Variational approach to gas flows in microchannelsCarlo Cercignani, Maria Lampis, and Silvia Lorenzani Citation: Phys. Fluids 16, 3426 (2004); doi: 10.1063/1.1764700 View online: http://dx.doi.org/10.1063/1.1764700 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v16/i9 Published by the American Institute of Physics. Related ArticlesNanoparticle collisions in the gas phase in the presence of singular contact potentials J. Chem. Phys. 137, 064316 (2012) High accuracy numerical solutions of the Boltzmann Bhatnagar-Gross-Krook equation for steady and oscillatoryCouette flows Phys. Fluids 24, 032004 (2012) Rare backflow and extreme wall-normal velocity fluctuations in near-wall turbulence Phys. Fluids 24, 035110 (2012) Binary scattering model for Lennard-Jones potential: Transport coefficients and collision integrals for non-equilibrium gas flow simulations Phys. Fluids 24, 027101 (2012) Admissible shock waves and shock-induced phase transitions in a van der Waals fluid Phys. Fluids 23, 086101 (2011) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Variational approach to gas flows in microchannelsCarlo Cercignani, Maria Lampis, and Silvia LorenzaniDipartimento di Matematica, Politecnico di Milano, Milano, Italy 20133

~Received 17 February 2004; accepted 28 April 2004; published online 2 August 2004!

Gas flow rates in microchannels have been rigorously evaluated by means of a variational techniquewhich applies to the integrodifferential form of the Boltzmann equation based on the Bhatnagar,Gross, and Krook model. The Maxwell scattering kernel has been used to describe the gas-wallinteractions in the most general case of two surfaces with different accommodationcoefficients. ©2004 American Institute of Physics.@DOI: 10.1063/1.1764700#

I. INTRODUCTION

Rarefied gas flows occur in many microelectromechani-cal systems~MEMS!,1,2 such as actuators, microturbines, gaschromatographs, and microair vehicles. A correct predictionof these flows is important to design and develop MEMS.Nanoscale design occurs for computer components as welland is no longer limited to chip technology but extends tomechanical devices as well. In a modern disk drive, the read/write head floats at distances of the order of 50 nm above thesurface of the spinning platter. The prediction of the verticalforce on the head~as obtained from the pressure distributionin the gas! is a crucial design calculation since the head willnot accurately read or write if it flies too high. If the headflies too low, it can catastrophically collide against the plat-ter. Microchannels may have further computer applicationsbecause they are supposed to dissipate the heat generated inmicrochips more effectively than fans, and may be used as amore practical cooling system in integrated circuit chips.3

Since, as these examples indicate, microdevices aregaining popularity both in commercial applications and inscientific research, there exists a rapidly growing interest inimproving the conventional design techniques related withthese devices. Microdevices are often operated in gaseousenvironments~typically air!, and thus their performances areaffected by the gas around them. The numerical simulationof all these flows cannot be performed with the Navier-Stokes equations~or the related Reynolds equation for aslider air bearing! because the smallest characteristic lengthof MEMS ~to be denoted byL) or of the thin air film occur-ring in a computer drive is comparable with~or smaller than!the mean free pathl of the gas molecules. For this reason thecontinuum equations are no longer valid and the Boltzmannequation must be considered4–6 to understand and computethe rarefied flows related with these devices.

The Boltzmann equation describes the time evolution ofthe distribution function of the gas molecules. Numericalmethods based on this equation are generally numericallyexpensive especially when the flow to be consideredprogresses from free molecular, through transitional, to con-tinuum regions. Since these flows, contrary to the flow pastspace vehicles, are usually at low Mach number, the use ofthe linearized Boltzmann equation is permissible and thisrevives old methods developed in the 1960s and 1970s to

deal with this equation.4–6 The Boltzmann equation must beused when the Knudsen number (Kn5l/L) is no longersmall enough to be negligible.

The flow between parallel plates has a simple geometryand has been studied by researchers analytically, experimen-tally, and numerically using different techniques.7–11 In thenumerical investigations, the direct simulation Monte Carlo~DSMC! method of Bird12 has proven to be a valuable tool,and various microchannel flows have been simulatedsuccessfully.13 Experimental data for flows inside channelsmuch larger than the microdevices are presented by Taguchiet al.7 for a wide range of Reynolds numbers covering mostof the transitional regime between collisionless flow andcontinuum limits.

Molecular based numerical schemes, such as the DSMCmethod,14 are physically appropriate for this kind of gasflows where noncontinuum, rarefied gas effects become im-portant. In the DSMC method, macroscopic observablequantities, such as velocity and temperature, are obtainedthrough averaging appropriate microscopic properties over asmall space region. The simulated results are therefore inher-ently accompanied by statistical noise due to finite samplesize. Some researchers9,15,16 applied the DSMC method tomicroscale gas flows like those in microchannels, and foundit very difficult to obtain statistically convergent results un-der experimental conditions.3 According to Ref. 13 thesignal-to-noise ratio in a dilute gas may be written as~seeRef. 17 for a more detailed discussion!

u/du5MaAgN,

whereu is the characteristic flow velocity,du is the statisti-cal fluctuation, Ma is the Mach number,g is the specific heatratio, andN is the sample size. A typical flow velocity inmicrochannel experiments3 is about 0.2 m/s that correspondsto Ma of 1024. The above relation indicates that if we re-quire the signal to be ten times larger than the noise,N mustbe 1010. Such an enormous sample size is extremely timeconsuming and beyond the capabilities of currentcomputers.18 According to Ref. 13 when Ma is small enoughfor compressibility effects to be negligible, favorable relativestatistical errors may be obtained by performing simulationsat a Ma increased to a level where compressibility is stillnegligible. This idea is consistent with the intuitive feeling

PHYSICS OF FLUIDS VOLUME 16, NUMBER 9 SEPTEMBER 2004

34261070-6631/2004/16(9)/3426/12/$22.00 © 2004 American Institute of Physics

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that, for small Ma, the deviation from a Maxwellian is pro-portional to the Mach number, no matter what Kn; this rea-sonable assumption justifies linearization but may bedoubted, as indicated in Ref. 19. While it is true that for lowMach number flows the number of statistical samples needsto be quite large in order to measure the flow velocity accu-rately, in most cases this can still be done with DSMC, giventhe speed of modern computers. Furthermore, the propertiesof interest, such as flow rate and skin friction, are integratedquantities, which reduce the statistical scatter.

