Fluctuation-induced pressures in fluids in thermal nonequilibrium steady states

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by  T. R. Kirkpatrick

Fluctuation-induced pressures in fluids in thermal nonequilibrium steady states
T. R. Kirkpatrick,1,2,∗ J. M. Ortiz de Za´rate,3 and J. V. Sengers1
1Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20877, USA
2Department of Physics, University of Maryland, College Park, Maryland 20877, USA
3Departamento de F´ısica Aplicada I, Facultad de F´ısica, Universidad Complutense, 28040 Madrid, Spain
(Dated: February 18, 2014)
Correlations in fluids in nonequilibrium steady states are long ranged. Hence, finite-size effects
4
have important consequences in the nonequilibrium thermodynamics of fluids. One consequence
1
is that nonequilibrium temperature fluctuations induce nonequilibrium Casimir-like pressures pro-
0
portional to the square of the temperature gradient. Hence, fluctuations cause a breakdown of
2
the concept of local thermal equilibrium. Furthermore, transport coefficients become dependent
b on boundary conditions and on gravity. Thus nonequilibrium fluctuations affect some traditional
e concepts in nonequilibrium thermodynamics.
F

7 PACSnumbers: 65.40.De,05.20.Jj,05.70.Ln
1

]
h
c I. INTRODUCTION
e
m
Fluctuation induced forces are common in nature [1]. A prototype of such a force is the Casimir force between
-
t conducting plates due to quantum fluctuations of the electromagnetic (EM) field [2]. Its overall strength is set by
a Planck’s constant, ~, and the force per unit area, or pressure, is [3]
t
s
t. π2 ~c
a pEM =−240L4, (1.1)
m
- where L is the distance between the plates, and c the speed of light. Equation (1.1) is valid for distances larger
d than a microscopic length L which depends on the actual metal of the conducting plates. The minus sign indicates
0
n
that the Casimir force is an attractive force for two identical conducting plates. More generally, the electromagnetic
o
Casimirforcemaybeattractiveorrepulsivefortwoplatesofdifferentmaterialswithappropriatelychosenintermediate
c
[ fluids [4, 5].
Other commonly discussed forces are those induced by thermal fluctuations in fluids, where the energy scale is set
2
by k T, with k is Boltzmann’s constant and T the temperature [1]. When the thermal fluctuations are large and
v B B
long range, they will induce forces detectable and satisfying universal laws at distances large compared to molecular
9
lengthscales[6]. InanalogytotheelectromagneticCasimirforce,suchfluctuationinducedforcesincondensedmatter
3
3 arealso referredto as Casimir orCasimir-likeforces. One importantcaseof this type is the Casimir force inducedby
1 criticalfluctuationsinfluids,originallypredictedbyFisheranddeGennes[7]. Inprinciple,oneshouldexpectCasimir
. forcesinducedbydensityfluctuationsinthevicinityofavapor-liquidcriticalpointorbyconcentrationfluctuationsin
1
0 aliquidmixturenearacriticalpointofmixing[8,9]. PracticalstudieshaveallbeendevotedtotheCasimireffectdue
4 to critical concentrationfluctuations in liquid mixtures, because of the experimental convenience of ambient pressure
1 at which the phenomenon can be observed[6, 10, 11]. One finds a scale-dependent force per unit area,to be denoted
: as a critical Casimir pressure p , which is given by [6]
v c
i
X k T
B
p = Θ(L/ξ), (1.2)
r c L3
a
where Θ is a finite-size scaling function with ξ being the (bulk) correlation length of the critical fluctuations. For
studying the magnitude of the effect one defines a (critical) Casimir amplitude, ∆ = lim Θ(L/ξ), which for the
L/ξ→0
universalityclassofIsing-likesystemsmayvaryfrom 0.01to+2dependingontheboundaryconditions[9,10]. That
−
is, the criticalCasimirforcecanbe either repulsiveorattractive[12]. Note thatatlargerL, p > p . Thatis, the
c EM
| | | |
criticalfluctuationsthatcausep areeffectivelyoflongerrangethantheelectromagneticfluctuationsthatcausep .
c EM
Casimir-like forces are to be expected in fluids, whenever long-range correlations are present, such as those induced
by Goldstein modes in superfluids and in liquid crystals [1].
