Brownian diffusion of a small particle in a suspension

Physwa II9A (1983) 307-316 North-Holland Pubhshmg Co

BROWNIAN DIFFUSION OF A SMALL PARTICLE IN A S U S P E N S I O N 11. HYDRODYNAMIC EFFECT IN A RANDOM FIXED BED

J-P CARTON Serv,ce de Phystque du Sohde et de R~sonance Magn~ttque, Centre d'Etudes Nucl~atres de Saclay, 91191 Gtf sur Yvette, C~dex, France

E DUBOIS-VIOLETTE Laboratmre de Phystque des Sohdes, Bdt 510, Unwerstt~ de Parts-Sud, Centre D'Orsay, 91405 Orsay, France

and J PROST Centre de Recherches Paul Pascal, 33405 Talence, France

Recetved 30 August 1982 The Browman&ffuslonof a small parUcle m a fired flowingthrough a random fixed bed of identical spheres is considered To lowest order m the volume fraction qb of the suspension, the &ffuslon constant is shown to be renormahzed m the &rectmn transverse to the velooty as Dl = D(I + 3(~/2)I/2) prowded the velocity is large enough

1. I n t r o d u c t i o n

In a previous paper1), hereafter referred to as I, we investigated the Browman dxffusion of a microscopic particle in a suspension of much larger particles without any motion relative to the fired, and which for slmphclty were assumed spherical and mono&sperse The overall effect was shown to be a reduction of the diffusion coefficient by a factor 1 - ~b/2, to lowest order m the volume fraction ~b. In the present paper, we consider the case of spheres moving with the same velocity V0 with respect to the fluid, the force exerted on each particle being random, or eqmvalently that of a fluid flowing through a random fixed bed This is different from the problem of a settling suspension, in which the particles move under the effect o f their weight (the force exerted on each particle is gwen) and their velocities are statistically &stributed due to their random positions and the hydrodynamic interaction For the sake o f convenience we choose a reference frame in which the spheres are fixed so that the fluid flows with velocity - V0 On the z direction) through a random fixed bed. The velocity field of the fluid is written as V ( r ) = v ( r ) - - V o where t, Is the hydrodynamic perturbation mduced by the presence of the spheres 0378-4371/83/0000-0000/$03.00 © 1983 North-Holland

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J - P CARTON et al

In the same spirit as I, we shall determine the behaviours for small concentration c and small sphere radius a o f the mean and mean-square displacements of the test particle. As seen for example in the well-known Stokes formula in the case o f an isolated sphere

Vo+-7-r)+

Vo-a- -r

r

the total velocity field v (r) IS o f order a, which allows for a perturbation expansion in v(r) The meaning o f this perturbation treatment regarding parameters c and a will be investigated below. Our starting point is the F o k k e r - P l a n c k equation satisfied by the retarded p r o p a g a t o r G(r, to, t, to), which gives the probability o f finding the test particle at (r, t) knowing the initial condition (to, to) c3

I

DV,2 +

V(r). V , ) G ( r , to, t, to) = 6(r - ro)6 (t - to)

(1)

together with the b o u n d a r y conditions G(r, to, t, to) = 0

for t < to,

(2)

and (VG) • hi{spheres}= 0.

(3)

F r o m the definition ( ( r ~ - r0~)n) = S d 3 r ( r ~ - ro~)"G(r, to, t, to), which assumes the particle to be at r 0 at time to, integrating by parts using (1) and (3), one derives the equations o f m o t i o n for the first and second m o m e n t s

-~(r -- ro) = - D ~t((r~ -- r0~)2) :

f

dS{spheres}(r)G(r,to, t, to) + (V(r)),

2DO(t)- 2D + 2((r~ -

f

(4)

- ro,)G(,, ,o, t, to)

ro~)V~(r))

(5)

One expects in the z direction, as the main effect, a driven m o t i o n due to velocity Vo whereas in the transverse direction such an effect IS ruled out by symmetry, therefore the quantities o f experimental interest are ( d / d t ) ( ( z - z o)) and ( d / d t ) ( ( x - x0)2) The time-Fourier transform G(r, ro, 09) will be expanded in powers o f v. Writing (1) as ( - l o 9 - DV 2 -- V0" V,)G(r, r0, co) = tS(r - r0) - v(r)" V,G(r, roog) leads to the following equation which involves the Green's funcUon Go of ( - - lo9 - - D V 2 -- Vo" V) #

G(r, ro, co) = Go(r, ro, co) - Id3r'Go(r, r', og)v(r')" VvG(r', ro, to) J

BROWNIAN DIFFUSION T H R O U G H A R A N D O M FIXED BED

309

Iterating n times it yields the perturbation expansion to order n m v; Go msthe non perturbated propagator. To second order it reduces to

