algebraic decay of velocity fluctuations near a wall

PHYSICAL REVIEW E DECEMBER 1998VOLUME 58, NUMBER 6

Algebraic decay of velocity fluctuations near a wall

I. Pagonabarraga,1 M. H. J. Hagen,1,* C. P. Lowe,1,2 and D. Frenkel11FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands2Computational Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

~Received 23 June 1998!

Computer simulations of the dynamics of a colloidal particle suspended in a fluid confined by an interfaceshow that the asymptotic decay of the velocity correlation functions is algebraic. The exponents of the long-time tails depend on the direction of motion of the particle relative to the surface, as well as on the specificnature of the boundary conditions. In particular, we find that for the angular velocity correlation function, thedecay in the presence of a slip surface is faster than the one corresponding to a stick one. An intuitive pictureis introduced to explain the various long-time tails, and the simulations are compared with theoretical expres-sions where available.@S1063-651X~98!00612-6#

PACS number~s!: 66.20.1d, 83.20.Jp, 05.40.1j, 82.70.Dd

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I. INTRODUCTION

The dynamical behavior of bulk colloidal suspensionson the whole, fairly well understood@1#. By means of dif-fusing wave spectroscopy~DWS! @2,3#, the time regime inwhich hydrodynamic modes relax has been studied, and ein this regime, experiments and theory are in agreemWhereas DWS has been applied to study the dynamicbulk suspension, the effect of confining surfaces on thenamics is less well understood, despite its relevance in sations of great practical interest, such as diffusion in pormedia.

In this paper we concentrate on the simplest possible cfining geometry, namely, a semi-infinite fluid confined byplanar interface. It is well known that the presence of a sinwall modifies the hydrodynamic response of the fluid, anhas been suggested that this feature is used by some morganisms to enhance their motility@4#. We will focus on thedynamics of colloidal particles close to a surface at tiscales short compared to the time it takes the particledisplace over a distance comparable to their own radius. Tcorresponds to the ‘‘short-time’’ regime in the dynamicscolloidal suspensions, in which hydrodynamic excitatiodecay. In this regime, the particles experience the influeof such modes. When, in what follows, we refer to ‘‘lontimes,’’ we mean the time scale of the asymptotic decaythis ‘‘short-time’’ regime. For a typical suspension, thetimes are of order;1028 s, while the time it takes a particle to displace over a distance equal to its radius is mlonger (;1023 s).

Until the early 1970s it was thought that the decay ofvelocity of a suspended particle would be exponential. Hoever, in their pioneering work, Alder and Wainwright@5#showed that the velocity autocorrelation function~VACF!,Cv(t)5^vW (t)•vW (0)&, of a tagged particle in a hard-sphefluid moving with velocity vW (t) exhibited an algebraicdecay at long times,Cv(t)5(d21)MCv(0)/@dr„4p(n

*Present address: Unilever Research, Port Sunlight LaboraQuarry Road East, Bebington, Wirral L63 3JW, U.K.

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1D)…d/2td/2#, whered is the dimensionality of the system,Mthe mass of the particle,r and n are the density and thekinematic viscosity of the solvent, respectively, andD is thediffusion coefficient of the tagged particle. As was alreaargued by Alder and Wainwright, this behavior is quite geeral as a consequence of the conservation of momentum.argument also implies that the angular velocity autocorretion function ~AVACF! of a tagged particle with angulavelocity v(t), Cv(t)5^v(t)•v(0)&, will decay algebra-ically at long times,Cw(t);1/td/211 @6#, and this has indeedbeen found in computer simulations@7#. The study of corre-lation functions is also interesting because their integralsrelated to transport coefficients, connecting thus the dynacal properties of the constituent particles of the system wits macroscopic response. In particular, the time-dependdiffusion coefficient is obtained as the integral of the VAC

