Table Of ContentThe no-slip condition for a mixture of two liquids
Joel Koplik
Benjamin Levich Institute and Department of Physics
City College of the City University of New York, New York, NY 10031
Jayanth R. Banavar
Department of Physics and Center for Materials Physics
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Pennsylvania State University, University Park, PA 16802
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9 (February 1, 2008)
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Whenamixtureoftwoviscous liquidsflows pastasolid wall thereisan ambiguityin theuseof
n the no-slip boundary condition. It is not obvious whether the mass-averaged velocity, the volume-
a averaged velocity, the individual species velocities, all or none of the above, or none of the above
J
should exhibit no-slip. Extensivemolecular dynamics simulations of the Poiseuille flow of mixtures
8 ofcoexistingliquidspeciespastanatomisticwallindicatethatthevelocityofeachindividualliquid
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species satisfies theno-slip condition and, therefore, so do mass and volumeaverages.
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t PACS Numbers: 47.10.+g, 47.11+j, 47.55.Kf, 83.20.Lr
f
o
s The no-slip boundary condition in fluid mechanics andconcludethatthemassratiowillenterinthebound-
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t states that the velocity of a fluid at a solid wall equals ary condition. Yet another line of reasoning is to assert
a
m the velocity of the wall. This condition has a lengthy that a fluid molecule moves in a potential due to the
history [1] and solid experimental support for the case nearly-fixed wall atoms and the nearly-randomly fluctu-
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d of Newtonian liquids, but its microscopic basis has been ating neighboring fluid molecules, in which there is little
n understood only relatively recently based on molecular difference between a homogeneous liquid and a mixture,
o dynamics simulations [2–5]. The validity of the no-slip andthereforetheindividualspeciesvelocitiesshouldsat-
c
condition is far from automatic, since for example gases isfy no-slip. Unfortunately, none of these heuristic argu-
[
[6] and non-Newtonian liquids [7] are known to exhibit ments is completely decisive in itself.
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slip,andinthispaperweconsiderthequestionoftheap- In order to provide an unbiased and fundamental cal-
v
4 propriate wall boundary condition for a mixture of two culationof the wallboundary conditionfor a liquid mix-
0 liquids. The question is relevant to any flow problem ture, we have conducted systematic molecular dynamics
3 involving miscible liquids, such as double diffusive con- (MD)calculations[14]ofthePoiseuilleflowofsystemsof
1 vection, where the composition is spatially varying. severalthousandliquidatomsinachannelmadeofatom-
0
8 Sincethereisnosatisfactoryanalyticargumentforthe istic solidwalls,in caseswithorwithout a concentration
9 origin of no-slip even in a one-component fluid, it is not gradient. Previous MD studies of this nature (see, [15]
/ feasible to obtain a theoretical derivation here from first forageneralreview)indicatethatthecontinuumaspects
t
a principles. Likewise, it is difficult to address the ques- offluidflowarewellreproducedinthisway,andwehave
m
tion experimentally because the absence of an accepted been able to obtain unambiguous numerical results for
- andcompletetheoryforfluidmixturespreventsonefrom, the showing that the individual species velocities satisfy
d
n for example, extracting a slip length by measuring flux the no-slip condition. We emphasize that this statement
o vs. pressure drop in Poiseuille flow. In recent work it is holds for liquids; in a mixture of gases one has the more
c common to assume that the mass-averagedvelocity sat- difficult question of studying the variation of a non-zero
v: isfies no slip; see, e.g., Perera and Sekerka [8], and Liao slip length with the properties of the components of the
i and Joseph [9]. Aside from some theoretical justifica- mixture.
