surface of the ellipsoid (7.1) Se Surface of a particle of arbitrary SP shape Surface of a sphere of unit radius Sl S ,,,l,Q...ntk Scalar surface spherical harmonic of degree k ‘“Sj k-adic surface spherical harmonic of degree j To, T;, T”o, 90 Vector hydrodynamic torques about point 0, experienced by particle in consequence of the Stokesian fields v, v’, vN,i Tr A transposition operator Stokesian flow field at large disU tances from a particle U Constant velocity vector Uo Velocity vector of point 0 Local Stokes velocity field V r”

NOTATION a Sphere radius aa Semi-axes of ellibsoid “Ao General n-adic ai any point 0 A(O) mlma Cartesian tensor of rank n wrres. . . rnR pending to above polyadic &, Ba, Bs, B4 Constant polyadics defined by (8.27)-(8.30) co Coupling dyadic at any point 0 ds !Zcalar element of surface area a!3 Directed element of surface area dR Element of surface area on a sphere of unit radius 02 A second-order scalar differential operator defined by (7.11) or (7.30) D2m Refers to m successive applications of above operator Right-handed triad of unit vectors el, e2, es parallel to the principal axes of an ellipsoid E A constant dyadic defined by (7.28) al, u2,

725

fields of flow

H. BRENNER Mean velocity of flow vectol through a circular tube Angle in spherical co-ordinates Shear-force triadic at point 0 m-adic force resistance coefficient at point 0 Value of above at centre of undeformed sphere of radius a Constant defined by (7.2) Homogeneous scalar function of degree k A scalar function Homogeneous scalar function of degree n Polyadic function Polyadic function, homogeneous of degree n Angular velocity vector of a particle Rotation resistance dyadic at 0

Local Stokes velocity field defined

by (2.6)

Local Stokes velocity field defined by (2.7) Local Stokes velocity field satisfying arbitrary boundary conditions on sp and vanishing at infinity Local Stokes velocity field satisfying VW. transitional boundary conditions (6.1) ovrot Local Stokes velocity field satisfying rotational boundary conditions (6.10) “Vo m-adic “velocity” field, dependent on choice of 0. defined by (3.4a) In@, ?nv(d) Zero-th and firstiorder pertdrbation fields for a slightly deformed sphere, defined by (8.13). n+1w Constant (n + l)-adic defined by (AII.20) 4 Y, = Cartesian co-ordinates Cartesian co-ordinates measured Xl, x2, x3 parallel to principal axes of an ellipsoid _ Scalar solid soherical harmonic of yk order k m-adic solid spherical harmonic of mYk order k nt1z Constant (n + 1)-adic defined by (AII.19)

&jkZ

e

0, h mA

x, x’, If,

,.

Tc,

CL

7&r.,07hJt.

P (Tk

Constants defined by (7.3) Radius of a sphere m-adic torque resistance coemcient at point 0 Value of above at centre of undeformed sphere of radius a Kronecker delta Function of h defined by (7.4) A small arbitrary parameter Unit isotropic triadic Permutation symbol Angle in spherical co-ordinates A scalar operator defined by (AI.7) Variable of integration An m-adic solid spherical harmonic function of any order Viscosity or cos 0 Newtonian stress dyadics deriving from the Stokesian velocity fields I v, v , VI, t, hr., Ovrot.,respectively m-adic stress polyadic deiined by (3.13) Stress triadic derived from vtr.; defined by (6.5) Stress triadic derived from ovrot.; defined by (6.12) Distancedefined by (7.5) Homogeneous scalar function of degree k Shear-torque triadic at point 0

Miscellaneous Ordinary vector “nabla” operator An n-adic differential operator denoting n successive applications of the nabla operator Ordinary (scalar) Laplace operator Denotes m successive applications of the viz operator A vector differential operator defined by (7.25) or (7.31) Subscripts Centroid

i, j, k, I, m, n Integers ml,mz,

. . . , mk

0

P rot. R tr.

Integers ranging over the values 1, 293 Arbitrary point; or centre of a sphere or an ellipsoid; or refers to evaluation at point 0; or means measured relative to an origin at point 0 Arbitrary point Field caused by a rotating particle Centre of reaction Field caused by a translating particle

Superscripts i, k, 1, m, n Integers. If they appear as presuperscripts they refer to the degree of polyadicity of the entity to which they are attached. As post-superscripts they are exponents m, (1) Zero-th and first-order perturbation fields Reciprocal dyadic -1 Transposition operator Marks over symbols rn ,.

, I

726

Derived from the Stokes fields v’, VI, Iv, respectively

The Stokes resistanceof an arbitrary particle--IV. Arbitrary fields of flow REFERENCES [l] [2] [3] [4] [.5j [6] [7] [8] [9] [lo] [l l] [12] [13] [14]

[IS]

[16]

BRENNER H., Chem. Engng. Sci. 1963 18 1. BRENNER H., Chem. Engng. Sci. 1964 19 599. BRENNER H., Chem. Engng. Sci. 1963 18 557. BRENNER H., Chem. Engng. Sci. 1964 19 631. FAXENH., Ark. Mat. Astr. Fys. 1927 20 no. 8. BATEMAN H., MURNACHAN F. P. and DRYDENH. L., Hydrodynamics p. 296. Dover reprint, New York 1956. CHAPMAN S. and COWLINGT. G., The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge 1953. GIBBSJ. W. and WILSONE. B., Vector Analysis. Dover reprint, New York 1960. BRENNER H., .7. FIuid Mech. 1962 12 35. DREWT. B., Handbook of Vector and Polyadic Analysis. Reinhold, New York 1961. MACROBERT T. M., Spherical Harmonics (2nd Ed.) pp. 96,98. Dover, New York 1947. BRENNER H., Chem. Engng. Sci. 1964 19 519. LAMBH., Hydrodynamics (6th Ed.) p. 604. Cambridge University Press, Cambridge 1932. HOBSON E. W., The Theory of Spherical and EIIipsoicLzE Harmonics. Cambridge University Press, Cambridge 1931. JEFFERY G. B., Proc. Roy. Sot. (Land.) 1922 A102 161. ARISR., Vectors, Tensors and the Basic Equations of Fluid Mechanics p. ix. Prentice-Hall, New Jersey 1962.

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