The effect that a bounding surface has on the decay ofvelocity of an initially moving particle has been analyzetheoretically by Gotoh and Kaneda@8#. These authors considered the VACF of a single sphere of radiusr at a distanceh from asticksurface, i.e., an interface where the velocitythe fluid and the surface are equal. They showed, by aturbative calculation that becomes exact for asymptoticalong times, that the wall generates both an anisotropic reation and that it modifies the power law of the long-timalgebraic decay. If the time needed for vorticity to diffuse tdistance between the particle and the wall istw[h2/n, theypredict for timest@tw ,

Cvuu~ t !5

Auu

t5/21OS 1

t7/2D , Cv'~ t !5

A'

t7/21OS 1

t9/2D ~1!

with the amplitudes given by

Auu5MCv

uu~0!

nr

h2

~4pn!3/2, A'5

MCv'~0!

4n2r

h4

~4pn!3/2,

~2!

whereCvuu(0) andCv

' are the initial values of the VACF fora particle moving initially parallel and perpendicular to thwall, respectively.

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7288 © 1998 The American Physical Society

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PRE 58 7289ALGEBRAIC DECAY OF VELOCITY FLUCTUATIONS . . .

One may expect that the change in boundary conditionthe surface will make a qualitative difference for the dynaics of the colloids. For example, one may consider aslipinterface, where the tangential component of the strescontinuous across the surface, and which is characterista fluid/fluid interface. In fact, there exist examples wheredynamics of colloidal particles is sensitive not only to tconfinement, but to the specific behavior of the velocity fieat the interface@9#.

Therefore, we have performed computer simulationsthe short-time dynamics of a particle in the presence of ba slip and a stick surface. We have focused both onVACF and AVACF, as these two functions are most direclinked to experimentally observable quantities. Before psenting our results in Sec. III, we first introduce an intuitipicture that will allow us to interpret our results, and prvides a physical way of understanding the modifications ta bounding surface induces in the relaxation of the hydronamic modes. We conclude the paper with a discussionour results.

II. THEORETICAL DESCRIPTION

The short-time dynamics of a colloidal particle is cotrolled by the decay of the longest-lived hydrodynamic moin the solvent. In the case of an unbounded fluid, the loamplitude of the sound modes excited initially by the partidecays much faster than the vorticity, which relaxes difsively. The algebraic tailCv(t);1/td/2 can be simply under-stood as a coupling of the velocity of the particle to tdiffusion of vorticity @10#. The velocity fieldvW of a lowReynolds number incompressible flow satisfies the diffusequation,

]vW ~rW,t !

]t5n“2vW ~rW,t !. ~3!

A particle of massM located originally atrW0 and withvelocity uW 0 , will create a flow field in the fluid. This field isequal to the one generated by a small fluid elementdV cen-tered in rW0 , with initial velocity vW 0 such that MuW 0

5rvW 0dV. In the limit dV→0, the initial velocity field is

vW (rW,t50)5(MuW 0 /r)d(rW2rW0). From Eq.~3!, it then follows

vW ~rW,t !5H expF2r 2

4nt G~4pnt !d/2

~12 r r !

21

2~pr 2!d/2gS d

2,

r 2

4nt D ~12dr r !J •vW 0 , ~4!

whereg is the incomplete gamma function. From the preous expression, it is easy to check that indeed the veloinduced by a localized perturbation decays asymptotically1/td/2. The presence of a single bounding surface will ndisturb the diffusive decay of the velocity at long times dto vorticity. The only feature we have to take into accounthe fact that the wall will modify the induced flow field. W

