Effects of Axial Corrugated Roughness on Low Reynolds Number Slip Flow and Continuum Flow in Microtubes

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1 Zhpeng Duan Department of Mechancal and Mechatroncs Engneerng, Unversty of Waterloo, Waterloo, ON, NL 3G, Canada Y. S. Muzychka Faculty of Engneerng and Appled Scence, Memoral Unversty of Newfoundland, St. John s, NL, AB 3X5, Canada Effects of Axal Corrugated Roughness on Low Reynolds Number Slp Flow and Contnuum Flow n Mcrotubes The effect of axal corrugated surface roughness on fully developed lamnar flow n mcrotubes s nvestgated. The radus of a mcrotube vares wth the axal dstance due to corrugated roughness. The Stokes equaton s solved usng a perturbaton method wth slp at the boundary. Analytcal models are developed to predct frcton factor and pressure drop n corrugated rough mcrotubes for contnuum flow and slp flow. The developed model proposes an explanaton on the observed phenomenon that some expermental pressure drop results for mcrochannel flow have shown a sgnfcant ncrease due to roughness. The developed model for slp flow llustrates the coupled effects between velocty slp and small corrugated roughness. Compressblty effect has also been examned and smple models are proposed to predct the pressure dstrbuton and mass flow rate for slp flow n corrugated rough mcrotubes. DOI: 0.5/.3854 Keywords: corrugatons, roughness, perturbaton, slp flow, mcrotubes Introducton Flud flow n mcrochannels has emerged as an mportant research area. Ths has been motvated by ther varous applcatons such as medcal and bomedcal use, computer chps, and chemcal separatons. The advent of mcro-electro-mechancal systems MEMSs has opened up a new research area where noncontnuum and surface roughness characterstcs are mportant. Mcrochannels are a fundamental part of mcrofludc systems. Understandng the flow characterstcs of mcrochannel flows s very mportant n determnng pressure drop, heat transfer, and transport propertes of the flow. Mcrochannels can be defned as channels whose characterstc dmensons are from m to mm. Above mm the flow exhbts behavor, whch s the same as contnuum flows. Some researchers have reported on devatons between mcroscale flow behavor and conventonal macroscale flow theory. For lamnar fully developed flow through mcrochannels, researchers have observed sgnfcant ncreases n the pressure drop from the macroscale flow theoretcal values, as data appear up to 50% above the theoretcal values 8. Some publcatons ndcate that flows on the mcroscale are dfferent from that on macroscale. Several theores and models have been proposed to explan the observed devatons, but an ndsputable concluson has not yet been reached. In macroscale flow theory, the frcton factor s ndependent of relatve roughness n the lamnar regon. However, some researchers proposed that the frcton factor depends on the relatve roughness of the walls of mcrochannels n the lamnar regon and as such the relatve roughness cannot be neglected,0,5 6. Due to lmtatons n current mcromachnng technology, the walls of mcrofabrcated mcrochannels typcally exhbt some degree of roughness. Roughness plays an ncreasngly mportant role n mcrochannel flows, but t s dffcult to characterze ts effects theoretcally or numercally. It can be characterzed usng a stylustype surface proflometer, optcal measurement, scannng electron Contrbuted by the Heat Transfer Dvson of ASME for publcaton n the JOUR- NAL OF HEAT TRANSFER. Manuscrpt receved June, 008; fnal manuscrpt receved January 4, 009; publshed onlne February 7, 00. Revew conducted by Satsh G. Kandlkar. mcroscope SEM, atomc force mcroscope AFM, or scannng tunnelng mcroscope STM. Thus, there s a need for a better understandng of the effects of wall roughness on flud characterstcs n mcrochannels. Mala and L 3 measured the frcton factor of water n mcrotubes wth dameters rangng from 50 m to 54 m. They proposed a roughness-vscosty model to explan the ncrease n the frcton factor. Klenstreuer