Taylor dispersion in systems of sedimenting nonspherical Brownian particles

Taylor Dispersion in Systems of Sedimenting Brownian Particles III. Time-Periodic

L. H. DILL’

Nonspherical

Forces

H. BRENNER*

AND

Department of Chemical Engineering, University of Rochester, Rochester, New York 14627, and *Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received September 21, 1982; accepted November 18, 1982 An Aris-type moment scheme is employed to calculate the time-average mean velocity vector and Taylor-Aris dispersion tensor for the Stokes sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles which simultaneously undergo translational and rotational Brownian motions while settling in an otherwise quiescent viscous fluid under the influence of a time-periodic external force. This generalizes to time-periodic forces prior dispersion results previously restricted to time-independent forces, Each harmonic component of the force independently gives rise to its own Taylor dispersion contribution, in which appears a dimensionless coefficient quantifying the effects of frequency and particle geometry-namely size and shape. This parameter vanishes identically for spherical and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.), whose translational hydrodynamic resistances are orientation independent. Advantages attaching to an experimental “wet” particle-shape characterization device based upon the principles developed herein are pointed out. In this context a novel experimental device is proposed based upon preventing any net (i.e., time-average) gravitational sedimentation of the nonneutrally buoyant particles by continuous rotation of the suspension about a horizontal axis. 1. INTRODUCTION

sumed to possess the quasistatic Stokes representation Following up the original analyses of Goren P.11 U(4) = M(4) - Fo, (1) and Brenner [(2) hereafter referred to as Part I] concerned with the special case of where F. is the (constant) external force vecspheroidal particles, Brenner [(3); hereafter tor and M the translational mobility dyadic referred to as Part II] has derived expressions of the ellipsoid with respect to an origin at for the long-time mean translational velocity its center. The argument O2 appearing in vector and dispersivity dyadic arising from [ 1. I] indicates that the respective quantities the Taylor-scale convective-diffusive trans- to which it is affixed are functionally depenport of a homogeneous Brownian ellipsoidal dent upon the particle’s orientation C#J relative particle slowly sedimenting in a viscous fluid to an observer fixed in space. under the action of a sready external force, * Characterization of a particle’s instantaneous oritypically gravity. The instantaneous, orientation-specific, translational velocity vector entation may be accomplished by specifying the orienof the center of the ellipsoid was there as- tations of a set of orthonormal body-fixed vectors (e,,

I Present address: Upjohn Company, Fine Chemicals Research and Development Unit, Kalamazoo, Mich. 4900 1.

eZ, es) relative to a comparable space-fixed trio (ir , iz, is). The orientation 6 of the ellipsoid may thus be regarded as functionally of the form 6 = (el, eZ, e3), wherein the e, lie along the principal axes ofthe ellipsoid. Orientation-space operations, including differential and integral operations, are explained fully in Part II.

430 002 1-9197183 $3.00 Copyright Q 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal oJCoiloid and Inrerjbce Science, Vol. 94. No. 2, August 1983

SEDIMENTING

431

PARTICLES

In an extension of generalized Taylor dispersion theory, Dill and Brenner (4) present a detailed scheme for calculating the timeasymptotic mean velocity vector and dispersion dyadic for situations wherein the “global” tracer velocity vector is time periodic, rather than steady. In the present context, this extension will be applied to instantaneous particle translational velocities satisfying

title then occurs, though there will exist a dispersion about this state of zero net transport. Such systems, as well as variations thereof, offer a unique opportunity for performing a “wet” characterization of both particle size and shape since a number of independently manipulable external parameters, such as frequency, force magnitude, and direction, are available to the experimentalist. Of course, U(4, t + 7) = U(4, a, r1.21 similar remarks pertain to the characterizawith 7 the period. Specifically, the present tion of (monodisperse) multiparticle systems contribution is addressed to situations that are sufficiently dilute to permit ignoring wherein the (center of the) ellipsoid translates particle-particle interactions, for each indiinstantaneously with the time-periodic ve- vidual particle in such a system may be relocity garded as isolated from the others. UC4 0 = W#J) - F(t), L1.31 with F a time-periodic external force vector, satisfying F(t + 7) = F(t). r1.41 Subsequent calculation of the Taylor dispersivity resulting from an otherwise arbitrary single-harmonic force F(t) = 2% F exp(iwt),

Ll.51

in conjunction with an appropriate superposition theorem (4), ultimately permits calculation of the Taylor dispersivity resulting from general time-periodic forces. Appearing in [ I.51 is the real operator L%, the frequency o = 2~jr, and the complex-valued time-independent constant vector F, defined by E = r-’

s0

T F(t) exp(-iot)dt.

I. I. Stochastic Description

Following the notation of Part II, the probability density P(X, Cp,tlV) provides the conditional probability Pd3Xd3$ that the center of the particle lies at time t within the physical-space volume element d3X centered at X, while simultaneously the particle possesses an orientation lying within the elementary orientational domain d34 centered at 4, given that the center of the particle was initially located (at time t = 0) at the physicalspace origin X = 0 and simultaneously possessed some prescribed orientation 9’. Conservation and continuity of probability density require that P satisfy the differential equation

[I.61 dP/dt + V-J + 0,-j

A neutrally buoyant particle will, for example, experience a time-periodic force if it is charged and subjected to an ac electric field. Alternatively, rotation about a horizontal axis of the container housing the fluidparticle system will give rise to a time-periodic particle velocity in the case of a nonneutrally buoyant body settling under the influence of gravity. In each of these two cases, circumstances are such that the timeaverage translational motion of the particle is identically zero. No net motion of the par-

= S(X)S(d, - &)6(t) in addition to the pre-initial P = 0

for

[ 1.71

condition

t < 0.

11.81

Herein, V = a/ax and V, = a/&$ are, respectively, physical- and orientation-space gradient operators. Vector fields J and j, respectively, denote the orientation-specific translational and rotational flux densities of P. Given appropriate constitutive expressions for the latter fluxes, the resulting system of equations describes the particle transport Journal of Colhd

and Inkyface Science. Vol. 94, No. 2, August I983

432

DILL

AND

within the abstract six-dimensional (X, 9) phase space. In lieu of a separate initial condition imposed upon P, the Dirac delta function instantaneous unit source term appearing on the right-hand side of [ 1.71 represents introduction of the particle at the point (0, 4’) at time t = 0. Under the mild assumption that P tends toward zero at least exponentially rapidly as 1x1 tends toward infinity, the quantities (P, J, j) will be required to satisfy the physical-space boundary condition jXl”(P, J, j) -

0

as

(XI -

cc

[1.9]

form=O, 1,2,. . . . This assures convergence of certain integrals that arise in the theory (cf. Part II). No orientation-space boundary conditions are imposed upon (P, J, j) except for the implicit requirement that they be continuous and single-valued for all orientations 4. Since the particle is centrally symmetric and internally homogeneous, the force [ 1.51 acts through the center of symmetry. No external torques act upon the particle tending to set it into a preferred orientation. In these circumstances the orientation-specific translational and rotational flux densities appearing in [ 1.71 are respectively given by the constitutive equations J = U(4, t)P - D((b).OP

[l.lO]

and j = -d(4).

V, P.

[l.ll]

In these expressions U, given explicitly by [1.3], [1,5], and [1.15], is the instantaneous translational velocity vector of the center of the ellipsoid; D and d respectively represent the translational and rotational diffusion dyadics with respect to an origin situated at the ellipsoid center. Diffusion dyadics D and d, though constant with respect to body-fixed axes, are, however, functions of orientation relative to space-fixed axes. Expressed in terms of the orthonormal vectors (ei , e2, e3) lying parallel to the three principal axes of the orthotropic particles, the diffusion dyadics for such cenJournal of Colloid and Inrerface Science, Vol. 94, No. 2, August 1983

BRENNER

trally symmetric particles possess the canonical representations D = elelDl

+ e2e2D2 + e3e3D3

[1.12]

d = eleldl

+ e2e2d2 + e3e3d3.