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account for this by using the standard method of imagesthis method, one places additional sources in the regionspace not accessible to the system to enforce the deboundary condition at the interface@11#. However, a simplepoint image, as is used when considering mass diffusionelectrostatic potentials, is not sufficient to enforce a bouary condition for a vectorial field, such as the velocity. Tomore precise, for slip boundary conditions, we need oimpose the condition that there is no flux of momentuacross the interface. This can be accomplished by a simimage particle moving with the appropriate velocity. Fstick boundary conditions, however, all the componentsthe velocity have to vanish at the surface. In this casesimple point image is not enough to fulfill the conditionsthe wall. Blake@4# worked out the complete set of imageneeded to reproduce the steady flow corresponding to antial point source. It has even been argued that at finitequency, the flow in the presence of a solid wall cannotdescribed with a finite set of image singularities@12#. None-theless, rather than attempting a rigorous description of vticity diffusion in the presence of a solid surface, in thsimplified theory we will consider the simplest image snecessary to provide the basic features of the long-timedrodynamic behavior. Therefore, we will characterize tsolid surface by imposing that no momentum can be traported along it. This description produces a boundary contion that resembles more a porous surface rather than asolid wall, because there can still be momentum flux acrthe wall, but it contains the basic feature that a solid surfresists the flow of fluid along it. The comparison with thsimulations will show that this assumption predicts tproper temporal decay at long times, although it is in genequantitatively inaccurate, as could be expected.

Let us first consider that the particle is placed at a distahz above a slip interface located atz50. If at time t50 theparticle has a velocityu0y, at later times the velocity at thesame point will be expressed as the sum of the velocity gerated by the particle itself plus the one generated byimage source located at2hz moving with the same velocityu0y, as displayed in Fig. 1~a!. As can be derived from Eq~4!, the long-time decay is given by

uuusl~ t !5

2~d21!

d

v0

~4pnt !d/21OS 1

td/211D . ~5!

FIG. 1. Image particle for a translating colloid above a s

surface, placed initially athz. ~a! Parallel motion,~b! perpendicularmotion.

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7290 PRE 58PAGONABARRAGA, HAGEN, LOWE, AND FRENKEL

In order to avoid confusion when we consider later a stsurface, we have used the superscript ‘‘sl’’ to indicate tthe particle is moving in the presence of a slip interface,opposed to the superscript ‘‘st’’ labeling a stick surface.

The interface does not modify the power of the tempodecay in this particular situation. However, if the particmoves perpendicularly to the interface, then the image pticle has to move in the opposite direction, as shown in F1~b! In this case, the velocity at long times is given by

u'sl~ t !5

4pv0h2

~d12!~4pnt !d/2111OS 1

td/212D , ~6!

which shows that the interface induces a faster decay ofVACF, due to the interaction with the image.

If we consider a wall where stick boundary conditions asatisfied, then the images should be modified accordinglythe particle is moving initially at a distancehz and parallel tothe wall with velocityu0y, the image, located at2hz, has tomove in the opposite direction with velocity2u0y. We thenget asymptotically,

uuust~ t !5

6p

~d21!~d12!

v0h2

~4pnt !d/2111OS 1

td/212D , ~7!

which for three dimensions~3D! coincides with the decaypredicted by Eq.~1!, and recovers the proper dimensionbehavior of the amplitude. However, as already mentionwith this intuitive approach the numerical factor of the tdoes not coincide with the prediction of Eq.~2!.

For motion perpendicular to the interface, we need to ta quadrupole in order to impose that there is no flux parato the wall. For example, in three dimensions, besidesimage at2hz with velocity 2u0z, one can enforce a vanishing parallel velocity at the wall by placing an image rinon the surface at a distance 2h from the origin, with astrength twice the strength of the initial source, pointiaway from the origin. In this case, the combination of tsources gives

u'st~ t !5A

v0h4

~4pnt !d/2121OS 1

td/213D , ~8!

whereA524p2/7 in 3D. It is worth noting that the algebraidecay is characterized by a large power, 7/2 in three dimsions. Clearly, the stick wall always induces a faster decathe velocity than the slip interface for motion in any diretion, since it damps momentum more efficiently than a liquinterface. For both kinds of interfaces, the motion perpdicular to the surface always decays faster.