and Koo 7 proposed a numercal model to consder the effect of wall roughness on lqud flow n mcrochannels. They modeled roughness by consderng a porous medum layer PML near the wall. The porous medum layer approach s able to mmc some detals of the velocty profles and of the effect of the roughness heght. Wang et al. 8 numercally nvestgated the frcton factors of sngle phase contnuum flow n mcrochannels wth varous roughness elements rectangular, trangular, and ellptcal. The two-dmensonal numercal soluton shows sgnfcant nfluence of surface roughness ncludng the heght and spacng of the roughness elements on the Poseulle number. The Poseulle number ncreases wth an ncrease n roughness heght and decreases wth an ncrease n the roughness spacng. Bahram et al. developed a model to predct the pressure drof fully developed lamnar contnuum flows n roughness mcrotubes. In ths model, the wall roughness s assumed to possess a Gaussan sotropc dstrbuton. Prezjev and Troan 9 nvestgated the nfluence of perodc surface roughness on the slp behavor of a Newtonan lqud n steady planar shear. However, the physcs of lqud slp s complcated and not completely understood. Rarefacton effects must be consdered n gases n whch the molecular mean free path s comparable to the channel s characterstc dmenson. The contnuum assumpton s no longer vald and the gas exhbts noncontnuum effects such as velocty slp and temperature jump at the channel walls. Tradtonal examples of noncontnuum gas flows n channels nclude low-densty applcatons such as hgh-alttude arcraft or vacuum technology. The recent development of mcroscale flud systems has motvated great nterest n ths feld of study. There s strong evdence to support the use of Naver Stokes and energy equatons to model Journal of Heat Transfer Copyrght 00 by ASME APRIL 00, Vol. 3 / 0400-

2 the slp flow problem, whle the boundary condtons are modfed by ncludng velocty slp and temperature jump at the channel walls. The Knudsen number Kn relates the molecular mean free path of gas to a characterstc dmenson of the duct. Knudsen numbers are very small for contnuum flows. However, for mcroscale gas flows where the gas mean free path becomes comparable wth the characterstc dmenson of the duct, the Knudsen number may be greater than 0 3. Mcrochannels wth characterstc lengths on the order of 00 m would produce flows nsde the slp regme for gas wth a typcal mean free path of approxmately 00 nm at standard condtons. The slp flow regme to be studed here s classfed as 0 3 Kn0, and the flow s assumed to be sothermal. Snce the pressure drop s a result of vscous effects and not the free expanson of the gas, the sothermal assumpton should be reasonable. L et al. 30 studed the effects of surface roughness on the slp flow n long mcrotubes. The rough surface was represented as a porous flm based on the Brnkman-extended Darcy model, and the core regon of the flow utlzed velocty slp to model the rarefacton effects. By usng the approprate matchng condtons at the gas/porous nterface velocty slp and stress contnuty, the governng equaton of pressure dstrbuton was derved. Sun and Faghr 3 nvestgated the effects of surface roughness on ntrogen flow n a mcrochannel usng the drect smulaton Monte Carlo method. The surface roughness was modeled by an array of rectangular modules placed on two surfaces of a parallel plate channel. The effects of relatve surface roughness, roughness dstrbuton, and gas rarefacton on flow were studed. It was found that the effect of surface roughness s more pronounced at low Knudsen numbers. The roughness dstrbuton represented by the rato of the roughness heght to spacng of the modules has a sgnfcant effect on the frcton factor. The frcton factor ncreases not only as the roughness heght ncreases but also as the dstance between the roughness modules decreases. Ths s consstent wth the conclusons of Wang et al. 8. The exact solutons of Hagen Poseulle flow can be obtaned theoretcally. However, when the radus of a