A STRICT KINETIC SOLUTION OF THE FINITE LENGTH MICROCHANNEL FLOW PROBLEM. Ching Shen

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1 Prceedings f ICMM005 rd Internatinal Cnference n Micrchannels and Minichannels June Trnt Ontari Canada ICMM A STRICT KINETIC SOLUTION OF THE FINITE LENGTH MICROCHANNEL FLOW PROBLEM Ching Shen High Temperature Gas Dynamics Labratry Institute f Mechanics Chinese Academy f Sciences 15 Zhngguancun Rad Beijing China Tel Fax cshen@imech.ac.cn ABSTRACT In finite length tube r channel gas flw the pressure gradient is determined by the glbal mass cnservatin law. In the cntinuum and slip flw the pressure distributin determined by the glbal mass cnservatin is given analytically. In the transitinal flw regime an equatin cntaining the flw rate f the Piseuille flw slved by the strict kinetic thery is btained and is shwn t be the degenerated Reynlds equatin fr the lubricatin thery. The integratin f the equatin is illustrated in the case f full diffuse reflectin f the channel wall. The pressure distributin thus btained is shwn t be in excellent agreement with experimental data f lng micrchannels and the simulatin results f the infrmatin preservatin methd. The results as having the strict kinetic theretical merit are used t cnfirm the unfeasibility f the Lattice Bltzmann methd in the transitinal flw regime. Keywrds : micrchannel flw pressure distributin MEMS infrmatin preservatin (IP) methd Lattice Bltzmann methd (LBM) INTRODUCTION Micrchannel is the basic cnstituent f the MEMS devices its gemetric frm is regular and simple but the gas flw in it is the basic flw pattern in MEMS and can reveal the specific features f the lw speed micr internal flws s it has been a subject f grwing intensive investigatin. In the classical fluid mechanics the steady Piseuille flw is cnsidered as a flw in lng tube r channel having cnstant pressure gradient G= p/ x (see e.g. Batchelr [1]). In the kinetic theretical cnsideratin a cnstant pressure gradient is als prpsed fr the linearized Bltzmann slutin f the Piseuille flw f gas (see e.g. Ohwada et al. []). But the cnstancy f pressure gradient in a tube r channel is true nly fr the flw f liquid (with cnstant density) and as shwn in the fllwing in any realistic tube r channel gas flw the pressure gradient is never a cnstant but varies frm ne crss sectin t anther and is determined by the glbal mass cnservatin law. The assumptin f the cnstancy f the pressure gradient in the Piseuille flw is nt necessary in fact the slutin f the Piseuille flw determines nly the lcal flw characteristics in dependence f the lcal pressure gradient. As it is shwn in the fllwing the glbal mass cnservatin defines the pressure distributin in the cntinuum flw regime and the slip flw regime. In the transitinal flw regime a strict kinetic theretical slutin f the channel flw pressure distributin is als prvided by the mass cnservatin acrss the channel given the kinetic theretical slutin f the mass flux fr the Piseuille flw is knwn. The channel pressure distributin s determined is cmpared with the experimental data f Png et al. and Shih et al.[4] and the numerical simulatin result by the infrmatin preservatin (IP) methd [5]. It is used als t test the validity f the calculatin f the channel flw by the Lattice Bltzmann methd accmplished by Nie et al. [6]. NOMENCLATURE h: height f the channel Kn: Knudsen number L: length f the channel p: pressure P: =p/p dimensiinless pressure Q m : mass flw rate Q m : dimensinless mass flw rate R: gas cnstant T: temperature u: x directin velcity x: Cartesian crdinate in the channel axis directin y: Cartesian crdinate nrmal t the channel surface z: the third Cartesian crdinate ς : slip cefficient = ( ) λ/ ς : ς = ς p= ( ) λp/ λ : mean free path µ : viscsity ρ : density 1 Cpyright 005 by ASME Dwnladed Frm: n 06/9/019 Terms f Use:

2 : tangential reflectin cefficient subscript C: cntinuum flw i: inlet ; utlet SLIP: slip flw TR: transitinal flw 1. The pressure distributin determined by the glbal mass cnservatin Cnsider the gas flw in a channel between tw plane plates at a crss sectin x with the lcal pressure gradient dp/dx which can vary frm ne crss sectin t anther (see Fig.1). First we cnsider the cntinuum flw case. In the case f lw speed the temperature variatin is neglected µ is a cnstant the Navier-Stkes equatin attains the frm d u dp dx =. (1) dy µ Fig. 1 The schematic f the Piseuille flw Equatin (1) with the slipless bundary cnditin u=0 at y=h/ yields 1 dp u = ( h 4y ) 8µ dx. () The mass flux Q mc flwn in unit time thrugh the gap between the tw plates (with unit length in z) is bviusly h h QmC = ρ udy =. () RT dx 1µ 0 The glbal mass cnservatin acrss any crss sectin requires that p(dp/dx)=cnstant (4) r in the dimensinless variables P = p/ p0 and X=x/L( ( p : the pressure at the utlet L : the length f the channel) PdP/dX=cnstant. (5) Frm the abve equatin with the inlet and utlet bundary cnditins P = Pi pi / p at X = 0 and P=1 at X=1 (6) the pressure distributin is easily determined 1 P P[1 (1 ) X] 1/ = i (7) Pi with the pressure gradient dp / = P(1 )[1 (1 ) X]. (8) dx P P i i i Obviusly the pressure distributin is apprximately linear if the value (1 1/ P i ) is small and its slpe is negative with Pi > 1 and the abslute value f the pressure slpe mntnusly increases with increasing X. In the case f slip flw the basic equatin (1) remains the same the velcity prfile in the case f slip bundary cnditin du u y= h/= ζ with ζ = λ is dy u 1 dp = ( 4 4 ) 8 h y + ζ µ h. (9) dx And the mass flux instead f Eq. () is btained as mslip h 1 1 ζ RT dx 4µ 0. (10) Q = ρ udy = h + h Nte that ς is dependent n p: ς = ( ) λ/ and as ρλ = cnstant ne can write ς = ς / p where ς = λp = λp. (11). S frm Eq. (10) the mass cnversatin can be written as dp ς dp p + 6 = cnstant dx h dx r in dimensinless variables dp dp P + 6 Kn0 = cnstant (1) dx dx where Kn = λ / h is Knudsen number at the utlet f the channel. The slutin f the abve equatin with the integratin cnstants determined frm the inlet and utlet bundary cnditins Eq. (6) is P = 6 Kn + (6 Kn) + Pi + 1 KnPi + [1 Pi + 1 Kn(1 Pi)] X (1) Nte that the mass flux Eq. (10) in slip flw als can be written as h Q mslip = (1 + 6 Kn) (14) 1µ RT dx r h QmSLIP = QmSLIP QmSLIP = 1+ 6 Kn 1µRT dx (15) if the first half f Eq. (11) is used fr ς and here Kn= λ /h is the lcal Knudsen number. This is the well knwn slutin f the mass flw rate f the Piseuille flw in the slip flw regime where p dp/dx and Kn are lcal values f pressure pressure gradient and Knudsen number. The abve expressin f pressure distributin in the slip flw regime seems t be btained the earliest by Arkilic Cpyright 005 by ASME Dwnladed Frm: n 06/9/019 Terms f Use:

