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 boundarylayer flow on an upward facing horizontal surface heated convectively is considered. The problem is reduced to similarity form, a necessary requirement for which is that the outer flow and surface heat transfer coefficient are spatially dependent. The resulting similarity equations involve, apart from the Prandtl number, two dimensionless parameters, a measure of the relative strength of the outer flow M and a heat transfer 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, including solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency shutdown, heat exchanges placed in a lowvelocity 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 becomes significant. Thermal buoyancy forces can play an important role in forced convection heat transfer when the flow velocity is relatively low or the temperature difference between with the free stream is relatively large. Convective flows, both free and mixed, on vertical or inclined surfaces have already received much attention. Somewhat less consideration has been given to mixed convection on horizontal surfaces, which is what we discuss here. Natural convection on a horizontal surface, sometimes referred to an ‘indirect convection’ arises through a different mechanism to that on a vertical surface. Here the buoyancy forces acting vertically, i.e. normal to the surface, generate a horizontal pressure gradient, i.e. parallel to the surface. 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 surface 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. Conversely above a cooled horizontal surface an adverse pressure gradient is generated precluding the development of a boundary-layer flow. There is intrinsic interest in convective flows over horizontal surfaces as they offer an alternative mechanism for driving a convective flow and have been studied 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 applications, 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 convective flow which can converge to form buoyant plumes. There can also be a wind giving an interaction between 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. Neither a prescribed temperature or heat flux would seem entirely appropriate, hence we take a convective condition, being in essence a combination of these two conditions. Although this could well be an over simplification, it should provide further useful insights. There has already been some work on mixed convection boundary-layer flows along horizontal flat surfaces. Here we mention specifically papers by Schneider [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 excellent review papers by Schneider [15] and Steinrück [16] which summarize the results previously reported on this problem. Previous treatments of mixed convection boundary-layer flows along horizontal surfaces have considered an isothermal or variable surface temperature 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 vertical flat plate immersed in a viscous (Newtonian) fluid. More recently, Aziz [18], see also the comments on this paper by Magyari [19], used the convective 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 conditions, see for example Ishak [20], Cortell Bataller [21], Makinde and Aziz [22], Makinde and Olanrewaju [23], Merkin and Pop [24] and Yao et al. [25], including non-Newtonian fluids [26], variable viscosity [27] and nanofluid convection within a porous material [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 differential 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 boundarylayer flow on a horizontal flat surface. We assume that the surface faces upwards and is within a uniform ambient temperature T∞. We also assume that the surface 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 approximation 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 00(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 (16) Since γ > 0, we have A > 0 and hence, when a solution 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 using 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 higherorder 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 . Numerical 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 condition 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 essentially unaltered except that now differentiation is with respect to η. The only change is in the boundary conditions (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 representative 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 moderate values of M. In each case the values of θ(0) decrease as M is increased appearing to have the same functional form for the larger values of M. This behaviour 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 introduce 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)
