Skip to main content
SpringerLink
Log in

Couette–Poiseuille flow of a gas in long microchannels

  • Research Paper
  • Published:
Microfluidics and Nanofluidics Aims and scope Submit manuscript

Abstract

The flow of a compressible, isothermal gas under slightly rarefied conditions in a 2D planar geometry is considered. The gas is shear driven and is also subject to an applied pressure gradient, which is also known as Couette–Poiseuille (CP) flow. In this paper, the full Navier-Stokes (NS) equations are solved using a perturbation expansion up to the first order. The pressure profile is solved numerically. On the basis of the solutions, effects of rarefaction and compressibility on the flow characteristics are investigated in detail. The results show the parallel flow assumption to be invalid for cases with slight rarefaction. The axial and vertical velocity components are found to depend on the degree of rarefaction, applied pressure gradient and wall velocity. The effects of rarefaction on the occurrence of back flow are also discussed. In addition, the results for the Poiseuille and CP flow with and without rarefaction can be easily obtained from our results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Abbreviations

H :

microchannel height

K n :

Knudsen number

K o :

outlet Knudsen number

L :

microchannel length

M :

Mach number

p :

pressure

P :

pressure ratio of inlet to outlet

R :

Reynolds number

T :

temperature

u :

streamwise velocity

u w :

wall velocity

υ:

vertical velocity

α:

parameter for simplicity, α  = (ɛ R)/(γ M 2)

γ:

ratio of specific heat

ɛ:

ratio of channel height to length (perturbation parameter)

μ:

viscosity

ρ:

density

σ:

streamwise momentum accommodation factor, σ  = (2  −  σ m ) /σ m

σ m :

tangential momentum accommodation coefficient

ℜ:

ideal gas constant

o:

outlet

0,1:

orders for perturbation solution

\(\widetilde{{}}\) :

nondimensional symbol

\(\bar{{}} \) :

area average symbol at outlet

References

  • Arkilic EB, Schmidt MA, Breuer KS (1997) Gaseous slip flow in long microchannels. JMEM Syst 6:167–178

    Google Scholar 

  • Arkilic EB, Breuer KS, Schmidt MA (2001) Mass flow and tangential momentum accommodation in silicon micromachined channels. J Fluid Mech 437:29–43

    Article  Google Scholar 

  • Barber RW, Emerson DR (2006) Challenges in modeling gas-phase flow in microchannels: from slip to transition. Heat Transf Eng 27(4):3–12

    Article  Google Scholar 

  • Beskok A, Karniadakis GE, Trimmer W (1996) Rarefaction and compressibility effects in gas microflows. ASME J Fluids Eng 118:448–56

    Google Scholar 

  • Burden RL, Faires JD (2001) Numerical analysis (7th). Prindle Weber and Schmidt, Boston

    Google Scholar 

  • Colin S (2005) Rarefaction and compressibility effects on steady and transient gas flows in microchannels. Microfluid Nanofluid 1:268–279

    Article  Google Scholar 

  • Colin S, Lalonde P, Caen R (2004) Validation of a second-order slip flow model in rectangular microchannels. Heat Transf Eng 25(3):23–30

    Article  Google Scholar 

  • Gad-el-Hak M (1999) The fluid mechanics of microdevices—the Freemann scholar lecture. J Fluids Eng 121:5–33

    Google Scholar 

  • Guderley KG, Hsu CC (1973) A special form of Galerkin’s method applied to heat transfer in plane Couette-Poiseuille flows. J Comput Phys 11:70–89

    Article  Google Scholar 

  • Homme RP, Chapman T, Bowen J (1986) Laminar compressible flow in a tube. Appl Sci Res 43:67–74

    Article  MathSciNet  Google Scholar 

  • Jang J, Wereley ST (2004) Pressure distributions of gaseous slip flow in straight and uniform rectangular microchannels. Microfluid Nanofluid 1:41–51

    Article  Google Scholar 

  • Karniadakis GE, Beskok A (2002) Microflows: fundamentals and simulation. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Lund KO, Bush WB (1980) Asymptotic analysis of plane turbulent Couette–Poiseuille flows. J Fluid Mech 96:81–104

    Article  Google Scholar 

  • Marques JW, Kremer GM, Sharipov FM (2000) Couette flow with slip and jump boundary conditions. Continuum Mech Thermodyn 12:379–386

    Article  Google Scholar 

  • Soong CY, Wang SH (2003) Theoretical analysis of electrokinetic flow and heat transfer in a microchannel under asymmetric boundary conditions. J Colloid Interface Sci 265:202–213

    Article  Google Scholar 

  • Van den Berg H, Seldam C, Gulik P (1993) Compressible laminar flow in a capillary. J Fluid Mech 246:1–20

    Article  Google Scholar 

  • Xue H, Ji HM, Shu C (2001) Analysis of micro-Couette flow using the Burnett equations. Int J Heat Mass Transf 44:4139–4146

    Article  Google Scholar 

  • Xue H, Ji H, Shu C (2003) Prediction of flow and heat transfer characteristics in micro-couette flow. Microscale Thermophys Eng7(2):51–68

    Google Scholar 

Download references

Acknowledgments

The authors are very grateful for the support of Natural Science Foundation of China under Grant No. 10472054 and SRFDP under Grant No. 20040003070.

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Wasif Ali Zahid, Youbing Yin & Ke-Qin Zhu

Authors
  1. Wasif Ali Zahid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youbing Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ke-Qin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ke-Qin Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zahid, W.A., Yin, Y. & Zhu, KQ. Couette–Poiseuille flow of a gas in long microchannels. Microfluid Nanofluid 3, 55–64 (2007). https://doi.org/10.1007/s10404-006-0108-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10404-006-0108-5

Keywords

Search

Navigation