Another method, the so-called information preservationmethod20–25 has been used to analyze subsonic, microscalegas flows. The statistically convergent results obtained in testproblems for characteristic velocities ranging from 0.01 m/sto 1 m/s, corresponding to Ma of 1025– 1023 and a normalsample size of 103– 104 were in excellent agreement with theexact solutions in the continuum and free molecular regimes,with those of the linearized Boltzmann equation24,26and alsowith the experimental data25 in the transition regime. Herewe shall adopt the linearized equation, since all the availablemathematical results27 indicate that this is correct for suffi-ciently low Mach numbers; how small these must is an openand interesting question.

An important aspect of the matter is to have an approxi-mate closed form solution for the flow rate of plane Poi-seuille flow in order to use it in applications. A typical useoccurs in the correction of the Reynolds equation in lubrica-tion theory proposed by Fukui and Kaneko,28 who in a sub-sequent paper provided a database29 for this flow rate. Thedatabase was interpolated by Robert with a formula given inRef. 17; this matter was further investigated by Veijolaet al.30,31 In order to develop a more accurate formula di-rectly from kinetic theory there is a particularly useful tech-nique, the variational method proposed by one of theauthors.32 We stress that this variational technique32 appliesdirectly to the integrodifferential form of the Boltzmannequation and can be used for any linearized Boltzmann equa-tion, while the variational methods used by other authorsapply to the integral form of the Boltzmann equation, whichis available only for special models, such as the Bhatnagar,Gross, and Krook~BGK! model.33 In this paper the varia-tional technique32 is used to compute the flow rate of planePoiseuille flow as a function of the Knudsen number by us-ing, for the sake of simplicity, the BGK model.33 The varia-tional results will be compared with accurate numerical so-lutions of the same equation.

We shall pay particular attention to the role of boundaryconditions. The boundaries will be permitted to have differ-ent gas-surface interaction properties, an aspect which, inthese days, is no longer academic and requires a particularattention. In fact, e.g., in the actuator considered by Veijolaet al.,30,31 the single-crystal silicon surface of the accelerom-eter’s proof mass is a potential candidate for noticeable re-flections close to specular, since it has a smooth, pure surfacehaving an ordered molecular structure, whereas the other sur-face ~a metallized electrode! is rough.

II. BOUNDARY-VALUE PROBLEM FOR POISEUILLEFLOW

Plane Poiseuille flow is the flow of a gas between twoparallel walls due to a pressure gradient parallel to thebounding surfaces.5,6,34 In the (x,z) plane, thez axis coin-cides with the direction of the fluid motion, while the twoplates, whose temperature is assumed to be constant, arefixed atx56d/2.

If the pressure gradient is small, it can be assumed thatthe velocity distribution of the flow is nearly the same as thatoccurring in an equilibrium state. This means that the Bolt-zmann equation can be linearized about a Maxwellianf 0 byputting

f 5 f 0~11h!, ~1!

where f (x,z,c) is the distribution function for the molecularvelocity c expressed in units of (2RT)1/2 (R being the gasconstant,T the temperature!, andh(x,c) is the small pertur-bation upon the basic equilibrium state. The above men-tioned Maxwellian is given by

f 0~z,c!5~11kz!r0p2 3/2exp~2c2!, ~2!

wherer0 is the density on the boundaries and

k51

p

]p

]z5

1

r

]r

]z,

with p andr being the gas pressure and density, respectively.If one assumes the linearized BGK model for the colli-

sion operator,33 the Boltzmann equation reads:34

1

2k1cx

]Z

]x5

1

u Fp2 ~1/2!E2`

1`

e2cx1

2Z~x,cx1

! dcx12Z~x,cx!G ,

~3!

where by definition

Z~x,cx!5p21E2`

1`E2`

1`

e2cy22cz

2czh~x,c!dcy dcz

and u is the collision time. Hence, the bulk velocity of thegas can be written as follows:

q~x!5p2 ~1/2!E2`

1`

e2cx1

2Z~x,cx1

!dcx1~4!

and the flow rateF ~per unit time through unit thickness! as

F5rE2d/2

d/2

q~x!dx. ~5!

The solution of Eq.~3! in integral form is35,36

Z~x,cx!5expF2S x1d

2sgncxD Y ~cxu!G

3ZS 2d

2sgncx ,cxD1E

2 ~d/2! sgncx

x

3expS 2ux2tuucxuu

D @q~ t !2ku/2#/~cxu!dt, ~6!

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with the values at the boundaryZ(2 (d/2)sgncx ,cx) depend-ing on the model of boundary conditions chosen. LetZ2(2d/2,cx), Z2(d/2,cx) be the distribution functions ofthe molecules impinging upon the walls andZ1(2d/2,cx),Z1(d/2,cx) the distribution functions of the molecules re-emerging from them. Enforcing Eq.~6! at the boundaries,one obtains two equations

Z2~2d/2,cx!5exp@d/~cxu!#Z1~d/2,cx!

1E2 ~d/2!

d/2

expS 2S d

21t D

ucxuuD

3@q~ t !2ku/2#

ucxuudt, ~7!