2154 Meccanica(2013)48:2149-2158 Table 2 Values related to the skin friction coefficient f"(0).the surface pressure P(0)and the local Nusselt number (0) P(0) a0) 0.1 0.443043 0.603379 0.26706 0.82184 137505 0.74781 0.981E3 1.68719 9 0.978190 1.73443 0.643695 0.043371 0.026939 0.1 0.78211 032348 0.20922 1.092786 1.016773 0.70886 1.226456 1338960 0.95939 1000 1.246939 1.389483 0.999575 0.01 19.648720 0.004485 0.008747 0.1 19.654808 0.041573 0.081086 19.687404 0.240214 0.468674 10723474 046016 0.898138 19.731927 0.511729 0.99886 100 0.01 621.323835 0.00045 0.00278 0.1 621324041 002714 621.325649 0035376 0.218178 621.30009 0.119369 0.73619 1000 621.33219 0.161565 0.996429 This equation has the solution as a consequence F1=yF1.We find for o =1.0 that F(0)=0.51745.Thus we have H=()ds f(0)~M3P(0.62132+1.72163yM-3+) yM-121o (28) 0~1+M-%+ (30 The constant Bo is determined from the boundary con- =2.46739yM-1/1-2.46739yM-12+) dition on=0as asM→oo. =1+yM-i6 As a check on our asymptotic analysis we com- Puted f"()M-32 and =e01+yM-1/2o) where6=。ewg)dg (29) from our numerical lutions.These are 3 fo entative values of y.Fro We note that in this case lo=lo()is a known func tion of and is determined from the solution to(26) limits of 0.62132 and 1.0.We see that,in each case For =1.0 we find that /o =2.46739 and a plot of these limits are approached as M is increased,provid lo against a is shown in Fig.4.Having determined ing a confirmation of our analysis.Expressions (30) H.at least to O(M-3).we are in a position to de- explain why the values of f"()rapidly approach the termine the next term F1 in expansion (25).To do same value.independent of y.as M is increased.with this we require Ho.the leading term in the expansion the correction to this value being of (M-3)for M implied in (28).(29).To calculate this term we put large.These results also show why(0)decreases only Ho=yHo in Eq.(27).where now Ho(0)=-1,and slowly,as M-12,for large M. Springer
2154 Meccanica (2013) 48:2149–2158 Table 2 Values related to the skin friction coefficient f (0), the surface pressure P(0) and the local Nusselt number θ(0) Mγ f (0) P(0) θ(0) 0 0.1 0.443043 0.603379 0.267069 1 0.821848 1.375053 0.747817 10 0.958138 1.687192 0.965724 1000 0.978190 1.734434 0.999642 1 0.01 0.643695 0.043371 0.026939 0.1 0.782114 0.323480 0.209222 1 1.092786 1.016773 0.708863 10 1.226456 1.338960 0.959394 1000 1.246939 1.389483 0.999575 10 0.01 19.648720 0.004485 0.008747 0.1 19.654808 0.041573 0.081086 1 19.687404 0.240214 0.468674 10 19.723474 0.460164 0.898138 1000 19.731927 0.511729 0.998867 100 0.01 621.323835 0.000451 0.002782 0.1 621.324041 0.004402 0.027148 1 621.325649 0.035376 0.218178 10 621.330009 0.119369 0.736192 1000 621.332199 0.161565 0.996429 This equation has the solution H = B0 ∞ ζ e−q0 (s)ds where q0(ζ) = 3σ 5 ζ 0 F0(s)ds is known (28) The constant B0 is determined from the boundary condition on ζ = 0 as B0 = γ 1 + γM−1/2I0 where I0 = ∞ 0 e−q0 (ζ)dζ (29) We note that in this case I0 = I0(σ) is a known function of σ and is determined from the solution to (26). For σ = 1.0 we find that I0 = 2.46739 and a plot of I0 against σ is shown in Fig. 4. Having determined H, at least to O(M−3), we are in a position to determine the next term F1 in expansion (25). To do this we require H0, the leading term in the expansion implied in (28), (29). To calculate this term we put H0 = γ H0 in Eq. (27), where now H 0(0) = −1, and as a consequence F1 = γ F1. We find for σ = 1.0 that F 1(0) = 0.51745. Thus we have f (0) ∼ M3/2 0.62132 + 1.72163γM−3 +··· θ(0) ∼ γM−1/2I0 1 + γM−1/2I0 +··· = 2.46739γM−1/2 1 − 2.46739γM−1/2 +··· (30) as M → ∞. As a check on our asymptotic analysis we computed f (0)M−3/2 and θ0 = θ(0)(1 + γM−1/2I0)/ γM−1/2I0 from our numerical solutions. These are plotted in Fig. 3 for representative values of γ . From (30) these should respectively approach the asymptotic limits of 0.62132 and 1.0. We see that, in each case, these limits are approached as M is increased, providing a confirmation of our analysis. Expressions (30) explain why the values of f (0) rapidly approach the same value, independent of γ , as M is increased, with the correction to this value being of O(M−3) for M large. These results also show why θ(0) decreases only slowly, as M−1/2, for large M.
Meccanica(2013)48:2149-2158 2155 1.15 08 5 5.0 Fig.4 Aplot of the integral o defined in (29)against f"(0)~0.62132M/2+0(M-52 lo (33) 0~1+8+0M-s）sM→0 097 for y of (M)and where a plot of lo=lo(a)is 02 shown in Fig.4. 09 To illustrate the general case we present veloc ity.temperature and pressure profiles in Figs.5.6.7 0.95 and8fora=0.72:M=0.1,10and100:y=0.01 0.1.1.10.100 and 1000.It is seen from Fig.1 that for the free convection case (M=0).the boundary layer thicknesses increase with y for all three pro don to files,velocity,temperature and pressure.This can be physically explained by the fact that,for large y.the plate becomes increasingly isothermal.Further,in the This asymptotic analysis has to be modified slightly mixed convection case (M>0)the thicknesses of all o() profiles increase with the increases in y(see Fig.6). with 8 of 0(1)and However,for small values of M (=1)the velocity napM罗nR profile presents an overshoot near the plate,while for large M(=10 and 100)the velocity profiles are very close each others (see Figs.7 and 8). uhcThe boundary conditions are still as in (24)except that now 4 Conclusions 0'=-8(1-6)on=0.An expansion of the form F=f+M52F+. We have considered the mixed convection boundary- 0=%+M-5/Pa+… (31) layer How on ar P is suggested.The leading-order term Fo is as before and (32) ed ved.apart from the Pra less parameters,namely M.a measure of the rela Springer
