PHYSICAL REVIEW A VOLUME 46, NUMBER 8

Slip length in a dilute gas

15 OCTOBER 1992

David L. MorrisDepartment ofPhysics, San Jose State University, San Jose, California 95192 01-06

Lawrence HannonIBM Scientific Center, 1530Pagemill Road, Palo Alto, California 94304

Alejandro L. GarciaDepartment ofPhysics, San Jose State University, San Jose, California 95192 010-6

(Received 24 February 1992)

We study the phenomenon of slip length using molecular dynamics and direct simulation Monte Carlosimulations of a dilute gas. Our work extends the range of Knudsen numbers that have been previouslystudied. In a recent paper, Bhattacharya and Lie [Phys. Rev. 43, 761 (1991)] suggest a logarithmicdependence of slip length on Knudsen number. By a simple redefinition of the mean free path, we obtain

good agreement between simulation results and Maxwell theory for slip length. The anomalies seen byBhattacharya and Lie appear to be due to their definition of the mean free path.

PACS number(s}: 47.45.6x

v(x =0) vo

(dv/dx ) —o 7'o

with the wall located at x =0. The dimensionless sliplength l,* is defined as

I,I*=——L (2)

where L is the characteristic length in the problem.In this work we investigate how I,' varies with Knud-

sen number for planar Couette and Poiseuille flow. TheKnudsen number is a dimensionless parameter defined as

Kn= —,L (3)

where A, is the mean free path. Maxwell theory predictsthat the slip length is related to the mean free path as

l,*=aKn, (4)

The velocity profile of a fluid very close to a wall(within a few mean free paths) exhibits a phenomenonknown as slip. The velocity of the fluid v(x) near the walldoes not equal the wall's velocity even if the wall is per-fectly thermalizing. A similar phenomenon, called jump,also occurs for temperature. The existence of velocityslip was first predicted by Maxwell [1]. Its effect is im-portant in microscopic flows (e.g., flow between the writehead and the platter on a hard disk) and in rarefied gasflows (e.g., very-high-altitude flight). In this short paperwe address the question: Is slip a strictly local effect ordoes it depend on the entire flow field'7 Physical intuitionsuggest the former but recent computer experiments indi-cate the latter [2,3).

%e define the slip length I„as the distance inside thewall at which the (extrapolated) fluid velocity wouldequal the wall's velocity [4]. The slip length may be writ-ten as

where a -=1.15 is the slip coefficient [5].From basic kinetic theory we know that the mean free

path A,z, in a hard-sphere gas is

l?l~h

&2nd'p' (5)

where p is the Quid density, and m and d are the massand diameter of the particles, respectively. Chapman-Enskog theory gives us the following expression for theviscosity g of a hard-sphere gas:

1/22~k~ T

g= —,',pA, (6)

Notice that A,„ is defined in terms of the viscosity. Wedefine the hard-sphere and viscous Knudsen numbers asKnl, =kz/L and Kn, =A,„/L, respectively. For a hard-sphere gas at equilibrium, the two definitions of meanfree path are equivalent [7). Far from equilibrium theydiffer, since A,„ includes boundary effects.

The two definitions of the mean free path A, I, and A,, arefunctions of density, the latter also depends on tempera-ture. %'e computed the mean free path using the fluiddensity and temperature as measured near the wall. Us-ing the average density and temperature instead had littleeffect on the results from our runs.

We measured slip length in two flow geometries. Inplanar Couette flow a fluid is confined between walls

where T is the temperature and k~ is Boltzmann's con-stant.

For particles with extended potentials, the definition ofthe mean free path is problematic. Cercignani suggeststhe following operational definition for the mean freepath [6]:

' 1/21rrn

p 2k~ T

46 5279 1992 The American Physical Society

5280 BRIEF REPORTS 46

moving in opposite directions with velocity v, the wallsare located at x =+L/2. By conventional hydrodynam-ics [8], the steady-state velocity profile is

2vpv(x)= x .

L(8)

In planar Poiseuille flow, a fluid between stationary wallsis acted on by an external force in a direction parallel tothe walls; the velocity profile is

0.7

0.6

0.5

c4 04C/l

P 3O

0.2A

~ B&L

o MD

o MD

DSM

81 DSM

~ 8

L 2

v(x)= x + v-2g 8g

(9)0. 1

00.001 0.01 0.1 10

P.y =—nXp . (10)

In the Poiseuille runs, the effective viscosity was a1socomputed from the velocity profile [see Eq. (9)] and thetwo methods gave similar values.

Bhattacharya and Lie [2,3] reported that for largeKnudsen number ()0.05) they observed a significantreduction in the expected slip length. Their analysis em-

ployed the simple, hard-sphere definition for the meanfree path. Using Knh, we reproduce their results in bothCouette and Poiseuille fiow (see Fig. 1). They suggestedthat the slip coefficient may vary as I,* ~ln( Kni, ). Ourruns extend the range of Knudsen number to lowervalues, and we find good agreement with Eq. (4) for

Note that by measuring the curvature of the profile wemay obtain the viscosity of the fiuid [9].