G (troOP) = Go(rrotO) - f d3r 'Go(rr 'co )v (r ')" V r,Go(r 'r0o9) + ld3r" d3r"Go(rr'to)v(r') • VeGo(r'r"og)v(r")" VeGo(r"roco). d

(6)

2. Transverse direction Clearly, as shown in I m the case V0 = 0, the surface term in (5) leads to corrections of order q~. The bulk contnbutlon ((x - Xo) Vx(r)) wdl be found below of order ~bl/2 so that terms o f order ~b can be neglected. Similarly to the pure excluded volume case, the free propagator Go contams a bulk part and a reflected part due to the presence of the spheres The latter, as shown in I, is lnfimtely small with a and is only essential m surface integrals on the spheres where both parts contribute the same order. Therefore m volume integrals only the bulk propagator, namely in the Fourier transform (~0(k, ~o) =

1

-loJ + Dk2 + lVo. k '

(7)

wdl be retained Using (5), (6), (7) and averaging over configurations {R,} (see I) one obtains

- 1o9( ( x - x0) 2) = 2D 1 + 2 f d3r(x - Xo)vx(r)Go(r - to, o~) co

2 fd3r d3r'(x - xo)Go(r - r', og)v~(r)v(r')" VG0(r' - r0, 09).

(8)

From symmetry considerations g~ -- 0 and the second term of the RHS drops out. One is led to evaluate the correlation of the velocity field F~p(r, r') = v~(r)v~(r') Some care must be taken to avoid divergences that would result from simply taking for the v's the superposiUon of Stokeslet fields u. In fact, it has been shown 6) that the influence of a gwen sphere is screened out at large &stances because of the presence of the others surrounding To lowest order m concentration v(r) = ~ u(r -- R,). t

Inserting (9) m the definition of F gaves

(9)

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J-P

CARTOI~I et al

F~,(r, r') = ~, u~(r -- R,)vtj(r' )

(10)

t

Let us introduce conditional averages noted one particle is present at R, Then

IR,, 1 e , averages knowing that

r,,(r, r') = ~ f dsRp(R,)u,(r -- R,)vo(r')lR,, F,,(r

-

(11)

t" r') = c J d 3 R u , ( r -- R ) U , ( r " - R ) ,

-

v(r')lR

where U(r' - R ) = is the screened velocity field as computed for example by Saffman 3) and which amounts to resummlng one class of contributions arising from all the spheres The probability P ( R ) that one particle is located at R is simply the reverse of the total volume [2 The reader might object that taking expression (9) for v(r) is inconsistent with screening In fact, since Y~u is weighted by U which smooths out the singularities, expression (11) is not divergent and is expected to be valid to lowest order in c A rigorous derivation of (11) is given in the appendix where the required truncation procedure is made explicit Now in Fourier transform, (11) becomes f ,¢(k ) = cfi,(k ) O~(- k)

(12)

Eq (8) reads - lO~((x - x0)2) - 2Dlto

2 f d 3 r d3r'(x - xo)Go(r -- r', tn)Fx¢ (r -- r')

x ~Go(r' -- r0, co) Setting r - r o = p, r ' - - r 0 = p ' and integrating over all space (in so doing we neglect corrections of order ~b)

The latter integral is computed in Fourier space to give - ltO((x - Xo)2) = - 2DI + 2!(27t) -3 1"d 3 k f ~ ( - k ) t T o ( + k , (O 60 J where use has been made of

o~),

(13)

f~, = (2n)3O,6(k) 1

Thus, in the long time limit, the random velocity field due to the backflow of the spheres, renormahzes the diffusion constant in the transverse direction

BROWNIAN DIFFUSION THROUGH A RANDOM FIXED BED

DA_ = D +

x(-k)ao(-[-k,

d tZn)

O) -~- O -Jf-O I.

311

(14)

This relation Is reminiscent o f a K u b o f o r m u l a m which geometrical and thermal averages have been decoupled It has the same structure as the one obtained by Bedeaux et al 5) when velocity thermal fluctuatmns are taken into a c c o u n t in a fluid otherwise at rest, that type o f formulation relies on a r a n d o m gaussmn noise hypothes~s. However, m our derivation the only a s s u m p t m n is that a power series e x p a n s m n as a function o f the spheres radms a, is meaningful. Take for the screened and unscreened potentmls U and u the expressmns gwen in ref 3

~ ( k ) = 6naV%(t$~ -- k~k~/k2)/k 2, O~(k) = 6rmV%(fi~, - k~ka/kE)/(k2+ x 2) The 8~3 factor &fference with ref 3 comes from the Fourier transform definltmn. /~ --1 lS the screening length due to the h y d r o d y n a m i c interactmns between spheres = x - 1 = ( 6 n a c ) - 1/2 which gwes DI = (6rra V0)2c

d k

2 2

6

2 + 1Vokz)(k 2 + x 2))}

or after te&ous m t e g r a t m n

Dl = 9rm2cax -112/3 + 0t(1/2 - 0t) - 0t(1 - ct2) ln{(1 + ct)/0t }]