∗Electronicaddress: [email protected]
2
In addition to the Casimir forces mentioned above, it is also known that velocity fluctuations in equilibrium fluids
induce forces on the boundaries that are comparable to Van der Waals forces and that decay as inverse cube of
wall separation [13, 14]. The present article is concerned with Casimir-like forces induced by thermal fluctuations in
fluids in thermal nonequilibrium steady states (NESS). Specifically, we shall demonstrate the existence of significant
nonequilibrium(NE)Casimir-likeforcesinfluidsinthepresenceofatemperaturegradient T. Long-rangecorrelation
∇
phenomena are ubiquitous in fluids as reviewed by Dorfman et al. [15]. For fluids in thermodynamic equilibrium,
couplings between hydrodynamic modes cause long-range dynamic correlations that are responsible for the presence
of algebraiclong-time tails in the correlationfunctions for the transportcoefficients [16–20] and for the divergence of
certain transport properties near critical points [21–23]. In nonequilibrium fluids not only the dynamic correlation
functions but also the static correlation functions become long range, yielding a pronounced enhancement of the
intensityofthermalnonequilibriumfluctuations. Themoststudiedcaseisaquiescentfluidinthepresenceofauniform
temperature gradient, T. Then the nonequilibrium contribution to the intensity of the temperature fluctuations is
∇
given by [15, 20]
2
k T k T
(δT (k))2 = B k∇ . (1.3)
NE ρDT (ν+DT)(cid:0) k6 (cid:1)
D E
Here the temperature gradient is in the z-direction in a fluid bounded by two horizontal plates, located at z = 0,L,
while k is the magnitude of the wave vector k = k ,k ,k and k the magnitude of its component k = k ,k
x y z k k x y
{ } { }
parallel to the plates, i.e., perpendicular to the direction of the temperature gradient. In this equation ρ is the mass
density, D the thermal diffusivity, and ν the kinematic viscosity. This result was first predicted by Kirkpatrick et
T
al.[24],whoextendedthemode-couplingtheoryfordealingwithlong-timecorrelationsinequilibriumsystems[16–22]
tosystemsoutofequilibrium[24–26]. Thesameresultwassubsequentlyshowntofollowfromaratherstraightforward
extensionoffluctuating hydrodynamicsto systems outofequilibrium[27–29]. These temperaturefluctuations canbe
measured by Rayleigh-scattering experiments which have accurately confirmed the validity of Eq. (1.3) [30, 31].
Equation (1.3) is valid for wave lengths (i.e., inverse wavenumbers), much smaller than the plate separation L,
suchasthoseaccessibleinlight-scatteringexperiments[30,31]. However,fromEq.(1.3)wesee that,forwavevectors
with a non-vanishing component parallel to the plates, the intensity of the NE temperature fluctuations will diverge
as k−4 when k 0, leading to large finite-size effects for fluid layers bounded between two surfaces [32, 33]. In this
→
paper we show that asa consequence the long-rangeNE fluctuations induce significantCasimir-likeforcesin fluids in
the presence of a temperature gradient. Specifically, for the scale-dependent NE fluctuation contribution, p (z;L),
NE
to the pressure, we find
c k T2(γ 1) 1 ∂c 1 ∂α T 2
p B p
p (z;L)= − 1 + F(z˜) L ∇ . (1.4)
NE 96πDT (ν+DT)" − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p# (cid:18) T (cid:19)
Here c is the isobaric specific heat capacity, γ the ratio of isobaric and isochoric heat capacities, and α the thermal
p
expansion coefficient. The NE pressure depends on the location in the fluid layer through the function
F(z˜)=6z˜(1 z˜), (1.5)
−
where z˜= z/L. Substituting F(z˜) = 1 in Eq. (1.5) yields the average fluctuation-induced NE pressure given in our
previous publication [34].
Note that for a fixed value of the temperature gradient, the NE pressure actually grows with increasing L. This
anomalousbehaviorisareflectionoftheverylong-rangedspatialcorrelationsinafluidinatemperaturegradient. We
note that the intensity of critical fluctuations diverges only as k−2 when k 0, which means that in real space the
→
critical correlations still decay as a function of the distance r [35]. However,the divergence of the NE fluctuations as
k−4 impliesthattheNEcorrelationsactuallyincreaseinrealspacewithincreasingdistancer,sothatfinite-sizeeffects
are always present in nonequilibrium systems. The finite-size effects depend on the actual boundary conditions for
the fluctuations at the surfaces ofthe plates. Equations(1.4)and (1.5) correspondto stress-freeboundary conditions
for which we are able to obtain an explicit analytic expression. We note that a fluctuation-induced NE pressure
contributionwill alwaysbe presentat anydensity and temperature of the fluid state, while the criticalCasimir effect
is only encountered at a restricted range of temperatures and densities or concentrations close to a critical point.
Hence, in contrast to the critical Casimir effect, the NE Casimir effects, and those caused by gauge fluctuations and
Goldstone modes, are generic effects existing in entire phases, and are not restricted to special states.
TounderstandthephysicaloriginoftheNEpressures,wenotethatatemperaturegradientcancausenormalstresses
or pressures if nonequilibrium thermodynamics is extended to include nonlinear effects. Specifically, we consider a
nonlinear Onsager-like cross effect causing a nonequilibrium contribution to the pressure induced by a temperature
gradient:
p =κ ( T)2. (1.6)
NE NL
∇
3
In Eq. (1.6) κ is a nonlinear kinetic coefficient commonly referred to as a nonlinear Burnett coefficient [36]. It
NL
is well known that Burnett coefficients, which go beyond the linear transport coefficients in the ordinary Navier-
Stokes equations, do not exist due to the presence of long-time-tail (LTT) effects in the corresponding molecular
time-dependent correlation functions [19, 36, 37]. Instead, one expects that κ contains a LTT contribution which
NL
causes κ to diverge linearly with the system size. To account for this phenomenon we need to consider κ as
NL NL
the sum of a bare molecular contribution κ(0) associated with short-range correlations and a long-range fluctuating
NL
hydrodynamics contribution Lκ(1) diverging as L :
NL →∞
κ =κ(0) +κ(1)L. (1.7)
NL NL NL
Substituting κ κ(0) into Eq. (1.6) yields a pressure contribution due to short-range correlations. Since the ratio
NL ≃ NL
κ(0)/κ(1)Lwillbe ofthe orderofσ/L,where σ is a typicalintermoleculardistance,this contributionis smallandcan
NL NL
be neglected. Also, such an effect is one of several at molecular scales, including accommodation of the velocity and
kinetic energy of the molecules with the wall, which do not satisfy universal laws and should not be characterized as
Casimir effects [6]. However, substitution of κ κ(1)L into Eq. (1.6) yields a genuine NE Casimir-like pressure
NL ≃ NL
due to long-range correlations that are present at L σ:
≫
p (L)=κ(1)L( T)2. (1.8)
NE NL ∇
Hence, from Eqs. (1.4) and (1.8) we conclude that the NE pressure is a direct consequence of the divergence of the
corresponding nonlinear Burnett coefficient κ .