We can also use this intuitive approach to analyze hthe interface will modify the decay of the angular velocautocorrelation functions. In this case, we should look atdecay of the angular velocity if initially a torque is appliedthe particle. If we calln a vector normal to the surface of thparticle, located atrW0 , the angular velocityvW induced by aninitially applied force distribution,FW (rW,t)5FW @d(rW2rW02nW )2d(rW2rW01nW )#d(t), can be shown to satisfy@13#

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vW 3rW5E e2nk2teikW•~rW2r 0W !~12 kk!•~ IVW 0!sin~kW•nW !dkW ,

~9!

where the vectorrW means a vector joining the center of thobject to a point on its surface. Moreover,I is the moment ofinertia of the particle andVW 0 is the initial angular velocity ofthe fluid, which is related to the total initial torqueTW appliedby the force distribution FW through TW 5*dnFW 3nW

[IVW 0d(t). Note that there is no net momentum transmittto the system. Similar to the case of the velocity, the magtude of the torque is related to the initial angular velocitythe particle in such a way thatVW 5vW 0 /r, wherevW 0 is itsinitial angular velocity.

Let us first consider that the colloid rotates in the preseof a slip surface. If the axis of rotation is parallel to thinterface, then in order to ensure that there is no componof the velocity field through the interface, we should placeimage vortex which rotates in the opposite sense, as depiin Fig. 2~a!. According to Eq.~9!, this initial perturbationgives rise to an asymptotic decay of the angular velocity

v uusl~ t !5

3~4p!2

d11

h2T

~4pnt !d/2121OS 1

td/213D . ~10!

If the axis of rotation is perpendicular to the interfacecorotating source and image suffice to ensure that there inet flux of momentum across the interface. This combinatleads to a slower decay of the vorticity than in the previogeometry, namely,

v'sl~ t !5

8p

d11

T

~4pnt !d/2111OS 1

td/212D . ~11!

In the case of stick boundary conditions, when the axisrotation is parallel to the interface, the velocity along twall is zero if we take a corotating image particle, as dplayed in Fig. 2~b! In this case the decay is then

v uust~ t !5

8p

d11

T

~4pnt !d/2111OS 1

td/212D , ~12!

FIG. 2. Image set for a disk above a surface rotating with

axis parallel to the wall, placed initially athz. ~a! Slip surface,~b!stick surface.

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which coincides with Eq.~11!, and has an amplitude whicis twice the one corresponding to a rotating particle inunbounded fluid. Finally, if the axis is perpendicular to tinterface, the image should rotate in the opposite sensorder to avoid the flow along the wall. This leads then to

v'st~ t !5

~4p!2

d11

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~4pnt !d/2121OS 1

td/213D . ~13!

In this particular case, due to the symmetry of the tsources, all the components of the velocity will vanish atsolid surface. This implies that the amplitude also has toasymptotically correct in this case.

The previous analysis shows that the AVACF behadifferently close to an interface than the VACF. For an ubounded fluid, the AVACF always decays with one powhigher than the corresponding VACF. A stick surface alwaproduces a faster decay on the VACF than a slip interfaThe behavior of the AVACF in the presence of an interfais not simply related to the corresponding VACF. In fact, warrive at the rather surprising prediction that, for rotatiparallel to the interface, slip boundary conditions yieldfaster decay of the AVACF than stick boundary condition

In Table I we have summarized the different exponentsthe VACF and AVACF as a function of the geometry and tboundary conditions.

Thus far we have focused on the decay of the velocfield generated by a moving particle, disregarding its Browian motion. Diffusion of the colloid during the decay of thflow field will lead to mode coupling between flow and difusion. For an unbounded fluid, this so-called mode-coupeffect modifies the amplitude of the correlation functions, bnot the power law characterizing the long-time tail@14#. Us-ing mode coupling theory, it is easy to show that, as inunbounded fluid, mode-coupling leads now to the samepressions obtained in this section ifn is substituted byn1D, where D is the diffusion coefficient of the colloidaparticle.