Since the phenomenon of slip occurs on a scale of onlya few mean free paths, it is difficult to measure experi-mentally. In this respect, computer simulations are auseful tool for this problem. The two commonly used al-gorithms are molecular dynamics [10] (MD) and direct-simulation Monte Carlo [11] (DSMC). We simulated theflow of argon using both methods. In MD we used theLennard-Jones potential (effective diameter o =3.405 A);in DSMC we used the variable hard-sphere potential [12].Results from the two simulation methods for similar runswere in very close agreement.

The system consisted of 1000 atoms enclosed in a rec-tangular box of size L X A, where A is the cross-sectionalarea. The distance between the walls was constant atL =30o, while the cross section varied from A =81o. to40000&v . The boundaries in the y and z directions wereperiodic and the boundaries in the x direction werethermal walls. When a particle reaches a thermal wa11,

the particle's velocity is reset randomly from the biasedMaxwell-Boltzmann distribution with temperatureT„=450K.

For the Poiseuille runs, the systems were subjected toan acceleration field of magnitude g=2.27X10' cm/sin MD runs and g =2.27 X 10' cm/s in the DSMC runs.For Couette flow, the thermal wall moved in oppositedirections with speeds of +4.36X 10 cm/s.

The effective viscosity was obtained from the shearstress P„„at the thermal wall. The shear stress was cal-culated as the net change of momentum of particles atthe walls per unit per unit time. The effective viscosity iscomputed from

Knudsen number (H S )

FIG. 1. Dimensionless slip length I,* vs hard-sphere (HS)Knudsen number Knh. The closed circles are from the Ref. [3].The open circles and squares are our molecular-dynamicsPoiseuille and Couette data, respectively. The cross-hatchedcircles and squares are our DSMC Poiseuille and Couette data,respectively. The solid line is given by (4) with a=1.15; thislinear relation appears as a curve in this graph due to the serni-

log scale.

0.7

0.6

0.5

G4 p4

0.3

& 0.2A

0.1

ell

01S.

ouette

C Pois.

C Couette

0.1 0.2 0.3 0.4Knudsen number (Vise.)

0.5 0.6 0.7

FIG. 2. Dimensionless slip length I,* vs viscous Knudsen

number Kn, . See Fig. 1 caption for details.

Kn„&0.05.Using the viscous definition for mean free path, we find

that the slip length is accurately given by Maxwell theoryfor the entire range of Knudsen numbers (see Fig. 2). Aleast-squares fit of the data gives the value a=0.97 forthe slip coefficients; for just the low-Knudsen-numberdata (Kn, (0.05) we obtain the slightly higher value ofa = 1.04. Given an estimated relative error in A,„ofabout10%, the data are in reasonable agreement with theory.

There remains the question of why the two definitionsof mean free path give similar results at low Knudsennumber but differ dramatically when the separation be-tween the walls is small. Maxwell theory was originallyformulated for the semi-infinite geometry, i.e., a fluidbounded by a single wall. It should not be surprising thatthe theory needs some modification when there is asecond boundary nearby.

The hard-sphere definition for the mean free path only

BRIEF REPORTS 5281

depends on the density and the particle diameter. On theother hand, the viscous mean free path also depends onthe distance between the walls, since the e6'ective viscosi-

ty decreases with increasing Knudsen number [4]. Heu-ristically, the viscous mean free path is the effective dis-tance traveled by a particle between collisions with parti-

cles or walls. It would be interesting to develop this pointof view along more rigorous lines.

The authors wish to thank D. BaganofF, Wm.Feireisen, and B. Alder for helpful discussions. Thiswork was supported in part by a CSU Research Award.

[1]J.C. Maxwell, Philos. Trans. Soc. London 170, 231 (1867).

[2) D. K. Bhattacharya and G. C. Lie, Phys. Rev. Lett. 62,897 (1989).

[3]D. K. Bhattacharya and G. C. Lie, Phys. Rev. 43, 761

(1991).[4] W. G. Vincenti and C. H. Kruger, Introduction to Physical

Gas Dynamics (Krieger, Malabar, 1965).[5] D. R. Willis, Phys. Fluids 5, 127 (1962); S. Ziering, ibid 6, .

993 (1963).[6] C. Cercignani, Mathematical Methods in Kinetic Theory

(Plenum, New York, 1969).[7] Actually they differ by a small multiplicative constant:

~p /A, „=16/5m =-1.02.

[8] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Per-gamon, Oxford, 1959).

[9] L. Hannon, G. C. Lie, and E. Clementi, Phys. Lett. A 119,174 (1986).

[10]M. P. Allen and D. J. Tildesley, Computer Simulation ofLiquids (Clarendon, Oxford, 1987).

[11]G. A. Bird, Molecular Gas Dynamics (Clarendon, Oxford,1976).

[12]G. A. Bird, in Rarefted Gas Dynamics: Technical Papersfrom the 12th International Symposium, Progress in As-tronautics and Aeronautics Series (AIAA„New York,1980), Vol. 74, Part 1, p. 239.