Microfluidics and Nanofluidics

, Volume 14, Issue 1–2, pp 31–43 | Cite as

The effect of Knudsen layers on rarefied cylindrical Couette gas flows

  • Nishanth Dongari
  • Robert W. Barber
  • David R. Emerson
  • Stefan K. Stefanov
  • Yonghao Zhang
  • Jason M. Reese
Research Paper


We investigate a power-law probability distribution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature-dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. The Navier–Stokes constitutive relations and the velocity-slip boundary conditions are then modified based on this power-law scaling through the transport property expressions in terms of the mean free path. Velocity profiles for isothermal cylindrical Couette flow are obtained using this power-law model and compared with direct simulation Monte Carlo (DSMC) data. We demonstrate that our model is more accurate than the classical slip solution, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomenon of velocity inversion. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution, and is in good agreement with available DSMC data. Our proposed constitutive scaling for non-planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations of microfluidic systems.


Molecular mean free path Knudsen layer Cylindrical Couette flow Velocity inversion Curvature effects Gas micro flows Rarefied gas dynamics 



The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement ITN GASMEMS no. 215504. Additional support was provided by the UK Engineering and Physical Sciences Research Council (EPSRC) under the auspices of Collaborative Computational Project 12 (CCP12).


  1. Agrawal A, Prabhu SV (2008a) Survey on measurement of tangential momentum accommodation coefficient. J Vac Sci Technol A 26: 634–645CrossRefGoogle Scholar
  2. Agrawal A, Prabhu SV (2008b) Deduction of slip coefficient in slip and transition regimes from existing cylindrical Couette flow data. Exp Therm Fluid Sci 32:991–996CrossRefGoogle Scholar
  3. Aoki K, Yoshida H, Nakanishi T, Garcia AL (2003) Inverted velocity profile in the cylindrical Couette flow of a rarefied gas. Phys Rev E 68:016302Google Scholar
  4. Bahukudumbi P, Park JH, Beskok A (2003) A unified engineering model for steady and unsteady shear-driven gas microflows. Microscale Thermophys Eng 7:291Google Scholar
  5. Barber RW, Emerson DR (2006) Challenges in modeling gas-phase flow in microchannels: from slip to transition. Heat Transf Eng 27(4): 3–12CrossRefGoogle Scholar
  6. Barber RW, Sun Y, Gu XJ, Emerson DR (2004) Isothermal slip flow over curved surfaces. Vacuum 76:73–81CrossRefGoogle Scholar
  7. Beskok A (2001) Validation of a new velocity-slip model for separated gas microflows. Numer Heat Transf Part B: Fundam 40(6):451–471CrossRefGoogle Scholar
  8. Bird GA (1994) Molecular gas dynamics and the direct simulation of gas flows. Oxford University Press, New YorkGoogle Scholar
  9. Burnett D (1935) The distribution of molecular velocities and the mean motion in a non-unifrom gas. Proc Lond Math Soc 40:382MathSciNetCrossRefGoogle Scholar
  10. Cercignani C (1988) The Boltzmann equation and its applications. Springer, New YorkzbMATHCrossRefGoogle Scholar
  11. Chapman S, Cowling TG (1970) Mathematical theory of non-uniform gases, 3rd edn. Cambridge University Press, CambridgeGoogle Scholar
  12. Dongari N, Sharma A, Durst F (2009) Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes. Microfluid Nanofluid 6(5):679–692CrossRefGoogle Scholar
  13. Dongari N, Durst F, Chakraborty S (2010) Predicting microscale gas flows and rarefaction effects through extended Navier Stokes Fourier equations from phoretic transport considerations. Microfluid Nanofluid 9(4):831–846CrossRefGoogle Scholar
  14. Dongari N, Zhang YH, Reese JM (2011a) Molecular free path distribution in rarefied gases. J Phys D Appl Phys 44:125502Google Scholar
  15. Dongari N, Zhang YH, Reese JM (2011b) Modeling of Knudsen layer effects in micro/nanoscale gas flows. J Fluid Eng 133(7):071101Google Scholar
  16. Dongari N, Zhang YH, Reese JM (2011c) Behaviour of microscale gas flows based on a power-law free path distribution function. AIP Conf Proc 1333:724–729Google Scholar