3 thrugh a different derivatin as a first apprximatin with accuracy f rder ε = H / L(see Ph.D. Thesis f E. B. Arkilic [7] and its publicatin versin [8]) and the similar results have been als published and recrded in [ ]. In the transitinal flw regime the flw rate f the plane Pisuille flw has been slved by the linearized Bltzmann equatin r its BGK [1] mdel versin by many authrs (see e.g. by Cercignani and Daneri [1] and Ohwada et al. []). Fukui and Kanek [14] in derivatin f the generalized Reynlds lubricatin equatin als calculated the flw rate f Piseuille flw numerically and later [15] they had used this flw rate calculated rigrusly t generate a database fr rapid calculatin f the generalized Reynlds equatin fr high Knudsen numbers. The flw rate f the Piseuille flw can be expressed in the frm h Q mtr = QmTR ( Kn) (16) 1µRT dx where Kn is the lcal Knudsen number and Q mtr ( Kn ) is the flw rate in transitinal regime (nrmalized by the slip-less value Q mc ) calculated frm the linearized Bltzmann equatin fr Piseuille flw. A tabled database f the calculated values f QmTR ( Kn ) fr = 1 = 0.9 = 0.8 and = 0.7 is prvided in [15] and a fitted frmula apprximatin fr diffuse reflectin ( = 1 ) by Rbert is recrded in [16] (there the secnd term n the right hand side is misprinted as 6A π Kn) 1 Q ( ) 1 6 lg(1 ) mtr Kn = + AKn + Kn + BKn (17) π where A = and B = The glbal mass cnservatin i.e. the cnstancy f the mass flux being equal at any crss sectin requires QmTR ( Kn) cns tant dx = (18) r in the dimensinal frm d dp QmTR ( Kn) P 0 dx = dx. (19) With the database incrprated this equatin is valid fr any surface cnditins f the plates and can be integrated numerically. But fr the illustrative purpse nly the case f cmplete diffuse reflectin = 1 is expunded here then the mass cnservatin is (frm Eqs. (17) and (19)) d 1 dp {[1 + 6AKn + Knlg(1 + BKn)] P } = 0. (0) dx π dx Fr the ease f integratin the lcal Knudsen number Kn is mst cnveniently expressed thrugh P λ λ p Kn = = = Kn / P (1) h h p and Eq. (0) becmes 1 BKn dp [ P + 6AKn + Knlg(1 + )] = D () π P dx where D is an unspecified cnstant t be determined frm the integratin and has the physical meaning f the flw rate acrss the channel nrmalized by the slip-less flw rate value. The equatins determining the pressure distributin (19) and (0) are in fact the generalized Reynlds equatin f the gas lubricatin thery degenerated fr the micrchannel flws (see [17 18]). They are integrated with the inlet and utlet bundary cnditins (6) t yield the pressure distributin in the transitinal regime.. Cmparisn with the experimental data and the simulatin results T cmpare the abve results with the experimental data f flw in the micrchannel with pressure sensrs imbedded in the channel accmplished by Png et al. [] and Shih et al.[4] we calculate the pressure distributin fr nitrgen in the µ m channel [] and fr helium in the µ m channel [4]. Under the experimental cnditin T0 = 94K the value f Kn fr nitrgen is and fr helium is The experimental data f nitrgen is cmpared with the results f the glbal mass cnservatin in the cntinuum regime Eq. (7) and the slip flw regime Eq. (1) in Fig.. The pressures p i at the inlet f the channel in psig are given in [] and [4] and the crrespnding values f Pi = P X = 0 are listed in Table 1. In Eq. (1) Kn is taken equal t and = 1. The cmparisn shws that the glbal mass cnservatin predicts the pressure distributin quite well and the slip flw result imprves the agreement f the cntinuum predictin with the experimental data. The experimental data are als cmpared with the predictin f the transitinal flw regime. Equatin () is integrated under the fllwing bundary cnditin P = X = 0 pi / p as given in Table 1 and P = X 1 p / p = = 1 Fig. Cmparisn f the pressure distributin btained frm slip slutin Eq.(1) (slid line) and slipless slutin Eq. (7) (dashed Line) with experimental data in a µ m micrchannel fr nitrgen [] (symbls). Kn = Cpyright 005 by ASME Dwnladed Frm: n 06/9/019 Terms f Use:

4 Table I The experimental inlet pressure data in psig and crrespnding values f P X = 0 fr nitrgen and helium fr nitrgen [] p (in psig ) i P = X 0 fr helium [4] p (in psig ) i P = X 0 The results f integratin are presented in Fig. and Fig.4. It is seen frm the figures that the results f the glbal mass cnservatin (degenerated Reynlds equatin) agree well with the experimental data. The finite length channel flws under the experimental cnditins have been calculated als by the infrmatin preservatin (IP) methd [190] in Shen Fan and Xie [1]. The IP simulatin results are als presented in Figs. and 4 by dashed lines. The cmparisn shws excellent agreement between the results calculated by the glbal mass cnservatin and by the IP methd. Fig.4 Cmparisn f the pressure distributin frm Eq.() (slid line) in a µ m with experimental data in a micrchannel fr helium [4] (symbls) and the the IP methd [1] (dashed lines nte that the slid lines and the dashed lines almst cincide). Kn = It is f fervent cncern fr the scientific cmmunity t knw whether LBM is capable t simulate crrectly the transitinal regime flws in MEMS. Shen Tian Xie and Fan examined the feasibility f using LBM in simulating MEMS flws by cmparisn with the DSMC calculatins []. Here the LBM results are cmpared with the calculatins by using the mass cnservatin equatin r the degenerated Reynlds equatin t attain the same cnclusins as in Reference [] but this time the cnclusin is cnfirmed by a test stne with the merit f strict kinetic thery. The equatin () is integrated under the fllwing cnditins (definitely in transitinal regime) fr a shrt 1 100µ m micrchannel that have been cnsidered by LBM in Nie Dlen and Chen [6] Fig. Cmparisn f the pressure distributin frm Eq.() (slid line) with experimental data in a µ m micrchannel fr nitrgen [] (symbls) and the result f IP methd [1] (dashed line). Kn = See als [18].. Examinatin f the feasibility f LBM in slving the micrchannel prblem in transitinal regime The Lattice Bltzmann methd (LBM see [] and references cited there) slves the simplified Bltzmann equatin n lattice pints. LBM slutin cnverges t the Navier-Stkes slutin fr small Kn. The ease f LBM in handling cmplex gemetry simplicity in implementatin and its high efficiency makes it tempting t be used in simulating gas flws in MEMS. Recently Nie Dlen and Chen [6] attempt t use it in transitinal regime. Kn = P = P = 1 () X= 0 X= 1 Kn = 0.88 P = P = 1. (4) X= 0 X= 1 The results f cmparisn f results f equatin () with thse f Nie Dlen and Chen by using the Lattice Bltzmann methd (LBM) [6] are shwn in Fig.5 and Fig.6. Als shwn in the figures are the DSMC and IP results. Frm the cmparisn it is seen that the DSMC methd the IP methd and the mass cnservatin equatin (degenerated Reynlds equatin) yield almst identical pressure distributin but the LBM results are quite different frm the calculatins f the degenerated Reynlds equatin and as well frm DSMC and IP results s it can be cncluded that the versin f LBM emplyed in Reference [6] is nt feasible fr simulating micrchannel flws in the transitinal regime. Fr the detailed examinatin f the feasibility f the LBM methd in simulating transitinal regime micrchannel flws readers are referred t Reference [18] where amng ther things the versin f LBM used [6] was briefly described. Als there the cmparisn f the pressure distributins was given in the frm f deviatin frm linearity s the differences between results f varius methds were revealed in mre relief. In this paper the quantities being 4 Cpyright 005 by ASME Dwnladed Frm: n 06/9/019 Terms f Use:

5 cmpared are the pressure distributins themselves and the same cnclusins are achieved. But here the cnclusin f unfeasibility f the LBM in simulating transitinal flws is recnfirmed by the cmparisn with calculatins having the strict kinetic theretical merit. Fig. 5 Pressure distributin in a micrchannel with Kn = at utlet ( h/ L = 100). Cmparisn f equatin () DSMC IP methds [1] and the LBM methd [6] 4. Cncluding remarks In the present paper the finite length micrchannel pressure distributin fr cntinuum flw slip flw and transitinal flw regimes is determined by the glbal mass cnservatin equatin. In the transitinal flw regime this equatin is the Reynlds equatin fr the gas film lubricatin prblem degenerated by the suggestin f the authr fr slving the micrchannel flw [18]. The cmparisn shws excellent agreement with the experimental data and the IP results in the lng micrchannels. Earlier the IP results fr lng channel were nly cmpared [5] with the experimental data (the DSMC was nt able t accmplish simulatin f the lng channel flw where the experimental data were available) nw they are verified by cmparisn with results having the strict kinetic thery merit. The glbal mass cnservatin equatin (the degenerated Reynlds equatin) prvides a means with the merit f strict kinetic thery t test varius methds in slving the micr scale rarefied gas dynamics flws in transitinal regime. The unfeasibility f LBM fr slving the transitinal flw prblem [] is recnfirmed by this means. Frm the practical applicatin pint f view database fr the flw rates f the Piseuille flw with varius cmbinatins f pssible surface prperties btained frm strict kinetic thery is an actual task fr the slutin f the micrchannel flw and als fr the thin film air bearing prblem and the gas damping prblem [4] in micrmechanical accelermeters. The database in the frm f fitting frmulas is especially desirable. ACKNOWLEDGMENTS The supprt by the NNSFC (grant ) is appreciated. The authr als appreciates the help f Dr Jiang JZ in carrying the calculatin and the drawings. Fig. 6 Pressure distributin in a micrchannel with Kn = 0.88 at utlet ( h/ L = 100 ). Cmparisn f equatin () DSMC IP methds [1] and the LBM methd [6] It is nted that in [] the cmparisn f LBM with DSMC and IP methds was als given fr smaller Knudsen numbers (e.g. Kn=0.0194) where gd agreement between them was btained s the failure f LBM in predicting micrchannel flw is verified fr transitinal flw regime ( Kn = ) but nt fr slip flw regime. Fr the relative errr f LBM readers are referred t [] where the results were shwn via the deviatin ( p pl) / p f the pressure distributin frm the linear distributed pressure pl = p0 + ( pi p)(1 X). REFERENCES 1. Batchelr G.K An Intrductin t Fluid Dynamics Cambridge University Press Cambridge.. Ohwada T. Sne Y. and Aki K 1989 Numerical analysis f the Piseuille and thermal transpiratin flws f a rarefied gas between tw parallel plates Phys. f Fluids A1 pp Png K.C. H C.M. Liu J.Q. and Tai Y.C Nn-linear pressure distributin in unifrm micrchannels ASME FED 197 Applicatin f Micrfabricatin t Fluid Mechanics pp Shih J.C. H C.M. Liu J.Q. and Tai Y.C Mnatmic and plyatmic gas flw thrugh unifrm micrchannels ASME-DSC 59 pp Shen C. Fan J. and Xie C. 00 Statistical simulatin f rarefied gas flws in micr-channels Jurnal f Cmputatinal Physics 189 pp Nie X. Dlen G.D. and Chen S. 00 Lattice Bltzmann simulatin f fluid flws in MEMS Jurnal f Statistical Physics 107 pp Arkilic E.B Measurement f the mass flw and tangential mmentum accmmdatin cefficient in silicn micrmachined channels FDRL TR 97-1 Ph.D. Thesis MIT. 8. Arkilic E.B. et al. 001 Mass flw and tangential mmentum accmmdatin cefficient in silicn micrmachined channels J. Fluid Mech. 47 pp Cpyright 005 by ASME Dwnladed Frm: n 06/9/019 Terms f Use:

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