NL
We shall proceed as follows. First, in Sec. II we shall derive a relationship between the NE pressure and the NE
temperature-temperature correlationfunction. This relationshipcan be derivedfromNE statisticalmechanics by ex-
pressingthenonequilibriumpressureintermsoftheintegraloveranunequal-timecurrent-currentcorrelationfunction
which can be evaluated by a method previously employed in [24–26] for nonequilibrium steady states (Sec. IIA). We
alsoshowthatthe samerelationshipcanbe obtainedfroma Taylorexpansionofthe pressurefluctuations (Sec.IIB).
The NE temperature-temperature correlation function for a fluid between two horizontal plates is then evaluated in
Sec. III. The NE temperature fluctuations and, hence, the NE pressure depend on the boundary conditions for the
fluctuations at the surfaces of the plates. In Section IIIA we evaluate the nonequilibrium temperature fluctuations
and the resulting NE pressure assuming stress-free boundary conditions, for which we are able to obtain an explicit
analytic solution. In Section IIIB we consider the alternative case of rigid boundaries for which we derive a Galerkin
approximationoftheNEpressure. InSec.IVwepresentestimatesforthemagnitudeofthe NEpressureforsomeliq-
uids. We find that the NE pressure becomes significantat distances muchlargerthan those atwhich electromagnetic
andcriticalCasimirforceshavebeenobserved. Furthermore,itturnsoutthatthethermalNEfluctuations,incontrast
to critical fluctuations, become sufficiently long range to be affected not only by the boundaries, but also by gravity.
GravityeffectsontheNEfluctuationshavebeenpredictedonthebasisofNEfluctuatinghydrodynamics[38–40],and
they havebeen confirmedby light-scattering[41] and shadow-graphexperiments [42]. In Sec.V we showhow gravity
alsoaffectstheNEpressure. Forfluidlayersheatedfromabove,gravityhasonlyamodestinfluenceonthemagnitude
of the NE pressure. However,for fluid layersheated from below, we find that the NE pressure divergesas the critical
value of the Rayleigh number, associated with the onset of convection, is approached. In Sec. VI we shall consider
some consequences of our results for the long-time and long-range behavior of correlation functions associated with
linear and non-linear transport coefficients. Our conclusions are summarized in Sec. VII.
II. NE PRESSURE AND NE TEMPERATURE FLUCTUATIONS
A. Statistical-Mechanical Derivation
Instatisticalmechanicsthe pressureisgivenby the diagonalelementofthe microscopicstresstensoraveragedover
the N-particle distribution function, ρ . Here we consider a fluid in a nonequilibrium steady state (NESS) that is
N
close to local equilibrium. We can then decompose ρ into a local-equilibrium part, ρ , and a part linear in the
N LE
macroscopic gradients, ρ . The explicit expression for the local-equilibrium part is
∇
exp[y⋆a]
ρ = (2.1)
LE
Trexp[y⋆a]
with a = ρ,mv,e the set of microscopic conserved quantities, y = βµ βu2/2,βu, β the macroscopic
{ } { } { } { − − }
conjugate variables, while y⋆a = dr y(r) a(r) denotes an integration over space. In these expressions, ρ is the
R
4
mass density, mv is the momentum density, e is the energy density, β = 1/k T is the inverse temperature, µ is the
B
chemicalpotential,anduisthefluidvelocitywithmagnitudeu. Inanonequilibriumsteadystateofaone-component
fluid with a temperature gradient, but no velocity gradient, Liouville’s equation gives for the gradient part of the
N-particle distribution function a time-dependent integral of the form [43]
∞ ∂y
ρ = dtexp( t)ρ J ⋆ e. (2.2)
∇ LE e
− −L ∂x
Z0
b
Here is Liouville’s operator, J is the part of the energy current that is orthogonal to the conserved quantities
e
L
(J = P J with J the energy current and P a projection operator), and y = β. With J being the diagonal
e ⊥ e e ⊥ e l
−
element of the microscopic stressbtensor, the nonequilibrium or gradientpart of the pressure tensor can be written as
b
p (r)= J (r)
NE h l iNE
∞ ∂y (2.3)
= dt J (r,t)J (0) ⋆ e.