III. RESULTS

We have performed computer simulations to studyVACF and AVACF of a spherical particle suspended influid, in contact with both a stick and a slip interface. To th

TABLE I. Powers of the algebraic tail for the VACF and thAVACF in d dimensions. Ford52, Cv

' is not defined.

slip stick unbounded

Cvuu d

2d

211

Cv' d

211

d

212

d

2

Cvuu d

212

d

211

Cv' d

211

d

212

d

211

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end, we have used the lattice-Boltzmann model to descthe fluid. The state of the fluid is specified by the averanumber of particles,n(c,r ,t), with velocity c, at each latticesite r at time t. The time evolution of the distribution functions is determined by the discretized analog of the Bomann equation@15#, in which then(c,r ,t) evolves in discretetime in two steps: propagation and collision. Collisions aspecified such that the time evolution of the hydrodynamfields satisfies the linearized Navier-Stokes equations foisothermal fluid@15#. In this method, the solid surface whicdefines the colloidal particle and the confining surfacetreated on the same footing, by modifying appropriatelycollision step of the nodes adjacent to the surfaces@15#. Stickboundary conditions are ensured by imposing bounce-bof the incomingn(c,r ,t) along the links joining nodes of thesurface and the fluid, while slip boundary conditions areforced by specular reflection. Stick boundary conditionsalways satisfied at the surface of the colloidal particle, whwe change from stick to slip boundary conditions for tsurface that bounds the fluid. The equations of motion ofparticle are integrated using the self-consistent methodscribed in Ref.@7#. We calculated the VACF by giving aninitial velocity, vW (0), to acolloidal particle in an otherwisequiescent fluid. In this description of the fluid, fluctuatioare absent. Nonetheless, correlating the initial velocity offreely moving particlevW (0) with its subsequent valuesvW (t)while it relaxes is, according to Onsager’s regression hypoesis, equivalent to calculating the VACF in a ‘‘real’’ fluctuating fluid @16#. The initial velocity given to the particle issufficiently small to ensure that its displacement is negligicompared to the lattice spacing. Our units are such thatmass of the lattice particles, the lattice spacing, and the tstep are all unity, the densityr has a value of 24, and thspeed of soundc has a value of 1/A2. The mass of the col-loidal particle is chosen to correspond to neutral buoyanIn all cases, the VACF is only calculated for times less ththe time it takes for a sound wave to cross the systemthere are no finite-size effects to consider. In most ofcases, the asymptotic algebraic decay is reached withinsimulation time, although in a few cases it is only the aproach to it that is observable. We will start by analyzing tdecay of the correlation functions of a disk in two dimesions, because a more detailed analysis is feasible sinccan reach longer times than in their 3D counterparts. Aftwards, we will focus on the more realistic situation ofsphere in three dimensions, where we can compare withact theoretical predictions@8#.

Figures 3 and 4 show a log-log plot of the VACF fordisk in a 2D fluid both for slip and stick boundary condtions. For the sake of comparison, we also plot the VACFthe same particle in an unbounded fluid. The analysis ofpreceding section assumed that only diffusion of velocityrelevant. This is indeed true in the asymptotic long-timegime, but at sufficiently short times, compressibility effecrelated to the emission and reflection of sound waves,modify the relaxation of the colloid. In Figs. 3 and 4, thdiffusive decay of the VACF is only recovered at timesorder tw , when momentum has diffused the distance btween the particle and the wall. This is also true for tparticle in the unbounded medium, although in this case m

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mentum has to diffuse over the particle diameter, definincharacteristic diffusive timet`[r 2/n. Once the relaxationof the VACF is controlled by the diffusion of vorticity, algebraic decay is obtained, with powers given by Eqs.~5!–~8!. This suggests that the intuitive approach described inpreceding section is able to predict the decay of the differVACF’s. For slip boundary conditions, we can also compthe amplitudes of the long-time tails. By fitting the algebradecay, for a particle moving parallel to the interface wean amplitudeAuu

sim,sl50.939, while the theoretical expressioEq. ~5! predictsAuu

th,sl50.938. For a colloid moving perpendicular to the interface, we should take into account thatactual distance between the colloidal particle and the wadefined up to the resolution of the lattice itself@15#. Thisindeterminacy is important in this case because the distato the wall is small, and a power of this distance enters in

FIG. 3. Log-log plot of the VACF of a disk of radiusr 51.5 ata distanceh55.5 from a slip surface, in a fluid of viscosityn50.6. The lines correspond to the theoretical prediction forasymptotic decay.

