Microfluidics and Nanofluidics

, Volume 1, Issue 3, pp 268–279 | Cite as

Rarefaction and compressibility effects on steady and transient gas flows in microchannels

  • Stéphane ColinEmail author


The main theoretical and experimental results from the literature about steady pressure-driven gas microflows are summarized. Among the different gas flow regimes in microchannels, the slip flow regime is the most frequently encountered. For this reason, the slip flow regime is particularly detailed and the question of appropriate choice of boundary conditions is discussed. It is shown that using second-order boundary conditions allows us to extend the applicability of the slip flow regime to higher Knudsen numbers that are usually relevant to the transition regime.

The review of pulsed flows is also presented, as this kind of flow is frequently encountered in micropumps. The influence of slip on the frequency behavior (pressure gain and phase) of microchannels is illustrated. When subjected to sinusoidal pressure fluctuations, microdiffusers reveal a diode effect which depends on the frequency. This diode effect may be reversed when the depth is shrunk from a few hundred to a few μm.

Thermally driven flows in microchannels are also described. They are particularly interesting for vacuum generation using microsystems without moving parts.


Microfluidics Gas microflow Rarefaction Microchannel 



Aspect ratio, h/b (dimensionless)


Widths of microdiffusers (m)


Coefficients for second-order slip flow models (dimensionless)


Width (m)


Mean-square molecular speed (m s−1)


Molecular diameter (m)


Diode efficiency (dimensionless)


Eckert number (dimensionless)


Microchannel depth (m)


Boltzmann constant (J K−1)


Knudsen number, λ/2h (dimensionless)


Characteristic length of the studied volume (m)


Lengths of diffusers parts (m)


Microchannel length (m)


Characteristic length of a sampling volume (m)


Mass of a molecule (kg)


Mach number (dimensionless)


Number density (m−3)

\(\ifmmode\expandafter\dot\else\expandafter\.\fi{N} \)

Molecular flux (s−1)


Pressure (Pa)


Fluctuating pressure (Pa)


Pressure gain (dimensionless)


Fluctuating pressure gain (dimensionless)


Prandtl number (dimensionless)


Mass flow rate (kg s-1)


Reduced mass flow rate, q/qNS0 (dimensionless)


Specific gas constant (J mol−1 K−1)


Reynolds number (dimensionless)


Schmidt number (dimensionless)


Temperature (K)


Tangential velocity (m s−1)


Inlet over outlet pressure ratio, Pi/Po (dimensionless)

Greek letters


Diffuser angle (rad)


Mean molecular spacing (m)


Phase (rad)


Ratio of specific heats (dimensionless)


Mean free path (m)


Density (m3s−1)


Tangential momentum accommodation coefficient (dimensionless)


Thermal accommodation coefficient (dimensionless)


Characteristic time of intermolecular collisions (s)