[1.13]

and

Scalar coefficients Dj and dj are constants. For ellipsoids they are given explicitly by Brenner (5). With respect to an origin at the center of symmetry, the translational mobility dyadic appearing in [ 1. l] and [ 1.31 is related to the translational diffusivity dyadic via the StokesEinstein relation M = D/kT,

[1.14]

where k and T are the Boltzmann constant and absolute temperature, respectively. Thereby, the mobility possessesthe canonical representation M = elelMl

+ e2e&f2

+ e3e$f3

[1.15]

with scalars M, = D,/kT,

[1.16]

Equations [ 1.31 and [ 1.51 to [ 1.141 uniquely define the probability density P. This system of equations automatically (6) fulfills the required normalization,

ss xx 00

Pd34d3X = 1 for =0

for

t

> 0,

t
[1.17]

as a result of the presence of the instantaneous unit source time in [ 1.71. Here, X, represents the entire infinite domain embodying all of physical space, while $I, represents the entire finite orientation-space domain. 2. GENERALIZED DISPERSION

TAYLOR THEORY

Two phenomenological coefficients of dominant interest in physical applications are

SEDIMENTING

*+r

u* = lim 7-l

st

*-a3

(dM,/dt)dt,

[2. l]

representing the long-time mean translational velocity vector of the particle, and D* = ;:527)-1

s’ t

(d/d) X (M2 - MrMJdt,

[2.2]

representing the comparable Taylor-Aris dispersion dyadic. Appearing in these expressions is the mth-order polyadic total moment M,, defined for m = 0, 1, 2, . . . , as Mm(tlQ’) =

ssxc do

X”P(X, 4, tb’)d3&i3X,

[2.3]

with P the probability density defined in Section 1. The phrase “long time” refers to times sufficiently long for the particle to effectively achieve orientational equilibrium (at 9 l), but yet sufficiently short such that physicalspace equilibrium is not attained. An explicit scheme for calculating this pair of coefficients for time-periodic systems is derived by Dill and Brenner (4). By specializing their general methods to the physical/ orientational-space transport of homogeneous orthotropic bodies considered in this paper, the following formulas are obtained for U* and 6*: 6* = ((8~‘)~’ +

=

5”

+

s U($, t)d’t#+ &I

z;jC 2

$’

= (844

D(4)d3#, s 90

(f) “Z 7-l [

f(t)&,

WI

of an arbitrary time-periodic tensorial function f. In the preceding formulas the coefficient 8a2 arises from the fact that (in the notation of Part II) PO” = (87~~))‘. This constancy of P$ reflects the fact that, asymptotically, all orientations become equally probable. The numerical coefficient 87r2 is equal to the volume Jm, d39 = 85r2 of orientation space. Vector field B = B(4, t) appearing in [2.7] corresponds to the solution of the secondorder partial differential equation

aBlat- AB = u - u*,

P.91

in which A denotes the second-order, scalar, orientation space, partial differential operator A = V,.d-V,. [2. lo] Although no boundary conditions are imposed upon the B-field (except that it be continuous and single valued at all points C#I within C&J, it must, however, be time periodic: [2.1 I] W&J, t + 7) = B(~J, 0.

The solution B of Eqs. [2.9] to [2.1 l] is unique only to within an arbitrary additive constant vector. However, since only V,B, rather than B itself, enters into the computation [2.7] of EC, the value of this additive constant is physically irrelevant. [2.4] The pair of distinct contributions bM and 5” to the dispersivity dyadic D* correspond P-51 to the respective effects of molecular diffusion alone and convection. In lieu of utilizing [2.7] to calculate EC, P-61 one may instead use the equivalent formula (4)

and ii” = ((W-’

433

PARTICLES

6” = sym s, (V,B)+ . d . V,Bd3~ ). f2.71

Angular brackets denote the time average,

X ((8~“))’ with “sym” defined by

l0 (U - U -* )Bd3~ ), the symmetrization

[2.12] operator,

434

DILL

symA

def.

=

1

5 (A + A+) 0

AND BRENNER

for any dyadic A. In much of the subsequent analysis the equations summarized in this section will be used in conjunction with the single-harmonic force [ 1.51 or, equivalently, the single-harmonic instantaneous translational tracer velocity vector [ 1.31. However, these tabulated equations apply equally well to more general situations, wherein the instantaneous translational tracer velocity vector is an arbitrary time-periodic function of orientation and time-in particular for circumstances in which the instantaneous velocity vector is given by [ 1.31 with an arbitrary time-periodic force. This observation will prove useful in constructing the mean velocity vector and dispersion dyadic for an arbitrary time-periodic force field. 3. PHENOMENOLOGICAL

COEFFICIENTS

3. I. Single-Harmonic Substitution

Force

of [ 1.31 into [2.4] gives u* = k(F)

for the mean translational

= (1/3)(Mr

[2.13]

(F) = 0.

k3.21

the mean translational mobility of [ 1.151 and the identity (7)

dyadic. Use

u* = 0,

(i = 1,2,3)

L3.31

iii = IM,

[3-41

thereby yields in which I is the dyadic idemfactor, whose representation in body coordinates is

and G is the invariant % = (1/3)tr M

l3.71

implying that, on average, the center of the ellipsoid remains at the same position in physical space. This may prove to be a useful property for the experimentalist to exploit, since experimental apparatus dimensions may then be reduced below those required (see Part II) for the case of steady external force, such as gravity. From [1.12], [2.6], and [3.3] the purely molecular contribution to the dispersion dyadic is found to be 6” zz 10 ,

13-a

where D = (1/3)tr D is the invariant D = (l/3)(0,

+ D2 + D3).

f3.51

scalar r3.91

In view of [ 1.161 and [3.5] this may be expressed alternatively as ii = kT%.

(8~~)~’ J ejejd34 = (3 Qb

I = elel + e2e2 + e3e3,

[3-61

It follows immediately that for a single-harmonic force the mean translational velocity possesses the value

ii=@,rr2)-’ sWd)d34 60

[3.5b]

Here, tr A = I:A denotes the trace of an arbitrary dyadic A. Equation [3.4] shows the mean mobility dyadic to be isotropic. Equation [3. l] applies for any time-periodic force. Time average the single-harmonic force [ I.51 to obtain

t3.11

velocity, with

+ M2 + M3).

[3.10]

Central to the problem of establishing the Taylor contribution I$ to the dispersivity D* is determination of the vector field B(4, t). From [1.3], [1.5], [2.9], [2.11], and [3.7] this field represents the time-periodic solution (with period T) of the differential equation aB/& - AB = 2% M 9E exp(iwt).