tube vares wth the axal dstance, the flow cannot be characterzed by Hagen Poseulle law. Langlos 3,33 employed the lubrcaton approxmaton to calculate the mean pressure drop. The predcton of the smple approxmaton method agrees well wth the exact value when the tube radus vares slowly. Snce Langlos paper, some authors nvestgated ths flow problem n tubes of slowly varyng radus. Tanner and Lnnett 34 extended the perturbaton analyss of Blasus to predct the knetc energy losses of vscometrc capllary tubes. However, they neglected second-order vscous terms n the momentum equatons. Manton 35 obtaned an asymptotc seres soluton for the low Reynolds number flow through an axsymmetrc tube whose radus vares slowly n the axal drecton. However, he also neglected frst and second-order vscous terms. Notcng ths and the applcatons of ths flow n bomechancs, Phan-Then 36 developed a perturbaton soluton and obtaned an mproved soluton up to the second-order O for the mean pressure gradent. Vasudevah and Balamurugan 37 tred to solve the correspondng problem for slp flow. However, ther dervaton and results are questonable, and wll be dscussed later n ths paper. Wang 38 analyzed the Stokes flow between corrugated plates usng a perturbaton method. Chu 39 studed Stokes flow between corrugated plates n the slp flow regme. Later, Chu 40 nvestgated the small Knudsen number flow n a tube wth corrugated wall by the perturbaton method. In vew of the mportance of ths flow problem, we develop a new perturbaton soluton for slp flow through axally corrugated rough mcrotubes. The no-slp boundary condtons are not vald n the slp flow regme 0.00Kn0., and a knetc boundary layer on the order of one mean free path 4,4, known as the ordnary Knudsen layer, starts to become domnant between the bulk of the flud and the wall surface. The flow n the Knudsen layer cannot be analyzed usng the Naver Stokes equatons, and t needs specal equatons of Boltzmann equaton. The contrbutons of the Knudsen layer to the velocty feld are of order Kn. However, for Kn0., the contrbuton of the Knudsen layer s small snce the Knudsen layer covers less than 0% of the tube dameter. Thus, the Knudsen layer can be neglected by extrapolatng the bulk gas flow toward the walls 4. For flow past a convex body, the knetc boundary sublayer due to curvature s present n the moments. Ths boundary sublayer s formed by ponts, whch cannot be reached from the boundary along straght lnes much longer than the mean free path. Sone 43 dscovered that the sublayer s of order the mean free path squared dvded by the radus of curvature of the boundary. Cercgnan suggests there s a smlar effect for concave surface. Detals can be found n Refs The knetc boundary sublayer s a porton of the Knudsen layer. For Kn0., the boundary sublayer covers much less than 0% of the tube dameter, and ths boundary sublayer can be neglected. Usng the frst-order slp boundary condtons s expected to yeld good approxmatons for Kn0.. Snce analytcal models derved usng the frst-order slp boundary condton have been shown to be relatvely accurate up to Knudsen numbers of approxmately 0. 46, the frst-order slp boundary condton s employed n ths paper. A varety of researchers have attempted to develop second-order slp models, whch can be used n the early transton regme. However, there are large varatons n the second-order slp coeffcent 4,46. The lack of a unversally accepted second-order slp coeffcent s a major problem n extendng the Naver Stokes equatons nto the transton regme 46. Theoretcal Analyss In ths paper we examne a smple approach to modelng surface roughness n the slp flow and contnuum flow Kn 0 regme. In order to smplfy the roughness problem, we can consder flow nsde a mcrotube wth a rough surface, whch s approxmately snusodal corrugaton, rr+r sn/l, as llustrated n Fg., where R s the mean radus of the rough mcrotube, relatve roughness b/r, and b and l are the ampltude and wavelength of the rough corrugated walls, respectvely. It s convenent to normalze all length varables wth respect to R. When the tubes are long enough L/D and the Reynolds number s relatvely low where the nerta terms are neglgble, the Stokes equaton n cylndrcal coordnates r,,z s 47 and Fg. An axal snusodal wave rough mcrotube E 4 0 E r r + r r where s the dmensonless stream functon normalzed by Q/. The boundary condtons are 0, 0 at r 0 3 r / Vol. 3, APRIL 00 Transactons of the ASME