l e
−Z0 D ELE ∂x
For reference we note that Eq. (2.3) also implies that the noneqbuilibrium contribution to the equal-time conserved
quantity correlation function is,
∞ ∂y
D (r ,r )= dt a (r ,t)a (r ,t)Jˆ (0) ⋆ e. (2.4)
αβ 1 2 α 1 β 2 e
−Z0 D ELE ∂x
Here denotes a nonequilibrium ensemble average and denotes a local-equilibrium ensemble average. Gen-
hiNE hiLE
erally, p (r) is a local NE pressure depending on the position r= x,y,z . Equation (2.3) has the structure of a
NE
{ }
Green-Kuboexpressionforatransportcoefficient,namely,anunequaltime current-currentcorrelationfunction,inte-
gratedoveralltimes t [44]. Note, however,thatthe currentsinthe integrandofthis equationaredifferent,unlike the
current-current correlation functions for the usual Navier-Stokes transport coefficients [17]. Hence, the NE pressure
originates from a cross Onsager-like effect, i.e., a normal stress or pressure is caused by a temperature gradient in
accordance with Eq. (1.6). This point will be further discussed in Sec. VI.
Techniques to evaluate the long-wavelength,or hydrodynamic-mode, contributions to local-equilibrium correlation
functions like Eq. (2.3) have been developed by Kirkpatrick et al. [24–26], who extended the methods of Ernst et al.
[17,18]tononequilibriumsteadystates. Thebasicideaisasfollows. FirstthelocalequilibriumaverageinEq.(2.3)is
writtenastwoaverages. ThefirstaverageisaconstrainedLEaveragewherethereisafixedinitialcurrentdistribution
Jˆ , and where the conserved quantities a are given specified values A at t = 0. The second average is over
{ } { } { }
all values of Jˆ and A . The unknown time dependence in Eq. (2.3) is then transferred into a time-dependent
{ } { }
constrainedensemble F (t). A local-equilibriumassumptionis used, F (t) F [y +δy(t)]+correctionsthat are
JA JA LE ss
≃
rapidly decaying or are higher order in the gradients and can be neglected. Here y =y +δy are the hydrodynamic
ss
fields in the constrained ensemble. In particular, in the formalism the average Jˆ appears. Expanding in
h eiFLE[yss+δy]
powers of δy⋆a and using that the projectedcurrents are orthogonalto a single conservedquantity, we find that the
leading non-zero contributions are the two-mode, or mode-coupling, terms δy(t)δy(t)⋆⋆ aaJˆ . Using all of this
e LE
∼ h i
and Eq. (2.4), we can write Eq. (2.3) in terms of a D and another mode-coupling amplitude J (r)s s , where
l 0 0 LE
∼ h i
s is a microscopic entropy fluctuation at zero wave number. The leading contribution in the case of a temperature
0
gradient is
2
1 ρ
p (r)= J s s D (r,r), (2.5)
NE l,0 0 0 ss
2 k c h i
(cid:18) B p(cid:19)
where D (r,r) is the nonequilibrium contribution to the entropy-entropy correlation function with s being the
ss
microscopicentropydensity. The subscripts0ofthe phasevariablesinthe statistical-mechanicalaverageinEq. (2.5)
indicate that they are all at zero wave number, so that in accordance with Eq. (A.15) in Ref. [17], including a
conversionof entropy per particle to entropy density:
1 ∂c 1 ∂α
J s s =ρc k2T(γ 1) 1 p + . (2.6)
h l,0 0 0i p B − " − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p#
Rather than the nonequilibrium entropy fluctuations we prefer to relate the NE pressure to the nonequilibrium
contribution to the temperature-temperature correlationfunction D (r,r) which is related to D (r,r) via
TT ss
c 2
D (r,r)= p D (r,r), (2.7)
ss TT
T
(cid:16) (cid:17)
5
since δs =(c /T) δT. Substitution of Eqs. (2.6) and (2.7) into Eq. (2.5) yields for the relationship between the NE
p
pressure and the NE temperature fluctuations
ρc (γ 1) 1 ∂c 1 ∂α
p p
p (z)= − 1 + D (z,z). (2.8)
NE 2T " − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p# TT
We note that in the parallel-plate configuration adopted, the NE pressure will only depend on the z coordinate
perpendicular to the plates.
