2150 Meccanica(2013)48:2149-215 [17]for the problem of free convection ve layer flow in cooled ho ntal surfoc Auid. aso the nressure gradient is a erated precluding the develop ment of a boundary-layer flow ents on thi There is intrinsic interest in convective flows ove tion horizontal surfaces as they offer an alternative mecha ayer flo nism for driving a convective dow and have been stud. ied both theoretically and experimentally.rotem and 、convective mp 1201.Cortell Claassen [3 showed experimentally the existence of [21,M a boundary-layer flow near the leading edge above a kinde and Azi [22].Makinde and Olanre tal.25 heated horizontal surface.This was result was con rkin and Pop [24]and Ya wtonian.variable firmed by Pera and Gebhart [4]who also treated a slightly inclined surface,also studied theoretically in ity [27]and nanofluid convection within a porous ma- terial [28]. more detail by Jones [51.The se fows can also play In the present paper.the effect of steady mixed an important r in the modelling of severa applic convection boundary laver flow over a horizontal ions,one ing large scale fire ere a nre spre flat surface is studied,when the upper face of the plate is heated convectively.Using pseudo-similarity as emai This sets up variables,the basic partial differential equations are y whie reduced to a coupled system of ordinary differen- tial equations.The resulting similarity equations are aural and fo givin the solved numerically and the results discussed with the on horiz limiting cases of free convection limit and a high free provide a useful insight into this complex problem stream velocity analyzed. This forms the basis for our.admittedly rather simple model.There is as well the question as to what sur boundary condition to apply on the temper ature nei ther a prescribed temperature or heat fux would seem 2 Equations entirely appropriate.hence we take a convective con dition,being in essence a combination of these two We consider the steady mixed convection boundary nditions.Although this could well be an over sim- on a horizon ntal flat surface.We plification,it should provide further useful insights d is within There has already been some work on mixed con- that the vection boundary-layer flows along horizontal flat sur ated by convectio om a ot fluid faces.Here we mention specifically pape ure wit [6].Dey [7].Afz an Unde Hong e see [18]fo 110].D. Stei sand the mple approx ma -layer equations can be written as papers by Schneid er 1 see [1]for example the results pre +=0 (1) ar' oth and heat flux s.The 2) ondition in this been considered previously and we show that this new 1 ap =88(T-T) (3) effect leads to some interesting and novel features o av The idea of using a convective (or conjugate) aT aT 27 boundary condition was first introduced by Merkin +"= Springer 2150 Meccanica (2013) 48:2149–2158 edge, i.e. to a favourable pressure gradient, and hence a boundary-layer flow starting at the leading edge. Con￾versely above a cooled horizontal surface an adverse pressure gradient is generated precluding the develop￾ment of a boundary-layer flow. There is intrinsic interest in convective flows over horizontal surfaces as they offer an alternative mecha￾nism for driving a convective flow and have been stud￾ied both theoretically and experimentally. Rotem and Claassen [3] showed experimentally the existence of a boundary-layer flow near the leading edge above a heated horizontal surface. This was result was con- firmed by Pera and Gebhart [4] who also treated a slightly inclined surface, also studied theoretically in more detail by Jones [5]. These flows can also play an important role in the modelling of several applica￾tions, one of which being large scale fires. Here a fire, for example a bush fire, can spread over a large area and, after the combustion front has passed, a region of heated ground can remain. This sets up a convec￾tive flow which can converge to form buoyant plumes. There can also be a wind giving an interaction be￾tween natural and forced convection. Thus the study of mixed convection flows on horizontal surfaces can provide a useful insight into this complex problem. This forms the basis for our, admittedly rather simple, model. There is as well the question as to what surface boundary condition to apply on the temperature. Nei￾ther a prescribed temperature or heat flux would seem entirely appropriate, hence we take a convective con￾dition, being in essence a combination of these two conditions. Although this could well be an over sim￾plification, it should provide further useful insights. There has already been some work on mixed con￾vection boundary-layer flows along horizontal flat sur￾faces. Here we mention specifically papers by Schnei￾der [6], Dey [7], Afzal and Hussain [8], De Hong et al. [9], Ramanaiah et al. [10], Daniels [11], Steinrück [12, 13], Rudischer and Steinrück [14]. There are ex￾cellent review papers by Schneider [15] and Steinrück [16] which summarize the results previously reported on this problem. Previous treatments of mixed convec￾tion boundary-layer flows along horizontal surfaces have considered an isothermal or variable surface tem￾perature and heat flux conditions. The application of a convective boundary condition in this context has not been considered previously and we show that this new effect leads to some interesting and novel features. The idea of using a convective (or conjugate) boundary condition was first introduced by Merkin [17] for the problem of free convection past a ver￾tical flat plate immersed in a viscous (Newtonian) fluid. More recently, Aziz [18], see also the com￾ments on this paper by Magyari [19], used the convec￾tive boundary condition to study the classical problem of forced convection boundary-layer flow over a flat plate. Since then, a number of boundary-layer flows have been revised with a convective boundary con￾ditions, see for example Ishak [20], Cortell Bataller [21], Makinde and Aziz [22], Makinde and Olanre￾waju [23], Merkin and Pop [24] and Yao et al. [25], including non-Newtonian fluids [26], variable viscos￾ity [27] and nanofluid convection within a porous ma￾terial [28]. In the present paper, the effect of steady mixed convection boundary layer flow over a horizontal flat surface is studied, when the upper face of the plate is heated convectively. Using pseudo-similarity variables, the basic partial differential equations are reduced to a coupled system of ordinary differen￾tial equations. The resulting similarity equations are solved numerically and the results discussed with the limiting cases of free convection limit and a high free stream velocity analyzed. 2 Equations We consider the steady, mixed convection boundary￾layer flow on a horizontal flat surface. We assume that the surface faces upwards and is within a uniform am￾bient temperature T∞. We also assume that the sur￾face is heated by convection from a hot fluid source of constant temperature Tf with a corresponding heat transfer coefficient hf , see [18] for example. Under these assumptions and the usual Boussinesq approxi￾mation the boundary-layer equations can be written as, see [10] for example, ∂u ∂x + ∂v ∂y = 0 (1) u ∂u ∂x + v ∂u ∂y = − 1 ρ ∂p ∂x + ν ∂2u ∂y2 (2) 1 ρ ∂p ∂y = gβ(T − T∞) (3) u ∂T ∂x + v ∂T ∂y = α ∂2T ∂y2 (4)