Normal direction


Navier-Stokes model with no-slip boundary conditions


Navier-Stokes model with first-order slip flow boundary conditions


Navier-Stokes model with second order slip flow boundary conditions


Quasihydrodynamic model with first-order slip flow boundary conditions




Tangential direction




  1. Arkilic EB, Breuer KS (1993) Gaseous flow in small channels. AIAA paper, 93–3270, pp 1–7Google Scholar
  2. Arkilic EB, Breuer KS, Schmidt MA (2001) Mass flow and tangential momentum accommodation in silicon micromachined channels. J Fluid Mech 437:29–43CrossRefGoogle Scholar
  3. Aubert C, Colin S (2001) High-order boundary conditions for gaseous flows in rectangular microchannels. Microscale Therm Eng 5(1):41–54CrossRefGoogle Scholar
  4. Aubert C, Colin S, Caen R (1998) Unsteady gaseous flows in tapered microchannels. In: Proceedings of the 1st international conference on modeling and simulation of microsystems, semiconductors, sensors, and actuators (MSM’98), vol 1, Santa Clara, California, Marriot Computational Publications, pp 486–491Google Scholar
  5. Beskok A, Karniadakis GE (1999) A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Therm Eng 3(1):43–77CrossRefGoogle Scholar
  6. Bestman AR, Ikonwa IO, Mbelegodu IU (1995) Transient slip flow. Int J Energ Res 19(3):275–277Google Scholar
  7. Bhatnagar P, Gross E, Krook K (1954) A model for collision processes in gasses. Phys Rev 94:511–524CrossRefGoogle Scholar
  8. Bird GA (1978) Monte Carlo simulation of gas flows. Annu Rev Fluid Mech 10:11–31CrossRefGoogle Scholar
  9. Bird GA (1998) Molecular gas dynamics and the direct simulation of gas flows. Clarendon Press, OxfordGoogle Scholar
  10. Caen R, Mas I, Colin S (1996) Ecoulements non permanents dans les microcanaux: réponse fréquentielle des microtubes pneumatiques. C R Acad Sci, Sér IIb 323:805–812Google Scholar
  11. Cercignani C, Illner R, Pulvirenti M (1994) The mathematical theory of dilute gases, vol 106. Springer, Berlin Heidelberg New YorkGoogle Scholar
  12. Chapman S, Cowling TG (1952) The mathematical theory of non-uniform gases. Cambridge University Press, CambridgeGoogle Scholar
  13. Chen S, Doolen G (1998) Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech 30:329–364CrossRefGoogle Scholar
  14. Chen CS, Lee SM, Sheu JD (1998) Numerical analysis of gas flow in microchannels. Numer Heat Transf A 33:749–762Google Scholar
  15. Colin S, Anduze M, Caen R (1998a) A pneumatic frequency generator for experimental analysis of unsteady microflows. In: Proceedings of the 1998 ASME International mechanical engineering congress and exposition, Anaheim, California, November 1998Google Scholar
  16. Colin S, Aubert C, Caen R (1998b) Unsteady gaseous flows in rectangular microchannels: frequency response of one or two pneumatic lines connected in series. Euro J Mech B–Fluids 17(1):79–104Google Scholar
  17. Colin S, Elizarova TG, Sheretov YV, Lengrand J-C, Camon H (2003) Micro-écoulements gazeux: validation expérimentale de modèles QHD et de Navier-Stokes avec conditions aux limites de glissement. In: CDROM de 16ème Congrès Français de Mécanique, Nice, France, September 2003Google Scholar
  18. Colin S, Lalonde P, Caen R (2004) Validation of a second-order slip flow model in rectangular microchannels. Heat Transfer Eng 25(3):23–30Google Scholar
  19. Deissler RG (1964) An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases. Int J Heat Mass Transf 7:681–694CrossRefGoogle Scholar
  20. Ebert WA, Sparrow EM (1965) Slip flow in rectangular and annular ducts. J Basic Eng 87:1018–1024Google Scholar
  21. Elizavora TG, Sheretov YV (2001) Theoretical and numerical investigation of quasi-gasdynamic and quasi-hydrodynamic equations. Comput Math Phys 41(2)219–234Google Scholar
  22. Elizarova TG, Sheretov YV (2003) Analyse du problème de l’écoulement gazeux dans les microcanaux par les équations quasi hydrodynamiques. La Houille Blanche 5:66–72Google Scholar
  23. Fan J, Shen C (1999) Statistical simulation of low-speed unidirectional flows in transition regime. In: Brun R, Campargue R, Gatigno Rl, Lengrand J-C (eds) Rarefied gas dynamics, vol 2. Cépaduès Editions, Toulouse, France, pp 245–252Google Scholar
  24. Gad-el-Hak M (1999) The fluid mechanics of microdevices—the Freeman scholar lecture. J Fluid Eng 121:5–33Google Scholar
  25. Grad H (1949) On the kinetic theory of rarefied gases. Commun Pure Appl Math 2:331–407Google Scholar
  26. Harley JC, Huang Y, Bau HH, Zemel JN (1995) Gas flow in micro-channels. J Fluid Mech 284:257–274Google Scholar
  27. Hash D, Hassan H (1997) Two-dimensional coupling issues of hybrid DSMC/Navier-Stokes solvers. AIAA paper 97-2507:6333–6336Google Scholar
  28. Hobson JP (1970) Accommodation pumping—a new principle for low pressure. J Vacuum Sci Technol 7(2):301–357Google Scholar
  29. Hobson JP (1971) Analysis of accommodation pumps. J Vacuum Sci Technol 8(1):290–293CrossRefGoogle Scholar
  30. Hobson JP (1972) Physical factors influencing accommodation pumps. J Vacuum Sci Technol 9(1):252–256CrossRefGoogle Scholar
  31. Hudson ML, Bartel TJ (1999) DSMC simulation of thermal transpiration and accommodation pumps. In: Brun R, Campargue R, Gatigno Rl, Lengrand J-C (eds) Rarefied gas dynamics, vol 1. Cépaduès Editions, Toulouse, France, pp 719–726Google Scholar