B. Taylor Expansion of Pressure Fluctuations
An alternative procedure for relating the NE pressure to the NE temperature fluctuations starts by considering a
fluctuating pressure as a function of a fluctuating mass density ρ+δρ and a fluctuating energy density e+δe and
writing the pressure in terms of the local mean values ρ,e and their fluctuations δρ,δe:
p(ρ+δρ,e+δe)=p(ρ,e)+δp. (2.9)
In the theory of fluctuations it is most simple to consider the pressure as a function of the conservedthermodynamic
quantities ρ and e [45]. Then, we apply a Taylor expansion up to terms quadratic in terms of δρ and δe:
p(ρ+δρ,e+δe)=
p(ρ,e)+ ∂p δρ+ ∂p δe+ 1 ∂2p (δρ)2+2 ∂2p δρδe+ ∂2p (δe)2 . (2.10)
∂ρ ∂e 2 ∂ρ2 ∂ρ∂e ∂e2
e ρ e ρ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
We know that the NE enhancement of the temperature fluctuations, given by Eq. (1.3), is caused by a coupling
of entropy or temperature fluctuations with transverse velocity fluctuations with vanishing linear pressure fluctua-
tions [24, 28, 29], so that
∂p ∂p
δρ+ δe=0, (2.11)
∂ρ ∂e
(cid:18) (cid:19)e (cid:18) (cid:19)ρ
and
δρ= ραδT. (2.12)
−
WenowsubstituteEqs.(2.11)and(2.12)into(2.10)anddeterminetheaverageNEcontributionp totheequilibrium
NE
pressure p:
(nα)2 ∂2p ∂2p ∂2p
p (r)= 2w +w2 (δT (r))2 (2.13)
NE 2 ∂n2 − ∂n∂e ∂e2 NE
(cid:20)(cid:18) (cid:19)e (cid:18) (cid:19) (cid:18) (cid:19)n(cid:21)D E
with
∂p ∂p
w = / . (2.14)
∂n ∂e
(cid:18) (cid:19)e (cid:18) (cid:19)n
We note that only the NE temperature fluctuations (δT)2 cause a renormalization of the pressure, since the
NE
equilibrium temperature fluctuations are already incorDporateEd in the un-renormalized pressure. Again with the help
of some thermodynamic relations [17], Eq. (2.13) can be converted into
ρc (γ 1) 1 ∂c 1 ∂α
p (z)= p − 1 p + (δT)2 . (2.15)
NE 2T " − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p#D ENE
We thus recover Eq. (2.8), since
(δT)2 = (δT (z)δT (z)) =D (z,z). (2.16)
NE h iNE TT
D E
6
III. BOUNDARY CONDITIONS AND NE PRESSURE
A. Stress-free boundary conditions
Nonequilibrium temperature fluctuations in a fluid layer between two horizontal plates have been evaluated in
previouspublications[32,33]. Thelong-rangeNEtemperaturefluctuationsdependontheactualboundaryconditions
for the fluctuations at the surfaces of the plates. Here we consider stress-free boundary conditions for the vertical
component of the velocity and perfectly heat-conducting walls [46]:
δT =0 at z =0,L, (3.1)
n δv=0= δv at z =0,L, (3.2)
z
· ±
( δv) n=0 at z =0,L, (3.3)
∇× ×
b
where n is the unit vector normal to the walls. For incompressible fluids condition (3.3) simplifies to [32, 46]
b
d2v
z
b =0 at z =0,L. (3.4)
dz2
The advantage of slip-free boundary conditions, as opposed to rigid boundaries, is that an explicit analytic solution
can be obtained for the NE temperature fluctuations.
Insteadofthetemperaturefluctuationsdirectly,previouspublications[32,33]evaluatedanonequilibriumstructure
factor S k ,z,z′ , which is related to the nonequilibrium spatial correlation function of equal-time temperature
NE k
fluctuations as [29, 32]
(cid:0) (cid:1)
1 1
δT (r)δT (r′) = dk exp ik (x x′) S k ,z,z′ , (3.5)
h iNE α2ρ24π2 k k· − NE k
ZZ
(cid:2) (cid:3) (cid:0) (cid:1)
wherexandx′ aretheprojectionsofrandr′ ontotheplaneparalleltotheboundariesatz =0,L. Forthestress-free
boundary conditions (3.1)-(3.4), this nonequilibrium structure factor is given by
ρ2κ c k (γ 1)
S k ,z,z′ = T p B − ( T)2S k ,z,z′ (3.6)
NE k D (ν+D ) γ ∇ NE k
T T
(cid:0) (cid:1) (cid:0) (cid:1)
with e
∞ k2L2sin(Nπz/L)sin(Nπz′/L)
S k ,z,z′ =2L3 k (3.7)
NE k 3
1 k2L2+N2π2
(cid:0) (cid:1) X k
e
(cid:16) (cid:17)
in accordance with Eqs. (20) and (21) in [32]. If we substitute Eq. (3.6) into Eq. (3.5) and use the relation,
ρκ c =α2Tγ/(γ 1), where κ is the isothermal compressibility, we obtain
T p T
−
k T ( T)2 1
δT (r)δT (r′) = B ∇ dk exp ik (x x′) S k ,z,z′ . (3.8)
h iNE ρD (ν+D )4π2 k k· − NE k
T T ZZ
(cid:2) (cid:3) (cid:0) (cid:1)
To obtain the NE pressure from Eq. (2.15), we now need the consider the nonequiliebrium part of the temperature
fluctuations autocorrelation function
k T ( T)2 1
δT2 = B ∇ dk S k ,z,z . (3.9)
NE ρD (ν+D )4π2 k NE k
T T
ZZ
(cid:10) (cid:11) (cid:0) (cid:1)
If we substitute Eq. (3.7) into (3.9) with z′ = z and introduce dimenesionless variables k = k L and z = z/L, we
