《高等传热学》课程教学资源（文献资料）Mixed convection boundary-layer flow on a horizontal flat surface with a convective boundary condition

6ia89o118a127958 amie20131482140215 Mixed convection boundary-layer flow on a horizontal flat surface with a convective boundary condition T.Grosan.J.H.Merkin.LPop cht 2013 Abstract The steady mixed conv vection boundary processes inengineering devices and innature,incud ayer flow on an upward rizon ng solar rec exposed to wind currents d to similarity form a necessary requirement for emergency shutdown.heat exchanges placed in a low which is that the outer flow and surface heat transfer velocity environment,etc.Such processes occur when coefficient are spatially dependent.The resulting sim- the effect of the buoyancy force in forced convection ilarity equations involve,apart from the Prandtl num or the effect of a forced flow in free convection be- ber,two dimensionless parameters,a measure of the comes significant.Thermal buoyancy forces can play relative strength of the outer flow M and a heat trans- an important role in forced convection heat transfer fer coefficient y.The free convection,M=0,case when the flow velocity is relatively low or the temper- is considered with the asymptotic limits of large and ature difference between with the free stream is rela- small y being derived.Results for the general,M>0. ively larg case are pre ented and the asymptotic limit of large M ctive flows,both free and mixed,on vertical being treated. or inclined surfac s have alr Somewhat less Keywords Boundary-layer flow:mixed conve to mixed convect ntal surface.Co nvective ion on horizontal surfaces,which is oundary conditior what we discuss here.Natural convection on a hori zontal surface,sometimes referred to an'indirect con- 1 Introduction vection'arises through a different mechanism to that on a vertical surface.Here the buoyancy forces act- ing vertically.i.e.normal to the surface,generate a Mixed convection flows,or the combination of both forced and free convection. arise in many transpor horizontal pressure gradient,i.e.parallel to the sur- face.It is this longitudinal pressure gradient that drives the ective e of this is that T.Grosan:I Pon nd. possible artson [1) Fo a boundary-layer f ow above a he J.H.Merkin☒) ed horizonta face the density is less than the ambient density.Thi gives rise to a decrease in the hydrostatic pressure at e-mail:amtjhm@maths.leeds.ac.uk the surface with increasing distance from the leading Springer

Meccanica (2013) 48:2149–2158 DOI 10.1007/s11012-013-9730-y Mixed convection boundary-layer flow on a horizontal flat surface with a convective boundary condition T. Grosan · J.H. Merkin ·I. Pop Received: 23 January 2013 / Accepted: 22 March 2013 / Published online: 5 April 2013 © Springer Science+Business Media Dordrecht 2013 Abstract The steady mixed convection boundary￾layer flow on an upward facing horizontal surface heated convectively is considered. The problem is re￾duced to similarity form, a necessary requirement for which is that the outer flow and surface heat transfer coefficient are spatially dependent. The resulting sim￾ilarity equations involve, apart from the Prandtl num￾ber, two dimensionless parameters, a measure of the relative strength of the outer flow M and a heat trans￾fer coefficient γ . The free convection, M = 0, case is considered with the asymptotic limits of large and small γ being derived. Results for the general, M > 0, case are presented and the asymptotic limit of large M being treated. Keywords Boundary-layer flow: mixed convection · Horizontal surface · Convective boundary condition 1 Introduction Mixed convection flows, or the combination of both forced and free convection, arise in many transport T. Grosan · I. Pop Department of Applied Mathematics, Babe¸s-Bolyai University, 3400 Cluj, CP 253, Romania J.H. Merkin () Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK e-mail: amtjhm@maths.leeds.ac.uk processes in engineering devices and in nature, includ￾ing solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency shutdown, heat exchanges placed in a low￾velocity environment, etc. Such processes occur when the effect of the buoyancy force in forced convection or the effect of a forced flow in free convection be￾comes significant. Thermal buoyancy forces can play an important role in forced convection heat transfer when the flow velocity is relatively low or the temper￾ature difference between with the free stream is rela￾tively large. Convective flows, both free and mixed, on vertical or inclined surfaces have already received much at￾tention. Somewhat less consideration has been given to mixed convection on horizontal surfaces, which is what we discuss here. Natural convection on a hori￾zontal surface, sometimes referred to an ‘indirect con￾vection’ arises through a different mechanism to that on a vertical surface. Here the buoyancy forces act￾ing vertically, i.e. normal to the surface, generate a horizontal pressure gradient, i.e. parallel to the sur￾face. It is this longitudinal pressure gradient that drives the convective flow. A consequence of this is that an attached boundary-layer flow is possible only on one side of a horizontal surface, as first pointed out by Stewartson [1], corrected by Gill et al. [2]. For a boundary-layer flow above a heated horizontal sur￾face the density is less than the ambient density. This gives rise to a decrease in the hydrostatic pressure at the surface with increasing distance from the leading