Z2~d/2,cx!5exp@2d/~cxu!#Z1~2d/2,cx!

1E2 d/2

d/2

expS 2S d

22t D

ucxuuD @q~ t !2ku/2#

ucxuudt.

~8!

These equations can be simply reexpressed in terms of theincoming stream adding the appropriate boundary conditionson the two plates. In the following, we will focus upon Max-well’s scattering kernel and specialize the analysis to wallshaving different physical properties so that two accommoda-tion coefficients (a1 ,a2) can be defined. In this case, theboundary conditions can be written as35,36

Z1~d/2,cx!5~12a1!Z2~d/2,2cx!, ~9!

Z1~2d/2,cx!5~12a2!Z2~2d/2,2cx!. ~10!

Substituting Eqs.~9! and ~10! into Eqs.~7! and ~8! one fi-nally obtains a system of two equations for the distributionfunctions of the impinging molecules. If we introduce twofunctionsS(cx) andT(cx) defined on the half spacecx.0 as

S~cx!5Z~2d/2,2cx!, T~cx!5Z~d/2,cx!, ~11!

such a system becomes

S~cx!5~12a1!exp@2d/~cxu!#T~cx!1x~cx!, ~12!

T~cx!5~12a2!exp@2d/~cxu!#S~cx!1f~cx!, ~13!

with

x~cx!5E2 d/2

d/2

expS 2S d

21t D

ucxuuD @q~ t !2ku/2#

ucxuudt, ~14!

f~cx!5E2 d/2

d/2

expS 2S d

22t D

ucxuuD @q~ t !2ku/2#

ucxuudt. ~15!

The system formed by Eqs.~12! and~13!, whose determinantis

D~cx!512~12a1!~12a2!exp@22d/~cxu!# ~16!

admits the solution

S~cx!5D~cx!21

†~12a1!exp@2d/~cxu!#f~cx!1x~cx!‡,~17!

T ~cx!5D~cx!21

†~12a2!exp@2d/~cxu!#x~cx!1f~cx!‡.~18!

If the two walls are identical (a15a25a), the problem issymmetric and the following relations hold:q(x)5q(2x),f(cx)5x(cx), S(cx)5T(cx). In the limiting cases of per-fectly reflecting surfaces (a→0) and of free molecular flow(d/u→0), the determinantD(cx) vanishes and the distribu-tion functionsS(cx), T(cx) are divergent. This divergencedoes not arise only ifa1Þa2 . This is because at least one ofthe accommodation coefficients must be necessarily differentfrom zero, which is to say, at least one of the walls slowsdown the flow.

Armed with the integral solution of the Boltzmann equa-tion @Eq. ~6!# subject to the boundary conditions explicitlygiven by Eqs.~17! and ~18!, the Poiseuille flow rate can beevaluated by means of a semianalytical procedure, based ona variational principle.5,6,37

III. THE VARIATIONAL APPROACH

Let us rewrite the Boltzmann equation given by Eq.~3!in a symbolic way35

~D2L !Z5S, ~19!

whereDZ5cx (]Z/]x) , S52k/2, and

LZ51

u Fp2 ~1/2!E2`

1`

e2cx1

2Z~x,cx1

! dcx12Z~x,cx!G .

The boundary conditions to be matched to Eq.~19! have thefollowing general expression:

Z15KZ2,

whereK is a generic operator whose explicit form dependson the scattering kernel used@see Eqs.~9! and ~10!#. Usingthe general technique described in Refs. 5, 6, and 37 weintroduce the following functionalJ of the test functionZ

J~ Z!5@„Z,P~DZ2LZ!…#22†~PS,Z!‡

1~ Z12KZ2,PZ2!B , ~20!

whereP is the parity operator in velocity space and@~,!#, (,)B

denote two scalar products defined as follows:

~h,g!5p21/2E2d/2

1d/2E2`

1`

exp~2cx2!

3h~x,cx!g~x,cx!dcxdx, ~21!

~h6,g6!B5p21/2E]VE

cx.0cx exp~2cx

2!

3h6~cx!g6~cx!dcxds. ~22!

Here g6, h6 are the restrictions of the functionsg and h,defined on the boundary, to positive, respectively negative,values ofcx . In the one-dimensional case, the integrationover the boundary]V reduces to the sum of the terms atx56d/2. The first variationdJ of J vanishes if and only if

3428 Phys. Fluids, Vol. 16, No. 9, September 2004 Cercignani, Lampis, and Lorenzani

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Z5Z, whereZ(x,cx) is the solution of Eq.~3! with appro-priate boundary conditions. Let us now consider the valueattained byJ whenZ5Z. Equation~20! gives

J~Z!52@~PS,Z!#5k

2@~1,Z!#. ~23!

Looking at the definitions~4! and~5!, it can be easily shownthat the stationary value ofJ is related to a quantity of physi-cal interest, the flow rate of the gas, through the relation

F52r

kJ~Z!. ~24!

It must be stressed that in the general formulation of thevariational principle@expressed by Eq.~20!# Z does not needto satisfy the same boundary conditions asZ. Nevertheless,taking advantage of the mathematical formulation presentedin Sec. II, where the values ofZ at the boundary have beenexplicitly evaluated, we can simplify the problem. Thus, inorder to obtain an expression for the trial functionZ(x,cx),the solution of the Boltzmann equation in integral form hasbeen considered

Z~x,cx!5expF2S x1d

2sgncxD Y ~cxu!G

3ZS 2d

2sgncx ,cxD1E

2 ~d/2! sgncx

x

expS 2ux2tuucxuu

D3@ q~ t !2ku/2#/~cxu!dt. ~25!