k k
obtain
k TL( T)2 e e
δT2 = B ∇ F(z) (3.10)
NE 48πρD (ν+D )
T T
(cid:10) (cid:11)
with e
∞ ∞ k3sin2(Nπz)
F (z)=48 dk =6z(1 z). (3.11)
3 −
Z0 1 k2+N2π2
X e e
e e (cid:16) (cid:17) e e
e
7
Next we substitute Eq. (3.10) into Eq. (2.15) so as to obtain for the NE pressure induced by a temperature gradient
c k T2(γ 1) 1 ∂c 1 ∂α T 2
p B p
p (z;L)= − 1 + F (z)L ∇ . (3.12)
NE 96πDT (ν+DT)" − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p# (cid:18) T (cid:19)
e
We note thatthe NEpressuredepends onthe verticallocationz andalsodepends (parametrically)onthe distance L
between the two horizontal plates. From Eq. (3.12) the average NE pressure induced in the fluid layer can be easily
calculated, resulting in [34]
c k T2(γ 1) 1 ∂c 1 ∂α T 2
p B p
p (L)= p (z;L) = − 1 + L ∇ . (3.13)
NE h NE iz 96πDT (ν+DT)" − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p# (cid:18) T (cid:19)
B. Rigid boundaries
While the stress-free boundary conditions are convenient for obtaining the simple and exact expression (3.12) for
the NE pressure, a fluid bounded by two free surfaces may not be a realistic representation of some experimental
situations [46]. For a fluid layer between two rigid walls one commonly imposes no-slip boundary conditions for the
local velocity. That is, Eq. (3.4) is replaced by [33, 46]
dv
z
=0 at z =0,L, (3.14)
± dz
but we can continue to assume perfectly heat-conducting walls implied by condition (3.1). For rigid boundary
conditionsitisnotpossibletoobtainanexplicitanalyticexpression. InsteadweadopthereaGalerkinapproximation
for the temperature fluctuations obtained elsewhere [33]. We then find that Eq. (3.7) for S˜ (k ,z,z′) needs to be
NE k

1.50
1.25
e
d
u
t1.00
i
l
p
m
0.75
A
0.50
0.25
0.00
0.0 0.2 0.4 0.6 0.8 1.0
z/L
FIG.1: Amplitudeof NEpressureas afunction ofze. Dashed curve: thefunctionF(ze)in Eq. (3.12) forthecaseof stress-free
boundary conditions. Solid curve: the function F(ze) evaluated numerically for the case of rigid boundaries in a Galerkin
approximation with Pr=6.
8
replacedbyEq.(26)in[33]. IfwethenfollowthesameprocedureasinSubsectionIIIA,weobtainfortheNE-pressure
inafluidlayerbetweentworigidboundariesanexpressionsimilartoEq.(3.12)before,butwherethefunctionF(z˜)for
free boundaries is replacedby an equivalentfunction F(z˜) for rigidboundaries. Itturns outthat for rigidboundaries
the amplitude function F(z˜) also depends on the Prandtl number Pr= ν/D , unlike F(z˜) for free boundaries. The
T
explicit expressionfor the amplitude function F can be readily evaluated on the basis of the Galerkin approximation
of [33], but it results in a somewhat complicated expression. Hence, we prefer here to discuss the function F(z˜)
graphically.
InFig.1 we showthe functions F (z) andF (z) asa functionofthe locationz inthe fluidlayerbetweenthe plates.
The major difference is that in the case of stress-free boundary condition the slope dp (z;L)/dz has a finite limit
NE
at z = 0,L, while in the case of rigid boundary conditions the slope dp (z;L)/dz vanishes at the boundaries. We
e e NE e
also observe in Fig. 1 that the amplitude function is somewhat smaller in the case of rigid boundaries, the average
through the layer being, roughly, 40% less compared to the case of free boundaries. For the average NE pressure we
find
p (L)= p (z;L)
NE h NE iz
c k T2(γ 1) 1 ∂c 1 ∂α T 2 (3.15)
p B p
= − F(z) 1 + L ∇ .
96πDT (ν+DT) z" − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p# (cid:18) T (cid:19)
(cid:10) (cid:11)
e
where the averageamplitude F (z) over the height of the layer is now a function of the Prandtl number, while for
z
the case of free boundaries it was a constant. An explicit expressionfor F (z) as a function of Pr can be obtained
(cid:10) (cid:11) z
from the expressions of [33]. We heave found that for Pr & 1 the average F (z) is well represented (within a few
(cid:10) (cid:11) z
percent) by the first two terms of its asymptotic expansion in powers of Pr−1e, namely:
(cid:10) (cid:11)
e
B 1
F (z) A+ + (3.16)
z ≃ Pr O Pr2
(cid:18) (cid:19)
(cid:10) (cid:11)
with e
1 2
A 0.40112, B =A π 2arctan .
≃ − 2√10" − r5!#
We emphasize that Eq. (3.16) for rigid boundary conditions is obtained on the basis of a zeroth-order Galerkin
approximation [33].
