Hydrodynamic fluctuation forces

Physica 105A (1981) 395-416 @ North-Holland Publishing Co.

HYDRODYNAMIC FLUCTUATION FORCES

R.B. JONES

Department of Physics, Queen Mary College, Mile End Road, London El 4NS, England

Received 1 August 1980

We consider the fluctuating hydrodynamics of Landau and Lifshitz for a fluid confined by hard walls at finite distance. By considering the non-linearity of the stochastic fluid equations of motion, we show that there can be an inhomogeneous average stress set up throughout the fluid. The average stress corresponds to a force density on the fluid which is expressed in terms of the Green’s function for the fluid in the linearized theory. For simple geometries we obtain the average stress explicitly as a long range pressure field. The effect can be interpreted as a long range effective force acting between the fluid boundaries. In this sense it might have observable consequences in thin films or in suspensions of hard colloid particles. The effect is strongest in incompressible fluids. It is greatly weakened by compressibility but relaxation of the fluid viscosity prevents the effect vanishing.

1. Introduction

Macroscopic hydrodynamics describes the behaviour of long wavelength low frequency modes in a fluid’). Because these slow modes are coupled microscopically to fast degrees of freedom we expect to see fluctuation effects in the macroscopic hydrodynamic description. Fluctuations may be included in the macroscopic description by use of a Fokker-Planck formalism as was done by Green’) or by the addition of a stochastic Langevin-like forcing term to the macroscopic fluid equations of motion as was done by Landau and Lifshitz’). One interesting application of this fluctuating hydrodynamics is the derivation of the Langevin equation for a Brownian particle immersed in a fluid. This idea, which was introduced by Zwanzig’) and developed further by Fox and Uhlenbeck4), describes the motion of the Brownian particle as being driven by the boundary condition force exerted by the fluctuating fluid. This picture leads to a generalized Langevin equation of motion for the Brownian particle in which memory effects due to the fluid inertia are incorporated5T6). This description has been further extended to include cdmpressibility’), rotational motion’), Brownian motion in a fluid near its critical point?, and Brownian motion of a polymer”‘).

All the above mentioned applications use the linearized equations of fluctuating hydrodynamics. In the present article we argue that the non-

395

396 R.B. JONES

linearity of the fluid equations of motion can lead to an additional effect. Specifically we show how non-linear terms in the momentum equation can lead to a static, repulsive long-range force between solid particles or boundaries immersed in a fluctuating fluid. In a sense this force is a wall effect, arising from the interplay between the fluid boundary conditions at a wall and the non-linearity of the stochastic Navier-Stokes equations of motion. This long-range static force has certain properties that are analogous to those of the Van der Waals force between dielectric materials which arises from electromagnetic fluctuations”).

The Van der Waals force arises from microscopic quantum mechanical fluctuations and there have long existed microscopic theories of such dispersion forces’*). In 1955, however, Lifshitz introduced an alternative macroscopic description”). The macroscopic approach of Lifshitz has certain advantages of simplicity as compared with microscopic theories”*‘3~‘4). One advantage is that the microscopic calculation of the interaction of the electromagnetic field with the individual atoms is simply lumped into macroscopic susceptibilities which give the fluctuation spectra of certain stochastic forces which drive the macroscopic Maxwell equations. A second advantage of the Lifshitz approach is that macroscopic geometric effects of shape are readily taken into account through an appropriate Green’s function of the Maxwell equations.

In the discussion which follows of fluctuating hydrodynamics we will work in the spirit of the Lifshitz approach to dispersion forces. This macroscopic point of view will enable us, firstly, to identify which macroscopic fluid properties are relevant, and secondly, to treat simple wall geometries explicitly. Although the predicted hydrodynamic fluctuation force is weak compared to an unretarded Van der Waals force, it may still have an observable effect over certain scales of distance in colloid suspensions or in thin fluid films.

In section 2 we introduce a simple perturbative method to take account of the non-linearity of the stochastic fluid equations of motion. In lowest order beyond the linearized theory we find that the fluctuations produce on average a non-homogeneous static force density acting on the fluid. This force density corresponds to a non-vanishing average stress throughout the fluid. We show how the force density may be expressed in terms of the Green’s function of the linearized equations of motion. In section 3 we find explicitly this Green’s function for the simplest geometry of a fluid bounded by a single plane hard wall. In section 4 we consider how the fluctuation force density and its corresponding pressure field depend on fluid compressibility and viscous relaxation effects. In section 5 we treat approximately the geometry of a single hard sphere immersed in an incompressible fluid. In section 6 we

HYDRODYNAMIC FLUCTUATION FORCES 391

summarize conclusions and indicate some shortcomings of the macroscopic approach.

2. Solution of the stochastic equations of motion

We consider the fluctuating hydrodynamics of a compressible Newtonian fluid, taking into account the mass and momentum equations but neglecting for simplicity heat transfer. The Landau-Lifshitz’) stochastic equations can be written as

TV% + (TJ/3 + &JV(V . v) - vp - p[&J/C% + (v * V)v] = -v * s, (2.1)

ap/at + v * (pv) = 0, (2.2)

where v(r, t), p(r, t), p(r, t) are velocity, pressure and density, respectively, 77 and 5 are shear and bulk viscosity coefficients, and S(r, t) is the Landau- Lifshitz random stress tensor which drives the fluctuations. The above equations are in terms of r and t while later it will be more convenient to work in terms of r and o where t and w are related by Fourier transform (f(t) = J e-‘(“‘f(w) do). In frequency language the random stress S(r, w) has the following assumed Gaussian stochastic properties:

(2.3a)

(Sij(r, w)Sk*I(r’, W’)> = !$ [Re ~(w)(&k~jl+ &lajk)

- Re(2r)(o)/3 - J’(o))8ijSklJfi(r - r’)S(o - w’). (2.3b)

The pointed brackets denote an average over an equilibrium ensemble at absolute temperature T, ke is Boltzmann’s constant, and q(o), l(o) are assumed to be frequency dependent to take account of relaxation phenomena. We emphasize at this point that the fields which occur above are macroscopic fields and that we assume local equilibrium in the fluid. Thus position r is defined down to macroscopically small distances which are still large compared with atomic dimensions. Frequency w is defined up to some maximum frequency which is still much less than atomic collision frequencies. In various equations which appear later there are integrations over inverse distance and frequency; these integrals must be understood as cut off at appropriate short distance and high frequency values.

