Abstract
The Darcy free convection boundary layer flow over a vertical flat plate is considered in the presence of volumetric heat generation/absorption. In the present first part of the paper it is assumed that the heat generation/absorption takes place in a self-consistent way, the source term q ′′′≡ S of the energy equation being an analytical function of the local temperature difference T − T ∞. In a forthcoming second part, the case of the externally controlled source terms S = S(x,y ) will be considered. It is shown that due to the presence of S, the physical equivalence of the up- and downflows gets in general broken, in the sense that the free convection flow over the upward projecting hot plate (“upflow”) and over its downward projecting cold counterpart (“downflow”) in general become physically distinct. The consequences of this circumstance are examined for different forms of S. Several analytical solutions are given. Some of them describe algebraically decaying boundary layers which can also be recovered as limiting cases of exponentially decayingones. This asymptotic phenomenon is discussed in some detail.
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Abo-Eldahab E.M. and El Aziz M.A. (2004). Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. Int. J. Thermal Sci. 43: 709–719
Abo-Eldahab E.M. and El Aziz M.A. (2005). Flow and heat transfer in a micropolar fluid past a stretching surface embedded in an non-darcian porous medium with uniform free stream. Appl Math. Comput. 162: 881–899
Bagai S. (2003). Similarity solutions of free convection boundary layers over a body of arbitrary shape in a porous medium with internal heat generation. Int. Comm. Heat Mass Transfer 30: 997–1003
Bejan A. (1995). Convective Heat Transfer 2nd ed. Wiley, New York
Bejan A., Dincer I., Lorente S., Miguel A.F. and Reis A.H. (2004). Porous and Complex Flow Structures in Modern Technologies. Springer, New York
Brown S.N. and Stewartson K. (1965). On similarity solutions of the boundary-layer equations with algebraic decay. J. Fluid. Mech. 23: 673–687
Chamkha Ali J. (1999). Hydromagnetic three-dimensional free convection on a vertical stretching surface with heat generation or absorption. Int. J. Heat Fluid Flow 20: 84–92
Chamkha Ali J. (2003). MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Int. Comm. Heat Mass Transfer 26: 1183–1191
Chamkha Ali J. and Quadri M.M.A. (2002). Combined heat and mass transfer by hydromagnetic natural convection over a cone embedded in a non-Darcian porous medium with heat generation/absorption effects. Heat Mass Transfer 38: 487–495
Cheng P. and Minkowycz W.J. (1977). Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res 82: 2040–2044
Crepeau J.C. and Clarksean R. (1997). Similarity solutions of natural convection with internal heat generation. ASME J. Heat Transfer 119: 183–185
Ene H.I. and Sanchez-Palencia E. (1982). On thermal equations for flow in porous media. Int. J. Engg. Sci. 29: 623–630
Goldstein S. (1965). On backward boundary layers and flow in converging passages. J. Fluid. Mech. 21: 33–45
Grosan T. and Pop I. (2001). Free convection over a vertical flat plate with variable wall temperature and internal heat generation in a porous medium saturated with a non-Newtonian fluid. Technische Mechanik 21: 313–318
Grosan T. and Pop I. (2002). Free convection of non-Newtonian fluids over a vertical surface in a porous medium with internal heat generation. Int. J. Appl. Mech. Engg. 7: 401–407
Hossain M.A. and Wilson M. (2002). Natural convection flow in a fluid-saturated porous medium enclosed by non-isothermal walls with heat generation. Int. J. Thermal Sci. 41: 447–454
Hossain M.A. and Rees D.A.S. (2005). Natural convection flow of water near its density maximum in a rectangular enclosure having isothermal walls with heat generation. Heat Mass Transfer 41: 367–374
Ingham, D. B. and Pop, I. (eds.): 1998, vol. II 2002, Transport Phenomena in Porous Media. Pergamon, Oxford
Ingham, D. B. and Pop, I. (eds.): 2005, Transport Phenomena in Porous Media III. Elsevier, Oxford.
Ingham, D. B., Bejan, A., Mamut, E. and Pop, I. (eds.): 2004, Emerging Technologies and Techniques in Porous Media. Kluwer, Dordrecht.
Kahn M.A. and Stewartson K. (1984). On natural convection from a cooling vertical sheet. Q. J. Mech. Appl. Math. 37: 325–338
Kuiken H.K. (1981a). On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small. IMA J. Appl. Math. 27: 387–405
Kuiken H.K. (1981). A backward free-convective boundary layers. Q. J. Mech. Appl. Math. 34: 397–413
Kuiken H.K. (1983). A class of backward free-convective boundary layer similarity solutions. Int. J. Heat Mass Transfer 26: 655–661
Magyari E. and Keller B. (2003). The opposing effect of viscous dissipation allows for a parallel free convection boundary layer flow along a cold vertical flat plate. Transport Porous Media 51: 227–230
Magyari E. and Keller B. (2004). Backward free convection boundary layers in porous media. Transport Porous Media 55: 285–300
Magyari, E. Rees, D. A. S. and Keller, B.: 2005, Effect of viscous dissipation on the flow in fluid saturated porous media, Chapter 9 in Handbook of Porous Media (2nd edition), Vafai, K. (ed.), Taylor & Francis, New York.
Merkin J.H. (1978). On solutions of the boundary-layer equations with algebraic decay. J. Fluid. Mech. 88: 309–321
Molla M.M., Hossain M.A. and Yao L.S. (2004). Natural convection flow along a vertical surface with uniform surface temperature in presence of heat generation/absorption. Int. J. Thermal Sci. 43: 157–163
Nield D.A. and Bejan A. (1999). Convection in Porous Media. Springer, New York
Pop I. and Ingham D.B. (2001). Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon, Oxford
Postelnicu A. and Pop I. (1999). Similarity solutions of free convection boundary layers with internal heat generation about a vertical and horizontal surfaces in porous media. Int. Comm. Heat Mass Transfer 26: 1183–1191
Postelnicu A., Grosan T. and Pop I. (2000). Free convection boundary layer over a vertical permeable flat plate in a porous medium with internal heat generation. Int. Comm. Heat Mass Transfer 27: 729–738
Rees D.A.S., Magyari E. and Keller B. (2003). The development of the asymptotic dissipation profile in a vertical free convective boundary layer flow in a porous medium. Transport Porous Media 53: 347–355
Tashtoush B. and Duwairi H.M. (2003). Transient mixed convection with internal heat generation and oscillating plate temperature. Acta Mechanica 174: 185–199
(2000). Handbook of Porous Media. Marcel Dekker, New York
(2005). Handbook of Porous Media, 2nd edition. Taylor & Francis, New York
Vajravelu K. and Nayfeh J. (1992). Hydromagnetic convection at a cone and a wedge. Int. Comm. Heat Mass Transfer 19: 701–710
Vajravelu K. and Hadjinicalaou A. (1993). Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation. Int. Comm. Heat Mass Transfer 20: 417–430
Vajravelu K. and Hadjinicalaou A. (1997). Convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. Int. J. Engg. Sci 35: 1237–1244
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Magyari, E., Pop, I. & Postelnicu, A. Effect of the Source Term on Steady Free Convection Boundary Layer Flows over an Vertical Plate in a Porous Medium. Part I. Transp Porous Med 67, 49–67 (2007). https://doi.org/10.1007/s11242-006-0012-1
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DOI: https://doi.org/10.1007/s11242-006-0012-1