This study is devoted to the investigation of thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined isothermal plane. It is assumed that the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. The governing non-linear equations for conservation of momentum and energy are obtained and solved by using a new computational approach based on a special type of Hermite-Padé approximation technique implemented in MAPLE. This semi-numerical scheme offers some advantages over solutions obtained with traditional methods such as finite differences, spectral method, and shooting method. It reveals the analytical structure of the solution function. Important properties of overall flow structure including velocity field, temperature field, thermal criticality, and bifurcations are discussed.
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Szeri, A. Z. and Rajagopal, K. R. Flow of a non-Newtonian fluid between heated parallel plates. International Journal of Non-Linear Mechanics 20, 91–101 (1985)
Siddiqui, A. M., Mahomood, R., and Ghori, Q. K. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons & Fractals 35(1), 140–147 (2008)
Cebeci, T. and Bradshaw, P. Physical and computational aspects of convective heat transfer, Springer-Verlag, New York (1984)
Makinde, O. D. Irreversibility analysis for gravity driven non-Newtonian liquid film along an inclined isothermal plate. Physica Scripta 74, 642–645 (2006)
Weinstein, S. J., Ruschak, K.J., and Ng, K. C. Developing flow of a power-law liquid film on an inclined plane. Physics of Fluids 15(10) 2973–2986 (2003)
Hayat, T., Asif Farooq, M., Javed T., and Sajid, M. Partial slips effects on the flow and heat transfer characteristics in a third grade fluid. Nonlinear Analysis: Real World Applications 10(2), 745–755 (2009)
Makinde, O. D. Laminar falling liquid film with variable viscosity along an inclined heated plate. Applied Mathematics and Computation 175, 80–88 (2006)
Massoudi, M. and Christie, I. Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe. International Journal of Non-Linear Mechanics 30, 687–699 (1995)
Frank-Kamenetskii, D. A. Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York (1969)
Bowes, P. C. Self-Heating: Evaluating and Controlling the Hazard, Elsevier, Amsterdam (1984)
Hunter, D. L. and Baker, G. A. Methods of series analysis III: integral approximant methods. Physical Review B 19, 3808–3821 (1979)
Landau, L. D. and Lifshitz, E. M. Fluid Mechanics (2nd ed.), Pergamon Press, New York (1989)
Bender, C. and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill (1978)
Guttamann, A. J. Asymptotic analysis of power-series expansions. Phase Transitions and Critical Phenomena (eds. Domb, C. and Lebowitz, J. K.), Academic Press, New York, 1–234 (1989)
Project supported by the National Research Foundation of South Africa Thuthuka Programme
(Communicated by Zhe-wei ZHOU)
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Makinde, O.D. Thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined plane. Appl. Math. Mech.-Engl. Ed. 30, 373–380 (2009). https://doi.org/10.1007/s10483-009-0311-6
- isothermal inclined plate
- third-grade fluid
- Hermite-Padé approximation
- thermal criticality
Chinese Library Classification
2000 Mathematics Subject Classification