[3.1 l]

By expressing B in the form B(4, t) = 2% B(4) exp(iwt) + const, [3.12]

scalar [3Sa]

Journal o~CoNoid and Interface Science. Vol. 94, No. 2, August 1983

the temporal and orientational

dependencies

SEDIMENTING

435

PARTICLES

of B may be separated, as indicated. Substi- with (CX,, LYE,01~)scalar constants to be detution of [3.12] into [3.11] reveals that the termined and complex-valued vector field B satisfies the [3.17] differential equation 13 131 This body-fixed dyadic may be written ex(A - iw)B = -M(4) - F, plicitly as [cf. [ 1.151 and [ 1.1611 together with the auxiliary requirement that rin = e,e&f, + e2e21G2+ e3e3G3, [3.18] it be continuous and single valued within 4,. Unlike the B-field, which is unique only with to within an additive constant vector, the Bfield is uniquely determined (4). tij=Mj-ii4 (j= 1,2,3) [3.19] Since Eq. [3.13], defining B, involves both body-fixed parameters M and d as well as the constant scalars. From [3.2] and 13.51 it is space-fixed parameter F, its traditional mode evident that fi is traceless, i.e., of solution suggests utilizing Euler angles tr fi = I:h = A$, + ti2 + &j = 0, [3.20] (~9,c$, $) relating body- and space-fixed coordinates. Such conventional solution tech- in addition to possessing a vanishing orientation-average value: niques, thought straightforward in principle, would entail much tedious and unnecessary (8a2)-’J lhd3rfJ = 0. [3.21] algebra in present circumstances. More sim60 ply, by invoking the ansatz employed in Part II, the search for the vector field B may in- [That the Taylor dispersivity too vanishes for stead be replaced by the search for a com- isotropic bodies, i.e., where A& = 0 (j = 1, parable complex-valued dyadic field b(4), 2, 3), constitutes the clue that originally sugrelated to B via the linear transformation gested the form [3.16] for b, involving I’& rather than M.] B=h*F. [3.14] In Part II is derived the orientation-space Since F is a constant space-fixed vector, identity it follows upon substitution into [3.13] that Aelel = 2[-e,e,(d, + d3) the b-field satisfies the differential equation (A - iu)h = -M.

[3.15]

This dyadic equation possesses a much simpler geometric structure than does the original vector equation [3.13] from which it derives. In particular, by virtue of having removed the space-fixed vector F from consideration, the solution of this equation is most readily effected in body-fixed, rather than space-fixed, axes. Uniqueness of the B field leads to a comparable property of the b field as an immediate consequence of its definition [3.14]. In consequence of the appearance of the two dyadics M and d in [3.15] it will be assumed, subject to a posteriori verification, that b possesses the invariant representation _b = cqd.lh

+ a,lh + cql,

[3.16]

+ e2ezd3 + e,e3d2]

[3.22]

along with similar identities for the 2 and 3 directions. In view of [3. IS] and the representation de &i = e,e,d,i@, + e2e2d2ti2 + e,e,d,tij,

the above identity may be employed to evaluate the dyadics Ad . a and Al$ ultimately yielding Ad-h = -2d,&i [3.23] and Arin = 6d. hi - 12&l

- 2d:k.

[3.24]

Invariant scalars d and d,, appearing above are defined by the relations d= (1/3)tr d = (1/3)(d, + d2 + d3)

[3.25]

Journnl of Co/bid and Inlerface Science, Vol. 94. No. 2, August 1983

436

DILL

AND

and

valid for any traceless body-fixed dyadic A (cf. [3.20] and [3.21]), it follows that

dr, = (1/2)[(tr d)2 - tr (d*)] = t&d2 + d2d3 + dxd,,

[3.26]

with d* = da d. Substitution of the trial solution [3.16] along with [3.17] into the differential equation [3.15], followed by application of the above identities, serves to determine the values of the three scalars (cy,, (Ye, (Ye).In turn, this eventually yields IJ = A-‘(6d.

BRENNER

Iii - 2d:hlI + icth) + (io)-‘%I,

[3.27]

since I:1 = 3. Consequently,

(87+ f hlli?d3c#J= ll%2(io)-1. 40

+=2%symF

[3.29]

Of the pair of alternative formulas [2.7] and [2.12] for computing DC from knowledge of the B-field, the latter will be employed. Equations [ 1.31 and [3.7] combine to give U-U*

=M.F.

s,, Wiid34]-t

[3.33]

[3.28]

Having calculated the dyadic Sr, the vector field B is easily determined from [ 3.121 and [3.14] to be

+ const.

[3.32]

Inasmuch as this integral is purely imaginary, it does not contribute to the Taylor dispersivity. Hence, [3.31] reduces to

-[(8r*)-I

A = 12du - W* + i12 dw.

= 0,

&I

with A the complex scalar

. E exp(&)

(6d. ti - 2d:til)d3d

(8a2)-’ j-

[3.30]

Substitute the last two displayed equations into [2.12] and exchange the order of integration and time averaging. After time averaging with use of [ 1.61 and [2.8] there results

The tetradic integral appearing in the latter expression depends upon both the geometry of the body and the scalar w, but not upon any (vector) directions fixed in space. As such, the integral is necessarily isotropic. A similar tetradic integral was evaluated in Part II. In particular, upon recognizing that b is symmetric and & traceless, the general result of Part II may be immediately employed to obtain (87~*)-’ j- B&d34

&J =

i 2 I iij&[-2&j& 1 , I

+

3(6idjr

+

36$j/JJ

X (1/30)tr nir.h.

Summations represented by the indices i, j, k, 1 are each over the integers 1,2, 3, while 6ij is the Kronecker delta. Space-fixed Carte- [(8r’)-’ s, BMd’+ti, L3.311 sian unit vectors (ir , i2, i3) appearing above are mutually orthogonal. Introduction of the last displayed equation in which the tilde denotes the complex coninto [3.33] yields jugate. Simplification in evaluating the preceding tetradic integral is achieved upon re+ 3lE.E] placing M in the integrand by ti + I%& In D” = (l/lS)[sym(@) view of the integral relation (cf. Part II) X tr(kl.% b). [3.34] iSC=2%symF

(8r2)-’

j- Ad34 = 0, 90

This relation may be expressed in an alternative form by utilizing the pair of identities

SEDIMENTING

(FF) = 2 sym(@)

[3.35a]

and

437

PARTICLES

(IFI’) = 2F.p = 21F[* derived from [ 1.51. In conjunction fact that the real part of [3.27] is

[3.35b] with the

~ b = (12dii - w2)(6cI - liil - 2d:ilI)

+ 12dw2n;r (124, - b.+* + (12&)2 ’

-

[3.36]

one thereby obtains EC = (1/.5)[(FF)

+ 31(IFi2)]

( 12d,, - w*)tr(M - d - &l) + 2&*tr#l. (12& - W2)2+ (12&)2

The latter expression constitutes the Taylor contribution to the dispersivity, corresponding to the single-harmonic force [ 1.51. It represents a canonical form in the sense that EC is expressed directly in terms of explicit operations to be performed upon the originally prescribed data, namely F(t), d, and M. Moreover, in the limit where w = 0 and F is time independent (so that (F) = F), this canonical form reduces directly to a comparable formula given in Part II for the special case of a steady force. In terms of the principal axes of the ellipsoid the invariant scalars appearing in [3.37] possess the Cartesian representations = d,#

+ d2&: + d,ti:

.

[3.37]

Corresponding to the functional form [3.40], the Taylor dispersivity [3.37] may now be written as 6” = (E + 3i)772y/5d.

[3.44]

Appearing herein is the dimensionless symmetric dyadic E of unit trace, defined in the plane of the force field by the expression [3.45]

E = (FF)/(IFl*).

Moreover, the dimensionless geometric/frequency parameter y is 12PK+ + G2K7

[email protected]@

fi)

=

5j2&12p

_

&2)2

+

(QG)2]

W61

[3.38] in which

and tr(fi.lin)

= f%?: + f@ + ti:.

[3.39]

Kf = tr[h.(ld

More suggestively, 6’ may be recast into the functional form

K- = tr[h.(ld-

(velocity)*/molecular

diffusivity

+ 6). 1\31], &.ti]

with the traceless dyadic d defined by

[3.40]

in order to emphasize the similarity of the present phenomenon with classical Taylor dispersion (8- 10) occurring in cylindrical tubes. In this context, define a characteristic translational velocity n, dimensionless frequency 5, and dimensionless geometric parameter p by the following equations: c = ((F12)1’2%,

[3.41]

; = w/d,

[3.42]

/kt= d&d2.