In the absence of the ralidom stress S eqs. (2.1) and (2.2) have the solution v = 0, p = PO, p = PO for a fluid at rest with unifrom pressure, density and temperature PO, PO and T. Since the driving term - V * S can be regarded as small (keT is macroscopically small) the resulting fluctuating fields v, p and p

398 R.B. JONES

are correspondingly small and one normally linearizes (2.1) and (2.2). This is done by setting u = u(l), p = p. + p(I), p = p. + p(I), where superscript 1 denotes a term of order S, and then keeping only first order terms in eqs. (2.1) and (2.2). For such small fluctuations p(‘) and p(l) are simply related by a sound velocity, p”’ = tip”‘.

In the present calculation we wish to take account of the non-linear nature of (2.1) (2.2). A simple minded way to do this is to imagine each field as being given by an infinite series in powers of the driving stress S. Thus we write

(2.4)

where j denotes a term of order j in S and u(O) = 0, p(O) = po, p(O) = po. The assumption of local equilibrium enables us to relate p and p by an equation of state p = p(p, T). If p = PO+ 6p and p = po+ 6p, we have a virial expansion 6p = &p + bg(Sp)*. . . . , where co’ = (dplap)~, b. = $(i?p/@~*)~ are an adiabatic sound velocity and virial coefficient, respectively. Combining the virial expansion with (2.4) gives

P (‘) = W”, (2Sa)

P (2) = &‘*’ + bo,,“+,“‘. (2Sb)

Inserting (2.4) and (2.5) in the equations of motion and equating corresponding powers of S gives in order 1 the usual linearized equations,

n v*u”’ + (n/3 + 5) V( v * u(l)) - VP(‘) - p&P/at = - v - s, (2.6a)

v - u(l) + (l/poC&!rp”‘/at = 0. (2.6b)

In second order we get corrections due to non-linearity. For the fields u(*), p’*’ we obtain the equations

nV*U$*)+(n/3+ c)Vi(V* U’*‘)~ViP’*‘-~O~U~*‘/~f

= poV’i(uj%j’)) + (l/cf)a(p’l’vl”)/at,

v * UC*) + (l/poco2)ap’*‘/a1

(2.7a)

= -(l/p,,&‘. (p”‘u”‘) + (bo/poc3a(p”‘p”‘)/at. (2.7b)

To obtain the right-hand side of (2.7a) we have used the first order equation (2.6b). The repeated subscript in (2.7a) is summed over.

One might suppose that the second order fields are far too small in comparison with the first order fields to have any possible observable effect. To see why this need not be so, recall that the Newtonian stress tensor u is

ai] = n(Viuj + Vjs) - (27/3 - [)( V * n)&j - p&j. (2.8)


We can think of u being expanded as in (2.4),

f7= 2 u(i),

I=0 (2.9)

where a”’ = - pal corresponds to the equilibrium uniform hydrostatic pressure. In the linear theory of eqs. (2.6a, b) the average values (TV) and (p”‘) of the first order fields vanish. Since u is linear in 2, and p, (a(‘)) vanishes as well. In the linear theory there is zero average first order stress in the fluctuating fluid. However, in eqs. (2.7a, b) the right-hand sides of the equations are bilinear in fluctuating quantities. In consequence (v’*‘), (p”‘) and hence (a(*)) need not vanish, giving a non-zero average stress in the fluid due entirely to the second order fields. To obtain a more concrete expression for the average stress (u(‘)) we use the following procedure. We first solve the linear eqs. (2.6a, b) according to the usual rules which tell us to apply normal stick boundary conditions at fluid boundaries while assuming a non-vanishing random S at all interior points of the fluid’j). The usual Green’s function method gives u(‘) and p(I) as functionals of the random stress S. Inserting u(l), p(l) into (2.7a, b) we can solve again for u(*), p’*’ which are bilinear in S. The averages (u(*)), (p’*‘) are then readily computed using the stochastic properties of S given in (2.3a, b).

To describe the Green’s function solution of the linear eqs. (2.6a, b) we work henceforth in a frequency representation so that these equations take the form

r)(w)V*u”‘(r, w) + (n(w)/3 + c(w))V(V * u(‘)(r, w))

- Vp”‘(r, w) + iwpou”‘(r, w) = B’*‘(r, w), (2.10a)

V * u(‘)(r, w) - io/(poc$p”‘(r, 0) = b”‘(r, w). (2. lob)

The fields B”‘(r, w) and b(‘)(r, w) represent inhomogeneous driving terms. To streamline notation we introduce a formal 4 component vector and tensor notation. The fields u and p are grouped together, and similarly B and b, to form 4-vectors V, and BP,

vex=(;), BP=(;). (2.11)

Greek indices take values 1, 2, 3, 4 where the values 1, 2, 3 refer to 3-vector field components and the value 4 refers to a scalar field. Latin indices i, j, k, etc. will take the values 1, 2, 3 only denoting ordinary 3 vector components. In this language we introduce a tensor differential operator D,,(r, o) defined in an obvious manner by

&=( [(rl(o)Vg iwp0)7 + (n(w)/3 + 5(0))VV] -v ) - Mp0c3 ’