(15)

w,th ct = xD/I v0lTh~s c u m b e r s o m e expression slmphfies m the two limiting cases

1)

o/Ivol ,> 1. D, = 9, 2ac, - 2l Vo I =

1V0la

(16)

In m o s t favorable cases, with D ,-~ 10-5 cm2 sec -1, a - 1 0 - 3 c m , ~b - 0 1 (volume fraction o f the spheres), the hmlt corresponds to V0 ~ 7 x 10- 3 cm s e c - ~ and D1 '~ 0 4D, which should be experimentally measurable F o r m u l a (16) Is remarkable m that D1 does not depend on the volume fraction ~b, and depends m a n o n analytical way on V0 ll) h~D/I v0l ~ 1

DI

=

6rra2x -icD = (3/x/2)D~b 1/2

(17)

This result Is again remarkable m that D~ does not depend on V0, and in a non analytical way on ~ N o t e that the correcUon to D, a l t h o u g h sizeable (for ~b -- 0 1, Dt ~- 0.67D), stays b o u n d e d as V0 increases This can be interpretated with the following heuristic arguments The h y d r o d y n a m i c effect o f a sphere is confined

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J - P C A R T O N et al

within a domain of radius ¢ = x - 1 In this region the Browman diffusion IS altered by the presence of a velocity gradient. As shown for instance m ref 4 the mean square displacement is of the form (6r 2) oc 2Ot(1 + ½G2t2) for a simple shear flow where G = IV V[ The mean shear rate G m the sphere of influence is of order a Vo/~ 2 whereas the particle crosses this region during a time ¢/V0 so that each sphere contributes D (a2/¢ 4) V02(~/ V0)3 to the mean square displacement D u n n g a time t the number of spheres explored is given by c~2Vot, so that

( x 2) = 2Dt + const Da2~ct, which explains eq (17) Let us determine how this hmlt behavlour is reached with increasing time Going to real times with

Go(k, t) = O(t) exp - ( D k 2 + IVokz)t gives

d

- Xo)

= 20(0

( (D- ~+n ) 3 1 -I d 3 k DJ~ ( +~ k ) ko[ 1

e - (m2 +,Vok~)t])

= 20(t)(D + Dt + D2(t)) The long time behavlour of D2(t) is obtained through a saddle point approximation m the sum over k One finally finds d ( ( x - Xo)2) ~ 2D± - const, x D(a/Vot ) The vall&ty of our expansion m v is now examined. It will be shown in the following that higher order contributions do not bring predominant terms m c The general form of the expansion is. 2DI

- - I(D ( ( X -- X0)2) = -

(D

-I-

2(-- 1)" E n=2p+l

fd3rl-I

d3r,(x - x0)

x Vx(r)Go(r - rOy(r1)" VGo(rl - r2)'" v(r~)" VGo(r~ - ro) Going to Fourier space one generalizes eq. 13: - - I g O ( ( X - - X0)2 ) = -2Di -+ 01

E - 2i D., n (D

n =2p+

1,

B R O W N I A N D I F F U S I O N T H R O U G H A R A N D O M F I X E D BED

313

where Dn symbolically writes D n = ( - - 1 ) n+l f

d3k I-I(-~)3vxGov, "lZGo v'eGov x

(18)

n - 1 terms

(integration is to be u n d e r s t o o d as a generalized convolution product) The existence o f a typical screening length ~ implies that the n - p o i n t correlation function is expected to read

F(kl,

, kn) V(kl) ~--

"

v(kn) = c(aVo)ny(~kl,

, ~kn;~).

Setting

Go(k) = Volg(k, D/Vo) it is readily seen that, a p a r t f r o m prefactors, the integral in (18) is a function only o f ¢ and D/Vo, namely o f a length ~ and a dimensionless p a r a m e t e r D/~Vo Dn oc c (a V0)n +1 Vonfn(¢, D / ~ V0) Dimensional h o m o g e n e i t y requires

Dn ~ cVoa~+l¢3-nhn(h/¢Vo), so that contributions for n t> 3 are negligible before O 1

3. Longitudinal direction Eq. (4) reads 1

Zo)

~

O

fdSz(r)a(r,,o, ~) + (vz(,)) - Vo.