[3.43]

and

il = d - Iii

Nonnegativity of y is assured by the comparable property of K-, demonstrated in Appendix A. (Comparable proof of the nonnegativity of Kf is trivial in view of the positive-definite nature of the rotary diffusion dyadic d, and is thus omitted.) Positivity, rather than the weaker condition of nonnegativity, of K+ and K- arises for nonisotropic bodies, where G # 0. All geometrical and frequency asymmetry effects upon the Taylor dispersivity are embedded in the parameter y. Journal o/Colloid

and Inrerfnce Scrence, Vol. 94, No. 2, August 1983

438

DILL

AND

Having established in [3.8] and [3.44], respectively, the mean molecular diffusivity and Taylor dispersivity appropriate to the time-periodic force [ 1S], the total dispersivity is given by their sum, as in [2.5]. Explicitly,

-* D

E” y .

- + (E + 31) T = ID

3.2. General Time-Periodic

The Taylor dispersivity calculated in the preceding section corresponds to translation of the ellipsoidal particle under the influence of the single-harmonic homogeneous force field [ 1.51. Comparable results for each higherorder harmonic in a general time-periodic force field may be analogously derived via an appropriate extension of the preceding analysis. Already available from Part II is the Taylor dispersivity corresponding to a steady, time-independent force. The relevant superposition theorem of Dill and Brenner (4) thereby permits calculation of the total Taylor dispersion dyadic ensuing from the action of a general time-periodic force. Every such time-periodic force may be expressed as the sum n=O

[3.48]

wherein the vector-valued periodic function F,(t) is the nth time-harmonic of the force F. These are determined via the formulas Fn(O = J@)Fo = 2B F, exp(inwt)

(n = 0)

(n >, l),

[3.49]

with the vector Fourier coefficients defined by the relations Fn = U/&)(F) = (F exp(-in&))

(l-2= 0)

(n > 1).

Let 6: designate the Taylor dispersion dyadic resulting from the action of the force component F,. According to the superposition theorem the total Taylor dispersivity dyadic 5’ resulting from the force [3.48] is given by the sum

[3.47]

Force

F(t) = i? F,(t),

BRENNER

[3.50]

For n > 1 the Fourier coefficient F, will generally be complex valued. Observe that the pair of identities [3.35], in which F and F are replaced by F, and F,, respectively, remain valid, even for y1= 0. Journal of Co/hid and Inrerfice Scknce, Vol. 94, No. 2, August 1983

Here, the Taylor dispersivity arising from the individual force component F,(t) is @ = (E, + 31)&,/5d

(?2=0,1,2;.*).

[3.51b]

Characteristic velocity u,, dimensionless frequency/geometrical parameter Y,,, and dyadic E, are here given by (cf. Eqs. [3.41], [3.46], and [3.45], respectively) v,, = ( IF,)2)1/2M E ,/(2)&]~,

[3.52]

12@K+ + Qz;)~K[3.53] Yn = @d[( 12p - r&2)* + (12&I)*] ’ and3 En = (F&)/(IFnl*) z sym (F&JllFn12.

[3.54]

The total dispersivity for a particle subject to the general time-periodic force [3.48] is given by the sum of Eqs. [3.8] and [3.5la], as in [2.5]. If a certain harmonic component of the force is zero, say F,(t) = 0,the corresponding Taylor dispersion dyadic D”, will also be zero. This follows immediately from the preceding formulas. In particular, even though the dyadic E, is undefined for F, = 0 (and hence F, = 0), the quantity DhE,,, appearing in [3.5 lb] vanishes in such circumstances. Introduction of [3.48] into [3. l] gives U* = Fog= 3 In circumstances constant, E, may = (#pn), wherein circumstances the may be written in E, + 31 = 4@,~,)

,/(2)F,M

[3.55]

where lFnl is a time-independent be wCvtten in the alternative form E, F, = FJIF,] is a unit vector. In such dyadic coefficient appearing in [3.5 1b] the transversely isotropic form (2, 3), + 3(l

-

@,pn).

SEDIMENTING

for the mean velocity of the center of the ellipsoid, induced by the general time-periodic force. Observe that only the steady, time-independent, force component F0 contributes to the mean particle velocity U*. Stated alternatively, the mean settling velocity of the ellipsoid is governed entirely by the mean, i.e., time-average force (F). 4. EXAMPLE: GRAVITY-INDUCED IN A UNIFORMLY ROTATING

SETTLING FLUID

Consider a homogeneous ellipsoidal Brownian particle undergoing gravitational settling in a viscous fluid confined within a cylindrical container that rotates with steady angular velocity vector 0. Centrifugal forces as well as other inertial forces will be systematically neglected throughout the subsequent development, as will wall effects. The sole purpose for incorporating container rotation into the analysis is to arrange matters such that, on average, no net translational motion of the particle occurs during any complete cycle-at least in circumstances where the rotation axis is perpendicular to gravity. By thereby preventing any mean sedimentation, the Taylor dispersion phenomenon may be leisurely studied in isolation-without any complicating effects (cf. Part II) arising from net center of mass motion during the process. Though this constitutes our primary motivation vis-a-vis rotation, we shall nevertheless analyze the more general case where the axis of rotation of the container is arbitrarily inclined relative to the direction of gravity, thereby allowing the possibility of net sedimentation. (This more general situation inadvertently includes the trivial case where the rotation axis is parallel to gravity, in which case the rotation is without physical effect upon either the sedimentation or dispersion processes.) It will prove convenient to analyze the sedimentation and Taylor dispersion processes from the viewpoint of an observer rotating with the fluid. Indeed, it is only in this reference frame that the fundamental assumptions underlying Taylor dispersion theory

439

PARTICLES

apply-namely that the phenomenological coefficients in the flux expressions [ 1. lo] and [ 1.1 I] be independent of the global position X. Having neglected any inertial, centrifugal, or other secondary effects, the particle will therefore settle with the instantaneous translational velocity [ I.31 relative to an observer rotating with the steady angular velocity w of the fluid. In that expression the buoyancycorrected gravitational force appears to the rotating observer to be time periodic. This force is given by F(t) = F&h

t4.11

wherein $j = g/g (with g = lg]) is a unit vector along the direction of the gravitational field g and F = (mp - mflg

14.21

is the scalar gravity force exerted on the particle. Here, mp and mf are, respectively, the masses of the particle and displaced fluid. (Note in [ 1.31 that since the mobility dyadic M is locked into the particle, and since the particle rotates with the fluid in which it is suspended, M will appear to the rotating observer to be a time-independent parameter, at least insofar as the particle orientation remains fixed relative to such an observer.) Although the unit vector $j is a time-independent constant with respect to a spacefixed observer, relative to the rotating observer it will appear to be a time-periodic unit vector, g(t), sweeping over the surface of a right-circular cone centered about the axis of revolution, as illustrated in Fig. 1. The equation of this cone is w-g = const = cos 69

L4.31

say, with the polar angle 0 (0 =G0 < r) a timeindependent constant and 2 = w/w a timeindependent unit vector along the axis of rotation, with o = (w] the angular speed of rotation. Suppose that at some time t = to, say to = 0, the vector g(t) possesses the value &to) Journal of Co/laid and Interface Science. Vol. 94, No. 2, August 1983

440

DILL

AND

BRENNER (b(t) = wt.

t4.61

From [4.1] to [4.6] the gravity force exerted on the ellipsoid is F(t) = F[&G + (I - ;;)

cos wt

+ 6-G sin wt] -&.