IV. ESTIMATED NE PRESSURES
Foragivenvalueofthetemperaturegradient T,theNEpressureincreaseswiththedistanceLbetweentheplates
∇
in accordance with Eqs. (3.13) and (3.15). Alternatively, one may study the NE pressure at a given temperature
difference ∆T =L T. For the case of free boundaries, from Eq. (3.13) one then obtains:
∇
p (L)= p (z˜;L)
NE NE z˜
h i
c k T2(γ 1) 1 ∂c 1 ∂α 1 ∆T 2 (4.1)
p B p
= − 1 + ,
96πDT (ν+DT)" − αcp (cid:18)∂T (cid:19)p α2 (cid:18)∂T(cid:19)p#L(cid:18) T (cid:19)
while for the case of rigid boundaries, the averagefluctuation-induced NE pressure at a given temperature difference
will be equal to the value for free boundaries (4.1) multiplied by a factor F (z) , that depends only on the
z
Prandtl number. Within the zeroth-orderGalerkinapproximationof [33], we obtain for this multiplicative factor the
(cid:10) (cid:11)
expression (3.16) given above.
e
In either case, the L dependence of the NE pressure p (L) L−1 at a given temperature difference ∆T may be
NE
compared with the EM Casimir pressure p L−4 in accorda∝nce with Eq. (1.1), and the critical Casimir pressure
EM
∝
p L−3 in accordance with Eq. (1.2). Hence, the fluctuation-induced NE pressures are present over a much larger
c
∝
range of distances L than either p or p . As an illustration, we present in Table I the estimated NE pressures for
EM c
liquid water and liquid heptane with ∆T = 25 K at an average temperature of 25◦C. In this paper we adopt the
−
conventionthat∆T ispositivewhenthe temperaturegradientisinthe samedirectionasgravity(heatedfrombelow)
and negative when the temperature gradient is opposite to the direction of gravity (heated from above). The values
quotedforp representthe averagevalueofthe NE pressureinduced inthe liquidlayeras calculatedfromEq.(4.1)
NE
9
TABLE I: Estimated Casimir pressures and NEfluctuation-induced pressures
pNEa,free boundaries pNEa,b, rigid boundaries
L pEMc pcd Water n-Heptane Water n-Heptane
10−6 m −1.3×10−3 Pa −6.1×10−4 Pa 1.21 Pa 0.40 Pa 0.50 Pa 0.16 Pa
10−4 m −1.3×10−11 Pa −6.1×10−10 Pa 12.1×10−3 Pa 4.0×10−3 Pa 5.0×10−3 Pa 1.7×10−3 Pa
aAtT =298Kand∆T =25K,forapressureof0.1MPa
bForwaterPr≃6.1,forn-heptanePr≃6.6
cEq.(1)
dpc=−0.15kBT/L3
for free boundaries, while for rigid boundaries the values for free boundaries have to be multiplied by the Prandtl
number-dependent factor F (z) . We have obtained the required thermophysical-property information for these
z
liquids from [47]. For comparison we also present the values of p calculated from Eq. (1.1) and for p calculated
(cid:10) (cid:11) EM c
from Eq. (1.2) with a Casimir amplitude ∆ = 0.15 corresponding to periodic boundary conditions [48], which are
e
−
conceptually closest to our case of stress-free boundary conditions. Indeed, the NE fluctuation-induced pressures are
orders of magnitudes larger than those induced by critical fluctuations especially at L = 0.1 mm and at L = 1 mm,
even for ∆T = 10 K [34]. In Table I we have presented the fluctuation induced NE pressures for ∆T = 25 K, so
− −
that we can make a comparison below with thermophoretic experiments obtained with this temperature difference.
ThesignoftheNEpressureisdeterminedbythe thermodynamicquantitybetweenthe squarebracketsinEqs.(3.13)
and (3.15), which can be either positive or negative. A review of the thermodynamic properties of a number of fluids
indicates that the NE fluctuation induced pressure is positive for fluids in the liquid state and negative for fluids in
the vapor state [47, 49].
Since our result for free boundaries is a rigorous one, while the one for rigid boundaries is an approximation, we
focusintheremainderofthispaperonNEpressuresinthecaseoffreeboundaries. WeneedtoemphasizethattheNE
pressurep (z;L),givenbyEq.(3.12),notonlydependsonthethicknessLofthefluidlayer,butalsoontheposition
NE
z inthefluidlayer. Animportantquestionis: whataretheconsequencesofthe tendencyofthe temperaturegradient
to induce a pressure gradient? To address this question, we first consider the case that there are no microparticles in
e
the fluid. Then mechanical equilibrium would require
e
dp dp dp
LE NE
= + =0. (4.2)
dz dz dz
In the absence of any particles in the fluid layer, the only mechanism for the pressure gradient to relax is through a
rearrangementof the density profile. We thus werite e e
ρ(z)=ρ (z)+ρ (z;L), (4.3)
LE NE
where ρ (z;L) is a nonequilibrium contribution to the local-equilibrium density profile ρ (z), such that
NE e e LE
ρ (z;L)= ρκ p (z;L). (4.4)
e NE − T NE e
We note that the relationship (4.4) is independent of the boundary conditions for the fluctuations.
