An Extended Langhaar’s Solution for Two-Dimensional Entry Microchannel Flows with High-Order Slip

  • R. Rasooli
  • B. ÇetinEmail author
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)


The tremendous advances in micro-fabrication technology have brought numerous applications to the field of micro-scale science and engineering in recent decades. Microchannels are inseparable part of microfluidic technology which necessitate knowledge of flow behavior inside microchannels. For gaseous flows, the mean free path of a gas is comparable with characteristic length of a microchannel due to the micro-scale dimension of the channel. So, no-slip velocity assumption on the boundaries of channel is no longer valid, and a slip velocity needs to be defined. Although rigorous modeling of rarefied flows requires molecular solutions, researchers proposed use of slip models for applicability of the continuum equations. In slip-flow regime (i.e. Knudsen numbers up to 0.1), well-known Maxwell’s first-order slip model is applicable. For higher Knudsen numbers, higher-order slip models can be implemented to extend the applicability limit of the continuum equations. In the present study, Langhaar’s assumptions for entrance region of two-dimensional microchannels (microtube, slit-channel and concentric annular microchannel) have been implemented using high-order slip models. Different slip models proposed in the literature have been used and velocity profile, entrance length and apparent friction factor have been obtained in integral forms.


Microchannel flow High-order slip Langhaar’s solution 


  1. 1.
    Cetin, B., Ozer, M.B., Solmaz, M.E.: Microfluidic bio-particle manipulation for biotechnology. Biochem. Eng. J. 92, 63–82 (2014)CrossRefGoogle Scholar
  2. 2.
    Guckenberger, D.J., de Groot, T.E., Wan, A.M.D., Beebe, D.J., Young, E.W.K.: Micromilling: a method for ultra-rapid prototyping of plastic microfluidic devices. Lab Chip 15, 2364–2378 (2015)CrossRefGoogle Scholar
  3. 3.
    Sugioka, K., Jian, X., Dong, W., Hanada, Y., Wang, Z., Cheng, Y., Midorikawa, K.: Femtosecond laser 3d micromachining: a powerful tool for the fabrication of microfluidic, optofluidic, and electrofluidic devices based on glass. Lab Chip 14, 3447–3458 (2014)CrossRefGoogle Scholar
  4. 4.
    Kerse, C., Kalaycioglu, H., Elahi, P., Cetin, B., Kesim, D.K., Akcaalan, O., Yavas, S., Asik, M.D., Öktem, B., Hoogland, H., Holzwarth, R., Ilday, F.O.: Ablation-cooled material removal with ultrafast bursts of pulses. Nature 537, 84–88 (2016)CrossRefGoogle Scholar
  5. 5.
    Yaman, M., Khudiyev, T., Ozgur, E., Kanik, M., Aktas, O., Ozgur, E.O., Deniz, H., Korkut, E., Bayindir, M.: Arrays of indefinitely long uniform nanowires and nanotubes. Nat. Mater. 10(7), 494–501 (2011)CrossRefGoogle Scholar
  6. 6.
    Gad-el Hak, M.: The fluid mechanics of microdevices–the freeman scholar lecture. J. Fluids Eng. 121(1), 5–33 (1999)CrossRefGoogle Scholar
  7. 7.
    Colin, S.: Rarefaction and compressibility effects on steady and transient gas flows in microchannels. Microfluid. Nanofluid. 1(3), 268–279 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Agrawal, A.: A comprehensive review on gas flow in microchannels. Int. J. Micro-Nano Scale Transp. 2(1), 1–40 (2011)CrossRefGoogle Scholar
  9. 9.
    Karniadakis, G., Beskok, A., Aluru, N.: Microflows and Nanoflows: Fundamentals and Simulation. Springer Science & Business Media (2006)Google Scholar
  10. 10.
    Dongari, N., Agrawal, A., Agrawal, A.: Analytical solution of gaseous slip flow in long microchannels. Int. J. Heat Mass Transf. 50(17–18), 3411–3421 (2007)CrossRefGoogle Scholar
  11. 11.
    Zhang, W.-M., Meng, G., Wei, X.: A review on slip models for gas microflows. Microfluid. Nanofluid. 13(6), 845–882 (2012)CrossRefGoogle Scholar
  12. 12.
    Colin, S., Lalonde, P., Caen, R.: Validation of a second-order slip flow model in rectangular microchannels. Heat Transfer Eng. 25(3), 23–30 (2004)CrossRefGoogle Scholar
  13. 13.
    Duan, Z.: Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels. Microfluid. Nanofluid. 12(5), 805–820 (2012)CrossRefGoogle Scholar
  14. 14.
    Hadjiconstantinou, N.G.: Comment on cercignani’s second-order slip coefficient. Phys. Fluids 15(8), 2352–2354 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hadjiconstantinou, N.G.: Validation of a second-order slip model for dilute gas flows. Microscale Thermophys. Eng. 9(2), 137–153 (2005)CrossRefGoogle Scholar
  16. 16.