3 , n n f R n at n n r + sn where n s the outer drecton normal and s the wave number R/l. The constant denotes tangental momentum accommodaton coeffcent, whch s usually between 0.87 and 48. Although the nature of the tangental momentum accommodaton coeffcents s stll an actve research problem, almost all evdence ndcates that for most gas-sold nteractons the coeffcents are approxmately.0. Therefore, may be assumed have a value of unty. It s convenent to ntroduce the Knudsen number Kn f R where f s the molecular mean free path. The characterstc length scale n the present analyss s defned as the mcrotube mean radus. The same procedure s vald even f, defnng a modfed Knudsen number as Kn Kn /. Usng perturbaton methods, we expand n terms of for the boundary condtons 4 we can expand + sn,z n a Taylor seres to obtan sn sn r r sn 0 r + O r 3 7 and r sn nr+ sn r sn cos r and n r+ sn cos 0 r + + O 3 rr+ sn r + sn 0 r cos 0 + r + sn r cos + sn 3 0 r 3 sn 0 r z cos 0 + O r r r + r + sn 0 3 r 3 cos 0 + r z cos r z + sn 4 0 r 4 r + sn 3 r 3 sn 3 0 r z + cos 0 cos 0 r The soluton of 0 s governed by 0 0, r E sn 0 + O r at r 0 0, Kn 0 0 r +Kn 0 at r r The zeroth-order 0 soluton s the usual slp Poseulle flow The -soluton s governed by 0 + Kn r r Kn 3 0, sn 0 r r E at r Kn +4 Knsn at r 6 Kn r +Kn r sn 0 r + cos 0 Kn sn 0 r 3 cos 0 r z 8 + Kn +4 Kn sn at r 7 Accordng to the boundary condtons, the s n the form where The soluton of s r,z rsn D 0, D r dr d d r dr r C ri r + C r I 0 r + B rk r + B r K 0 r where I x and K x are the modfed Bessel functons of the frst and second knd, respectvely, of order. Due to the boundedness of the velocty feld, B B 0. Therefore, C ri r + C r I 0 rsn Applyng boundary condtons 6 and 7 C I + C I 0 8 Kn +4 Kn C I 0 + Kn I + C + Kn I Kn I 8+ Kn + 4 Kn 3 Solvng Eqs. and 3, we have Journal of Heat Transfer APRIL 00, Vol. 3 /

4 8 C +4 Kn +4 Kn + Kn I 0 + Kn + Kn I I I 0 I + Kn I 8 Kn I Kn + Kn I C +4 Kn I I 0 I + Kn I 4 5 It s seen that the frst-order soluton s perodc n z and cannot be related to a mean pressure gradent along z. Next, the -soluton s governed by 0, r E 4 0 sn r sn 0 r C I 0 C I 0 + I at r Kn +4 Kn cos at r 8 Kn r +Kn r sn r + cos sn 3 0 r 3 + sn 0 r z + cos 0 r Knsn 0 r 3 cos r z + sn 4 0 r 4 sn 3 0 r z + cos 0 cos 0 r 3 sn 0 C +C C 3 Kn + C Kn I 0 + 3C + C Kn +C Kn C 3 Kn I + 6+Kn Kn +4 Kn +4 Kn + cos at r 9 The approprate soluton s n the form r,z a r + perodc soluton a r + cos 30 It s seen that the second-order soluton may cause a mean pressure gradent along z. Snce perodc soluton does not contrbute to the mean pressure gradent, only a needs to be determned. The a soluton s governed by a 0, r E 4 a 0 a 3 0 at r 0 3 a C I 0 C I 0 + I Kn +4 Kn at r 33 Kn a r +Kn a r C +C C 3 Kn + C Kn I 0 + 3C + C Kn +C Kn C 3 Kn I + 6+Kn Kn +4 Kn + 4 Kn at r 34 The soluton for a should be of the form a C 3 r C 4 r 4 In terms of boundary condtons 33 and 34, we obtan C 3 C I 0 C I 0 + I + C 4 where 4 4 Kn +4 Kn + C Kn C +C + C 3 Kn C Kn I 0 C + C Kn +C Kn C 3 Kn I +5 Kn Kn + Kn +4 Kn +4 Kn 5 Kn + Kn Kn I 0 + I 0 I + 3 I I I 0 I + Kn I 37 Kn Kn 6 Kn 4 Kn Kn Kn 4 Kn Kn 4 Kn + Kn + Kn + 4 Kn 3 The total flow rate s gven by where Q R4 dp/dz R4 dp/dz 8 l R l/r 0 r r E dz + 4 Kn +B + O 4 B +4 Kn C B s a functon of wave number and the modfed Knudsen number Kn. Fgure shows the effect of and Kn on B,Kn. The B values ncrease wth an ncrease n and Kn. It s seen that the perodc soluton cannot be related to the mean pressure gradent along z as ts contrbuton to the ntegral s zero. Only 0 and a contrbute to the mean pressure gradent Q Q sm + 4 Kn +B + O 4 43 where Q sm s the flow rate for contnuum flow n smooth mcrotubes. The flow rate decreases wth an ncrease n. Snce B s / Vol. 3, APRIL 00 Transactons of the ASME

