Skip to main content
SpringerLink
Log in

Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical flat plate

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady laminar free convection from a non-isothermal vertical flat plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature variations with position and time are derived. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. The heat-transfer characteristics for finite values of the Prandtl number Pr are presented, as temperature and velocity distributions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ames, W.F., Nonlinear partial differential equations in engineering, Vol. II, Chapter 2, Academic Press, New York (1972).

    Google Scholar 

  2. Ames, W.F., Similarity for the nonlinear diffusion equation, I & EC Fundamentals 4 (1965) 72–76.

    Google Scholar 

  3. Ames, W.F. and Nucci, M.C., Analysis of fluid equations by group methods, J. Engg. Math. 20 (1985) 181–187.

    Google Scholar 

  4. Birkhoff, G., Mathematics for engineers, Elect. Eng. 67 (1948) 1185.

    Google Scholar 

  5. Birkhoff, G., Hydrodynamics, Princeton Univ. Press, Princeton, New Jersey (1960).

    Google Scholar 

  6. Bluman, G.W. and Cole, J.D., Similarity methods of differential equations, Springer, New York (1974).

    Google Scholar 

  7. Boisvert, R.E., Ames, W.F. and Srivastava, U.N., Group properties and new solutions of Navier-Stokes equations, J. Engg. Math. 17 (1983) 203–221.

    Google Scholar 

  8. Brindley, J., An approximate technique for natural convection in a boundary layer, Int. J. Heat Mass Transfer 6 (1963) 1035–1048.

    Google Scholar 

  9. Burmeister, L.C., Convective heat transfer, John Wiley and Sons, New York (1983).

    Google Scholar 

  10. Callahan, G.D. and Marner, W.J., Transient free convection with mass transfer on an isothermal vertical flat plate, Int. J. Heat Mass Transfer 19 (1976) 165–174.

    Google Scholar 

  11. Eisenhart, L.P., Continous groups of transformation, Dover Publications, Inc., New York (1961).

    Google Scholar 

  12. Gabbert, C.H., Similarity for unsteady compressible boundary layers, AIAA J. 5 (1967) No. 6.

    Google Scholar 

  13. Gaggioli, R.A. and Moran, M.J., Group theoretic technique for the similarity solution of the systems of partial differential equations with auxiliary conditions, Math. Res. Center, U.S. Army, Univ. of Wisconsin, Tech. Summary Report No. 693 (1966).

  14. Gebhart, B., Transient natural convection from vertical elements, Trans. ASME: J. Heat Transfer 83 (1961) 61–70.

    Google Scholar 

  15. Heinisch, R.P., Viskanta, R. and Singer, R.M., Approximate solution of the transient free convection laminar boundary layer equations, Z. Angew. Math. Phys. 20 (1969) 19–33.

    Google Scholar 

  16. Hellums, J.D. and Churchill, S.W., Transient and steady state free and natural convection, numerical solutions, AIChE J. 8 (1962) 690–695.

    Google Scholar 

  17. Illingworth, C.R., Unsteady laminar flow of gas near an infinite flat plate, Proc. Camb. Phil. Soc. 46 (1950) 603–613.

    Google Scholar 

  18. Kulkarni, A.K., Jacobs, R.H. and Hwang, J.J., Similarity solution for natural convection flow over an isothermal vertical wall immersed in thermally stratified medium, Int. J. Heat Mass Transfer 30 (1987) 691–698.

    Google Scholar 

  19. Manhor, R., Some similarity solutions of partial differential equations of boundary layer, Math. Research Center, U.S. Army, Univ. of Wisconsin, Tech. Summary Report No. 375 (1963).

  20. Mankabadi, R.R., On the use of natural convection flow for solar pumping, Energy Sources 10 (1988) 173–182.

    Google Scholar 

  21. Menold, E.R. and Yang, K.T., Asymptotic solutions for unsteady laminar free convection on a vertical plate, Trans. ASME: J. Appl. Mech 29 (1962) 124–126.

    Google Scholar 

  22. Michal, A.D., Differential invariants and invariant partial differential equations under continuous transformation groups in normal linear spaces, Proc. Nat. Acad. Sci. U.S.A. 37 (1952) 623–627.

    Google Scholar 

  23. Mitchell, J.W. and Linkhan, J.H., A simplified general approach for obtaining similarity solutions to the partial differential equations of heat transfer and fluid mechanics, Univ. of Wisconsin, internal report (1965).

  24. Moran, M.J. and Gaggioli, R.A,, Similarity analysis via group theory, AIAA J. 6 (1968) 2014–2016.

    Google Scholar 

  25. Moran, M.J. and Gaggioli, R.A., Similarity analysis of compressible boundary layer flows via group theory, U.S. Army Math. Research Center, Tech. Summary Report No. 838, Madison, Wisconsin (1967).

  26. Moran, M.J. and Gaggioli, R.A., A new systematic formalism for similarity analysis, with application to boundary layer flows, U.S. Army Math. Research Center, Tech. Summary Report No. 918, (1968).

  27. Moran, M.J. and Gaggioli, R.A., Reduction of the number of variables in systems of partial differential equations with auxiliary conditions, SIAM J. App. Math. 16 (1968) 202–215.

    Google Scholar 

  28. Morgan, A.J.A., The reduction by one of the number of independent variables in some systems of partial differential equations, Quart. J. Math. 3 (1952) 250–259.

    Google Scholar 

  29. Siegel, R., Transient free convection from a vertical flat plate, Trans. ASME C: J. Heat Transfer 80 (1958) 347–359.

    Google Scholar 

  30. Slater, L.J., Confluent hypergeometric functions, Cambridge Univ. Press, London (1960).

    Google Scholar 

  31. Sparrow, E.M., Gregg, J.L., Similar solutions for free convection from a nonisothermal vertical plate, Trans. ASME C: J. Heat Transfer 80 (1958) 379–386.

    Google Scholar 

  32. Williams, J.C., Mulligan, J.C. and Rhyne, T.B., Semisimilar solutions for unsteady free-convective boundary-layer flow on a vertical flat plate, J. Fluid Mech. 175 (1987) 309–332.

    Google Scholar 

  33. Yang, K.T., Possible similarity solutions for laminar free convection on vertical plates and cylinders, Trans. ASME E: J. Appl. Mech. 82 (1960) 230–236.

    Google Scholar 

  34. Yang, K.T., Novotny, J.L. and Cheng, Y.S., Laminar free convection from a nonisothermal plate immersed in a temperature stratified medium, Int. J. Heat Mass Transfer 15 (1972) 1097–1109.

    Google Scholar 

  35. Zettl, G., An algorithm for minimization of a function of many variables, Numerical Methods 15 (1970) 115–432.

    Google Scholar 

Download references

Author information

Author notes

  1. M. B. Abd-El-Malek

    Present address: Department of Science, Mathematics Unit, The American University in Cairo, P.O. Box 2511, Cairo, Egypt

Authors and Affiliations

  1. Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt

    Y. Z. Boutros & N. A. Badran

Authors
  1. M. B. Abd-El-Malek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Z. Boutros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. A. Badran
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abd-El-Malek, M.B., Boutros, Y.Z. & Badran, N.A. Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical flat plate. J Eng Math 24, 343–368 (1990). https://doi.org/10.1007/BF00130688

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00130688

Keywords

Search

Navigation