    Beskok, A., Karniadakis, G.E.: Report: A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3(1), 43–77 (1999)CrossRefGoogle Scholar
  17. 17.
    Xue, H., Fan, Q.: A new analytic solution of the navier-stokes equations for microchannel flows. Microscale Thermophys. Eng. 4(2), 125–143 (2000)CrossRefGoogle Scholar
  18. 18.
    Arkilic, E.B., Schmidt, M.A., Breuer, K.S.: Gaseous slip flow in long microchannels. J. Microelectromech. Syst. 6(2), 167–178 (1997)CrossRefGoogle Scholar
  19. 19.
    Zohar, Y., Lee, S.Y.K., Lee, W.-Y., Jiang, L., Tong, P.: Subsonic gas flow in a straight and uniform microchannel. J. Fluid Mech. 472, 125–151 (2002)CrossRefGoogle Scholar
  20. 20.
    Gat, A., Frankel, I., Weihs, D.: Gas flows through constricted shallow micro-channels. J. Fluid Mech. 602, 427–442 (2008)CrossRefGoogle Scholar
  21. 21.
    Langhaar, H.L.: Steady flow in the transition length of a straight tube. J. Appl. Mech. 9(2), 55–58 (1942)Google Scholar
  22. 22.
    Sparrow, E.M., Lin, S.H., Lundgren, T.S.: Flow development in the hydrodynamic entrance region of tubes and ducts. Phys. Fluids 7(3), 338–347 (1964)MathSciNetCrossRefGoogle Scholar
  23. 23.
    McComas, S.T.: Hydrodynamic entrance lengths for ducts of arbitrary cross section. J. Basic Eng. 89(4), 847–850 (1967)CrossRefGoogle Scholar
  24. 24.
    Chen, R.-Y.: Flow in the entrance region at low reynolds numbers. J. Fluids Eng. 95(1), 153–158 (1973)CrossRefGoogle Scholar
  25. 25.
    Renksizbulut, M., Niazmand, H., Tercan, G.: Slip-flow and heat transfer in rectangular microchannels with constant wall temperature. Int. J. Therm. Sci. 45(9), 870–881 (2006)CrossRefGoogle Scholar
  26. 26.
    Duan, Z., Muzychka, Y.S.: Slip flow in the hydrodynamic entrance region of circular and noncircular microchannels. J. Fluids Eng. 132(1), 011201 (2010)CrossRefGoogle Scholar
  27. 27.
    Darbandi, M., Schneider, G.E.: Numerical study of the flow behavior in the uniform velocity entry flow problem. Numer. Heat Transf. Part A Appl. 34(5), 479–494 (1998)CrossRefGoogle Scholar
  28. 28.
    Berman, N.S., Santos, V.A.: Laminar velocity profiles in developing flows using a laser doppler technique. AIChE J. 15, 323–327 (1969)CrossRefGoogle Scholar
  29. 29.
    Molki, A., Khezzar, L., Goharzadeh, A.: Measurement of fluid velocity development in laminar pipe flow using laser doppler velocimetry. Euro. J. Phys. 34(5), 1127 (2013)CrossRefGoogle Scholar
  30. 30.
    Shah, R.K., London, A.L.: Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. Academic Press, Cambridge (1978)Google Scholar
  31. 31.
    Sugino, Eitaro: Velocity distribution and pressure drop in the laminar inlet of a pipe with annular space. Bull. JSME 5(20), 651–655 (1962)CrossRefGoogle Scholar
  32. 32.
    Han, L.S.: Hydrodynamic entrance lengths for incompressible laminar flow in rectangular ducts. J. Appl. Mech. 27(3), 403–409 (1960)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Harley, J.C., Huang, Y., Bau, H.H., Zemel, J.N.: Gas flow in micro-channels. J. Fluid Mech. 284, 257–274 (1995)CrossRefGoogle Scholar
  34. 34.
    Arkilic, E.B., Breuer, K.S., Schmidt, M.A.: Mass flow and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437, 29–43 (2001)CrossRefGoogle Scholar
  35. 35.
    Cercignani, C., Lorenzani, Silvia: Variational derivation of second-order slip coefficients on the basis of the boltzmann equation for hard-sphere molecules. Phys. Fluids 22(6), 062004 (2010)CrossRefGoogle Scholar
  36. 36.
    Cetin, B., Yazicioglu, A.G., Kakac, S.: Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation-An extended Graetz problem. Int. J. Therm. Sci. 48, 1673–1678 (2009)CrossRefGoogle Scholar
  37. 37.
    Cetin, B., Bayer, O.: Evaluation of Nusselt number for a flow in a microtube using second-order slip model. Therm. Sci. 15(Suppl. 1), 103–109 (2011)CrossRefGoogle Scholar
  38. 38.
    Çetin, Barbaros: Effect of Thermal Creep on Heat Transfer for a Two-Dimensional Microchannel Flow: An Analytical Approach. J. Heat Transf. 135(10), 101007–101008 (2013)CrossRefGoogle Scholar
  39. 39.
    Çetin, B., Zeinali, S.: Analysis of heat transfer and entropy generation for a low-Peclet-number microtube flow using a second-order slip model: an extended-Graetz problem. J. Eng. Math. 89(1), 13–25 (2014)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Microfluidics and Lab-on-a-chip Research Group, Mechanical Engineering Department İ.D. Bilkent UniversityAnkaraTurkey

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