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)

Meccanica(2013)48:2149-2158 2151 on 0<x<oo,0<y<oo subject to the boundary f”+9+产9s)ds)+f-2 conditions aT =v=0.kay=-hf()(-T)ony=0 (12) 4→Ux).T→Too asy→o on applying the boundary conditions as.We can then put P(n)=(s)ds to obtain where x and y are respectively the cartesian coordi nates measured along the surface and normal to it,u and v are the velocity components in the x and y di- f"+三0+P)+2ff"+(M2-f)=0(13) rections.T is the fluid temperature.p is the pre ssure c p'=-9 with then P→0asn→o (14) is the acceleration due to gravity.a is the thermal dif- fusivity.8 is the coefficient of thermal expansion.v is It is the problem given by Eqs.(1).(13).(14)subject the kinematic viscosity,k is the thermal conductivity to boundary conditions(11)that we now consider. and x)is outer flow.We can eliminate the Before discussing the solution this problem in de sure p from Fas ().(3)by differentiating ()with tail we can make some general observations about the nature of the solution.If we put g(n)=of(s)ds, 03 we can formally express the solution to Eq.(10)as 2u 6 gm=-Ae-9(15 To reduce (1).(4).(5).(6)to similarity form we need to specify specific functional forms for both h for some constant A.Expression(15)shows that 'is and U as of one sign on 0sn<oo.Boundary conditions (11) hf(x)=kCox-2/5. Uo(x)=/5 give We then put A=1+yloo where loo= 6 e-9mdm>0(16 =(u2g△T)5x35fm) Since y>0,we have A>0 and hence,when a solu- tion to (10).(13).(14).(11)exists,it must have> T-T=△T 2 and0'<0for0≤n<o. where AT=T-To.This gives fm+2+f"+5f”=0 3 Results (9) +0-0 (10) We start by considering the free convection,M=0. limit before considering the general problem where primes denote differentiation with re on 3.1 Free convection limit,M=0 ditions (5)beco f=f'=0. 0'=-y(1-0)0nn=0 Equations (10).(13).(14)with M=0 subject to olved n f→M, →0as0→0 nditions(11) ing a standard bo the sults,shown by plots of f(and0) =10 re given in Fig.1.We nd e(0)ir non M U y=Co 21s appe (w(g△T)P)5 for large y. to zero asy de This lead o consider We can integrate Eq.(9)to get Springer