(2.12)

400 R.B. JONES

where 7 is the unit 3-tensor, V is regarded as a column 3-vector, and d denotes its transpose as a row 3-vector. The eqs. (2.10a, b) can be written simply as

D&t-, o) V$‘( r, o) = B6f’( r, w). (2.13)

Next we introduce the Green’s function G associated with D. We denote it

by

G&r, r’; o) = G(r, r’; co) P(r, r’; w) R(r, r’; w) g(r, r’; 0) ’

(2.14)

where G is a 3-tensor, P and R are 3-vectors and g is a scalar. The Green’s function satisfies the differential equation

D&r, o)G&-, r’; w) = 6,,6(r - r’), (2.15)

and, as a function of r, Gi,(r, r’; o) vanishes on all boundaries and at infinity in accord with stick boundary conditions. Since the transport coefficients T(O), c(o) are defined as analytic functions of o with no singularities in the upper half complex w-plane, we have that G+(r, r’; o) is also analytic in the upper half w-plane by the usual causality argument. The function G&r, r’; o) obeys a number of identities which are listed in the appendix together with a brief derivation. The solution to eq. (2.13) takes the form

V!$(r, w) = I

G&r, r’; o)Bg)(r’, w) dr’. (2.16)

If we now go back to eqs. (2.6a, b) we can express the first order fields as linear functionals of the random stress S,

u$“(r, w) = I

V;Gik(r, r’; w)S,(r’, w) dr’, (2.17a)

p”‘(r, w) = I

V:Ri(r, r’; o)S,(r’, w) dr’, (2.17b)

where an integration by parts has been performed. In a similar manner we could formally solve (2.7a, b) for u@), p (2) However, since we are interested in .

the average values of these second order fields (uC2’>, (P’~‘), it is more convenient to first average the eqs. (2.7a, b) and then solve them. From the form of the bilinear inhomogeneous terms on the right-hand sides of (2.7a, b), which we denote by B’“(r, w) and b”‘(r, w), we see that we require averages of the form (v~“uj’)), (p”‘v?) and (p”‘p”‘). In time description these fields are at equal time so in frequency language the averages become averages of


(B$“(r, w)) = po J Vi(uf”(r, w + c.o’)v$‘)*(r, to’)) do’

-iw/ci I

(p”‘(r, o + w’)uj’)*(r, co’)) do’,

(bc2’(r, w)) = -(l/p&) 1 Vj(p”‘(r, w + o’)uj’)*(r, w’)) do’

(2.18a)

- (iobdpod) j (P”‘( r, w + u’)p(‘)*(r, w’)) do’. (2.18b)

In these equations we have used the reality of u(‘)(r, t) to introduce u(“*(r, w)

instead of u”‘(r, -co). If we now use (2.17a, b) together with the fluctuation spectrum of S given in (2.3a, b) we have

r

(u?(r, OJ + w’)uf’)*(r, 0’)) = -(k~7’/~)8(~) 1 dr’Gi’(r, r’; 0’)

x {Re q(o’)V”G$(r’, r; w’)

+ Re(q(w’)/3 + J(w’))V;V~G$(r’, r; o’)},

(2.19a)

(p”‘(r, w + w’)V, (‘)*(r, co’)) = -(kBT/n)tS(co) I

dr’R’(r, r’; o’)

x {Re q(o’)Vr2G$(r’, r; w’)

+ Re(v(w’)/3 + [(ti’))VPinG$(r’, r; to’)},

(2.19b)

(p”‘(r, w + td’)p”‘* (r, 0’)) = -(kBT/n)8(ti) j dr’R’(r, r’; w’)

X {Re q(W’)V’2P;F(r’, r; w’)

+ Re(q(W’)/3 + [(w’))VFi,J’t(r’, r; to’)}.

(2.19~)

As shown in the appendix, these expressions may be greatly simplified by use of bilinear Green’s function identities such as are given in eq. (A-8). As a result eqs. (2.19) become

(u?(r, w + to’)ui’)*(r, co’)) = -(kBT/p)tS(w) Re Gii(r, r; o’),

(p”‘(r, 0 + o’)uf’)* (r, to’)) = -(ikBT/m)S(o) Im Rj(r, r; o’),

(p”‘(r, 0 + o’)p(‘)* (r, 0’)) = -(keT/m)G(w) Re g(r, r; 0’).

(2.20a)

(2.20b)

(2.2Oc)

In eqs. (2.18) the expressions (2.20) are integrated over all frequencies w’. By

402 R.B. JONES

the reality of G&r, r’; t - t’), Re Gii(r, r; co’) is an even function of W’ while Im Rj(r, r; w’) is an odd function of 0’. By symmetric integration over frequency we have that Im Rj integrates to zero. If we use also that 08(o) = 0, we get finally from (2.18) the result

r

(Bj*‘(r, w)) = -(mkBT/T)S(ti) J do Re ViGij(r, r; o’), (2.21a)

(b’*‘(r, w)) = 0. (2.21b)

In eqs. (2.21) the only dependence on frequency o is through a delta function S(o). This tells us that the average value (B'*') is a static time independent driving term. It follows that the solution to the averaged second order equations of motion (2.7a, b) should also be time independent. To take account of this we write

(v(*)(r, 0)) = 6(o)V(r) 3 (p’*‘(r, w)> = G(wVVr), (2.22)

and we define

Hi(r) = (pokB’I’/n) 1 do’ Re ViGij(r, r; w’). (2.23)

The averaged eqs. (2.7a, b) can now be expressed as a pair of static equations

q0V*V(r) - VP(r) + H(r) = 0, (2.24a)

V - V(r) = 0 (2.24b)

where v. = q(0) is the low frequency shear viscosity_ In form these equations describe the static response of an incompressible fluid to an external force density H(r). Such an external force leads to a non-zero pressure distribution P(r) and possibly to a steady flow V(r) as well (if H(r) is not irrotational). In consequence, (a$]) is non-vanishing and a static average stress is present throughout the fluid.