Assuming Maxwell’s boundary conditions imposed on walls

with different accommodation coefficients, the values at theboundary,Z(2 (d/2)sgncx ,cx), defined on the half spacecx

.0, are given by

Z~2d/2,2cx!5D~cx!21E

2d/2

1d/2

@ q~ t !2ku/2#/~cxu!

3$exp@2~d/21t !/~cxu!#1~12a1!

3exp@2~3d/22t !/~cxu!#%dt, ~26!

Z~d/2,cx!5D~cx!21E

2d/2

1d/2

@ q~ t !2ku/2#/~cxu!

3$exp@2~d/22t !/~cxu!#

1~12a2!exp@2~3d/21t !/~cxu!#%dt,

~27!

as forZ @Eqs.~17! and~18!#. The numerical results presentedin Ref. 36 have shown that for nonsymmetric gas-wall inter-actions the bulk velocity profile can be approximated by aparabola asymmetric with respect to the channel axis. Thisenables us to assume the following test function for the bulkvelocity of the gas

q~x!5Ax21Bx1C, ~28!

whereA, B, andC are adjustable constants to be varied inorder to obtain the best value ofJ(Z). Putting in Eq.~25! theexpressions ofZ(2 (d/2)sgncx ,cx) andq(x), given by~26!–~28!, by means of analytical calculations, the trial functionZcan be explicitly written as follows:

Z~x,cx!5D~cx!21expF2S x1

d

2sgncxD Y ~cxu!G$@12F~a1 ,a2!#†~C2ku/2!1A/4~d224dcxu sgncx18cx

2u2!

1B/2~2d sgncx12cxu!2exp@2d sgncx /~cxu!#@~C2ku/2!1A/4~d214dcxu sgncx18cx2u2!

1B/2~d sgncx12cxu!#‡1~12a1!~12a2!†exp@2d sgncx /~cxu!#

3@~C2ku/2!1A/4~d224dcxu sgncx18cx2u2!2B/2~2d sgncx12cxu!#

2exp@22d sgncx /~cxu!#@~C2ku/2!1A/4~d214dcxu sgncx18cx2u2!2B/2~d sgncx12cxu!#‡%1~C2ku/2!

1A~x222xcxu12cx2u2!1B~x2cxu!2expF2S x1

d

2sgncxD Y ~cxu!G@~C2ku/2!1A/4~d214dcxu sgncx

18cx2u2!2B/2~d sgncx12cxu!#, ~29!

where

F~a1 ,a2!5H a2 , cx.0

a1 , cx,0.

Since this test function satisfies the same boundary con-ditions asZ, the last term on the right-hand side of Eq.~20!can be dropped because it is identically zero, leading to a

much simpler variational formulation. It is convenient nowto rescale all variables appearing in Eq.~29! as follows:

d5d/u; w5t/u; u5x/u; A5A/u2; B5B/u.

SubstitutingZ, given by Eq.~29!, in the reduced expressionof J(Z), we obtain the following polynomial of the secondorder with respect to the constantsA, B, andC, that are tobe determined

3429Phys. Fluids, Vol. 16, No. 9, September 2004 Variational approach to gas flows in microchannels

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J~ Z!5~Ap!21H c11

2A21

c22

2B21

c33

2C21c12AB1c13AC

1c23BC2c1A2c2B2c3C11

2~c32c33/4!J , ~30!

where the same symbolJ(Z) has been used to denote thedimensionless quantityJ(Z)/(ku)2. The coefficients in non-dimensional form are given by

c115d4

82

Ap

6d31d21242S d4

412d2DT1~d!2~2d3

18d!T2~d!2~8d2116!T3~d!216dT4~d!216T5~d!

1~22a12a2!F2S d4

81d2DS1~0!1S d4

412d2DS1~d!

2S d4

81d2DS1~2d!1~d314d!S2~0!

2~d314d!S2~2d!2~4d218!S3~0!

116S3~d!2~4d218!S3~2d!18dS4~0!28dS4~2d!

28S5~0!116S5~d!28S5~2d!G12~12a1!~12a2!F2S d4

81d2DS1~3d!

2S d4

81d2DS1~d!1S d4

412d2DS1~2d!

2~4d218!S3~d!116S3~2d!28S5~d!116S5~2d!

1~d314d!S2~d!18dS4~d!2~d314d!S2~3d!

2~4d218!S3~3d!28dS4~3d!28S5~3d!G12

ApI 1 ,

~31!

c125~a22a1!F2S d3

41d DS1~0!1S d3

41d DS1~2d!

1S 3

2d212DS2~0!1S 3

2d212DS2~2d!24dS3~0!

14dS3~2d!14S4~0!14S4~2d!1~d224!S2~d!

28S4~d!G12

ApI 2 , ~32!

c135d2

22Apd162~d214!T1~d!24dT2~d!28T3~d!

1~22a12a2!F2S d2

212DS1~0!1~d214!S1~d!

2S d2

212DS1~2d!12dS2~0!

22dS2~2d!24S3~0!18S3~d!24S3~2d!G12~12a1!~12a2!

3F2S d2

212DS1~3d!2S d2

212DS1~d!

1~d214!S1~2d!24S3~d!18S3~2d!

12dS2~d!22dS2~3d!24S3~3d!G12

ApI 3 , ~33!

c152Ap

12d31

d2

22Apd152~d212!T1~d!

24dT2~d!28T3~d!2~22a12a2!

3F S d2

211DS1~0!2~d212!S1~d!

1S d2

211DS1~2d!22dS2~0!12dS2~2d!14S3~0!

28S3~d!14S3~2d!G22~12a1!~12a2!

3F S d2

211DS1~3d!1S d2

211DS1~d!2~d2

12!S1~2d!14S3~d!28S3~2d!22dS2~d!