Meccanica (2013) 48:2149–2158 2151 on 0 ≤ x < ∞, 0 ≤ y < ∞ subject to the boundary conditions u = v = 0, k ∂T ∂y = −hf (x)(Tf − T ) on y = 0 u → U∞(x), T → T∞ as y → ∞ (5) where x and y are respectively the Cartesian coordi￾nates measured along the surface and normal to it, u and v are the velocity components in the x and y di￾rections, T is the fluid temperature, p is the pressure, g is the acceleration due to gravity, α is the thermal dif￾fusivity, β is the coefficient of thermal expansion, ν is the kinematic viscosity, k is the thermal conductivity and U∞(x) is outer flow. We can eliminate the pres￾sure p from Eqs. (2), (3) by differentiating (2) with respect to y and using (1) to get u ∂2u ∂x∂y + v ∂2u ∂y2 = −gβ ∂T ∂x + ν ∂3u ∂y3 (6) To reduce (1), (4), (5), (6) to similarity form we need to specify specific functional forms for both hf and U∞ as hf (x) = kC0x−2/5, U∞(x) = U0x1/5 (7) We then put ψ = ν3gβ T 1/5 x3/5f (η) T − T∞ = T θ, η = gβ T ν2 1/5 y x2/5 (8) where T = Tf − T∞. This gives f  + 2 5 ηθ + 3 5 ff  + 1 5 f  f  = 0 (9) 1 σ θ + 3 5 f θ = 0 (10) where primes denote differentiation with respect to η and where σ is the Prandtl number. The boundary con￾ditions (5) become f = f  = 0, θ = −γ(1 − θ) on η = 0 f  → M, θ → 0 as η → ∞ (11) where M is the constant positive (assisting flow) mixed convection parameter and γ is the Biot num￾ber, which are defined as M = U0 (ν(gβ T )2)1/5 , γ = C0  ν2 gβ T 1/5 We can integrate Eq. (9) to get f  + 2 5  ηθ +  ∞ η θ(s)ds + 3 5 ff  − 1 5 f  2 = constant = −1 5 M2 (12) on applying the boundary conditions as η → ∞. We can then put P(η) =  ∞ η θ(s)ds to obtain f  + 2 5 (ηθ + P) + 3 5 ff  + 1 5 M2 − f  2 = 0 (13) P = −θ with then P → 0 as η → ∞ (14) It is the problem given by Eqs. (10), (13), (14) subject to boundary conditions (11) that we now consider. Before discussing the solution this problem in de￾tail we can make some general observations about the nature of the solution. If we put q(η) = 3 5σ  η 0 f (s)ds, we can formally express the solution to Eq. (10) as θ(η) = A  ∞ η e−q(s) ds, θ (η) = −Ae−q(η) (15) for some constant A. Expression (15) shows that θ is of one sign on 0 ≤ η < ∞. Boundary conditions (11) give A = γ 1 + γI∞ where I∞ =  ∞ 0 e−q(η) dη> 0 (16) Since γ > 0, we have A > 0 and hence, when a solu￾tion to (10), (13), (14), (11) exists, it must have θ > 0 and θ < 0 for 0 ≤ η < ∞. 3 Results We start by considering the free convection, M = 0, limit before considering the general problem. 3.1 Free convection limit, M = 0 Equations (10), (13), (14) with M = 0 subject to boundary conditions (11) were solved numerically us￾ing a standard boundary-value problem solver [29] and the results, shown by plots of f (0) and θ(0) against γ for σ = 1.0, are given in Fig. 1. We see that both f (0) and θ(0) increase monotonically as γ is increased with both appearing to approach a finite asymptotic limit for large γ . Also both f (0) and θ(0) appear to reduce to zero as γ decreases to zero. This leads us to consider the asymptotic limits of both large and small γ .

2152 Meccanica(2013)48:2149-215s Table 1 Comparison results obtained for y1:(.results reported by [1]:[.results reported by [5] -8'(0 f"(0 P(0) Present results Ramanaiah et al.[1 Present results Ramanaiah et al.10]Present results Ramanaiah et al.1 0.357406 0.3574 0.978400 0.9784 1.734930 1.7349 (0.358) (0.971 (1.73 [1.73492 0.36022 0.976978 0.977 1.708942 1.7089 0.425113 0.4251 1.247154 1.2472 1.39001 1.3900 10 1.134254 1.3432 19.732022 19.7320 0.51230 0.5123 1003.583404 3.5834 621.332230 621.3332 0.162144 0.1621 pmlc es f"(0)= 0.8646,90 The highe for =1. 09 f"(0)0.86446-0.20256v-+.. (17) 0.3 00)~1-0.39054y-1+0.18302y-2+. asyoo.Asymptotic expressions(17)are shown in Fig.1 by broken lines.Both expressions in(17)are in very good agreement with the numerically determined values even at relatively small values of y.Numeri- cal solutions have been obtained for Eqs.(10).(11). (13).(14)both for the free convection,M=0.limit and for other values of M.namely M=0.1.1.10 and 100 when is large.i.e.applying the boundary condi- tion(0)=1.The results iven in Table 1 which shows 牌cgk的oa川a时 3.1.2 Small y Fig1 Plots of (a))and (b)0)againstyfor For this case we have to scale the variables by writing with M=0subject to oundary conditions (1).Asy f=y/67 0=y5/6a. P=y2/P. n=yl/6n (18) 3.1.1 Large y aveE.u0.B0wi地Mo pt that now For this case we leave the equations unscaled with respect to元.The only change is in the boundary con- ditions(11)which is now the leading-order problem still given by (10).(13). (14)but now subject to the condition that 0(0)=1. 日=-1+y5/6aom万=0 (19) Springer