[4.7]

Equation [3.50] may be utilized to derive the Fourier coefficients for the present case, namely Eo = (F/V2)iZ

* go,

t4.81

& = (F/2)[(1-

~26,)- it-G]-&,

r4.91

and

CD 6

E, = 0

FIG. 1. Rotation of the space-fixed gravity vector g = kg as viewed by an observer fixed in the rotating fluid reference frame. The spherical polar angle 0 = cos-’ D * jj represents the half apex angle of the rightcircular cone over whose surface the gravity vector sweeps.

= &,, say. At all other times t, g may then be obtained from the formula iat) = R(t) * &I 3

14.41

Evaluation of the Taylor dispersivity [3.5 I] relative to the rotating reference frame fixed in the fluid requires determination of the scalars a, and dyadics E,. These may be calculated from [ 3.521 and [ 3.541 in conjunction with the identity (cf. [4.3] in the form ii-go = cos 8) 1 I-&i,=--sin2 0 x ([(I - ix+&#

wherein R(t) = &iG + (I - is) cos f#J(t)

+ c - G sin 4(t)

[4. lo]

(n 2 2).

+ (G x j&o>‘}. [4.1 l]

This yields [4.5]

is the rotation dyadic (3, 7). Here c = -I X I is the unit alternating isotropic triadic, whose Cartesian tensor representation is the permutation symbol Etik. (Observe that the algebraic sign of the last term in [4.5] is opposite to that which normally appears in the expression for the rotation dyadic. This is a consequence of the fact that the angular velocity vector, -w, with which the gravity vector in [4.4] appears to rotate relative to the fluid-fixed observer, is opposite in direction to the angular velocity w with which the fluid rotates relative to the space-fixed observer, for whom g is a constant.] Appearing in [4.5] is the azimuthal angle

u. = FM cos 0 = (G - F(%,

[4.12]

U, = FM sin 0 = ](I - Gcj) +FIG,

[4.13]

and u, = 0

(n 2 2),

[4.14]

for the characteristic speeds, in addition to E,, = t&,

[4.15]

E1 = (l/2)(1 - &),

[4.16]

and ??;E, = 0

for the unit dyadics.

(n > 2)

[4.17]

SEDIMENTING

Introduction of the preceding into [3.5 l] yields the following expression for the Taylor dispersion dyadic of an ellipsoidal particle subject to the periodic force [4.7]: 6” zz $

+ EC1,

cessful in finding a comparable proof for more general, triaxial particles.) Its maximum value, attained at w = 0, is

[4.18]

in which

whereas its minimum w- cc.is

6‘ = [4&i; + 3(1 - G)] x (FM cos 8)2y,/5d

441

PARTICLES

[4.19]

value, attained

as L5.31

In rationalizing these results it should be understood that the relevant physical parameter is the dimensionless frequency cj = w/ @ = [3;; + (7/2)(1 - ;;)] 2. For small frequencies, where a < 1, the X (FM sin 0)‘y,/5d. [4.20] particle can-as a consequence of its rotary Brownian motion-sample all possible oriAs follows from [2.5] the total dispersion entations and, hence, all possible instantadyadic is thus neous velocities [ 1.31 many times during each cycle. It is this fact that results in the jj* = $4 + @ + DC1 2 [4.2 l] relatively large dispersion at small frequenwith EM given by [3.8]. cies. Conversely, at large frequencies, where To complete this example, the mean set- a % 1, during each cycle the particle is able tling velocity, obtained by substituting [4.8] to sample only those orientations lying in into [3.55], is close proximity to its initial orientation at the outset of the cycle (due to relative smallness U* = ijF%i cos 8. [4.22] of the rotary Brownian motion). ConseAlternatively, in canonical form, in terms of quently, the particle fails to achieve a very the originally prescribed data, broad distribution of settling velocity vectors [ 1.31 and hence does not display much disU* = ~2;.FM. [4.23] persion. Judged from the criterion of accuracy of This shows that, in general, net particle mothe physics entering into its computation, the tion can occur only along the axis of rotation. large frequency results must be regarded cauConsequences of the formulas derived in tiously-indeed skeptically-since the quathis section are elaborated upon in the next sistatic formula [ 1.31 is based upon the posection. tentially inconsistent assumption that the 5. DISCUSSION Reynolds number ua2/u is small compared with unity (a = characteristic linear particle 5.1. Frequency Dependence dimension, v = kinematic viscosity). Furof the Dispersivity thermore, at high frequencies, explicit recDefine the scalar nth-order Taylor disper- ognition would need be given not only to sivity coefficient (cf. [3.5 lb]) as fluid inertia, but to translational and rotational particle inertia as well. In this con0: = U&,/52. L5.11 nection it should be noted that nonlinear effects upon roAppendix B furnishes a demonstration that particle moment-of-inertia for n > 0 this scalar is a monotone decreasing tary Brownian motion have recently been function of the frequency o, at least for bod- reviewed by McConnell (11). Fluid-mechanies of revolution. (We have not been suc- ical inertia effects upon translational Brownand

Journal oJColloid and Inter/ace Science, Vol. 94, No. 2, August 1983

442

DILL

AND

ian motion have been studied by Chow and Her-mans ( 12, 13) and Hermans ( 14), among others. Finally, in the opposite, quasistatic limit o/d 4 1, we recognize the possibility of employing Parseval’s Fourier series identity, as in related time-periodic developments (4, 15), to relate the time-average dispersion dyadic b* to comparable steady-state dispersion phenomena (3)-assumed to prevail instantaneously at each stage of the quasistatic process. 5.2. Ellipsoid of Revolution The dispersion dyadic adopts a particularly simple form in the case of a spheroidor, more generally, a body of revolution possessing fore-aft symmetry. For the sake of definiteness, suppose that the body is one of revolution about the e3 axis, so that M, = M2 = ML, M3 = y,, d, = d2 = dL, and d3 = d,,. In this notation, Eq. [3.53] ultimately reduces to the remarkably simple form

in which

’

X = (M,, - MJIM

is a dimensionless parameter measuring the departure of the spheroid from the spherical shape, and dL = d, is the so-called rotary diffusion coefficient. Here z and d are now given respectively by the appropriate formulas specialized from [3.5] and [3.25] for an axisymmetric body. The preceding formula for -y,, can be laboriously derived either by formal reduction of [3.53] for bodies of revolution, or more simply (and elegantly), by first noting from the nth-order generalization of equations [3.34] and [3.44] (cf. [3.51]) that -rn may generally be regarded as deriving from a “complex dispersivity” via the formula 2 Yn=s.W

tr(6h-cl.fi + in&l-h) [ 12dir - n2w2 + i 12 dno

For bodies of revolution the latter reduces to the form %I=~

x2 d,a2 1 0 [ 1 + (ino/6dl)

1

by virtue of a common factor appearing in both numerator and denominator of the general triaxial expression. Here, X is given as above, or, equivalently, by the expression X = -3tiJM. Unfortunately, a similar simplification does not obtain for general triaxial ellipsoids. Among other things, the first displayed equation of this subsection explicitly confirms the inequalities [B.5] and [B.6] set forth in Appendix B. This same equation, in conjunction with [3.51] and [3.54], yields the following very simple expression for the convective contribution to the Taylor dispersion dyadic: EC = $j

dily,

- Ml12 5 (E, + 31) n=O

x (181’)1 + @zWTD)2 ’

W)~*@/&) Yn = 36 + (nw/dJ*

BRENNER

1

Journal of Colloid and Inrerjace Scmxe, Vol. 94, No. 2. August 1983

*

in which 7D = (6dJ’ is the rotational diffusion relaxation time ( 11, 16). (Observe that the spheroid dispersivity is independent of d,,, as it must be, since rotation about the symmetry axis constitutes a “dead” degree of freedom; that is, d,, cannot contribute to the physics of the dispersion process.) In the limit where w - 0 the preceding reduces to the steady-state result of Brenner (2, 3) for the sedimentation of a spheroid in a timeindependent field of force. Finally, having utilized only the real portion of 7n in the calculations of this subsection, by extension the possibility must be recognized of assigning physical significance to its imaginary portion too, thereby introducing the concept of a “complex dispersivity.” The latter may be regarded as existing on a par with other complex phenomenological transport coefficients, namely the complex viscosity, complex permittivity, etc.