FIG. 4. Log-log plot of the VACF of a disk of radiusr 52.5 ata distanceh55.5 from a stick surface, in a fluid of viscosityn51/6. The lines corresponding to the asymptotic decays areplaced to facilitate the comparison.

a

ente

t

eis

cee

expressions for the amplitudes. In Fig. 3, the nominal dtance between the disk and the interface ish55.5. The simu-lation gives an amplitudeA'

sim,sl59.85, while Eq.~6! predictsA'

th,sl510.16 for a real distanceh55.1. Note that the deviation with respect to the nominal distance is within half-lattispacing. If we move away from the surface, this indetermnacy becomes less relevant. For example, for a nominaltanceh513.5, A'

sim,sl569.0, whileA'th,sl571.2. This shows

that the deviations of a few percent that we get with respto the theoretical predictions can be attributed to the discrness of the lattice on which we simulate the fluid.

We can also give initially a certain angular velocity to aotherwise quiescent colloidal particle and inspect its decIn Fig. 5 we show the log-log plot of the AVACF for bothkinds of boundary conditions. The short-time decay is npurely diffusive due to the presence of the interface, budoes not lead to large distortions at short times. Surprisinit clearly shows that a slip surface induces a faster decathe angular velocity than a stick interface, in agreement wthe predictions of the preceding section, Eqs.~10! and ~12!.

We now turn to the more realistic case in which a sphecal colloid of radiusa50.5 is confined by a planar interfaceIn Fig. 6, we show the log-log plot of the VACF’s when thsurface satisfies stick boundary conditions. The figure pthe absolute value of the VACF, because the velocityverses its direction temporarily at short times. This is duethe small size of the particle and to the viscosity, whimakes the particle more sensitive to the sound reflectifrom the wall at short times. The asymptotic long-time decis algebraic and positive, and the exponents are againagreement with Eqs.~7! and~8! of the preceding section, anwith the predictions of Gotoh and Kaneda, Eq.~1!. In thiscase, we can also compare with the theoretical predictionEq. ~2!. Again, we should take into account the uncertainin the distance to the surface due to the discreteness olattice. The simulations reported in Fig. 6 correspond todecay of a particle placed at a distanceh52.5 from the wall.However, in order to compare with the theoretical pred

e

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FIG. 5. Log-log plot of the AVACF of a disk of radiusr 52.5 ata distanceh55 from the interface, in a fluid of viscosityn51/6.The lines corresponding to the asymptotic decays are displacefacilitate the comparison.

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tions for the amplitudes, we get the best agreement foseparationh52.42, which is quite close to the nominal ditance. By fitting the amplitudes of the asymptotic decaythe VACF’s in Fig. 6, we getAuu

sim,st50.253 andA'sim,st

50.583, while the theoretical expressions of Eq.~2! predictAuu

st50.2469 andA'st50.6024. Both simulation results agre

with their theoretical counterparts to within 4%. If we rduce the distance toh51.5, we find agreement with the theoretical estimates for an effective separation ofh51.4, butwith an error of 7%. In Fig. 6, the asymptotes drawn corspond to the theoretical predictions of Gotoh and KaneAlthough in the figure the asymptote corresponding toperpendicular motion does not seem to agree well, this isto the fact that it takes longer to reach the algebraic regiTo the best of our knowledge, this is the first direct testthe predictions of Ref.@8#.