  32. Jie D, Diao X, Cheong KB, Yong LK (2000) Navier-Stokes simulations of gas flow in micro devices. J Micromech Microeng 10(3):372–379CrossRefGoogle Scholar
  33. Karniadakis GE, Beskok A (2002) Microflows: fundamentals and simulation. Springer, Berlin Heidelberg New YorkGoogle Scholar
  34. Kennard EH (1938) Kinetic theory of gases, 1st ed. McGraw-Hill, New YorkGoogle Scholar
  35. Lalonde P (2001) Etude expérimentale d’écoulements gazeux dans les microsystèmes à fluides. PhD thesis, Institut National des Sciences Appliquées, Toulouse, FranceGoogle Scholar
  36. Lengrand J-C, Elizarova TG (2004) Microécoulements gazeux. In: Colin S (ed) Microfluidique, chapter 2. Hermès, Paris, FranceGoogle Scholar
  37. Liu J, Tai Y-C, Ho C-M (1995) MEMS for pressure distribution studies of gaseous flows in microchannels. In: Proceedings of the 8th IEEE annual international workshop on micro-electro-mechanical systems (MEMS’95), an investigation of micro structures, sensors, actuators, machines, and systems, Amsterdam, The Netherlands, January/February 1995, pp 209–215Google Scholar
  38. Loyalka SK, Hamoodi SA (1990) Poiseuille flow of a rarefied gas in a cylindrical tube: solution of linearized Boltzmann equation. Phys Fluids A 2(11): 2061–2065CrossRefGoogle Scholar
  39. Maurer J, Tabeling P, Joseph P, Willaime H (2003) Second-order slip laws in microchannels for helium and nitrogen. Phys Fluids 15(9):2613–2621CrossRefGoogle Scholar
  40. Mavriplis C, Ahn JC, Goulard R (1997) Heat transfer and flowfields in short microchannels using direct simulation Monte Carlo. J Thermophys Heat Transf 11(4):489–496Google Scholar
  41. Maxwell JC (1879) On stresses in rarefied gases arising from inequalities of temperature. Philos Trans R Soc 170:231–256Google Scholar
  42. Mitsuya Y (1993) Modified Reynolds equation for ultra-thin film gas lubrication using 1.5-order slip-flow model and considering surface accommodation coefficient. J Tribol 115:289–294Google Scholar
  43. Morini GL, Spiga M (1998) Slip flow in rectangular microtubes. Microscale Therm Eng 2(4):273–282CrossRefGoogle Scholar
  44. Muntz EP (1989) Rarefied gas dynamics. Annu Rev Fluid Mech 21:387–417CrossRefGoogle Scholar
  45. Muntz EP, Vargo SE (2002) Microscale vacuum pumps. In: Gad-el-Hak M (ed) The MEMS handbook. CRC Press, New York, pp 29.1–29.28Google Scholar
  46. Norberg P, Ackelid U, Lundstrom I, Petersson LG (1997) On the transient gas flow through catalytically active micromachined channels. J Appl Phys 81(5):2094–2100CrossRefGoogle Scholar
  47. Oran ES, Oh CK, Cybyk BZ (1998) Direct simulation Monte Carlo: recent advances and applications. Annu Rev Fluid Mech 30:403–441CrossRefGoogle Scholar
  48. Pan LS, Liu GR, Lam KY (1999) Determination of slip coefficient for rarefied gas flows using direct simulation Monte Carlo. J Micromech Microeng 9(1):89–96CrossRefGoogle Scholar
  49. Pan LS, Ng TY, Xu D, Lam KY (2001) Molecular block model direct simulation Monte Carlo method for low velocity microgas flows. J Micromech Microeng 11(3):181–188CrossRefGoogle Scholar
  50. Piekos ES, Breuer KS (1996) Numerical modeling of micromechanical devices using the direct simulation Monte Carlo method. J Fluid Eng 118:464–469Google Scholar
  51. Roveda R, Goldstein D, Varghese P (1998) Hybrid Euler/particle approach for continuum/rarefied flows. J Spacecraft Rockets 35(3):258–265Google Scholar
  52. Sharipov F, Seleznev V (1998) Data on internal rarefied gas flows. J Phys Chem Ref Data 27(3):657–706Google Scholar
  53. Shih JC, Ho C-M, Liu J, Tai Y-C (1996) Monatomic and polyatomic gas flow through uniform microchannels. ASME DSC 59:197–203Google Scholar
  54. Sreekanth AK (1969) Slip flow through long circular tubes. In: Trilling L, Wachman HY (eds) Proceedings of the 6th international symposium on rarefied gas dynamics. Academic Press, New York, pp 667–680Google Scholar
  55. Stefanov S, Cercignani C (1994) Monte Carlo simulation of a channel flow of a rarefied gas. Eur J Mech B–Fluids 13(1):93–114Google Scholar
  56. Vargo SE, Muntz EP (1997) An evaluation of a multiple-stage micromechanical Knudsen compressor and vacuum pump. In: Proceedings of the 20th international symposium on rarefied gas dynamics (RGD-20). Beijing, China, pp 995–1000Google Scholar
  57. Vargo SE, Muntz EP (1999) Comparison of experiment and prediction for transitional flow in a single-stage micromechanical Knudsen compressor. In: Brun R, Campargue R, Gatignol R, Lengrand J-C (eds) Rarefied gas dynamics, vol 1. Cépaduès Editions, Toulouse, France, pp 711–718Google Scholar
  58. Vargo SE, Muntz EP, Shiflett GR, Tang WC (1999) Knudsen compressor as a micro- and macroscale vacuum pump without moving parts or fluids. J Vacuum Sci Technol A 17(4):2308–2313CrossRefGoogle Scholar
  59. Wu J-S, Tseng K-C (2001) Analysis of micro-scale gas flows with pressure boundaries using direct simulation Monte Carlo method. Comput Fluids 30(6):711–735CrossRefGoogle Scholar
  60. Xue H, Fan Q (2000) A new analytic solution of the Navier-Stokes equations for microchannel flow. Microscale Therm Eng 4(2):125–143CrossRefGoogle Scholar
  61. Young RM (1999) Analysis of a micromachine based vacuum pump on a chip actuated by the thermal transpiration effect. J Vacuum Sci Technol B 17(2):280–287CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.LGMT—Institut National des Sciences AppliquéesToulouse Cedex 4France

Personalised recommendations