T o first order in v and c, using (6)

(vz(r)) --

V0 = vz -- Vo = - V0(1 - ~b) - ' ,

since volume conservation implies ( ~ term reads

Vo)(1 - q g ) +

cfd3RfdS,(r-R)G(r-R, ro-R,~o)=O because

fd3x'G(x,x',og)=(l/-io~)

and fdSz=O

¢# × 0 = - Vo T h e surface

314

J-P CARTON et al

and we have to lowest order m ~b _a ( z - z0) = - v0(1 + ¢ )

dt

In summary, the non trivial effect is expected in the transverse direction and results from the sampling of the inhomogeneous velocity field around the spheres Formula (17) provides a macroscopic experimental tool for evaluating the screening length and its concentration dependence

Acknowledgements We would like to thank J. Brady, P G de Gennes, E Guyon and P Pincus for fruitful discussions. This work has been done partly at L H M P. of E. S P. C. I in Paris and partly at the Centre de Recherche Paul Pascal m Bordeaux; we are Indebted to Professors G u y o n and Pacault for their hospltahty.

Appendix. Velocity correlation in the Debye approximation The Instantaneous velocity field v(r) for an incompressible fluid satisfies the Navler-Stokes equation"

tIV2v(r) = V P ( r ) - Fp(r)

(A 1)

and the condition V • v(r) = 0 The force density Fp represents the perturbation due to the suspension In a point particle approximation (see for instance Mou and Adelman 6) Fp(r) = 6zr~la ~ { V, - ~,(R3}3(r - R,).

(A 2)

!

V, ts the velocity of the tth particle of the suspension, i e In the case considered here V, = 110 and ~,(r) is the part of v(r) which IS regular at r = R, ~,(r) = v ( r ) -

T ( r - R,)" { V, - ~,(R,)},

where T is the Oseen tensor 3a/

1

\

T~(r) = ~ r~b~, + ~r~r, ) Defining the velooty correlation as a tensor

r(r, r') = v(r) ® v(r'),

BROWNIAN DIFFUSION THROUGH A RANDOM FIXED BED

315

eq. (A.I) and (A.2) readily yield the equation of motion of F ,Tv~,r(r,

r')

= v,P(r)

® v ( r ' ) - 6n~la ~ { V® - ~,(R,)} ®

v(r')6(r

- R,)

(A.3)

I

In the Debye-Bueche approxlmatmn, which amounts to truncating hierarchies of equations such as (A 3)6), one replaces { V0 - ~,(r)} ® v ( r ' ) 6 ( r - R,) ~- { V® - v(r)} ® v ( r ' ) 6 ( r - R,).

U(r, r')= f2v(r)6(r'-gl)

IS the mean velooty m r given that one suspension particle ~s present at r', f2 the total volume U obeys the equation of motion - R , ) - 67t)laf2 ~= { V o - g , ( R , ) } f ( r - R , ) f ( r ' - R1)

~IV2,U(r, r ' ) = f 2 V , P ( r ) 6 ( r '

i

In the same spirit one replaces g,(r)6(r - R ,) 6 ( r " - R1) ~- ~ ( r ) 6 ( r " - R I ) 3 ( r - R,) = f2 - 2 U ( r , r')

for 1 # 1

and ~ , ( r ) 6 ( r -- Rl)J(r' - R , ) = 6 ( r - r ' ) g l ( r ) 6 ( r - R l ) ~-- 6 ( r -- r ' ) v ( r ) 6 ( r -- Rl) = fl - '3 (r -- r')~

Then o V2,U(r, r ' ) = " ' V , P ( r ) 6 ( r ' tl

- R1) "~ ~2U(r, r') -~ 6 n a 6 ( r -- r ' ) ( ~ -- V0) - x2V0

1

V2,F(r, r ' ) = "-V,P(r) ® v ( r ' ) - x~(V0 - ~) ® U ( r ' , r),

(A 4)

where the screening length ~ -2 = x2 = 6rrac has been mtroduced In the reference frame of the fired g = 0, eqs. (A 4) are solved m Fourier transform to give I~(k ) = 6 1 r a s H

1

(k, Vo),

K2

f(k) = ~rt(k, Vo) @ ~(k)

(A.5)

w~th the help of the p r o j e c t o r / - / o n the set of fields of non zero dwergence II~(k, Vo) = ~ ( f ~

- ~k~k~)Vo~

Recall that, since U must vamsh at lnfimty, the constant term x2V0 m (A 4) cancels out with the mean pressure gradient With eq. (A5) the p r o o f o f (12) Is achieved

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J - P CARTON et al

References 1) 2) 3) 4) 5) 6)

J - P Carton and E Dubols-Violette and J Prost, Phys Lett 86A (1981) 407 G K Batchelor, J Flmd Mech 52 (1972) 245 P G Saffman, Studies m Appl Math L i l , no 2 (1973) 115 R T Folster and T G M Van de Ven, J Flmd Mech 96 0980) 105 D Bedeaux and P Mazur, Physlca 73 (1974) 431 and 75 (1974) 79 Chung Yuan Mou and S A Adelman, J Chem Phys 69 (1978) 3135