12dS2~3d!14S3~3d!G11

ApI 3 , ~34!

c225d2

2221d2T1~d!14dT2~d!14T3~d!2~22a1

2a2!Fd2

2S1~0!1d2S1~d!1

d2

2S1~2d!22dS2~0!

12dS2~2d!12S3~0!24S3~d!12S3~2d!G12~1

2a1!~12a2!Fd2

2S1~3d!1

d2

2S1~d!1d2S1~2d!

12S3~d!24S3~2d!22dS2~d!12dS2~3d!

12S3~3d!G12

ApI 4 , ~35!

c235~a22a1!@2dS1~0!1dS1~2d!12S2~0!

12S2~2d!24S2~d!#12

ApI 5 , ~36!

c25~a22a1!@2dS1~0!1dS1~2d!12S2~0!12S2~2d!

24S2~d!#11

ApI 5 , ~37!

3430 Phys. Fluids, Vol. 16, No. 9, September 2004 Cercignani, Lampis, and Lorenzani

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c335224T1~d!2~22a12a2!@2S1~0!24S1~d!

12S1~2d!#22~12a1!~12a2!@2S1~3d!

12S1~d!24S1~2d!#12

ApI 6 , ~38!

c352Apd1224T1~d!2~22a12a2!@2S1~0!

24S1~d!12S1~2d!#22~12a1!~12a2!

3@2S1~3d!12S1~d!24S1~2d!#11

ApI 6 , ~39!

whereTn(x) is an Abramowitz function defined by

Tn~x!5E0

1`

tn exp~2t22x/t !dt

andSn(x) is a generalized Abramowitz function defined by

Sn~x,d,a1 ,a2!5E0

1` tn exp~2t22x/t !

12~12a1!~12a2!exp~22d/t !dt.

The symbolsI 1 , I 2 , I 3 , I 4 , I 5 , I 6 stand for integral expres-

sions involving theTn(x) and Sn(x) functions, which are

negligible in the free molecular flow regime. The explicit

form of these integrals is given in the Appendix.

The derivatives ofJ(Z) with respect toA, B, C vanishin correspondence of the optimal values of these constants.

The resulting expression for the minimum ofJ(Z) is

minJ~ Z!5~8Ap!21@c132 c2222c12c13c231c12

2 c331c11~c232 2c22c33!#

21†28c11c2c23c314c11c23

2 c324c122 c3

214c11c22c32

14c11c22c332c11c23

2 c3314c122 c3c33

24c11c22c3c332c122 c33

2 1c11c22c332 28c1c12~c2c332c23c3!24c1

2~c232 2c22c33!12c13~4c1c2c2314c12c2c3

24c1c22c324c12c23c31c12c23c33!2c132 @4c2

21c22~c3324c3!#‡. ~40!

Thus, the computation of the optimal value of the functionalJ(Z) ~Eq. 40! will lead to an accurate estimate of the flowrate of the gas, which in nondimensional form reads

Q~d!5F

2r

2kd2

524

d2 J~Z!, ~41!

where d is the rarefaction parameter~inverse Knudsennumber!.

IV. RESULTS AND DISCUSSION

The rigorous computation of the Poiseuille flow rateQ,presented in this paper, is mainly motivated by its relevanceto several applications. In the 1980s Fukui and Kaneko de-rived a generalized Reynolds equation for lubricationproblems,28 based on the linearized Boltzmann equation,which holds for arbitrary Knudsen number gas films andarbitrary accommodation coefficients. The solution of thisequation requires that the Poiseuille flow rate has to be ac-curately calculated in advance. To this end, Fukui andKaneko gave a database,29 created by both numerical andvariational calculations38 ~based on an integral approach tothe BGK model!, for the values of the gas flow rateQ versusthe inverse of the Knudsen numberd, with d ranging from0.01 to 100.0 and the accommodation coefficienta of thebounding surfaces~supposed physically identical! varying inthe interval@0.7, 1#. This database is reported in Tables I and

II along with our results obtained by means of the variationalapproach presented in Sec. III and the numerical methoddescribed in Ref. 36, for a comparison.

As revealed by these tables, the agreement between ournumerical and variational outputs is very good~the relativeerror is less than 0.1%!. It is to be noted that our numericalfindings have been compared in Ref. 36 with those publishedby other authors,39–41 in the case of Maxwell’s boundaryconditions on two physically identical walls, showing a basicgood agreement. Of course, the variational approach carriedout here has the enormous advantage of reducing drasticallythe computational CPU time, giving at one time the sameaccuracy for each value of the rarefaction parameterd. Forthis reason, only our variational datas are listed in Tables Iand II for d>10.0. If one wanted to reach the same highaccuracy using the finite difference numerical technique de-scribed in Refs. 34 and 36, the number of mesh points shouldbe considerably increased asd increases, with the loss in theefficiency of the calculation. Some considerations are in or-der here concerning the results obtained by Fukui andKaneko which compare poorly with our outputs. Their varia-tional findings are inaccurate for every value ofd as a de-creases, while the numerical outputs are in fair agreementonly in a narrowd range which reduces decreasing the ac-commodation coefficient. This is demonstrated in Fig. 1which shows as a function of the rarefaction parameterd theflow rateQ evaluated by Fukui and Kaneko. For comparison,our variational results are drawn in the same figure. Theinaccuracy of the tabulated values in Ref. 29 is likely relatedto the high accuracy required in the computation of the gen-

3431Phys. Fluids, Vol. 16, No. 9, September 2004 Variational approach to gas flows in microchannels

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eralized Abramowitz functionsSn especially for small valuesof d. In our numerical method and variational analysis solu-tion, theSn functions have been numerically evaluated usingthe SLATEC libraries. We note in passing that a likely reasonfor the slight deviation registered between the DSMC datasobtained by Alexanderet al.17 and the Fukui and Kanekomodified Reynolds equation fora50.7 can be related to theinaccuracy of the tabulated values in Ref. 29.