We have displayed in Fig. 7 the AVACF for a sphereradiusa50.5. As opposed to the two-dimensional case, nthe particle can rotate with its axis either parallel or perpdicular to the interface. For both kinds of boundaries,powers obtained are again in agreement with the predictof the preceding section, Eqs.~10!–~13!. Contrary to the in-tuition, one can clearly see how the AVACF for a particrotating with its axis parallel to a stick surface approachthe same algebraic decay as the one corresponding tunbounded colloid, while the one corresponding to a sinterface decays faster. For the particular case in whichparticle rotates with its axis perpendicular to a solid wall,can also compare the theoretical prediction for the amplitwith the simulation results, because in this case the relarotation of the source and image ensures that all the comnents of the velocity cancel at the surface. The nominaltance in this case ish52.5, and we get the best agreemefor h52.46. In this case, the amplitudeA'

sim,st52.58, whilethe theoretical prediction givesA'

th,st52.56. If we wouldhave usedh52.42 as for the VACF, thenA'

th,st52.5, whichagrees with the simulation result within 3%, consistent wthe prediction for the VACF. In Fig. 7 one can see that

FIG. 6. Log-log plot of the VACF of a sphere of radiusr50.5 at a distanceh52.5 from a solid interface, in a fluid of viscosityn50.6. The straight lines correspond to the asymptotic, lotime behavior predicted by Gotoh and Kaneda.

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AVACF for rotation perpendicular to the stick surface aproaches the theoretical prediction, Eq.~13!, asymptotically.

Finally, the presence of an interface will induce rotatiin an initially translating disk parallel to the wall, and tranlation in an initially rotating disk. The decay of these cromotions for a 2D disk is shown in Fig. 8 for both kinds oboundary conditions. Initially, the corresponding velocityzero, which explains the short-time increase in the plots. Tdecay at long times is also algebraic, with the same exnent, irrespective of the particular boundary condition cosidered. In fact, the angular velocity in Fig. 8~a! is deter-mined by the vorticity induced by the image source. Onother hand, the velocity observed in Fig. 8~b! is due to theflow field generated by the image dipole. In both casesdecays ast22 in 2D, in general ast2d/211, i.e., the algebraicdecay due to a dipolar source. Note that this is independof the specific boundary conditions, and it arises purely aconsequence of the confinement. However, for stick bouary conditions the induced velocity reverses its directionmotion at long times due to the influence of the soundflected back at the boundary. Correspondingly, in Fig. 8have displayed the absolute value of the induced velocitya stick interface.

IV. DISCUSSION

In this paper we have studied the effect of an interfacethe short-time dynamics of a suspended particle. The rtional and translational diffusion become anisotropic duethe presence of a surface, which induces, in general, a fadecay of motions perpendicular to it.

We have also shown that the asymptotic algebraic deof the velocity of a particle is sensitive to the specific bounary conditions considered. In most cases, stick boundconditions induce a faster decay of the hydrodynamic extations than slip boundary conditions, and the AVACF dcays with one power faster than the corresponding VACHowever, we also find that for rotation with the axis paral

-

FIG. 7. Log-log plot of the AVACF of a sphere of radiusr50.5 at a distanceh52.5 from the interface, in a fluid of viscosityn50.6. All curves correspond to stick boundary conditions, excfor the one in which it is explicitly indicated. The liney;t25/2 isdisplaced to facilitate the comparison.

ng times.


FIG. 8. Log-log plot of the induced cross motion of a disk of radiusr 52.5 at a distanceh54.5 from the interface, in a fluid of viscosityn51/6. For stick boundary conditions we plot the absolute values, because the induced velocities reverse their direction at loVelocities and angular velocities are expressed in the units introduced in Sec. III.~a! Angular velocity of an initially translating disk.~b!Velocity of an initially rotating disk. The lines corresponding to the asymptotic decays are displaced to facilitate the comparison.

a

eth

ee

llr-

icds

e

oidhif

io

r-t

i

t

ffi-renglel andres-

fea-t at

shorts ofr-les.

di-

s

f ra-f

to the interface the behavior is unexpected: for a stick surfthe AVACF decays slower than for slip boundary condition