2152 Meccanica (2013) 48:2149–2158 Table 1 Comparison results obtained for γ  1; (.) results reported by [1]; [.] results reported by [5] M −θ (0) f (0) P(0) Present results Ramanaiah et al. [10] Present results Ramanaiah et al. [10] Present results Ramanaiah et al. [10] 0 0.357406 0.3574 0.978400 0.9784 1.734930 1.7349 (0.358) (0.971) (1.73) [0.35741] [0.97840] [1.73492] 0.1 0.360227 0.3602 0.976978 0.9770 1.708942 1.7089 1 0.425113 0.4251 1.247154 1.2472 1.390015 1.3900 10 1.134254 1.3432 19.732022 19.7320 0.512309 0.5123 100 3.583404 3.5834 621.332230 621.3332 0.162144 0.1621 Fig. 1 Plots of (a) f (0) and (b) θ(0) against γ for σ = 1.0 and obtained from the numerical solution to Eqs. (10), (13), (14) with M = 0 subject to boundary conditions (11). Asymptotic expressions (17), (20) for large and small γ are shown by broken lines 3.1.1 Large γ For this case we leave the equations unscaled with the leading-order problem still given by (10), (13), (14) but now subject to the condition that θ(0) = 1. Our numerical solution to this problem gives f (0) = 0.86446, θ (0) = −0.39054 for σ = 1.0. The higher￾order terms are found by expanding in inverse powers of γ . The details are straightforward and we find that, for σ = 1, f (0) ∼ 0.86446 − 0.20256γ −1 +··· θ(0) ∼ 1 − 0.39054γ −1 + 0.18302γ −2 +··· (17) as γ → ∞. Asymptotic expressions (17) are shown in Fig. 1 by broken lines. Both expressions in (17) are in very good agreement with the numerically determined values even at relatively small values of γ . Numeri￾cal solutions have been obtained for Eqs. (10), (11), (13), (14) both for the free convection, M = 0, limit and for other values of M, namely M = 0.1, 1, 10 and 100 when γ is large, i.e. applying the boundary condi￾tion θ(0) = 1. The results are given in Table 1 which shows a very good agreement with the previous results reported by Ramanaiah et al. [10], Stewartson [1] and Jones [5]. 3.1.2 Small γ For this case we have to scale the variables by writing f = γ 1/6f, θ = γ 5/6θ, P = γ 2/3P, η = γ 1/6η (18) This leaves Eqs. (10), (13), (14) with M = 0 essen￾tially unaltered except that now differentiation is with respect to η. The only change is in the boundary con￾ditions (11) which is now θ  = −1 + γ 5/6θ on η = 0 (19)