5 Fg. B,Kn as a functon of and Kn always postve ndcatng a decrease n the flow rate wth wall roughness. Settng 0 n Eq. 43, we obtan the correspondng expresson for the flow rate n smooth mcrotubes as Q/Q sm +4 Kn. However, Vasudevah and Balamurugan 37 gave the followng msleadng expresson: Q +5 Kn Q sm +Kn 44 After ntegratng Eq. 4, the pressure drop along the length of the ppe L may be determned to be p 8LQ +B + O 4 R 4 +4 Kn 45 It can be also shown that the effect of wall roughness on the pressure drop s gven by the followng equaton: p p p sm +B + O 4 +4 Kn 46 where p sm s the pressure drop for contnuum flow n smooth mcrotubes. The mean fcton factor Reynolds product can be obtaned smply by substtutng Eq. 4 nto the defnton of f Re. A dp h P dzd A dp h A P dzd f Re ū Q 6+B + O 4 +4 Kn 47 Fgure 3 demonstrates the effect of wave number, relatve roughness, and Knudsen number Kn on pressure drof mcrotubes for slp flow. Velocty slp decreases pressure drop and corrugated roughness ncreases pressure drop. Pressure drop depends on,, and Kn, and t can be less than, equal to, or greater than unty. The coupled effects between small corrugated roughness and velocty slp suggest a possble explanaton for the observed phenomenon that Chung et al. 49 and Kohl et al. 50 found that the frcton factors for gas flow n mcrochannels can be accurately determned from conventonal theory for large channels.. Contnuum Flow. The densty of lquds s tmes that of the typcal gaseous state. The molecules are closely packed and surrounded by other molecules. Snce the molecules Fg. 3 Effect of relatve roughness ε, wavenumber, andknudsennumber Kn on pressure drof mcrotubes for slp flow Journal of Heat Transfer APRIL 00, Vol. 3 /

6 Fg. 4 Effect of relatve roughness ε and wave number on pressure drop of mcrotubes for contnuum flow are contnuously n collson, the concept of a mean free path s not used for lquds. The lqud partcles contactng the wall must essentally be n equlbrum wth the sold. For most mcroscale lqud flows, the Naver Stokes equatons and no-slp boundary condtons stll hold. In the lmt of Kn 0, Eq. 4 reduces to ts correspondng contnuum flow soluton 36 4I B c 3+ I I 0 I 48 For practcal applcatons, a smple expresson, whch s vald for 3 can be used B c The mean fcton factor Reynolds product s gven by f Re 6 +B c + O Fgures 4 and 5 demonstrate the effect of wave number and relatve roughness on pressure drof mcrotubes for contnuum flow. The pressure drop ncreases wth an ncrease n. Snce B s always postve, the pressure drop ncreases wth wall roughness.. Compressblty Effects. Now we take account of the compressblty of the gas. The flow s assumed to be locally fully developed. The locally fully developed flow assumpton means that the velocty feld at any cross secton s the same as that of a fully developed flow at the local densty, and the wall shear stress also takes on locally fully developed values. Compressblty effects enter through the state equaton and contnuty equaton. If the cross-sectonal area slghtly vares, as t s the case for a corrugated wall, the locally fully developed assumpton stll approxmately holds snce we only consder the case of. Sun and Faghr 3 demonstrated that the locally fully developed flow model can be used to predct gas slp flow n a mcrochannel wth low values of relatve surface roughness 3 5%. For compressble flow, the mass flow rate n the mcrotube s gven by employng the equaton of state prt ṁ ūa Kn R4 8RT B +O 4 dzp dp +4 p 5 Fg. 5 Effect of relatve roughness ε and wave number on pressure drop of mcrotubes for contnuum flow / Vol. 3, APRIL 00 Transactons of the ASME