  17. Einzel D, Panzer P, Liu M (1990) Boundary condition for fluid flow: curved or rough surfaces. Phys Rev Lett 64:2269Google Scholar
  18. Emerson RD, Gu XJ, Stefanov SK, Yuhong S, Barber RW (2007) Nonplanar oscillatory shear flow: from the continuum to the free-molecular regime. Phys Fluids 19:107105Google Scholar
  19. Guest PG (1961) The solid angle subtended by a cylinder. Rev Sci Instrum 32:164Google Scholar
  20. Guo ZL, Shi BC, Zheng CG (2011) Velocity inversion of micro cylindrical Couette flow: a lattice Boltzmann study. Comput Math Appl 61: 3519–3527MathSciNetzbMATHCrossRefGoogle Scholar
  21. Grad H (1949) Note on N-dimensional hermite polynomials. Commun Pure Appl Math 2:325-330MathSciNetzbMATHCrossRefGoogle Scholar
  22. Kennard E H (1938) Kinetic theory of gases with an introduction to statistical mechanics. McGraw-Hill, New YorkGoogle Scholar
  23. Kim S (2009) Slip velocity and velocity inversion in a cylindrical Couette flow. Phys Rev E 79:036312Google Scholar
  24. Lilley CR, Sader JE (2007) Velocity gradient singulary and structure of the velocity profile in the Knudsen layer according to the Boltzmann equation. Phys Rev E 76:026315Google Scholar
  25. Lockerby DA, Reese JM, Emerson DR, Barber RW (2004) Velocity boundary condition at solid walls in rarefied gas calculations. Phys Rev E 70:017303Google Scholar
  26. Lockerby DA, Reese JM, Gallis MA (2005) The usefulness of higher-order constitutive relations for describing the Knudsen layer. Phys Fluids 19:100609Google Scholar
  27. Lockerby DA, Reese JM (2008) On the modelling of isothermal gas flows at the microscale. J Fluid Mech 604:235–261MathSciNetzbMATHCrossRefGoogle Scholar
  28. Maxwell JC (1879) On stresses in rarefied gases arising from inequalities of temperature. Philos Trans R Soc 170:231–256Google Scholar
  29. Montroll EW, Scher H (1973) Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries. J Stat Phys 9 (2):101–135CrossRefGoogle Scholar
  30. Myong RS, Reese JM, Barber RW, Emerson DR (2005) Velocity slip in microscale cylindrical Couette flow: the Langmuir model. Phys Fluids 17:087105Google Scholar
  31. Schlichting H (1979) Boundary-layer theory, 7th edn. McGraw-Hill, New YorkGoogle Scholar
  32. Sone Y (2002) Kinetic theory and fluid dynamics. Birkhauser, BostonzbMATHCrossRefGoogle Scholar
  33. Stefanov SK, Gospodinov P, Cercignani C (1998) Monte Carlo simulation and Navier-Stokes finite difference calculation of unsteady-state rarefied gas flows. Phys Fluids 10:289Google Scholar
  34. Stefanov SK, Barber RW, Emerson DR, Reese JM (2006) The critical accommodation coefficient for velocity inversion in rarefied cylindrical Couette flow in the slip and near free-molecular regimes. In: Ivanov MS, Rebrov AK (eds) Proceedings of the 25th International Symposium on Rarefied Gas Dynamics, St. Petersburg, Russia, pp 1146–1151. Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, RussiaGoogle Scholar
  35. Stops DW (1970) The mean free path of gas molecules in the transition regime. J Phys D Appl Phys 3:685–696CrossRefGoogle Scholar
  36. Tibbs KW, Baras F, Garcia AL (1997) Anomalous flow profile due to the curvature effect on slip length. Phys Rev E 56:2282Google Scholar
  37. Veijola T, Kuisma H, Lahdenpura J, Ryhanen T (1995) Equivalent-circuit model of the squeezed gas film in a silicon accelerometer. Sens Actuators A 48:239–248CrossRefGoogle Scholar
  38. Yuhong S, Barber RW, Emerson DR (2005) Inverted velocity profiles in rarefied cylindrical Couette gas flow and the impact of the accommodation coefficient. Phys Fluids 17:047102Google Scholar
  39. Zhang YH, Gu XJ, Barber RW, Emerson DR (2006) Capturing Knudsen layer phenomena using a lattice Boltzmann model. Phys Rev E 74:046704Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Nishanth Dongari
    • 1
    • 2
  • Robert W. Barber
    • 2
  • David R. Emerson
    • 2
  • Stefan K. Stefanov
    • 3
  • Yonghao Zhang
    • 1
  • Jason M. Reese
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of StrathclydeGlasgowUK
  2. 2.STFC Daresbury Laboratory, Centre for Microfluidics and Microsystems ModellingWarringtonUK
  3. 3.Institute of Mechanics, Bulgarian Academy of SciencesSofiaBulgaria

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