Beyond the lubrication problems, the Reynolds equationin linearized form has been applied in modeling the gasdamping effects acting on oscillating microstructures in amicromechanical accelerometer.30,31In ultrathin films the gasrarefaction effects have been taken into account by replacingthe gas viscosity with an effective viscosity which has beenexpressed as a function of the Poiseuille flow rate coefficientQ.31 Again a rigorous evaluation ofQ is required. In order to

TABLE I. Poiseuille flow rateQ vs d for Maxwell’s boundary conditions on identical walls. Comparison between our results~CLL! and Fukui and Kaneko’sresults Ref. 29.

d

a50.9 a51

Num. ~Ref. 29! Var. ~Ref. 29! Num. ~CLL! Var. ~CLL! Num. ~Ref. 29! Num. ~CLL! Var. ~CLL!

1.31025¯ ¯ 8.194 8.194 ¯ 6.855 6.855

1.31023¯ ¯ 5.042 5.042 ¯ 4.274 4.274

0.01 3.564 3.566 3.557 3.557 3.060 3.050 3.0500.03 2.923 2.942 2.924 2.924 2.522 2.524 2.5240.06 2.570 2.578 2.570 2.570 2.228 2.227 2.2270.09 2.384 2.391 2.383 2.383 2.071 2.071 2.0710.1 2.338 2.345 2.338 2.338 2.033 2.033 2.0330.3 1.947 1.946 1.946 1.946 1.703 1.702 1.7020.5 1.826 1.821 1.826 1.826 1.602 1.602 1.6020.7 1.774 1.729 1.773 1.773 1.559 1.559 1.5591.0 1.746 1.750 1.745 1.745 1.539 1.539 1.5392.0 1.789 1.795 1.793 1.793 1.595 1.595 1.5953.0 1.886 1.909 1.908 1.908 1.711 1.710 1.7105.0 2.141 2.189 2.190 2.189 1.991 1.990 1.9907.0 2.431 2.494 2.495 2.494 2.292 2.294 2.294

10.0 2.898 2.969 2.970 2.969 2.768 2.767 2.76720.0 4.494 4.598 ¯ 4.600 4.398 ¯ 4.39430.0 6.102 6.250 ¯ 6.253 6.049 ¯ 6.04560.0 10.737 ¯ ¯ 11.237 11.033 ¯ 11.02980.0 13.725 ¯ ¯ 14.567 14.363 ¯ 14.358

100.0 16.779 ¯ ¯ 17.898 17.693 ¯ 17.689

TABLE II. Poiseuille flow rateQ vs d for Maxwell’s boundary conditions on identical walls. Comparison between our results~CLL! and Fukui and Kaneko’sresults~Ref. 29!.

d

a50.7 a50.8

Num. ~Ref. 29! Var. ~Ref. 29! Num. ~CLL! Var. ~CLL! Num. ~Ref. 29! Var. ~Ref. 29! Num. ~CLL! Var. ~CLL!

1.31025¯ ¯ 11.923 11.923 ¯ ¯ 9.841 9.841

1.31023¯ ¯ 7.147 7.147 ¯ ¯ 5.978 5.978

0.01 4.932 5.868 4.929 4.929 4.174 4.388 4.169 4.1690.03 4.004 4.599 4.005 4.005 3.406 3.540 3.407 3.4070.06 3.497 3.922 3.496 3.496 2.984 3.084 2.983 2.9830.09 3.232 3.572 3.232 3.232 2.762 2.842 2.762 2.7620.1 3.168 3.487 3.168 3.168 2.708 2.783 2.708 2.7080.3 2.620 2.781 2.620 2.620 2.245 2.279 2.245 2.2450.5 2.452 2.570 2.452 2.452 2.102 2.124 2.102 2.1020.7 2.376 2.475 2.376 2.376 2.039 2.054 2.039 2.0391.0 2.330 2.417 2.330 2.330 2.002 2.009 2.002 2.0022.0 2.324 2.450 2.360 2.360 2.026 2.059 2.041 2.0413.0 2.357 2.572 2.472 2.472 2.096 2.175 2.155 2.1555.0 2.540 2.864 2.757 2.756 2.320 2.458 2.438 2.4377.0 2.794 3.175 3.066 3.065 2.595 2.765 2.746 2.744

10.0 3.246 3.654 3.546 3.544 3.054 3.242 3.223 3.22120.0 4.803 5.290 ¯ 5.181 4.633 4.875 ¯ 4.85530.0 6.432 6.945 ¯ 6.836 6.251 6.528 ¯ 6.50960.0 11.024 ¯ ¯ 11.823 10.869 ¯ ¯ 11.49580.0 13.935 ¯ ¯ 15.153 13.823 ¯ ¯ 14.825

100.0 17.051 ¯ ¯ 18.485 16.906 ¯ ¯ 18.156

3432 Phys. Fluids, Vol. 16, No. 9, September 2004 Cercignani, Lampis, and Lorenzani

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model the damping produced by a squeezed gas film in anaccelerometer where one surface is rough~metallized elec-trode! while the other surface is smooth~single-crystal sili-con!, Veijola et al.31 solved the linearized Boltzmann equa-tion imposing different boundary conditions on the twowalls. In particular, they suggested that the metallized sur-face reflects the molecules diffusely (a251) while the otherwall reflects specularly a noticeable part of the particles(a1>0). In Table III we list the Veijolaet al. results31 ob-tained by combining a variational analysis and a numericalapproach, along with our variational outputs. As shown bythe table, the agreement is very good in the transitional re-gime as well as in the continuum flow limit.