SEDIMENTING

5.3. Sedimentationand Dispersion in a Rotating Fluid It may appear surprising to observe from Eq. [4.22] that net particle motion occurs only in a direction parallel to the axis of rotation, since the external force driving this particle motion lies parallel to gravity. And it is natural to expect the latter to provide the direction of net motion. Inspection of Fig. 1, however, makes it clear via symmetry considerations that, on average, only that component of the 2 vector projected along the CZaxis can result in net particle motion; hence, the appearance of this projection, e. & = cos 0, in the expression [4.22] for the time-average mean velocity. (See also the example discussed in connection with Fig. 4 for the case of a nonuniform rotation.) This same symmetry argument explains why the Taylor dispersivity embodied in Eqs. [4.18] to [4.20] is transversely isotropic with respect to the axis of rotation ii of the fluid. Though Section 4 emphasized description of the local transport processes with respect to the rotating observer, owing to the timeaverage nature of the mean coefficients a* and DC the latter coefficients pertain equally to experiments performed by a space-fixed observer. (In this context, note that the vectors Tj and g are bothconstant relative to such an observer.) Two limiting cases arise according to whether the axis 2 of the rotation w is colinear with, or perpendicular to, the direction g of gravity. In the former case 6’ = 0 or 7r, in either of which circumstances the mean settling velocity [4.22] adopts the expected form U* = gFM. 15.41 So too does the Taylor dispersivity [4.18] which, since b? = 0, adopts the expected form ii” = [4&$ + 3(1 - j$~)](F%&,/5d

[5.5]

appropriate to gravitational Taylor dispersion in a nonrotating fluid (cf. Part II). Thus, as was to be anticipated on physical grounds,

443

PARTICLES

container rotation is wholly devoid of observable consequences when the rotation axis is either parallel or antiparallel to gravity. Of greater interest in applications is the situation where the rotation axis is perpendicular to gravity, in which case 2 * 2 = 0, corresponding to 13= 7r/2. In this event [4.22] becomes c* = 0 > L5.61 whereas [4.18] to [4.20] yields 6” = 6” 1 7

L5.71

wherein @ is given by [4.20] with sin B = 1. Equation [5.6] shows that no net settling of the particle occurs in such circumstances. This permits the dispersion phenomena to be studied in a leisurely fashion, in contrast with the conventional nonrotating gravitational sedimentation-dispersion case studied in Part II, where the particles tend to settle rapidly to the container bottom. The ability to offset normal gravity settling via container rotation constitutes a sufficiently novel effect to warrant separate examination of the phenomenon, independently of its possible utility in simplifying dispersion experiments. To examine this “weightlessness” phenomenon in its simplest possible setting, consider the case of a nonBrownian spherical particle settling under the influence of gravity through an otherwise quiescent fluid undergoing a rigid body rotation. Upon neglecting inertial, centrifugal, and wall effects, the position vector X(t) of the center of the sphere relative to an origin lying along the axis of rotation obeys the differential equation

dX/dt = w x X + gFM,

L5.81

where 2 = 1/(67rpa) is the sphere mobility. Since this equation adopts the perspective of a space-fixed observer, the unit gravity vector g is here regarded as a time-independent constant. Solution of this vector equation is facilitated by the decomposition Journal ofCo//oid

and Inlerface Science, Vol. 94, No. 2, August 1983

444

DILL

w> = X,,(t) + X,(t),

AND

BRENNER

P-91

with x,, = 6ii.x

[5. lo]

XI = (I - &2)-X,

[5.1 l]

r

and

W

respectively, representing the axial and planar transverse position vectors of the sphere center. From [5.8] these obey the differential equations dX,,fdt = iZ.~FM

[5.12]

= w x XI + (I - &+$F%.

[5.13]

k

and dX,Jdt

This pair of uncoupled equations possess the respective solutions X,,(t) = &.(X0

+ gFMt)

[5.14]

and

X,(t)- xs = R+(t). [X,(O)

- X,].

[5.15]

Here, X0 = X(0) denotes the particle’s initial position, whereas X, = & X $FM/o

[5.16]

denotes the steady-state solution of [5.13]. Observe from the above equations that the particle settles parallel to the axis of rotation with the constant velocity G(i;. gF%), in agreement with [4.22]. Simultaneously it executes a time-periodic circular motion in the transverse plane about the stagnation point Xs.4 Of course in any real experiment the fluid is externally limited in extent, whence the consequences of this boundedness upon the solutions [5.14] and [5.15] must be examined. Suppose, as illustrated in Fig. 2, that the spherical particle is contained within a (rotating) circular cylinder of radius R,. Two conditions must then be satisfied if collision

FIG. 2. Cross-sectional view of the orbits of a nonneutrally buoyant sphere within a rotating tube of radius R,. The direction G of the rotation axis is out of the plane of the page. Pointing vertically downward, and lying perpendicular to t, is the unit gravity vector g. Inasmuch as the angular velocity o exceeds wO,the stagnation point Xs lies within the tube interior. Curves (a) and (b) represent the circular trajectories of the two spheres initially located at points A and B, respectively. Trajectory (a) is unacceptable since it intersects the wall.

with the wall is to be avoided. First, the stagnation point must lie within the tube interior; that is, [5.17] I&l < R,. From (5.16) this requires that w ’ WC,

[5.18]

wc = [sin BIF%f/R,

[5.19]

wherein

is the critical angular velocity at which the stagnation point lies upon the tube circumference. Second, the circular trajectories X,(t) - Xs must not intersect the wall. From [5.15] this is equivalent to the initial-position restriction,

IX,(O) - &I < Ro - &I.

[XV

Inasmuch as 4 In the simplest case, where the rotation takes place about a horizontal axis, this stagnation point occurs at the location where the upward, rigid-body fluid velocity wX (X = distance from centrifuge axis to sphere center) exactly balances the downward settling velocity MF of the sphere relative to the surrounding fluid. Journal o/Co/bid

and Interface Science, Vol. 94, No. 2, August 1983

lXsl = lsin SIF%?/w

[5.21]

a wider range of initial positions X,(O) is acceptable at higher rotational speeds than at lower ones. Obviously, the faster is the fluid

SEDIMENTING

rotation the smaller is the distance traversed by the particle in the transverse plane during any one cycle before it reverses direction. 5.4. Nonuniform

Rotation

Though

emphasis has been placed upon fluid rotation (about a horizontal axis) as a simple means of achieving a state of no net sedimentation, this zero net transport condition can also be attained alternatively by any one of an infinity of nonuniform rotations. It is possible that one of these may prove to be experimentally superior to uniform rotation. As a simple nonuniform rotation example, consider a container that is intermittently turned about a horizontal axis. Specifically, suppose (as illustrated in Fig. 3 for a spherical particle) that for half of some period, T, say, uniform

445

PARTICLES

the container is maintained at rest, at the end of which time (t = 7/2) it is instantaneously turned through 180”, following which it is once again maintained at rest for the remainder (7/2 < t < 7) of the period, at the end of which time (t = 7) it is again rotated through half a turn, etc., with the cycle endlessly repeated. Analysis of this nonuniform rotation state may be accomplished by recognizing that the time-periodic gravitational force here takes the form

F(t) = Fi:

(0 < t < r/2)

= -Fg

[5.22]

(7.12 < t < T),

which describes the state of affairs from the viewpoint of an observer fixed in the fluid. Here, F is given by [4.2]. Expand [5.22] into a Fourier series and identify the various harmonics of [3.48] as

F. = 0,

[5.23]

F, = 0

GfaVity

(n even)

= (4F$nr)

sin nwt

(n odd).