The expression (2.23) for the force density H(r) appears highly singular since Gij(r, r; w’) has both its spatial arguments coincident. Similar apparent singularities occur in the Lifshitz approach to electromagnetic fluctuations where they are readily removed to give finite results”*‘3*‘4). In the present case also the apparent singularity is easily resolved. To see this, note that in (2.23) we have ViGij(r, r; co’) with the derivative Vi = alari acting on both arguments of Gij. In an infinite homogeneous fluid with all boundaries at infinity Gij is translationally invariant so that Gij(r, r; w’) = Gij(O, 0; o’) is an infinite but position independent quantity. As a result the derivative Vi acting on an infinite constant gives zero. Thus for a translation invariant fluid H(r)

vanishes identically and there is no effect. We conclude that boundaries must


be present to produce any effect at all. Suppose now that there are boundaries at finite distance. It is still true that the leading singularity in the Green’s function is the same as that of the translation invariant Green’s function or so-called fundamental solution”). The derivative Vi again removes this leading singularity while the wall correction terms to the fundamental solution are sufficiently less singular that Re ViGij(r, r; 0’) is finite and well behaved. We will show this explicitly for a simple case in the next section. To close this section we observe that, if we integrate the non-vanishing second order stress (a’*‘) over the fluid boundaries, we obtain what appear to be long range forces acting between the walls.

3. Single wall geometry

To make the discussion of section 2 more concrete we turn now to a simple special case. We assume the fluid to be semi-infinite with a single infinite wall taken to be a plane at x = 0 in a suitable coordinate system. The fluid lies in

the region x > 0. Note that in the directions of the y and z axes the fluid is still translationally invariant. For this geometry we will calculate the force density H(r) and the static fields V(r) and P(r). By symmetry each of these fields can depend only on the variable x. Since V - V(r) = 0, while V(r) itself vanishes at x = 0 by the boundary conditions, we have that V(r) is identically zero. The force density H(r) must then have the form H(r) = (H(x), 0,O) and is balanced by a pressure field P(x) obeying

aP(x)/dx = H(x). (3.1)

In the expression (2.23) for N(r) we see that the derivatives VZ, VJ give zero by translation invariance so that ViGir becomes simply aGrr/ax.

To find the Green’s function we use the translational symmetry in y and z, G&r, r’; w) = G&x, x’; y - y’; z - z’; o) to Fourier transform the y and z dependences. We write

G&r, r’; w) = I

eik ‘(r-r’)GaS(~, x’; k; o) dk2 dkJ, (3.2a)

6(r _ ,.l) = “;-j” 1 eik ‘(r--I’) dk2 dk3,

T (3.2b)

where k = (0, k2, k3) is a two-dimensional vector and k = jkl. The partial differential equations (2.15) now become ordinary differential equations with a parameter k.

To compute H(x) we need only the single component Grr(x, x’; k; w). However, the eq. (2.15) mixes together different components of the Green’s

404 R.B. JONES

function. Thus from the Q! = 1, y = j component of (2.15) we have

1 a Grj(x, X’; k; o)--- Rj(x, X’; k; O) = 6,j

6(x -x’) w(w) ax 7hJ)(WZ’

The (Y = n, y = j components where n = 2,3 give

G&x,x’; k; w)- ik 2 Rj(x, x’; k; o) = Snj ‘(’ - “)

7)1(w) n(w)(2#

The (Y = 4, y = j component gives

5 Glj(x, x’; k; O) f ikZGZi(x, x’; k; O) + ik3G3j(x, x’; k; W)

-sRj(X,X'; k;o)=O.

In these equations we have introduced the notation

I2 = k2 - L&n(w),

llr)l(w) = l/n(o)]1 - iw(n(o)/3 + 5(~))/(~0c31.

(3.3a)

(3.3b)

(3.3c)

(3.4a)

(3.4b)

By elimination among eqs. (3.3) we can obtain for Ri(x, x’; k; w) the equation

(-$- m2) 1 a ~(0) Rl(x, x’; k; WI= rlto)(2T)2 E _ 1 ax 6(x - ~‘1, (3.5)

where

E = iwrll(o)/(p0c% (3.6a)

m2 = k2- 02[cC!- iw(4r)(o)/3 + LJo))/pO]-‘. (3.6b)

Using (3.5) together with (3.3a) for j = 1 gives finally for Gr, a single fourth order equation,

(-$- 12)(-$- m")G,W'; k; w)

= T(W)mT) 1 p&$-mqs(x-x?. (3.7)

There are two sorts of boundary condition that must be used to solve (3.7). The first sort arises from the vanishing velocity condition at the fluid boundaries at x = 0, x = 00. This gives simply

G,,(O, x’; k; w) = 0, G,,(m, x’; k; w) = 0. (3.8)

The pressure field Rr(x, x’; k; w) vanishes at x = CC but need not vanish at the wall (x = 0). By eliminating R,(O, x’; k; o) from eqs. (3.3) one can obtain the


further condition at x,= 0,

[m*+ cl*] $ G,,(O, x’; k; co) = l -$ G,,(O, x’; k; 0). (3.9)

The second sort of boundary condition gives the behaviour of Gil at x = x’. This follows in the usual way by integrating the generalized function on the right-hand side of (3.7). One finds that Gil and its second derivative are continuous at x = x’ but the first and third derivatives have a discontinuity. In an obvious notation these conditions can be expressed as

Gu(x’+ 0, x’) - G,,(x’- 0, x’) = 0,

-& Gdx’+ 0, x7 - 6 GW- 0, x’) = q(o):2Ty (5) ~Gll(x'+O,~')-~G,,(~'-O,~')=o,

). (3.10)