Since walls with different physical structures are fullyused in designing micromachines and other devices withvery tiny gas gaps, Table IV shows the Poiseuille flow rateQ, computed by our variational analysis, versus the rarefac-tion parameterd in the general, nonsymmetrical case,a1

Þa2 . A comparison with the numerical results presented inRef. 36 reveals that the agreement is of the same quality as

found fora15a25a ~see Tables I and II!. Therefore a com-plete and general database of practical usefulness can be cre-ated exploiting the ability of our variational method to givehighly accurate results in the free molecular flow limit aswell as in the continuum regime.

V. CONCLUDING REMARKS

In the present paper a variational approach has been usedto solve the plane Poiseuille problem between two parallelplates as an issue of relevance for applications. The varia-tional method turns out to be quite precise in evaluating theflow rateQ progressing from free molecular, through transi-tional, to continuum regions. Since the implementation ofour variational solution is computationally straightforward,an accurate database can be created for the gas flow rates atdifferent Knudsen numbers.

Knowing that walls with different physical structures arefully used in designing micromachines, different accommo-dation coefficients (a1Þa2) have been considered in the

FIG. 1. Poiseuille flow rateQ vs d for Maxwell’s boundary conditions on identical walls. Comparison between the numerical~circles! and variational~squares! data tabulated by Fukui and Kaneko~Ref. 29! and our variational results~triangles!.

3433Phys. Fluids, Vol. 16, No. 9, September 2004 Variational approach to gas flows in microchannels

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TABLE III. Poiseuille flow rateQ vs d for Maxwell’s boundary conditions on different walls.a251. Com-parison between our results~CLL! and Veijolaet al.’s results~Ref. 31!.

d

a150.0 a150.25 a150.5 a150.75

CLL Reference 31 CLL Reference 31 CLL Reference 31 CLL Reference 31

1.31025 12.929 ¯ 11.410 ¯ 9.892 ¯ 8.373 ¯

1.31023 7.790 ¯ 6.908 ¯ 6.028 ¯ 5.150 ¯

0.01 5.422 5.422 4.819 4.819 4.222 4.222 3.633 3.6330.03 4.455 4.455 3.953 3.953 3.464 3.464 2.988 2.9880.07 3.835 3.835 3.386 3.386 2.959 2.959 2.553 2.5530.1 3.616 3.616 3.181 3.182 2.774 2.774 2.392 2.3920.2 3.282 3.282 2.854 2.854 2.470 2.470 2.123 2.1230.5 3.077 3.077 2.596 2.596 2.204 2.203 1.877 1.8770.7 3.095 3.095 2.564 2.564 2.153 2.153 1.826 1.8261.0 3.190 3.189 2.576 2.576 2.134 2.134 1.800 1.8003.0 4.281 4.279 3.014 3.013 2.368 2.367 1.975 1.9747.0 6.826 6.821 3.969 3.966 3.039 3.037 2.575 2.572

10.0 8.790 8.784 4.617 4.613 3.549 3.545 3.055 3.05120.0 15.409 15.402 6.556 6.551 5.233 5.228 4.692 4.68850.0 35.378 35.370 11.813 11.806 10.249 10.243 9.672 9.66770.0 48.705 48.697 15.207 15.201 13.586 13.580 13.001 12.996

100.0 68.701 68.692 20.257 20.250 18.589 18.583 17.997 17.992

TABLE IV. Poiseuille flow rateQ vs d for Maxwell’s boundary conditions on different walls.

d a2 a150.1 a150.3 a150.5 a150.7 a150.8 a150.9

1.31025 0.3 48.084 32.598 23.412 17.285 14.918 12.8820.5 30.125 23.412 18.371 14.432 12.763 11.2590.8 17.256 14.918 12.763 10.778 9.841 9.303

0.01 0.3 16.804 11.921 8.918 6.852 6.036 5.3250.5 11.161 8.918 7.200 5.832 5.246 4.7110.8 6.896 6.036 5.246 4.515 4.169 3.837

0.05 0.3 12.137 8.582 6.446 4.998 4.431 3.9390.5 8.073 6.446 5.223 4.264 3.856 3.4860.8 5.070 4.431 3.856 3.333 3.090 2.857

0.1 0.3 10.733 7.520 5.634 4.375 3.887 3.4660.5 7.096 5.634 4.556 3.724 3.375 3.0600.8 4.466 3.887 3.375 2.918 2.708 2.508

0.5 0.3 8.906 6.013 4.446 3.459 3.090 2.7780.5 5.786 4.446 3.544 2.896 2.635 2.4060.8 3.660 3.090 2.635 2.263 2.102 1.954

1.0 0.3 8.683 5.763 4.257 3.337 3.000 2.7170.5 5.673 4.257 3.368 2.758 2.520 2.3140.8 3.655 3.000 2.520 2.154 2.002 1.865

5.0 0.3 9.493 6.141 4.742 3.970 3.700 3.4790.5 6.825 4.742 3.773 3.210 3.009 2.8410.8 5.075 3.700 3.009 2.590 2.437 2.309

10.0 0.3 10.694 6.945 5.600 4.903 4.666 4.4730.5 8.297 5.600 4.569 4.020 3.831 3.6770.8 6.754 4.666 3.831 3.379 3.221 3.093

60.0 0.3 20.315 15.247 14.011 13.448 13.265 13.1210.5 18.707 14.011 12.860 12.333 12.163 12.0280.8 17.743 13.265 12.163 11.658 11.495 11.366

100.0 0.3 27.258 21.910 20.690 20.143 19.967 19.8280.5 25.793 20.690 19.522 18.998 18.830 18.6970.8 24.928 19.967 18.830 18.320 18.156 18.027