The decay of the perturbations is dominated by momtum diffusion at long times. For this reason, the onset ofasymptotic regime only occurs for times of ordertw . Fromthe expressions for the amplitudes in Sec. II, it would sethat the largerh is, the larger is the amplitude of the algbraic tail. However, since it is necessary to wait longerenter the asymptotic regime, the actual amplitude is smaAs can be seen in Table II, if we express the velocity corlation functions in units oft/tw , the amplitude always decreases withh ~ash2d for rotation andh2(d12) for rotation!.Of course, the velocity and angular momentum of a partin an unbounded fluid would exhibit the usual algebraiccay for times larger thant` . This implies that for distanceh@r , we will first observe the usualt2d/2 decay and, onlyafter a time that is a factor (h/r )2 larger, a crossover to thbehavior induced by the presence of the interface.

We focused our analysis on the single-particle case. Hever, our results should also apply to the case of a collosuspension close to an interface. Due to the linearity ofdrodynamics, the power law of the VACF will not be modfied. Only its amplitude will differ from the prediction oGotoh and Kaneda@8#. If the particles can move in randomdirections, at long times the lowest decaying perturbatwill be the dominant one. As an example, for the casetranslation in 3D, thet25/2 decay related to the motion paallel to the surface will be dominant at long times. In orderextract the decay of the perpendicular motion, it will be neessary to restrict the motion of the particles, e.g., by applyexternal fields.

From the experimental point of view, it may be easieranalyze the mean-square displacement,D[^@rW(t)

TABLE II. Scaling of the amplitudes of the VACF withh at theonset of the asymptotic regime, when time is expressed in unitthe diffusive timetw . For d52, Cv

' is not defined.

Cvuu Cv

' Cvuu Cv

'

Slip h2d h2d h2(d12) h2(d12)

Stick h2d h2d h2(d12) h2(d12)

ces.n-e

m-toer.e-

lee-

w-aly--

nof

oc-ng

o

2rW(0)#2&, which is related to the integral of the VACF,

D~ t !52dtS E0

t

Cv~ t8!dt821

t E0

t

t8Cv~ t8!dt8D . ~14!

In Fig. 9 we show the time-dependent diffusion coecient, D(t)[d„D(t)…/dt, for the motion of a single spheboth in an unbounded fluid and in the presence of a siwall. We have considered separately the case of paralleperpendicular motion with respect to the interface. The pence of the wall does not introduce major qualitativetures, except for the saturation of the diffusion coefficiena smaller value and for the more pronounced peak attimes. Therefore, it is the analysis of the time derivativeD(t) which will show more clearly the qualitative diffeences induced by the wall on the dynamics of the particAt finite volume fractions the asymptotic value will be mofied, but the qualitative features will remain.

of

FIG. 9. Mean-square displacement of a spherical particle odius r 50.5 at a distanceh52.5 from a stick surface, in a fluid oviscosityn50.6.

hit

hem

twa

y

lide

u-c

Or-

deen

-


In this paper we have considered the situation in whicsingle surface is present. We have also presented an intupicture along with the simulations, which relies on tmethod of images. One might be tempted to apply the samethod for a colloidal suspension confined betweenwalls, or inside a cylinder. However, in this case there mbe situations where sound modes propagate diffusivelstick boundary conditions apply@9#. Then, the assumptionthat the dynamics is controlled by vorticity is no longer vaand the algebraic decays for the VACF will differ from thpredictions based on the image scheme obtained here.

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aive

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ACKNOWLEDGMENTS

We thank J. Polson for a careful reading of the manscript. The work of the FOM Institute is part of the scientifiprogram of FOM and is supported by the Nederlandseganisatie voor Wetenschappelijk Onderzoek~NWO!. Com-puter time on the CRAY-C98/4256 at SARA was maavailable by the Stichting Nationale Computer Faciliteit~Foundation for National Computing Facilities!. I.P. ac-knowledges E.U. for financial support~Contract No.ERBFMBICT-950433!, and the FOM Institute for its hospitality.

kel,

ev.

s

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algebraic decay of velocity fluctuations near a wall

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