The solution of (3.7) is straightforward now. For x <x’ we have

Gu(x, x’; k; w) = A eL + B e-‘” + C emx + D e-““,

while for x > x’ we have

(3.11)

Gll(x, x’; k; o) = E e-‘” + F edmx. (3.12)

In these equations 1 and m are the roots of (3.4a) and (3.6b) such that Re I > 0, Re m > 0. The boundary conditions (3.8), (3.9), (3.10) give six conditions to fix A, B, C, D, E, F. The solution for x <x’ can be written as

G,,(x, x’; k; o) = [2(27r)* iop,J(k* - Im)]-‘{- k*(k* - Im) e-‘(“-‘) + lm(k* _ lm) e-“‘(x’-x) + k*(k* + Im) e-‘(X’+X)

_ 2imk*le -(mx’+W + e -W+mx)]

+ Im(k* + Im) e-m(x’+x)}.

Now letting x = x’ and differentiating gives

(3.13)

5 Gii(x, x; k; o) = -[(2v)* iwpo(k* - fm)]-‘{k’(k* + lm) e-*’

- 2k*m(m f I) e-cr+m)x + m*(k* + Im) e-2mx}. (3.14)

Using this result in (2.23) and integrating over the angular dependence of the two-dimensional vector k = (0, k2, k,) gives

H(x)=2poksT [kdk 1 doRe$Gii(x,x;k;o). 0 --m

(3.15)

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Since the imaginary part of G,i is an odd function of frequency w we can, if we desire, remove the Re from (3.15) and simply integrate the analytic function aG,,/ax over all frequencies in a symmetric way to obtain H(x).

4. Effects of compressibility and relaxation

With the explicit result (3.14) for single wall geometry we can now try to evaluate H(x) and to relate its magnitude to fluid properties. In eq. (3.15) there are still two integrations to perform which are not trivial. Although the k

and w integrations run formally to k = 03, w = +CQ, in reality these integrals are cut off in our macroscopic treatment. These macroscopic cutoff lengths and frequencies must be kept in mind below when for mathematical convenience we will formally integrate to infinite upper limits.

The double integration in (3.15) can be performed analytically for an incompressible fluid without relaxation effects. For this special case we first put ci = m in (3.14) and assume the shear viscosity to be frequency independent, q(w) = no. For such a fluid m = k and (3.15) reduces to

m m

H(x) = -g 1 k3 e-2kx dk 1 do [’ -l”Lk’“]2, (4.1) 0 -m

where we have omitted to take the real part as explained earlier. The integrand of the frequency integral I(w, k) is analytic in the upper half w-plane and as o +CQ for fixed k behaves as l/o. Thus, by pushing the o contour from the real axis to a semicircle at infinity, we get

j dw [‘-i~;k”l’= j I(o,k)dw=qor/po.

-m -m

From this follows m

H(x) = -g I

k3e-2’“dk = _3kBT ’ 167r ;;4’

0

By (3.1) this force density gives rise to a static pressure field

P(x)=$g

(4.2)

(4.3)

(4.4)

We note that P(x) is a long range field which is quite small (order kBT) in most of the fluid. As one approaches the wall, x + 0, P(x) appears to diverge. This divergence is spurious and relates to the fact that we neglected the short


distance cutoff in our evaluation of the integrals above. In our macroscopic treatment we could not expect (4.4) to be correct for x smaller than perhaps 10 nm. The value of P(x) at such a cutoff distance is still small. In our evaluation above we made also an unjustified assumption about the behaviour of the fluid at infinite frequency when we pushed a contour to infinity. To understand the validity of (4.4) a little better, let us now examine the o integration in (4.2) by keeping w on the real axis and trying to identify contributions coming from different frequency ranges.

Remembering that all non-vanishing contributions to (4.2) come from the real part of I(o, k) we can show the following. First of all, the real part of I(w, k) is positive and of order x2 for w in the range 0 s lolpo/(qok2) s 1. For 1 < lolpo/(q-,k2) < 10, Re I(o, k) passes through zero and becomes negative. For 10 < lo/po/(qok2), Re I(w, k) stays negative and vanishes like Iw~-~‘* as loI+ m. In fact one can estimate that for 100 < lwlpd(~ok2) the contribution to the integral (4.2) is about -q&2&. These results show that the main contribution to (4.2) is from the low frequency range 0 G lolpd(~ok*) s 1. Higher frequency contributions serve partly to cancel and thus reduce the low frequency contribution. If we look at the k integration in (4.1) we note that the bulk of the contributions in k come from k values such that k = l/x. Combining these observations we conclude that at a distance x from the wall the chief contribution to (4.3) comes from frequencies in the range

0 s [WI c n0/(p0x2). (4.5)

To get a feel for magnitudes we note that for a low viscosity fluid such as water with a distance x = 1 pm, (4.5) becomes 0 6 IwJ G IO6 s-l, a frequency range over which a hydrodynamic description should certainly be valid.

Although the incompressible result (4.4) is a low frequency effect we expect that at higher frequencies the finite sound velocity and the compressibility of a real fluid should modify (4.4). To examine this we return to the expression (3.14) but assume c,! to be finite. We still neglect relaxation by assuming n(w) = no, 4’(w) = lo to be frequency independent. We can again evaluate (3.15) analytically by pushing the o-contour to infinity in the upper half plane. However, with ci finite we find from (3.4a) and (3.6b) that as o + ~0, Re I + UJ

and Re m + 03 and hence the frequency integrand vanishes exponentially at w = m giving the result H(x) = 0. This result is analogous to the result for the Van der Waals force that retardation effects due to finite light velocity exactly cancel out the non-retarded contributions leaving a much weaker effective force. We note from the argument just above that the incompressible limit ci+ 03 does not commute with the limit o + m’). The conclusion that H(x) = 0 for finite ci again depends on pushing a contour to infinite frequency in a theory with a physical frequency cutoff. For that reason let us look at the

408 R.B. JONES

integrand in (3.15) for o real to see which frequency range is cancelling out the incompressible effect at low o.