3434 Phys. Fluids, Vol. 16, No. 9, September 2004 Cercignani, Lampis, and Lorenzani

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frame of Maxwell’s model for boundary conditions. This as-pect deserves a particular attention since in ultrathin films,when gas rarefaction effects dominate, the molecular inter-actions with the surfaces affect strongly the velocity profileand the gas flow rate. If two identical walls are considered,the Poiseuille flow rate increases significantly when goingfrom the complete Maxwell diffusion (a51) to the entirelyspecular reflection (a50). Previous works already pointed

out that, for Maxwell reemission, the flow rate diverges asd→0 ~free molecular flow limit!. In addition, our resultsshow that this divergence is enhanced in the limit of entirelyspecular reflection: for free molecular flow,Q(d) divergesfaster asa decreases. If the symmetry of the problem isbroken by assuming two different accommodation coeffi-cients (a1 , a2) for the bounding surfaces, the wall whichreflects specularly less molecules slows down the flow.

APPENDIX

I 15E0

d/2

duH 22Fd2

4~T0~d/22u!1T0~d/21u!!1d@T1~d/22u!1T1~d/21u!#12@T2~d/22u!1T2~d/21u!#G2

1Fd2

2@T0~d/22u!1T0~d/21u!#12d3@T1~d/22u!1T1~d/21u!#14@T2~d/22u!1T2~d/21u!#GF S d2

4L 0

22dL 11

12L 22D ~a1 ,a2!1S d2

4L 0

22dL 1112L 2

2D ~a2 ,a1!G12

ApF S d2

4L 0

22dL 1112L 2

2D 2

~a1 ,a2!

1S d2

4L 0

22dL 1112L 2

2D 2

~a2 ,a1!G J , ~A1!

I 25E0

d/2

duH Fd2

4@T0~d/22u!1T0~d/21u!#1d@T1~d/22u!1T1~d/21u!#12@T2~d/22u!1T2~d/21u!#G

3F S d

2A 0

12A 12D ~a1 ,a2!2S d

2A 0

12A 12D ~a2 ,a1!G1Fd2 @T0~d/21u!2T0~d/22u!#1@T1~d/21u!2T1~d/22u!#G

3F2S d2

4L 0

22dL 1112L 2

2D ~a1 ,a2!1S d2

4L 0

22dL 1112L 2

2D ~a2 ,a1!G12

ApF S d

2A 0

12A 12D ~a1 ,a2!

3S d2

4L 0

22dL 1112L 2

2D ~a1 ,a2!2S d

2A 0

12A 12D ~a2 ,a1!3S d2

4L 0

22dL 1112L 2

2D ~a2 ,a1!G J , ~A2!

I 35E0

d/2

duH Fd2

4@T0~d/22u!1T0~d/21u!#1d@T1~d/22u!1T1~d/21u!#123@T2~d/22u!1T2~d/21u!#G

3@22@T0~d/22u!1T0~d/21u!#1L 02~a1 ,a2!1L 0

2~a2 ,a1!#1@T0~d/22u!1T0~d/21u!#

3F S d2

4L 0

22dL 1112L 2

2D ~a1 ,a2!1S d2

4L 0

22dL 1112L 2

2D ~a2 ,a1!G12

ApFL 0

2~a1 ,a2!

3S d2

4L 0

22dL 1112L 2

2D ~a1 ,a2!1L 02~a2 ,a1!S d2

4L 0

22dL 1112L 2

2D ~a2 ,a1!G J , ~A3!

I 45E0

d/2

duH 21

2†d@T0~d/21u!2T0~d/22u!#12@T1~d/21u!2T1~d/22u!#‡21†d@T0~d/21u!2T0~d/22u!#

12@T1~d/21u!2T1~d/22u!#‡F2S d

2A 0

12A 12D ~a1 ,a2!2S d

2A 0

12A 12D ~a2 ,a1!G

12

ApF S d

2A 0

12A 12D 2

~a1 ,a2!1S d

2A 0

12A 12D 2

~a2 ,a1!G J , ~A4!

3435Phys. Fluids, Vol. 16, No. 9, September 2004 Variational approach to gas flows in microchannels

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I 55E0

d/2

duH @T0~d/22u!1T0~d/21u!#F S d

2A 0

12A 12D ~a1 ,a2!2S d

2A 0

12A 12D ~a2 ,a1!G

1Fd2 @T0~d/21u!2T0~d/22u!#1@T1~d/21u!2T1~d/22u!#G@2L 02~a1 ,a2!1L 0

2~a2 ,a1!#12

ApFL 0

2~a1 ,a2!

3S d

2A 0

12A 12D ~a1 ,a2!2L 0

2~a2 ,a1!S d

2A 0

12A 12D ~a2 ,a1!G J , ~A5!

I 65E0

d/2

duH 22@T0~d/22u!1T0~d/21u!#212@T0~d/22u!1T0~d/21u!#@L 02~a1 ,a2!1L 0

2~a2 ,a1!#

12

Ap@~L 0

2!2~a1 ,a2!1~L 02!2~a2 ,a1!#J , ~A6!

where

L n6~a1 ,a2!5~12a1!Sn~d/22u!1~12a2!Sn~d/21u!6@~12a1!Sn~3d/22u!1~12a2!Sn~3d/21u!#

1~12a1!~12a2!@Sn~3d/22u!1Sn~3d/21u!#6~12a1!~12a2!@Sn~5d/22u!1Sn~5d/21u!#,

A n6~a1 ,a2!5~12a1!Sn~d/22u!2~12a2!Sn~d/21u!6@~12a1!Sn~3d/22u!2~12a2!Sn~3d/21u!#

1~12a1!~12a2!@Sn~3d/21u!2Sn~3d/22u!#6~12a1!~12a2!@Sn~5d/21u!2Sn~5d/22u!#.

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