7 We can use p Kn Kn o snce p Kn s constant for sothermal flow. Integratng Eq. 5, we obtan ṁ ūa R4 6RTz B +O 4 p p p z o +8 o Kn p p z 5 where B denotes the average value of B,Kn and B,Kn z. Lettng zl gves ṁ ūa R4 B +O 4 p 6RTL p +8 o The contnuum flow mass flow rate s gven by Kn o p 53 ṁ c ūa R4 B +O 4 p 6RTL p o 54 The effect of slp may be llustrated clearly by dvdng the slp flow mass flow equaton 53 by the contnuum mass flow equaton 54 ṁ ṁ c + 8 Kn o p + 55 It s seen that the rarefacton ncreases the mass flow and that the effect of rarefacton becomes more sgnfcant when the pressure rato decreases. Ths could be nterpreted as a decrease n the gas vscosty. Combnng Eqs. 5 and 53, we obtan the expresson for pressure dstrbuton p z 4 Kn o +4 Kn o + p p p +8 o Kn o p z L The pressure dstrbuton exhbts a nonlnear behavor due to the compressblty effect. The pressure drop requred s less than that n a conventonal channel. The devatons of the pressure dstrbuton from the lnear dstrbuton decrease wth an ncrease n Knudsen number. The nonlnearty ncreases as the pressure rato ncreases. The effects of compressblty and rarefacton are opposte, as Karnadaks et al. 4 demonstrated. The ncompressblty assumpton lnear pressure dstrbuton s vald for a small pressure drop n the mcrotubes. The devatons of the nonlnear pressure dstrbuton from the lnear dstrbuton s gven by p z p p L z 4 Kn o +4 Kn o + p p p +8 o o Kn p z L p + p z 57 L 56 Takng the dervatve of Eq. 57 and settng t equal to zero, we obtan the locaton of maxmum devaton from lnearty as z L 3 p + 6 Kn o + +8 Kn o + 4 p 58 It s seen that the locaton of maxmum devaton from lnearty s between 0.5 and The locaton approaches 0.5 for low pressure rato and approaches 0.75 for hgh pressure rato. 3 Summary and Concluson The nfluence of axal corrugated surface roughness on fully developed lamnar flow n mcrotubes s studed, and models are proposed to predct frcton factor and pressure drop for contnuum flow and slp flow. Compressblty effect has also been examned, and smple models are proposed to predct the pressure dstrbuton and mass flow rate for slp flow n corrugated rough mcrotubes. It s observed that the normalzed pressure drop, p, s a functon of relatve roughness and wave number for contnuum flow,.e., p F,. The present model exhbts the nfluence of axal corrugated roughness. For most conventonal mcrotubes assumng and 0 0, the present model provdes an explanaton on the observed phenomenon that pressure drop results for contnuum flow have shown an ncrease due to roughness. For slp flow, p s a functon of relatve roughness, wave number, and Knudsen number Kn,.e., p F,,Kn. There exst coupled effects between velocty slp and corrugated roughness. Velocty slp decreases pressure drop and ncreases flow rate. Corrugated roughness ncreases pressure drop and decreases flow rate. These two effects can have a cancelng effect n some systems. The frcton factor Reynolds product depends on the relatve roughness of the walls of the mcrochannels also n lamnar regon and the relatve roughness cannot be neglected for mcrochannels n the lamnar regon. Common practce s to specfy surfaces wth a sngle parameter, average roughness as t s well establshed and understood. It s obvous that one parameter s not enough to descrbe the complete nature of a surface. The roughness spacng s another mportant parameter to descrbe surface roughness. The frcton factor Reynolds product ncreases not only as the roughness heght ncreases but also as the roughness spacng decreases. The present paper extends Phan-Then s excellent work to slp flow regme. The range of valdty of the analytcal models developed s 0.00Kn0. slp flow regme and. The developed smple models may be used by the research communty to estmate roughness and velocty slp effects for the practcal engneerng desgn of mcrotubes. Acknowledgment The authors would lke to express our apprecaton to professors Mchael Yovanovch and Rchard Culham for ther suggestons. The authors acknowledge the support of the Natural Scences and Engneerng Research Councl of Canada NSERC. Journal of Heat Transfer APRIL 00, Vol. 3 /