X
tion by Camacho and Brenner [10], and some support- The mechanics of the MD simulations are very similar
r
a ing molecular dynamics simulations by Mo and Rosen- tothoseusedbyusinastudyoftheslidingplateproblem
berger [11] for gases, there is some intuitive plausibility [16]. Wefirstconsiderthechannelflowofahomogeneous
forfocusingonmass-averagedvelocitybecauseinthemi- Newtonian liquid made of two types of atom. The simu-
croscopic interactions between fluid and wall atoms, the latedsystemconsistsof8000fluidatomsflowingbetween
momentum is the relevant dynamical variable. On the two solid plane atomic walls. The atoms interact with
other hand, one might say that local volume averaging each other via two-body Lennard-Jones (LJ) potentials,
is anappropriatescheme to treatmixture problems, and Vij(r)=4ǫ(cid:2)(r/σ)−12−cij(r/σ)−6(cid:3), cutoff at separation
boundary conditions should involve this variable. Alter- r = 2.5, where c (i,j=1,2,w) is an adjustable inter-
ij
natively, one might be influenced by the kinetic theory species interaction strength. Hereafter, all quantities in
arguments of Maxwell[12]for slip in gases,andconsider thispaperarenon-dimensionalizedusingtheenergyscale
the momentum exchange in inter-species collisions [13], ǫ, the length scale σ, and a reference mass m, giving a
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naturaltimescale τ =σ(m/ǫ)1/2. The walls are tethered In Fig. 1, we show typical results for the tangential
to fixed lattice sites by harmonic springs [5], and other- velocity u(y) and the shear stress s(y). The three sets
wiseinteractwithallotherwallandfluidatomsviaanLJ of curves are a reference one-species system (unit mass
potential with an appropriate c , allowing them to ex- and unit fluid-wall coupling) plotted as a solid line, and
ij
changeenergyandmomentumwiththefluid. Thesystem then the two distinct species for the mixture case shown
is three dimensional, with the walls parallel to the x-z as dashedordotted lines. In this case there are anequal
planeseparatedbyagapofwidth18σ inthey-direction, number of atoms of each species, and their parameters
periodic boundaries in x and z, and average flow in the have been chosen as m1 = 1.5, m2 = 0.75, c12 = 0.9,
x-directionproducedby applyinga gravitation-likeforce c1w =0.75andc2w =0.25. They-coordinateissuchthat
m gxˆtoeachfluidatom,withg =0.01. Inmostcases,we y=0and21arethecentersofthetwoinnermostlayersof
i
extract the heat generated by viscous friction by a local wallatomsconfiningthefluid,andthereare20sampling
Nos´e-Hooverthermostat[17]whichcoupleseachatomto bins for the fluid velocity and stress, centered at integer
a heat bath, providing a constant temperature system. values of y. All velocity profiles are parabolas which go
to zero in the vicinity of the wall. The pure liquid has
a stronger wall attraction, so its extrapolated velocity
vanishes slightly further away from the wall. In other
mixture simulations of this type, if c = 1 the species
iw
velocities vanish at approximately the same position as
the pure liquid, while when c > 1 the species veloc-
iw
ities vanish further away. The difference in the height
of the parabola of pure liquid compared to the mixture
arisesfromthelatterhavingadifferenteffectiveviscosity.
The stress profiles are linear in the interior of the chan-
nel, as expected in Poiseuille flow, but the component
of the mixture which is more (less) strongly attracted
to the wall has a higher (lower) stress value there, ac-
companiedby a sharpincrease(decrease)in density. We
haverepeated this calculationfor a number ofuniformly
mixed systems with different parameter values, and in
all cases the two components have essentially the same
velocityprofile. Whenthemixtureconcentrationis50%,
the shear stress profiles agree, but for other concentra-
tionsweobservedistinctslopes: thecomponentofhigher
concentration has a steeper stress gradient, and a corre-
spondingly higher effective viscosity, and a higher pres-
sure as well. In one test case we omitted the thermostat
andallowedthe system’stemperatureto rise. Theresult
is again a parabolic velocity profile which goes to zero
near the walls, but with larger numerical values reflect-
ing the decrease of viscosity with temperature.
Althoughwedonothaveaprecisemathematicalmodel
to elucidate the numerical observations, the physics is
reasonably clear. The molecular origin of no-slip is that
fluidmoleculesnearthewallareontheonehanddragged
along in the direction of the net flow by their neighbors
further away from the wall, and on the other hand are
pushed up againstthe wallby the crowding of molecules
inadenseliquid, wheretheyinteractwiththeeffectively
corrugated potential of the wall. While the molecules
near the wall are not literally stuck there, their trans-
lation speed is much reduced as they interact with the
almost-fixed wall atoms, and a typical trajectory [4] is
FIG.1. (a)Velocityand(b)shearstressprofilesforasingle
a random walk with a drift in the direction of the ap-
liquid (solid line), and fortheindividualspecies of auniform
plied forceand some transientlocalizationnear the wall.
mixture(dashed and dotted lines.