Once again the chief k contributions come from k = l/x. Putting k = l/x we get exactly the same contribution as in the incompressible case for those frequencies where m = k = l/x. This condition holds for o such that

w4 -=C [c”o + p;*w*(4,rlo/3 + &)1x-4. (4.6)

For a low viscosity fluid like water (4.6) simplifies to the statement that, for (WI < CO/X, compressibility has no effect. However, in the range

co/x < 10 I < p0&(47)ll/3 + Lo)-‘, (4.7)

we have m*= -o’/c$ so that the last term in (3.14) is growing and oscillating as o sin ox/co. Finally, for

10 I > poc%47)0/3 + 50)-l (4.8)

we have tn*= -iop0(4qo/3 + &)-‘, so that Re mx +=m and the integrand of (3.15) becomes exponentially small as o + m. For water with x = 1 pm we can say that the incompressible effects from the range 0 < (oj< lo6 s-’ are almost exactly cancelled by the growing and oscillating contributions due to compressibility in the range lo9 s-’ < [WI < lo’* s-‘. Since the shear viscosity of water does not relax”j) until o = lo’* SC’, we can expect no observable effect in water at such a value of x.

We have seen above that compressibility greatly reduces the magnitude of the field H(r) because of relatively high frequency effects (up to some cutoff frequency). However, in a real fluid there can also be relaxation effects in the viscosity coefficients over this same frequency range. Since some fluids can show shear relaxationr6) at w values as low as lo8 s-’ we must expect the relaxation to modify the compressibility effects in such fluids. To form some idea of what may happen in such a case let us assume that we have a finite c?, but that limit,,, 071(w) = constant, limit,,, OS(O) = constant. A simple model of such behaviour would be to assume single relaxation times TV, t2 where

It then follows that a growing and oscillating term as noted above (w sin ox/cl with c: = c!, + 4~o/(3potl) + &/(pot2)) persists at all frequencies as o + m. In consequence the w integration in (3.15) is now divergent and the field H(r) is strongly dependent on the high frequency cutoff. In the simple macroscopic theory used here we can say little more in detail about the effect of relaxation other than to note that the field H(r) is not zero but may even change sign as a result of the interplay between compressibility and relaxation.


We can summarize the discussion of this section in the following manner. Near a wall in a fluctuating fluid there is an inhomogeneous pressure field whose magnitude has as an upper bound the incompressible fluid result (4.4). Compressibility reduces the magnitude of this pressure field but relaxation effects prevent it from vanishing. The example treated above would suggest that for x > ~O/(pOcO) compressibility causes the field P(x) to be vanishingly small while for x < nO/(pocO) (but still macroscopic) one may have a result comparable to (4.4) but dependent in detail on the fluid relaxation properties at high frequencies.

5. Single sphere geometry

The calculation of section 3 extends straightforwardly to the case of two parallel hard walls with fluid between. However, the analytic expression which results is too cumbersome for analysis in any detail. A rather different geometry that is also of interest is the case of an infinite fluid surrounding a single hard sphere of radius R. According to section 2, we should calculate the exact Green’s function GaS(r, r’; w) as a function of o. This is not possible in analytic closed form. Nevertheless, if we assume incompressibility, neglect relaxation, and work far from the sphere (It-1 D R), then a method of reflections”) can be used to estimate GoS and to obtain the pressure field P(r)

approximately. We sketch the argument here and quote the final expression for the

pressure field at large distances. Let the origin of coordinates be placed at the centre of the sphere. We need the solution Gij(r, r’; o) of the linearized eqs. (2.15) subject to vanishing velocity boundary conditions on the surface of the sphere and at infinity. Suppose first that the sphere is not present. Then the translationally invariant Green’s function G$” for an infinite incompressible fluid is easily obtained”), it is

Gf"(r, r’; CO) = -6ij e-qlr-r’l

47Tqolr - r’l ’ (5.1)

where

q2 = - iop0/7)0, Req>O. (5.2)

G$)(r, r’; w) is the i component of the velocity field at r due to an in- stantaneous unit force acting in the negative j direction on the fluid at r’. If we place a hard sphere at the origin into this flow, an infinite set of force multipoles’**‘~ are induced on the sphere which act on the fluid to modify the response at r. The leading correction comes from the lowest force multipole

410 R.B. JONES

which is just the total force induced on the sphere. This force acts on the fluid and is propagated to r by the zero order Green’s function G(O).

A sphere placed at r = 0 in the velocity field G$“(r, r’; o) experiences a net force F(w) which is given by the FaxCn theorem for time dependent linearized flow 19*M) as

E:(w) = 47rno{[AoS(w) - (q’/lO)A~(o)]G$?(O, r’; O)

+ (l/10)A,P(o)V2G!?(0, r’; w)}, (5.3)

where the coefficients AZ(o) and AZ(w) are defined in ref. 20. At a distance Irl, large compared with the sphere radius (Irl+ R), the fluid will respond as though a point force of magnitude -F(w) acts on it at r = 0. This produces an additional velocity and pressure Au and Ap at point r determined by the linearized equations

qoV*Av(r, w) - VAp(r, o) - F(w)G(r) = -iopoAv(r, o),

V*Av =O.

