# Pressure distributions of gaseous slip flow in straight and uniform rectangular microchannels

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## Abstract

This paper presents analytical derivations of the pressure distribution in straight and uniform rectangular microchannels in the slip flow regime and new experimental data in those channels. The flow is to be steady state, two-dimensional, isothermal, and to have negligible transverse velocities with a first order slip boundary condition. The measured pressure distributions of airflows are compared with newly derived analytical results. There is close agreement between the measurements and calculation by the slip flow formula. The dimensionless location of the maximum deviation from the linear pressure distribution is found analytically and compared with the measurements. This dimensionless location of the maximum deviation increases with the increasing pressure ratios in the slip flow regime. The effect of several parameters such as the channel aspect ratio and the Knudsen number on the locations of maximum deviation from linearity are investigated. The nonlinearity of the pressure distribution is also discussed.

### Keywords

Gaseous slip flow Pressure distribution Rectangular microchannel DRIE Nonlinearity### Nomenclature

*a*channel aspect ratio (=height/width)

*D*_{h}hydraulic diameter of a noncircular channel (m)

*H*height of a channel (m)

*h*half of the height of a channel (m)

*Kn*Knudsen number,

*Kn*=*λ*/*H*- \( \overline{{Kn}} \)
modified Knudsen number \( \overline{{Kn}} = Kn{{\left( {2 - \sigma } \right)}} \mathord{\left/ {\vphantom {{{\left( {2 - \sigma } \right)}} \sigma }} \right. \kern-\nulldelimiterspace} \sigma \)

*L*channel or tube length between upstream and downstream (m)

*Ma*Mach number

*ṁ*mass flow rate (kg/s)

*p*pressure (Pa)

*p**(*ζ*)dimensionless pressure with respect to

*p*_{o}at*ζ**R*_{s}specific gas constant (J/(kg K))

*Re*Reynolds number

*T*temperature (K)

*u*velocity vector

- \( v \)
*z*direction velocity (m/s)- \( \bar{v} \)
dimensionless

*z*component velocity*W*width of a rectangular channel (m)

*w*_{c}half of the width of a channel (m)

*x*,*y*rectangular Cartesian coordinates at each cross-section (m)

*z*streamwise coordinate (m)

*γ*specific heat ratio

*ζ*dimensionless variable for

*z*direction (=*z*/*L*)*η*dimensionless variable for

*y*direction (=*y*/*h)**κ*eigenvalue

*λ*mean free path of a gas (m)

*μ*dynamic viscosity (kg/(m s))

*ξ*dimensionless variable for

*x*direction (=*x*/*w*_{c}*)*- Π
pressure ratio (Π=

*p*_{i}/*p*_{o}*)**ρ*density (kg/m

^{3})*σ*tangential momentum accommodation coefficient (TMAC)

### Subscript

- avg
cross-section average

- i
inlet or upstream

- o
outlet or downstream

### Superscript

- *
nondimensionalization

## Notes

### Acknowledgement

The authors wish to thank Yabin Zhao for his helpful discussions and the microfabrication laboratory at Purdue University, West Lafayette. We also wish to acknowledge the financial support of the Indiana 21st Century Research and Technology Fund.

### References

- Arkilic EB, Schmidt MA, Breuer KS (1996) TMAC measurement in silicon micromachined channels. Rarefied Gas Dynamics, Beijing, ChinaGoogle Scholar
- Arkilic EB, Schmidt MA, Breuer KS (1997) Gaseous slip flow in long microchannels. J Microelectromech Syst 6:167–178CrossRefGoogle Scholar
- Ayon AA, Chen KS, Lohner KA, Spearing SM, Sawin HH, Schmidt MA (1999) Deep reactive ion etching of silicon. Proc Mater Res Soc Symp 546:51–61Google Scholar
- Beskok A, Karniadakis GE, Trimmer W (1996) Rarefaction and compressibility effects in gas microflows. J Fluids Eng 118:448–456Google Scholar
- Bird GA (1983) Definition of mean free path for real gases. Phys Fluids 26:3222–3223CrossRefGoogle Scholar
- Ebert WA, Sparrow EM (1965) Slip flow in rectangular and annular ducts. J Basic Eng 87:1018–1024Google Scholar
- Gad-el-Hak M (1999) The fluid mechanics of microdevices—the Freemann scholar lecture. J Fluids Eng 121:5–33Google Scholar
- Harley JC, Huang Y, Bau HH, Zemel JN (1995) Gas flow in micro-channels. J Fluid Mech 284:257–274Google Scholar
- Jang J, Wereley ST (2004) Gaseous slip flows in long rectangular microchannels. J Fluid Mech (submitted in 2004)Google Scholar
- Jie D, Diao X, Cheong KB, Yong LK (2000) Navier–Stokes simulations of gas flow in micro devices. J Micromech Microeng 10:372–379CrossRefGoogle Scholar
- Karniadakis GE, Beskok A (2002) Micro flows: fundamentals and simulation. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
- Pong KC, Ho CM, Liu J, Tai YC (1994) Non-linear pressure distribution in uniform microchannels. Proc ASME FED-197: 51–56Google Scholar
- Shah RK, London AL (1978) Laminar flow forced convection in ducts. Advances in heat transfer series. Academic Press, New YorkGoogle Scholar
- Wu P, Mai J, Zohar Y, Tai YC, Ho CM (1998) A suspended microchannel with integrated temperature sensors for high-pressure flow studies. In: Proceedings of the 1998 IEEE MEMS Workshop, pp 87–92Google Scholar
- Zhao Y (2003) Design and characterization of micro and molecular flow sensors. MS thesis, Purdue University, West LafayetteGoogle Scholar
- Zohar Y, Lee SYK, Lee WY, Jiang L, Tong P (2002) Subsonic gas flow in a straight and uniform microchannel. J Fluid Mech 472:125–151CrossRefGoogle Scholar