[5.24]

It is a consequence of [3.55] that u* = 0,

[5.25]

corresponding to zero net particle motion. Similarly, [2.5], [3.51] to [3.54], and [5.24] combine to yield

a Settle l-2-3-4-l

N2

R@p 0

Settle

V2

Tgp

is* = ggq for the transversely dyadic, in which

+ (I - &)DT isotropic

[5.26] dispersion

0

FIG. 3. Gravitational settling of a spherical particle in a horizontal cylinder which is intermittently and instantaneously rotated through 180” twice per cycle. The four different steps in the overall cycle are as follows. (Step 1 - 2) The particle settles in the quiescent fluid for a time period T/2. (Step 2 - 3) The fluid-filled cylinder is instantaneously rotated through 180”, thereby carrying the sphere center with it to point 3. (Step 3 - 4) The particle settles in the quiescent fluid for a time period T/2. (Step 4 - 1) The fluid-filled cylinder is instantaneously rotated through 180”, thereby carrying the sphere center with it to point 1. This completes the overall cycle, whose period is T. Note that no net particle motion occurs during this period.

jj;=~+%~

Y2m+l

5d m=,, (2m + 1)2

[5.27]

and &5+3”:

5

Y2m+l

5d m=O(2m + 1)2

[5.28]

are the respective components of the Taylor dispersivity parallel and perpendicular to the direction of gravity. Appearing in these formulas is the characteristic velocity U, = 4F%&rJ(2), Journal of

[5.29]

Co/hid and Interface Science, Vol. 94, No. 2, August 1983

446

DILL

AND

corresponding to the fundamental harmonic translational velocity. Referring to Fig. 4, intermittent cylinder rotation of a closely related type, performed in an inclined (i.e., nonhorizontal) tube, nicely illustrates the general fact that the direction of mean sedimentation is parallel to the tube axis, rather than parallel to gravity. This fact was discussed in connection with Eqs. [4.22]-[4.23], and in the paragraph following [5.16], but only for the less general case where the fluid rotation is uniform in time, rather than intermittent. 5.5. Alternating

Current Fields

Neutrally buoyant, electrically charged particles in a steady dc (direct current) external field will undergo electrophoresis, whereby they migrate along the field in the direction of diminishing electric potential. By contrast, in an ac (alternating current) field, such particles will suffer no net translational motion. Nonspherical particles will, how-

BRENNER

ever, because of the time-periodic nature of the ac field, experience a Taylor dispersion that differs from that which would obtain during the comparable dc experiment conducted at, say, the same mean field intensity. In turn, both dispersivities will be greater than the orientation-average mean molecular diffusivity. Experimental measurement of the degree of diffusional enhancement (beyond the molecular value) occurring via the action of such ac fields offers the possibility of establishing and quantifying the nonspherical geometry of the suspended corpuscles by relatively simple means-in particular via just such a “wet” characterization experiment. Of special experimental interest is the lack of any net solute transport, since for the simple-harmonic ac force field [ 1.51 it follows from [3.7] that U* = 0. The Taylor dispersivity arising from this ac field is readily calculated from [3.44]. In the high-frequency limit, G = o/z % 1, the latter equation shows that EC tends to zero inversely with G2. For the same reaGravity

Time-average particle motion /

Sedimept$n -P

stepJ

FIG. 4. Example illustrating the fact that the direction of mean particle settling motion is along the axis of rotation, rather than parallel to gravity. In the example depicted, the cylinder (and hence the fluid) is maintained at rest for a fixed time interval, thereby allowing the sphere center to undergo the sedimentation process A - B parallel to gravity. At the end of this period the fluid is (effectively) instantaneously rotated through an appropriate angle about the transverse cylinder axis, thereby carrying the particle from B to A. Following completion of this, the fluid is maintained at rest once again, allowing the gravitational settling process A - B to occur. This cycle is repeated ad infinitum. Clearly, net motion of the sphere center occurs only in a direction parallel to the cylinder axis, in a downward direction. Journal of CoNoid and Inrerfbce Science. Vol. 94, No. 2, August 1983

SEDIMENTING

sons outlined in the last paragraph of Section 5.1, this limiting high-frequency result must be regarded with some reservation. 5.6. Inertial Efects Fluid and particle inertial effects have been systematically neglected throughout the analysis. In addition to the conventional types of translational and rotational inertial effects discussed in Refs. ( 1 l-14) there exist others of a centrifugal nature. Among the latter are Coriolis effects (17) and classical centrifugal force effects (7), the latter tending to cause particle migration toward or away from the axis of rotation, according to whether the particle is more or less dense than the fluid. These particular centrifugal effects exist even for spherical particles. In the realm of nonspherical particles, additional centrifugal phenomena obtain-including the existence of preferred particle orientations (7, 18) arising from the couple engendered by the nonconstancy of the centrifugal field strength over the particle. Such orienting tendencies destroy the orientational randomness induced by the rotary Brownian motion, thereby complicating the analysis significantly. [Other types of noninertial orienting effects have heretofore also been neglected (2, 3), such as those arising from possible internal mass inhomogeneities or the lack of a center of symmetry in the case of particles devoid of fore-aft symmetry, e.g., hemispheres.] It lies beyond the scope of the present elementary analysis to consider the importance of inertial and/or centrifugal effects upon the feasibility of the proposed shapedetermination scheme, though an order-ofmagnitude analysis of errors arising from their neglect will obviously be necessary if the scheme is to be reduced to laboratory practice. This will be left for the future. APPENDIX OFK-

A: NONNEGATIVITY = tr[kl.(ldti).lin]

It will be demonstrated herein that the invariant scalar K- satisfies the inequality

447

PARTICLES

K- 2 0.

LA.11

To effect a proof, observe that introduction of the identity d = d - Id, along with [ 1.131, [3.18], and [3.25], into the definition of Kp yields Kp = (l/3)

;; d,(;: k=l

2ti;

- 3&f;).

[A.21

j=l

Use [3.20] and set k = 1 to eliminate &?, from the parenthetical term of [A.2], thereby obtaining ;: 2ti; j=l

- 3ti:

= (ii& - &y2.

Similar identities hold for k = 2 and 3. Thereby, [A.21 may be written as K- = (1/3)[d,(&

- &)2

+ d2(&

- tiJ2

+ dj(h& - &2)2].

[A.31

Since each rotary diffusion coefficient d, is necessarily positive, the scalar K- obviously satisfies the inequality [A.l]. Equality holds only for translationally isotropic bodies, i.e., those for which $l = 0. APPENDIX

B: MONOTONICDIMINUTIONOFTAYLOR DISPERSIVITYWITH INCREASING FREQUENCY

For isotropic bodies @I = 0), the Taylor dispersion dyadic 6’ is identically zero, whence the total dispersivity b* is equal to its purely molecular value EM. On the other hand, for nonisotropic bodies @I # 0), Eqs. [3.5 I] to [3.54] show that EC is functionally dependent upon the frequency w (through the 7n parameter). Since [3.53] reveals that the independent variables invariably appear in the expression for yn in the combination nwf2, one need only examine the dependence of y, on iZ to establish the general frequency dependence. Excluding the degenerate case where ti = 0, differentiate [ 3.531 (for n = 1) with respect to L2 to obtain Journal qf Colioid and Interface Science,

Vol. 94, No. 2, August 1983

448

DILL

AND

& d(G2) =--

(A, + A&c2 + A3G4) x$d[(l2/3 - 2)2 + (12&)2]2 * IB*l1

Here, (A,, AZ, A3) are constant scalars defined by the relations A, = 12p*[ 12K+ - /3(2K+ + K-)],

K+ = 2&$d,[

LB.31 and

A3 = K-.