Meccanica(2013)48:2149-2158 2153 35 attaining almost the same value even at quite moder- ate values of M.In each case the values of 6(0)de 2.5 crease as M is increased appearing to have the same functional form for the larger values of M.This be haviour is also supported by the numerical values of f"(0).(0)and P(0)which are given in Table 2 for 5.0 0.2 05 3.2.1 Asymptotic solution for M large 5.0 for M large we expect the solution to approach the Zo 1.0 f=MRF. 5=M/2n 0.5 0=M-IPH. P=M-Ip (21) Equations(1).(13).(14)become 0.3 0.2 0.2 F+FF+-F+2gGH+p=0 0.9 2 4 (22) p'=-H HFH=0 (23) Eqs.()(1)(1)subject to boundary conditions (11) subject to F=F=0, H=-y+yM-1H on5-04) Expression(19)suggests an expansion in powersy5/6 F→1， p→0. H→0as5→o The numerical solution of the resulting equations where primes now denote differentiation with respect gives,for a =1.0. f"(0~y1/2(1.38329-1.51414y5/6+ Equation(22)suggests an expansion for F in the (20) for 8(0)=y5/6(2.18918-3.99376y56+)】 F(G:M0=F(G)+M-3F(G)+… (25) nd shov The leading-order problem for the velocity becomes 20) ct heha 0)a 00) independent of the temperature,as expected,and is 0and are i given by F”+FoF+1-F)=0 (26 We now consider the F)=F0)=0. Fo→1as→ 3.2 General case.M0 Our numerical solution of (6)gives F)=0.62132. Up to O(M-3).Eq.(23)is In Fig.2 we H”+号6H=0 ave va H'=-y+yM-IPH on=0 (27) H→0as5→oo Springer

Meccanica (2013) 48:2149–2158 2153 Fig. 2 Plots of (a) f (0) and (b) θ(0) against M for σ = 1.0 and γ = 0.2, 1.0, 5.0 obtained from the numerical solution to Eqs. (10), (13), (14) subject to boundary conditions (11) Expression (19) suggests an expansion in powers γ 5/6. The numerical solution of the resulting equations gives, for σ = 1.0, f (0) ∼ γ 1/2 1.38329 − 1.51414γ 5/6 +··· θ(0) = γ 5/6 2.18918 − 3.99376γ 5/6 +··· (20) as γ → 0. Asymptotic expressions (20) are also shown in Fig. 1 by broken lines and show that expressions (20) give the correct behaviour of f (0) and θ(0) as γ → 0 and are in good agreement with the numerical values for γ up to about 0.2 after which they rapidly diverge. We now consider the general, M = 0, case. 3.2 General case, M > 0 In Fig. 2 we plot f (0) and θ(0) against M for rep￾resentative values of γ and for σ = 1.0. We see that the values of f (0) start at their corresponding M = 0 values, see Fig. 1(a), and increase as M is increased, attaining almost the same value even at quite moder￾ate values of M. In each case the values of θ(0) de￾crease as M is increased appearing to have the same functional form for the larger values of M. This be￾haviour is also supported by the numerical values of f (0), θ(0) and P(0) which are given in Table 2 for σ = 0.72, M = 0, 1, 10 and 100 and γ = 0.01, 0.1, 1, 10, 100 and 1000. This leads us to consider the asymptotic solution for M large. 3.2.1 Asymptotic solution for M large For M large we expect the solution to approach the forced convection limit and this suggests that we in￾troduce the scaling, on assuming that γ is of O(1), f = M1/2F, ζ = M1/2η θ = M−1/2H, P = M−1p (21) Equations (10), (13), (14) become F + 3 5 FF + 1 5 1 − F 2 + 2M−3 5 (ζH + p) = 0 (22) p = −H, H + 3σ 5 FH = 0 (23) subject to F = F = 0, H = −γ + γM−1/2H on ζ = 0 F → 1, p → 0, H → 0 as ζ → ∞ (24) where primes now denote differentiation with respect to ζ . Equation (22) suggests an expansion for F in the form F(ζ ;M) = F0(ζ) + M−3F1(ζ) +··· (25) The leading-order problem for the velocity becomes independent of the temperature, as expected, and is given by F 0 + 3 5 F0F 0 + 1 5 1 − F 2 0 = 0 F0(0) = F 0(0) = 0, F0 → 1 as ζ → ∞ (26) Our numerical solution of (26) gives F 0 (0) = 0.62132. Up to O(M−3), Eq. (23) is H + 3σ 5 F0H = 0 H = −γ + γM−1/2H on ζ = 0 H → 0 as ζ → ∞ (27)