8 Nomenclature A flow area, m b ampltude, m D h hydraulc dameter, 4A/ P f Fannng frcton factor, //ū Kn Knudsen number, f /R Kn modfed Knudsen number, Kn / L tube length, m l wavelength, m ṁ mass flow rate, kg/s n outer drecton normal P permeter, m p pressure, N/m p normalzed pressure drop Q volume flow rate, m 3 /s R mean radus of a rough mcrotube, m R specfc gas constant, J/kg K Re Reynolds number, ūr/ r dmensonless radal coordnate n the cylndrcal coordnate system, r/r r radal coordnate n the cylndrcal coordnate system, m T temperature, K u velocty, m/s ū average velocty, m/s z dmensonless coordnate n flow drecton, z/r z coordnate n flow drecton, m Greek Symbols relatve roughness, b/r angular coordnate n the cylndrcal coordnate system, rad wave number, R/l f molecular mean free path, m dynamc vscosty, N s/m knematc vscosty, m /s tangental momentum accommodaton coeffcent wall shear stress, N/m dmensonless stream functon, Q/ Subscrpts c contnuum nlet o outlet sm smooth References Wu, P., and Lttle, W. 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C., and Ho, C. M., 998, A Suspended Mcrochannel Wth Integrated Temperature Sensors for Hgh Pressure Flow Studes, Proceedngs of the IEEE Workshon Mcro Electro Mechancal Systems, Hedelberg,Germany,pp Urbanek, W., Zemel, J. N., and Bau, H., 993, An Investgaton of the Temperature Dependence of Poseulle Numbers n Mcrochannel Flow, J. Mcromech. Mcroeng., 3, pp Papautsky, I., Brazzle, J., Ameel, T. A., and Frazer, A. B., 998, Mcrochannel Flud Behavor Usng Mcropolar Flud Behavor, Proceedngs of the 998 IEEE th Annual Internatonal Workshon Mcro Electro Mechancal Systems, Hedelberg,Germany,pp Pfund, D., Rector, D., Shekarrz, A., Popescu, A., and Welty, J., 998, Pressure Drop Measurements n a Mcrochannel, Proceedngs of the 998 ASME Internatonal Mechancal Engneerng Congress and Exposton: DSC Mcro- Electro-Mechancal Systems, Vol.66,pp Qu, W., Mala, G. M., and L, D., 999, Pressure-Drven Flows n Trapezodal Slcon Mcrochannels, Int. J. Heat Mass Transfer, 43, pp Bahram, M., Yovanovch, M. M., and Culham, J. R., 006, Pressure Drof Fully Developed, Lamnar Flow n Rough Mcrotubes, ASME J. Fluds Eng., 8, pp Mala, G. M., and L, D., 999, Flow Characterstcs of Water n Mcrotubes, Int. J. Heat Flud Flow, 0, pp Kulnsky, L., Wang, Y., and Ferrar, M., 999, Electrovscous Effects n Mcrochannels, SPIE Conference on Mcro- and Nanofabrcated Structures and Devces for Bomedcal Envronment Applcatons II, SanJose,CA,Vol.3606, pp Dng, L. S., Sun, H., Sheng, X. L., and Lee, B. D., 000, Measurement of Frcton Factor for R34a and R Through Mcrochannels, Proceedngs of the Symposum on Energy Engneerng n the st Century, Vol.,pp L, Z. X., Du, D. X., and Guo, Z. Y., 000, Characterstcs of Frctonal Resstance for Gas Flow n Mcrotubes, Proceedngs of the Symposum on Energy Engneerng n the st Century, Vol.,pp Kandlkar, S. G., Josh, S., and Tan, S., 00, Effect of Channel Roughness on Heat Transfer and Flud Flow Characterstcs at Low Reynolds Numbers n Small Dameter Tubes, Proceedngs of the 35th Natonal Heat Transfer Conference, Anahem,CA,PaperNo L, Z. X., Du, D. X., and Guo, Z. Y., 003, Expermental Study on Flow Characterstcs of Lqud n Crcular Mcrotubes, Mcroscale Thermophys. Eng., 7, pp Yu, D., Warrngton, R., Barron, R., and Ameel, T. A., 995, Expermental and Theoretcal Investgaton of Flud Flow and Heat Transfer n Mcrotubes, Proceedngs of the 995 ASME/JSME Thermal Engneerng Jont Conference, Mau, HI, Vol., pp Pfund, D., Shekarrz, A., Popescu, A., and Welty, J. R., 000, Pressure Drop Measurements n a Mcrochannel, AIChE J., 46, pp Celata, G. P., Cumo, M., Guglelm, M., and Zummo, G., 000, Expermental Investgaton of Hydraulc and Sngle Phase Heat Transfer n 0.30 mm Capllary Tube, Proceedngs of the Internatonal Conference on Heat Transfer and Transport Phenomena n Mcroscale, G.P.Celata,ed.,BegellHouse,New York, pp L, Z. X., Du, D. X., and Guo, Z. Y., 000, Expermental Study on Flow Characterstcs of Lqud n Crcular Mcrotubes, Proceedngs of the Internatonal Conference on Heat