P-41

for all frequencies. Thus y1 attains its maximum value at i3 = 0, decreasing monotonically thereafter with increasing 6. Given the relationship between yI and +rn, it also follows from these facts that the hierarchal sequence >

Y2

>

73

>

- * *

WI

obtains for a specified dimensionless frequency ~3> 0. Though it appears on physical grounds (see Section 5.1) that the inequality A, > 0 holds for general triaxial ellipsoids, we have been able to prove this contention only for ellipsoids of revolution. In the case of a body of revolution about, say, the e3 axis, one may write ti, =G2=&,

ti3=ti,,,

[B.7]

for the components of the traceless translational mobility dyadic I$ and dl=d2=dl,

d,=d,,,

P*81

for the comparable rotary diffusivity components. The traceless property of fi is here equivalent to the relation ^ ^ 2kf,+q,=o. P-91 and Inl~rface Science, Vol. 94, No. 2, August 1983

1 + 2(d,,/dJ,

K- = 6&:d,,

A2 = 24PK+,

Inasmuch as K+ > 0 and K- > 0 for nonisotropic bodies, the coefficients A2 and A3 are clearly positive. It will subsequently be demonstrated that A, > 0 (at least for a limited class of such bodies), whence [B. l] shows that dy,/d(G2) < 0 F-51

Journal o/Co/bid

Evaluate K+, K-, and ,L3for a body of revolution by substituting [B.7] and [B.8] into the appropriate defining expressions. Use of [B.9] to eliminate a,, from the resulting expressions eventually yields the respective values

[B.2]

and

Yl

BRENNER

P = 9[ 1 +

[B. lo] [B.ll]

2(d,,/ddl/P+ (4,/&>12- [B.12]

Introduction of these expressions into [B.2] gives rise to the formula A, = 72/3*@;dJl

+ 2(d,,/dl)13/ P

+ (d,,/&)12. LB.131

The latter, in conjunction with the positivity of the rotary diffusivities appearing therein, thereby leads to the requisite inequality Al > 0

[B. 141

for spheroidal particles. APPENDIX

a

A A P B I3 4

4, d3X d39 d d Di D D 6

C:

NOMENCLATURE

particle radius scalar function dyadic field complex-valued dyadic field time-periodic vector field harmonic of B rotational diffusion component along a principal axis orientation-average rotational diffusion coefficient second scalar invariant of d volume element in physical space volume element in orientation space rotational diffusion dyadic traceless rotational diffusion dyadic translational diffusion component along a principal axis mean isotropic translational diffusivity translational diffusion dyadic mean translational diffusion dyadic

SEDIMENTING

6”

u

EM E* ECn

Gl

convective contribution to 6* molecular contribution to 6* dispersion dyadic contribution of nth-order force harmonic to EC (e, , e2, e3) orthogonal body-fixed unit vectors of an orthotropic body E “force-derived” dyadic nth-order “force-derived” dyadic 6 f tensorial function F force magnitude, IFI F force vector total ah-order force vector harmonic F, unit vector parallel to F, RI complex-valued constant determining F single-harmonic force F F-n complex-valued constant determining nth-order harmonic force F, magnitude of gravity g gravitational acceleration vector g unit vector parallel to gravity i imaginary unit ;-rl, i2, i3) orthonormal space-fixed vectors I dyadic idemfactor orientation-specific rotational flux j density vector J orientation-specific translational flux density vector k Boltzmann constant K+, K- invariant scalars mass of displaced fluid mass of particle translational mobility component along a principal axis lllr mean isotropic translational mobility M mobility dyadic M mean translational mobility dyadic ril traceless translational mobility dyadic mth total moment of P WI Reynolds number NRe P probability density ‘5% real operator on a complex-valued tensor R rotation dyadic vm symmetric portion of dyadic t time tr trace operator T absolute temperature

449

PARTICLES

characteristic velocity magnitude nth-order characteristic velocity magnitude instantaneous orientation-specific translational particle velocity vector physical-space position vector unbounded domain comprising all of physical space

U

x xm

GreekLetters scalar coefficient dimensionless geometric parameter dimensionless geometric/frequency parameter Dirac delta function Kronecker delta unit alternating isotropic triadic permutation symbol Euler angle dimensionless mobility difference ratio for a body of revolution orientation-space scalar differential operator viscosity kinematic viscosity period rotational diffusion relaxation time for a body of revolution Euler angle orientation-space positional pseudovector finite domain comprising all of orientation space Euler angle frequency angular velocity vector of a centrifuge dimensionless frequency unit vector parallel to the vector w

SpecialSymbols

v, (

e

)

physical-space gradient operator orientation-space gradient operator time-average value (over symbols) complex conjugate

Journal o/Co/lord and Interface Science, Vol. 94, No. 2, August 1983

450

II I

DILL

AND

(over symbol) unit vector, traceless dyadic, or dimensionless quantity parallel perpendicular ACKNOWLEDGMENT

This research was funded by the United States Public Health Service (Grant 2POl HL 18208-06) at the University of Rochester. REFERENCES 1. 2. 3. 4.

Goren, S. L., J. Colloid IntefaceSci. 71,209 (1979). Brenner, H., .I. Colloid IntevfaceSci. 71, 189 (1979). Brenner, H., J. Colloid Interface Sci. 80,548 (198 1). Dill, L. H., and Brenner, H., PhysicoChem. Hydrodyn. 3, 267 (1982). 5. Brenner, H., J. Colloid Interface Sci. 23,407 (1967). 6. Dill, L. H., and Brenner, H., J. ColloidInterfaceSci. 85, 101 (1982). 7. Brenner, H., and Condiff, D. W., J. Colloidlnterface Sci. 41, 228 (1972).

Journal o/Colloid

and Interface Science, Vol. 94, No. 2, August 1983

BRENNER 8. Taylor, G. I., Proc. Roy. Sot. London Ser. A 219, 186 (1953). 9. Taylor, G. I., Proc. Roy. Sot. London Ser. A 255, 473 (1954). 10. Aris, R., Proc. Roy. Sot. London Ser. A 235, 61 (1956). 11. McConnell, J., “Rotational Brownian Motion and Dielectric Theory.” Academic Press. New York/ London, 1980. 12. Chow, T. S., and Hermans, J. J., J. Chem. Phys. 56, 3150 (1972). 13. Chow, T. S., and Hermans, J. J., Koninkl. Nederl. Akad. Wetenschappen-Amsterdam (Ser. B) 77, 18 (1974). 14. Hennans, J. J., J. Colloid Interface Sci. 71, 427 (1979). 15. Dill, L. H., and Brenner, H., PhysicoChem. Hydrodyn., 1983, in press. 16. Debye, P., “Polar Molecules.” Chem. Catalog Co. (Dover reprint), 1929. 17. Herron, I. H., Davis, S. H., and Bretherton, F. P., J. Fluid Mech. 68, 209 (1975). 18. Brenner, H., and Condiff, H., J. Colloid Interface Sci. 47, 199 (1974).