Skip to main content

Near Continuum Velocity and Temperature Coupled Compressible Boundary Layer Flow over a Flat Plate


The problem of a compressible gas flows over a flat plate with the velocity-slip and temperature-jump boundary conditions are being studied. The standard single- shooting method is applied to obtain the exact solutions for velocity and temperature profiles when the momentum and energy equations are weakly coupled. A double-shooting method is applied if these two equations are closely coupled. If the temperature affects the velocity directly, more significant velocity slip happens at locations closer to the plate’s leading edge, and inflections on the velocity profiles appear, indicating flows may become unstable. As a consequence, the temperature-jump and velocity-slip boundary conditions may trigger earlier flow transitions from a laminar to a turbulent flow state.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14


  1. H. Blasius, Grenzschichten in flüssigkeiten bei sehr kleiner reibung. Z. Angew. Math. Phys. 56, 1534–1535 (1908)

    Google Scholar 

  2. H. Schlichting, K. Gersten, Boundary-layer theory, 8th edn. (Springer) (2003)

  3. C.M. Ho, Y.C. Tai, Micro-electro-mechanical-systems (MEMS) and fluid flows. Annu. Rev. Fluid Mech. 30, 579–612 (1998)

    Article  ADS  Google Scholar 

  4. M. Gad-el-Hak, The fluid mechanics of microdevices —- the Freeman scholar lecture. J. Fluid. Eng-T. ASME. 121, 5 (1999)

    Article  Google Scholar 

  5. M.J. Martin, I.D. Boyd, Momentum and heat transfer in a laminar boundary layer with slip flow. J. Thermophys Heat Tr. 20(4), 710–719 (2006)

    Article  Google Scholar 

  6. M.H. Yazdi, S. Abdullah, I. Hashim, A. Zaharim, K. Sopian, N.E. Mastorakis, M. Poulos, V. Mladenov, Z. Bojkovic, D. Simian, Friction and heat transfer in slip flow boundary layer at constant heat flux boundary conditions. Math. Comput. Sci. Engin. 10, 207 (2008)

    Google Scholar 

  7. J. Lahjomri, A. Oubarra, Hydrodynamic and thermal characteristics of laminar slip flow over a horizontal isothermal flat plate. J. Heat Transf. 135, 021704 (2013)

    Article  Google Scholar 

  8. J.C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature. Proc. Royal Soci London. 27, 185–189 (1878)

    MATH  Google Scholar 

  9. C. Cai, Near continuum boundary layer flows at a flat plate. Theor. Appl. Mech. Lett. 5(8), 134–139 (2015)

    Article  Google Scholar 

  10. A. Aziz, Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Commun. Nonlinear Sci. 15(3), 573–580 (2010)

    Article  Google Scholar 

  11. M.R. Safaei, G. Ahmadi, M.S. Goodarzi, A. Kamyar, S.N. Kazi, Boundary layer flow and heat transfer of FMWCNT/water nanofluids over a flat plate. Fluids. 1(4), 31 (2016)

    Article  Google Scholar 

  12. N.K. Vedantam, N. Parthasarathy, N. Ramkumar, Effects of slip on the flow characteristics of laminar flat plate boundary-layer, 2nd US-European Fluids Engineering Meeting. p. 1551 (2006)

  13. F.M. White, Viscous fluid flow (McGraw-Hill, 2006)

  14. E. Pohlhausen, Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung, Vol. 1. Wiley (1921)

  15. C.R. Illingworth, Some solutions of the equations of flow of a viscous compressible fluids. Proc. Cambridge Pilos. Soc. 46, 469–478 (1950)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. C.B. Cohen, E. Reshotko, The compressible laminar boundary layer with heat transfer and arbitrary pressure gradient. Advances Heat Transf. 4, 317 (1967)

    Article  Google Scholar 

  17. L. Rayleigh, Sci. Pap. 1, 474 (1964)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Chunpei Cai.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

He, X., Cai, C. Near Continuum Velocity and Temperature Coupled Compressible Boundary Layer Flow over a Flat Plate. Braz J Phys 47, 182–188 (2017).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Boundary layer
  • Boundary conditions
  • Velocity slip
  • Flow instability