Transfer and Transport Phenomena n Mcroscale, G. P. Celata, ed., Begell House, New York, pp Bucc, A., Celata, G. P., Cumo, M., Serra, E., and Zummo, G., 003, Flud Flow and Sngle-Phase Flow Heat Transfer of Water n Capllary Tubes, Proceedngs of the Internatonal Conference on Mnchannels and Mcrochannels, Rochester,NY,PaperNo.ICMM Tu, X., and Hrnjak, P., 003, Expermental Investgaton of Sngle-Phase Flow Pressure Drop Through Rectangular Mcrochannels, Proceedngs of the Internatonal Conference on Mnchannels and Mcrochannels, Rochester,NY, Paper No. ICMM Bavere, R., Ayela, F., Le Person, S., and Favre-Marnet, M., 004, An Expermental Study of Water Flow n Smooth and Rough Rectangular Mcro- Channels, Proceedngs of the Second Internatonal Conference on Mnchannels and Mcrochannels, Rochester,NY,PaperNo.ICMM Tang, G. H., L, Z., He, Y. L., and Tao, W. Q., 007, Expermental Study of Compressblty, Roughness and Rarefacton Influences on Mcrochannel Flow, Int. J. Heat Mass Transfer, 50, pp Klenstreuer, C., and Koo, J., 004, Computatonal Analyss of Wall Roughness Effects for Lqud Flow n Mcro-Conduts, ASME J. Fluds Eng., 6, pp Wang, X. Q., Yap, C., and Mujumdar, A. S., 005, Effects of Two- Dmensonal Roughness n Flow n Mcrochannel, ASME J. Electron. Packag., 7, pp Prezjev, N. V., and Troan, S. M., 006, Influence of Perodc Wall Roughness on the Slp Behavour at Lqud/Sold Interfaces: Molecular-Scale Smulatons Versus Contnuum Predctons, J. Flud Mech., 554, pp L, W. L., Ln, J. W., Lee, S. C., and Chen, M. D., 00, Effects of Roughness on Rarefed Gas Flow n Long Mcrotubes, J. Mcromech. Mcroeng.,, pp Sun, H., and Faghr, M., 003, Effects of Surface Roughness on Ntrogen Flow n a Mcrochannel Usng the Drect Smulaton Monte Carlo Method, Numer. Heat Transfer, 43, pp Langlos, W. E., 958, Creepng Vscous Flow Through a Two-Dmensonal Channel of Varyng Gap, Proceedngs of the Thrd Natonal Congress of Appled Mechancs, pp Langlos, W. E., 958, Slow Vscous Flow, Macmllan,NewYork. 34 Tanner, R. I., and Lnnett, I. W., 965, Pressure Losses n Vscometrc Capllary Tubes of Varyng Dameter, Second Australan Conference on Hydraulcs and Flud Mechancs A, pp / Vol. 3, APRIL 00 Transactons of the ASME

9 35 Manton, M. J., 97, Low Reynolds Number Flow n Slowly Varyng Axsymmetrc Tubes, J. Flud Mech., 49, pp Phan-Then, N., 980, On the Stokes Flow of Vscous Fluds Through Corrugated Ppes, ASME J. Appl. Mech., 47, pp Vasudevah, M., and Balamurugan, K., 999, Stokes Slp Flow n a Corrugated Ppe, Int. J. Eng. Sc., 37, pp Wang, C. Y., 979, On Stokes Flow Between Corrugated Plates, ASME J. Appl. Mech., 46, pp Chu, K.-H. W., 996, Stokes Slp Flow Between Corrugated Walls, Z. Angew. Math. Phys., 47, pp Chu, K.-H. W., 999, Small-Knudsen-Number Flow n a Corrugated Tube, Meccanca, 34, pp Karnadaks, G. E., Beskok, A., and Aluru, N., 005, Mcroflows and Nanoflows, Sprnger,NewYork. 4 Hadjconstantnou, N. G., 005, Valdaton of a Second-Order Slp Model for Dlute Gas Flows, Nanoscale Mcroscale Thermophys. Eng., 9, pp Sone, Y., 973, New Knd of Boundary Layer of a Convex Sold Boundary n ararefedgas, Phys.Fluds, 6, pp Sone, Y., 000, Knetc Theory and Flud Dynamcs, Brkhauser,Boston. 45 Cercgnan, C., 988, The Boltzmann Equaton and Its Applcatons, Sprnger, New York. 46 Barber, R. W., and Emerson, D. R., 006, Challenges n Modelng Gas-Phase Flow n Mcrochannels: From Slp to Transton, Heat Transfer Eng., 7, pp Goldsten, S., 938, Modern Developments n Flud Dynamcs, OxfordUnversty Press, Oxford, London. 48 Rohsenow, W. M., and Cho, H. Y., 96, Heat, Mass, and Momentum Transfer, Prentce-Hall,EnglewoodClffs,NJ. 49 Chung, P., Kawaj, M., and Kawahara, A., 00, Characterstcs of Sngle- Phase Flow n Mcrochannels, Proceedngs of the ASME 00 Fluds Engneerng Dvson Summer Meetng, Montreal,Canada. 50 Kohl, M. J., Abdel-Khalk, S. I., Jeter, S. M., and Sadowsk, D. I., 005, An Expermental Investgaton of Mcrochannel Flow Wth Internal Pressure Measurements, Int. J. Heat Mass Transfer, 48, pp Journal of Heat Transfer APRIL 00, Vol. 3 /