Microfluidics and Nanofluidics

, Volume 9, Issue 4–5, pp 847–853 | Cite as

Rarefaction effects on gas viscosity in the Knudsen transition regime

  • Vasilis K. Michalis
  • Alexandros N. Kalarakis
  • Eugene D. Skouras
  • Vasilis N. Burganos
Research Paper


The effects of rarefaction on gas viscosity are investigated through the simulation of isothermal, low speed flow in a long straight channel using the Direct Simulation Monte Carlo (DSMC) method. Following convergence to the flow field inside the channel, the effective viscosity is calculated directly from its definition using shear stress calculations in each individual cell assuming that the gas flow is close to a local equilibrium state. Averaging over the cross-sectional area at different positions down the pressure gradient allows the determination of the gas viscosity as a function of the local Knudsen number (Kn) along the channel. Following an extensive investigation of this dependence over a wide range of Kn values, it was conveniently found that a Bosanquet-type of approximation describes very satisfactorily the Knudsen number dependence of the viscosity over the entire transition regime, i.e., from the slip-flow to the free-molecular flow limit. Such a simple functional dependence is expected to facilitate significantly phenomenological descriptions and numerical computations of rarefied flows that rely on the notion of an effective viscosity in the transition regime.


Rarefied flow Effective viscosity DSMC Straight channel 


  1. Agrawal A, Prabhu SV (2008a) Deduction of slip coefficient in slip and transition regimes from existing cylindrical Couette flow data. Exp Thermal Fluid Sci 32:991–996CrossRefGoogle Scholar
  2. Agrawal A, Prabhu SV (2008b) Survey on measurement of tangential momentum accommodation coefficient. J Vacuum Sci Technol A 26:634–645CrossRefGoogle Scholar
  3. Alexander FJ, Garcia AL (1997) The direct simulation Monte Carlo method. Comput Phys 11:588–593CrossRefGoogle Scholar
  4. Alexander FJ, Garcia AL, Alder BJ (1998) Cell size dependence of transport coefficients in stochastic particle algorithms. Phys Fluids 10:1540–1542CrossRefGoogle Scholar
  5. Barber RW, Emerson DR (2006) Challenges in modeling gas–phase flow in microchannels: from slip to transition. Heat Transf Eng 27:3–12CrossRefGoogle Scholar
  6. Beskok A, Karniadakis GE (1999) A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys Eng 3:43–77CrossRefGoogle Scholar
  7. Bird GA (1994) Molecular gas dynamics and the direct simulation of gas flows. Clarendon Press, OxfordGoogle Scholar
  8. Bird GA (1998) Recent advances and current challenges for DSMC. Comput Math Appl 35:1–14MATHCrossRefMathSciNetGoogle Scholar
  9. Cai CP, Boyd ID, Fan J, Candler GV (2000) Direct simulation methods for low-speed microchannel flows. J Thermophys Heat Transf 14:368–378CrossRefGoogle Scholar
  10. Chapman S, Cowling TG (1970) The mathematical theory of non-uniform gases, 3rd edn. Cambridge University Press, New YorkGoogle Scholar
  11. Dongari N, Sambasivam R, Durst F (2009) Extended Navier–Stokes equations and treatments of micro-channel gas flows. J Fluid Sci Technol 4:454–467CrossRefGoogle Scholar
  12. Evans A, Bieberle-Hutter A, Rupp JLM, Gauckler LJ (2009) Review on microfabricated micro-solid oxide fuel cell membranes. J Power Sources 194:119–129CrossRefGoogle Scholar
  13. Fan J, Shen C (1999) Statistical simulation of low-speed unidirectional flows in transition regime. In: Brun R, Campargue R, Gatignol R, Lengrand JC (eds) Rarefied gas dynamics, vol 2. Cepadus-Editions, ToulouseGoogle Scholar
  14. Fan J, Shen C (2001) Statistical simulation of low-speed rarefied gas flows. J Comput Phys 167:393–412MATHCrossRefGoogle Scholar
  15. Guo ZL, Shi BC, Zheng CG (2007) An extended Navier–Stokes formulation for gas flows in the Knudsen layer near a wall. Europhys Lett 80:24001–24006CrossRefGoogle Scholar
  16. Guo ZL, Zheng CG, Shi BC (2008) Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys Rev E 77:036707CrossRefGoogle Scholar
  17. Hadjiconstantinou NG (2000) Analysis of discretization in the direct simulation Monte Carlo. Phys Fluids 12:2634–2638CrossRefGoogle Scholar
  18. Karniadakis G, Beskok A, Aluru N (2005) Microflows and nanoflows: fundamentals and simulation. Springer, New YorkMATHGoogle Scholar
  19. LeBeau GJ, Lumpkin FE III (2001) Application highlights of the DSMC analysis code (DAC) software for simulating rarefied flows. Comput Methods Appl Mech Eng 191:595–609MATHCrossRefGoogle Scholar
  20. Lilley CR, Sader JE (2009) Velocity profile in the Knudsen layer according to the Boltzmann equation. Proc R Soc A 464:2015–2035CrossRefGoogle Scholar
  21. Pollard WG, Present RD (1948) On gaseous self-diffusion in long capillary tubes. Phys Rev 73:762–774CrossRefGoogle Scholar
  22. Roohi E, Darbandi M (2009) Extending the Navier–Stokes solutions to transition regime in two-dimensional micro- and nanochannel flows using information preservation scheme. Phys Fluids 21:082001CrossRefGoogle Scholar
  23. Stops DW (1970) The mean free path of gas molecules in the transition regime. J Phys D 3:685–696CrossRefGoogle Scholar
  24. Sun Q, Boyd ID (2002) A direct simulation method for subsonic microscale gas flows. J Comput Phys 179:400–425MATHCrossRefGoogle Scholar
  25. Wagner W (1992) A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation. J Stat Phys 66:1011–1044MATHCrossRefGoogle Scholar
  26. Wu JS, Tseng KC (2001) Analysis of micro-scale gas flows with pressure boundaries using direct simulation Monte Carlo. Comput Fluids 30:711–735MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Vasilis K. Michalis
    • 1
    • 2
  • Alexandros N. Kalarakis
    • 1
  • Eugene D. Skouras
    • 1
  • Vasilis N. Burganos
    • 1
  1. 1.Institute of Chemical Engineering and High Temperature Chemical ProcessesFoundation for Research and TechnologyPatrasGreece
  2. 2.Department of Chemical EngineeringUniversity of PatrasPatrasGreece

Personalised recommendations