When the atoms are sorted by sampling bins (or aver-
aged over position by the finite spatial resolution of any
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realistic laboratory measurement) the average velocity for the mean velocity, but in this case the concentration
decreases systematically as the wall is approached, and profile is unsuitable. In Lennard-Jones systems a liquid
tends to zero near the wall. The precise value of “near” freezesifthetemperatureistoolow,correspondingtoan
depends on the details of the wall-fluid interaction, but O(1) minimum liquid thermal velocity, and if the mean
forreasonablechoices,thedistancefromthenominalwall velocities are much below this value they are obscured
positionisatmostoneortwoatomicsizes,andzerofrom by thermal noise. However, if the mean velocities are
amacroscopicviewpoint. Thekeypointaboutamixture too large, the resulting concentration profile approaches
ofliquidsisthatthequalitativeargumentjustgivendoes a step, with a very narrow transition region, and within
not depend in any way on the details of the interactions each step the species concentrations are constant.
between the fluid atoms or whether there is more than
onefluidspecies. Thus,inadenseliquidingoodcontact
with the wall, any species should satisfy the no-slip con-
dition. (The qualification concerning the wall contact is
to distinguish Newtonian from non-Newtonian fluids or
suspensions: in a polymer melt only part of a molecule
adjacenttoawallneedbeinatomiccontact,whileinthe
case of a polymer solution or suspension there can be a
depletion region near the wall.)
A more interesting case arises when the relative con-
centration of the species varies with position along the
wall. Theintuitiveargumentjustpresentedisunchanged
by this complication, but the numerical verification is
rather more involved. If the variation is weak, one
may treat a local region by the homogeneous simulation
just described, using the appropriatelocal concentration
value. More generally, we wish to consider a rapid lin-
ear concentration gradient along the direction channel;
it is trivial to initialize the system in this way, but an
additional ingredient is needed to prevent the gradient
from diffusing away with time. To this end, we divide FIG.2. Concentrationprofilesforonespeciesinamixture
atrest(dashedline)andunderPoiseuilleflow(solidline).The
the channelalongthe (x) flow directioninto 40 “concen-
firstcurveisnoisierbecauseitresultsfromamuchshorterav-
tration bins”, where in bin 1 (21) the concentration of
eraging interval.
species 1 is fixed at 0.75 (0.25) at all time. These values
are maintained by changing the identity of atoms cho-
TheprofileshowninFig.2wasobtainedwitharather
sen at random from species 1 to 2, or vice versa, until
weakforcing,g =0.001,andin orderto observea mean-
the concentration has the desired value. In these two
ingful velocity profile we have simply averaged over a
bins a kindofMaxwelldemonis operating,andthe fluid
very long time interval of 2500τ (and also doubled the
and flow properties there may not be fully realistic, but
length of the system so as to have 16000 fluid atoms).
the other 38 bins are untouched andpresumably faithful
The resulting computation required about 2 months on
models of two-fluid coexistence at whatever local value
an HP-735/125workstation, and gave the concentration
of concentration occurs. A more sophisticated compu-
profileaboveandthesubsequentvelocityplots,whichare
tation [18] would fix the concentration in the two refer-
still somewhat noisy, but clearly display no-slip velocity
ence bins by coupling them to particle reservoirsusing a
profiles.
grand canonical Monte Carlo procedure, which has real-
istic fluid behavior everywhere, but the simpler method InFig.3,weshowsometypicalfluid1velocityprofiles,
suffices here. u1(x,y), as a function of y for selected x-bins. The con-
Intheabsenceofnetflowtheprocedurejustdescribed centration at each x may be found from the solid curve
indeedproducesasawtoothconcentrationprofile(Fig.2, in Fig. 2. Each curve is roughly parabolic and tends to
dashedline)butwhenmotionoccursthereservoirvalues zeronearthe wallpositions. Becauseeachpointinvolves
of concentration are convected downstream and the re- aaverageofonly5-15atoms,comparedto200intheuni-
sultisaprofilewhichisnotpiecewiselinear(Fig.2,solid form mixture case in Fig. 1, the statistical fluctuations
line). In fact, as one might expect, the observed profiles are much stronger. If these curves are ¡ averaged, one
are similar to solutions of the convection-diffusion equa- finds a smoothparabolaresemblingFig.1a. The profiles
tion. Onethenhasanumericaldilemma: inordertohave of the second fluid and the other half of the channel are
a statistically significant velocity profile, it is preferable qualitatively the same. Note that there is no qualitative
to have a strong forcing acceleration and large values difference as a function of concentration, nor between
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bin1,wherethe atoms’identitywasvariedunphysically, We thank E. J. Hinch, R. Jackson, R. F. Sekerka and
and the other bins. This insensitivity is analogous to M.G. Worsterforstimulating discussions. This research
that found as the method of thermostatting varies, and was supported by the NASA Microgravity Science and
is characteristic of MD simulations of this type. Applications Division.
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