Comparing with (2.19, one sees that

(5.4)

Ari(r 7 W) = G!?(r 0. o)F”(o). 7 I (5.5)

From (5.3) and (5.5) we have an approximate expression for the Green’s function Gii(r, r’; w) valid for r 9 R, r’ * R,

Gg(r, r’; co) = Gj?(r, r’; CO)

+ 4~no{[A$(w) - (q2/10)Ac(w)]G$?(r, 0; o)G$,,‘(O, r’; w>

+ (l/lO)Af(w)G$?(r, 0; o)V 2 (0) G, (0, r’;f.o)}+*... (5.6)

Remembering that G{F) contributes nothing to ViGii(r, r’, o) because of translation invariance, we calculate

ViGg(r, r; CO) = - *$f’(qr)-*jf(qr), (5.7a)

where

(eeq’ - 1) f(qr)=e-q’+3F+3 (qr)Z . (5.7b)

For a hard sphere with stick boundary conditions, the coefficients AZ(w), A;(o) are

(5.8a)


B(z) = (3/z)[z cash z - sinh z]. (5.8~)

In (5.7a, b) B’(z) means dB(z)/dz. On examining the o dependence above, we can once again evaluate the

frequency integral in (2.23) by moving the o contour into the upper half plane. The result is

m

I 9qoR3 r doViGij(r, r; O) = --

4p, +Y J -m

Solving (2.24) leads to a

3R3kBT 1 R(r) = 7% r

(5.9)

fluctuation pressure field

(5.10)

(5.11)

which is the leading contribution when r 9 R.

6. Discussion

We have argued that fluctuations in a fluid with boundaries can produce average static long range pressure and perhaps velocity fields in the fluid. From the simple examples above we see that for boundaries of moderate symmetry the velocity field V(r) certainly vanishes everywhere in the fluid as a consequence of the stick boundary conditions and the incompressibility (2.24b). We are unable to show that V(r) must vanish in general when there is no symmetry of the fluid boundaries, however. If there were situations in which V(r) # 0, the static flow would transport some heat and our initial neglect of the heat transfer equation would no longer be justified. For situations where there is only a pressure field this neglect of heat transfer is at least consistent.

The predicted pressure fields like (4.4) and (5.11), although quite small, could possibly have observable consequences since they lead to effective forces between the boundaries. To estimate the magnitude of such forces, consider a sphere of radius R with its centre at distance L (I_. >> R) from an infinite wall like in section 3. To get the exact pressure field near the sphere would require the impossible task of finding the Green’s function for sphere plus plane geometry. However, we might estimate the net force on the sphere by simply integrating the pressure field (4.4) due to the wall alone over the surface of the sphere. On doing this, we find that the sphere feels a repulsive

412 R.B. JONES

force from the wall of magnitude

F = bTR3 4L4 ’

L 9 R.

Similarly for two spheres of radii R, R’ at distance L apart there is a repulsive force estimated from (5.11) to be

F = 3R3Rt3kr,T L’ ’

L.% R, R’. (6.2)

The dependence on distance L of these forces is the same as for the corresponding Van der Waals forces. However, the magnitude of the hydrodynamic force is given by kaT which is much smaller than the strength of unretarded Van der Waals forces which depends on RwO with o. an atomic absorption frequency. This suggests that experimental observation of such forces in thin film experiments or as a weak repulsion between colloid particles is perhaps impossible because they would be masked completely by Van der Waals forces. One chance of evading this might be to work with a fluid in which the characteristic distance no/(poco) is so large that retardation weakens the dispersion forces.

Quite apart from experimental possibilities, the problem of how to deal with non-linearities in a stochastic macroscopic description is of continuing theoretical interest”). In the present work we have considered only the non-linearities in the macroscopic mass and momentum equations. There are certainly additional non-linearities in the energy equation as well as the possibility of non-Newtonian behaviour. Additional non-linear effects would also arise in a more microscopic mode-mode coupling**) calculation. Since we have shown that compressibility and relaxation in the fluid viscosity can alter considerably the magnitude of the effects discussed, it may be necessary to study this problem more microscopically to understand the influence of the high frequencies which have been simply cut off in the macroscopic treatment here. Also worth further study is the possibility that, for a fluid near its critical region, the enhanced fluctuations might make the hydrodynamic fluctuation forces much stronger than estimated here.

Appendix

Here we collect properties of the Green’s function G,,(r, r’; o) which are used in the calculations of section 2. In the notation introduced there, Gap satisfies the differential equation (2.14)

II&r, w)G&r, r’; to) = 6,,6(r - r’).


Because G&r, r’; o) is the Fourier transform of a real function we have, for real o, the property

GZp(r, r’; w) = G&r, r’; -0). (A. 1)

The transport coefficients n(o), c(w) also share this property so that we also have

D&(r, w) = Do(r, -w). (A.2)

The differential operator Dtis is a symmetric operator in the following sense. Let Us(r), W,(r) be suitably differentiable functions defined everywhere in the fluid but satisfying the boundary condition that Ui(r), WI(r) (i, j = 1,2,3) vanish at all boundaries and at infinity. Then we have the following identity

I U,(r)D,,(r, o)W,(r) dr = I W&P&, 0) U,(r) dr, V V

(A.3)

where the integration extends over the entire fluid volume. This identity follows by a simple integration by parts observing that all surface terms vanish because of the boundary conditions on Ui(r), W,(r).

Next we use (A.3) in the case that U,, Wp are taken to be the Green’s function itself. Since G has a singularity, the following equations have to be interpreted in the sense of generalized functionsz3). The classical method15) of justifying the following equations would be to cut out infinitesimal spheres about the singularities, then perform the integrations, then finally let the infinitesimal spheres shrink to zero volume. We have illustrated this method in a previous article for incompressible fluids”). The interested reader can follow the same method here if he wishes. To do so, he will need to know that the fundamental solution G?’ of (2.15) for an infinite fluid without boundaries is given by the following:

Gf)(r, r’; O) = -6ii e-qlr-r’l -q(r-r’l _ e-“lr-r’l

4nq )r - r’l 47rqq2)r - r’l I ’

R$O’(r, r’; w) = $ Vi (4~~~~r~,,),

g’O’(r, r’; 0) = $ [7)q2 - (4n/3 + &)V2](4($~‘~rJ,l).