e e
Moreinterestingistoconsiderthepresenceofmicroparticlesinthefluid. ThenwemaycomparetheNEfluctuation
inducedforceswiththermophoreticforcesonmicroparticlesinatemperaturegradientwhichcanbeestimatedfrom[50]
F = 6πηRD T, (4.5)
th th
− ∇
where η is the dynamic viscosity of the liquid, R the radius of the particle, and D its thermophoretic mobility in
th
the liquid. For silica particles D = 22 10−12 m2s−1K−1 in water and D = 5.7 10−12 m2s−1K−1 in n-heptane
th th
× ×
as measured by Regazetti et al. [51]. The experiments were performed on particles with a radius R = 3 µm in a
temperaturegradient T =25K/0.1mminaliquidlayerwithL=0.1mm. Withη =0.890 10−3Pasforwaterand
η =0.387 10−3Pas∇forn-heptane,wefindfromEq.(4.5) F =0.28 10−12Ninwateran×d F =0.03 10−12N
th th
× | | × | | ×
in n-heptane. On the other hand, we find that the fluctuations induce a NE force, F = πR2p , such that from
NE NE
the information in Table I, F 0.34 10−12 N in water and F 0.12 10−12 N in n-heptane. However, these
NE NE
≃ × ≃ ×
values are only averages and they vary strongly with the position z in the liquid as can be seen from Eq. (1.6) and
Fig. 1. Thus in water F will vary from 0 to 0.51 10−12 N in n-heptane from 0 to 0.18 10−12 N inducing NE
NE
× ×
force differences over a height of 3µm of about some tens of fN. Hence, depending on the size of the particles, the
magnitude of the temperature gradient, and of the choice of the liquid solvent, NE fluctuation-induced forces may
10
become comparableto thermophoretic forces,and we may havediscovereda new fluctuation induced thermophoretic
force.
We notethatNajafiandGolestanian[52]haveconsideredanalternativenonequilibriumCasimireffectdue toinho-
mogeneous noise correlations in a medium that is otherwise in local equilibrium. However, it has been demonstrated
thatnonequilibriumfluctuationsarisingfrominhomogeneousnoisecorrelationsareordersofmagnitudelesssignificant
than those arising from hydrodynamic couplings in the fluid by a temperature gradient [53].
V. EFFECT OF GRAVITY ON THE NE PRESSURE
TheNEtemperaturefluctuationsarerelatedtoNEdensityfluctuationsthroughEq.(2.12). Thesethermalnonequi-
libriumfluctuationsbecomesolargethattheyarenotonyaffectedbyfinite-sizeeffects,butalsobygravity. Kirkpatrick
and Cohen [54, 55] have pointed out that the structure factor in a fluid layer heated from below is strongly affected
by gravity near the Rayleigh-B´enard convective instability. Even more surprisingly, Segr`e et al. [38] concluded that
NE temperature fluctuations are strongly affected by gravity in fluid layers heated from above, i.e., very far away
from any hydrodynamic instability. These predictions have been confirmed by light-scattering [41] and shadowgraph
experiments [42]. Hence, we should also expect that the NE pressure will be affected by gravity.
The combinedfinite-size effects andgravityonthe NE temperaturefluctuations ina bounded fluidlayerhavebeen
analyzed by Ortiz de Za´rate and Sengers [39, 40]. The effect of the gravitational field g enters through the Rayleigh
number
αL4
Ra= g ∇T. (5.1)
νD ·
T
TheRayleighnumberisnegativeforfluidlayersheatedfromaboveandpositiveforfluidlayersheatedfrombelow. In
this paper we only consider fluids in a stable quiescent state for which Ra Ra , where Ra is the critical Rayleigh
c c
≤
number associated with the onset of thermal convection (for the case of free boundaries, Ra =27π4/4).
c
Influctuatinghydrodynamicsthetemperaturefluctuationscanbecalculatedasasolutionoftwocoupleddifferential
equations, namely a fluctuating heat equation and a fluctuating Navier-Stokes equation [29]. If we follow again this
procedure, but also include the gravitational contribution to the fluctuating Navier-Stokes equation, we find from
Eq. (27) in [39] that the expression (3.7) for the NE structure factor S (k ,z,z′) changes into:
NE k
∞ k2L2sin(Nπz/eL)sin(Nπz′/L)
S k ,z,z′ =2L3 k . (5.2)
NE k 3
1 k2L2+N2π2 Rak2L2
(cid:0) (cid:1) X k − k
e
(cid:16) (cid:17)
Notice that, as expected, when Ra = 0, Eq. (5.2) reduces to Eq. (3.7) above. If we substitute Eq. (5.2) with z′ = z
into Eq. (3.9) and follow the same procedure as in Sec. IIIA, we obtain
k TL( T)2
δT2(z) = B ∇ F (z;Ra) (5.3)
NE 48πρD (ν+D )
T T
(cid:10) (cid:11)
with e
∞ ∞ k3sin2(Nπz)
F(z;Ra)=48 dk . (5.4)
3
Z0 1 k2+N2π2 Rak2
X e −e
e e (cid:16) (cid:17)
Equation (5.3) replaces Eq. (3.10) with F(z;0)= F (z). IneFig. 2 we show ploets of the function F (z;Ra) for three
values of the Rayleigh number. The function F(z;Ra) diverges when Ra approaches the critical value Ra such that
c
e e e
4π
F (z;Ra) e as Ra Ra . (5.5)
c
→ √Ra Ra →
c
−
For negative Ra, i.e., heatingefrom above for regular fluids, the quiescent conductive state is always stable and its
limiting behavior is given by
6 3π
F (z;Ra) as Ra . (5.6)
→ Ra1/4 − Ra →−∞
| | | |
e p