We defined q in section 5 while c and v are defined by

c2 = cf - io(47)(w)/3 + I(wNlp0,

v2 = - w21c2, RevSO.

(A.4)

414 R.B. JONES

The short distance singularities of G&r, r’; 6.1) are the same as those of GFj(r, r’; w).

With the above proviso we sketch the proof of some identities. First, let us use (A.3) choosing U,,(r) = G&r, r”; w), W,(r) = G,(r, r’; w). The differential equation (2.15) for G,, then gives delta functions within the integrands of (A.3) and we obtain at once the reciprocity relation”)

GPp(r’, r”; co) = GpJr”, r’; 0). (A.3

In section 2 we used certain bilinear identities for GaB to pass from eqs. (2.19) to eqs. (2.20). Such identities arise in the following manner. From the complex conjugate of (2.15) we have

D&(r”, o)G&,(r”, r’; o) = &,S(r”- r’).

Multiply by G,(r, r”; w) to obtain

Gyu(r, r”; w)D$(r”, o)G&,(r”, r’; w) = G,(r, r”; w)6(r”- r’).

Integrating over the fluid volume gives

I G&r, r”; w)D&(r”, w)G&,(r”, r’; OJ) dr” = G,(r, r’; 0). (A.@ V

If one now uses the reciprocity relation (AS) on both sides of (A.6), takes the complex conjugate, uses the symmetry (A.3) of Da@, and finally relabels variables and indices, one also has

I Gyu(r, 1”; w)Qdr”, w)G&( r”, r’; o) dr” = G$,(r, r’; 0). 67) V

Adding (A.6) and (A.7) we have

I G,,(r, r”; to)[D,b(r”, co) + DS(r”, w)]G&(r”, r’; co) dr”

V

= GYP( r, r’; o) + G $,( r, r’; a). (A.8)

Let us illustrate how (A.8) is used to get (2.20b) from (2.19b). Take the y = 4, p = i component of (A.8) which is explicitly

Re Ri(r, r’; o) = I

R,(r, r”; w)[Re q(o)V”‘G,*i(r”, r’; w) V

+ Re(n(w)/3 + l(o))VgVqGz(r”, r’; o)] dr”


+ {g(r, r”; w)VyG,T(r”, r’; o) V

- R,(r, r”; w)V’LR?(r”, r’; o)} dr”.

From the Green’s function eq. (2.15), V’/Gg(r”, r’; o) = -[i&&J] R T(r”, r’, w), while by the reciprocity relation (AS), Rn(r, r”; w) = P,(r”, r ; w).

After integrating by parts the second integral above becomes

I {V,lP,(r”, r; 6.1) - [iw/(p,gi)]g(r”, r; w)}RT(r”, r’; w) dr”. V

But, by (2.15), the. bracket inside this last integral is just 6(r” - r). On performing this integration we get

I Rn(r, r”; o)[Re r)(~)V”*G~(r”, r’; o)

V

+ Re (n(w)/3 + 5(w))ViV9G,?(r”, r’; w)] dr

= Re Ri(r, r’; w) - RT(r, r’; OJ) = i Im Ri(r, r’; w).

Letting r’ = r now gives (2.20b). To obtain (2.20a) from (2.19a) one uses the y = i, p = i component of (A.@ in a similar manner.

References

1) MS. Green, J. Chem. Phys. 22 (1954) 398. 2) L. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1959). 3) R. Zwanzig, J. Res. Natl. Bur. Std. B 64 (1964) 143. 4) R.F. Fox and GE. Uhlenbeck, Phys. Fluids 13 (1970) 1893. 5) T.S. Chow and J.J. Hermaris, J. Chem. Phys. 56 (1972) 3150. 6) E.H. Hauge and A. Martin-Liif, J. Stat. Phys. 7 (1973) 259. 7) D. Bedeaux and P. Mazur, Physica 76 (1974) :247; 78 (1974) 505. 8) B.P. Hills, Physica 8oA (1975) 360. 9) P. Mazur and G. van der Zwan, Physica 92A l(l978) 483.

G. van der Zwan and P. Mazur, Physica 98A (1979) 169. 10) R.B. Jones, Physica lOOA (1980) 417.

(Eq. (A.8) of this reference should have a plus rather than a minus sign.) 11) J. Mahanty and B.W. Ninham, Dispersion Forces (Academic Press, London, 1976).

D. Langbein, Theory of Van der Waals Attraction, Ergebnisse der exacten Naturwissen- schaften 72 (Springer, Berlin, 1974).

12) R. Eisenschitz and F. London, Z. Phys. 60 (1930) 491. H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360.

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416 R.B. JONES

15) R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience. New York, 1953) p. 363.

16) T.A. Litovitz and C.M. Davis, Structural and Shear Relaxation in Liquids, in Physical Acoustics vol. HA, W.P. Mason, ed. (Academic Press, New York, 1%5).

17) B.U. Felderhof, Physics 89A (1977) 373. R.B. Jones, Physica 92A (1978) 545.

18) B.U. Felderhof, Physica 84A (1976) 557, 569. 19) P. Mazur and D. Bedeaux, Physica 76 (1974) 235. 20) R.B. Jones, Physica 95A (1979) 104. 21) N.G. van Kampen, Phys. Rep. 24C (1976) 171. 22) Y. Pomeau and P. Resibois, Phys. Rep. 19C (1975) 63. 23) I.M. Gelfand and G.E. Shilov, Generalized Functions Vol. 1 (trans. Saletan) (Academic

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Hydrodynamic fluctuation forces