Meccanica (2013) 48:2149–2158 2155 Fig. 3 Plots of (a) f (0)M−3/2 and (b) θ0 = θ(0)(1 + γM−1/2I0)/γM−1/2I0 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) This asymptotic analysis has to be modified slightly when γ is large, particularly when γ is of O(M1/2). To this end we put γ = δM1/2, with δ of O(1) and still scale f and η as in (21). The modification is to now leave θ unscaled and to write P = M−1/2 p. This leaves Eqs. (22), (23) essentially unaltered except that now the correction term in (22) is now M−5/2. The boundary conditions are still as in (24) except that now θ = −δ(1 − θ) on ζ = 0. An expansion of the form F = F0 + M−5/2F1 +··· θ = θ0 + M−5/2θ1 +··· (31) is suggested. The leading-order term F0 is as before and θ0 = δ 1 + δI0 ∞ ζ e−q0 (s)ds (32) following from (28), (29). The next order terms F1, θ1 in (31) both involve δ. Thus we have Fig. 4 A plot of the integral I0 defined in (29) against σ f (0) ∼ 0.62132M3/2 + O M−5/2 θ(0) ∼ δI0 1 + δI0 + O M−5/2 as M → ∞ (33) for γ of O(M1/2) and where a plot of I0 = I0(σ) is shown in Fig. 4. To illustrate the general case we present velocity, temperature and pressure profiles in Figs. 5, 6, 7, and 8 for σ = 0.72; M = 0, 1, 10 and 100; γ = 0.01, 0.1, 1, 10, 100 and 1000. It is seen from Fig. 1 that, for the free convection case (M = 0), the boundary layer thicknesses increase with γ for all three pro- files, velocity, temperature and pressure. This can be physically explained by the fact that, for large γ , the plate becomes increasingly isothermal. Further, in the mixed convection case (M > 0) the thicknesses of all profiles increase with the increases in γ (see Fig. 6). However, for small values of M (= 1) the velocity profile presents an overshoot near the plate, while for large M (= 10 and 100) the velocity profiles are very close each others (see Figs. 7 and 8). 4 Conclusions We have considered the mixed convection boundarylayer flow on an upward facing horizontal surface heated convectively by being in contact with a fluid held a constant temperature Tf . We reduced the problem to similarity form, Eqs. (10), (13) and (14) subject to boundary conditions (11), requiring that the outer flow U∞ and the surface heat transfer coefficient hf varied spatially, see expression (7). The problem involved, apart from the Prandtl number σ , two dimensionless parameters, namely M, a measure of the rela-
2156 Meccanica(2013)48:2149-215 12 0.6 =0.01,0.11.10,1000 0.9 0 20.6 0.01.0.11.10.100 0 14 1.2 ￥=0.01，a.1,1,10,1000 1 0,6 0.01,0.1,1.10,1000 0.4 0.2 0.8 0 0.6 =0.01.01.1,10.1000 0.6 E 7=0.01,0.1,1,10,100 0.4 0. 4 Fig.5 Velocity,pressure and temperature profiles forM=0 Fig6 Velocity,pressure and temperature profiles for M= tive strength of the outer flow,and y a dimensionless remained of (1),in our nondimensionalization(8) heat transfer coefficient. However,in the smally limit these both became small We started by considering the free convection limit. respectively of (y2)and of (y5/6). i.e.assuming that M=0.Here we found that the solu- We then treated the general,M>0,case.Here tion progressed smoothly from the prescribed surface we found that the solution for the velocity became heat flux,y=0,case to the prescribed wall temper- independent of the heat transfer coefficient y as the ature,yoo,case,see Fig.1,with the asymptotic value of M increased,see Fig.2(a).This was also seer limits of small and large y being derived,see expres- when we derived the large M limit,with to leading sions (20)and (17).For the large y limit the dimen- order the flow field being independent of y,see ex- sionless skin friction f"(0)and wall temperature(0) pression (0).though the temperature field did depend Springer
2156 Meccanica (2013) 48:2149–2158 Fig. 5 Velocity, pressure and temperature profiles for M = 0 tive strength of the outer flow, and γ a dimensionless heat transfer coefficient. We started by considering the free convection limit, i.e. assuming that M = 0. Here we found that the solution progressed smoothly from the prescribed surface heat flux, γ = 0, case to the prescribed wall temperature, γ → ∞, case, see Fig. 1, with the asymptotic limits of small and large γ being derived, see expressions (20) and (17). For the large γ limit the dimensionless skin friction f (0) and wall temperature θ(0) Fig. 6 Velocity, pressure and temperature profiles for M = 1 remained of O(1), in our nondimensionalization (8). However, in the small γ limit these both became small, respectively of O(γ 1/2) and of O(γ 5/6). We then treated the general, M > 0, case. Here we found that the solution for the velocity became independent of the heat transfer coefficient γ as the value of M increased, see Fig. 2(a). This was also seen when we derived the large M limit, with to leading order the flow field being independent of γ , see expression (30), though the temperature field did depend
Meccanica(2013)48:2149-2158 2157 12 100 60 Y=Q.01,0.1,1,10,1000 B=0.01.0.1,1.10.1000 1 02 Q15 Y=0.01,0.1,1,10,1000 =0.01.0.11.10.1000 a.05 Y=0.01,01,1,101000 7=0.01,0.1,1,10,1000 0.6 2 Fig.Velocity.pressure and temperature profiles forM-1 Fig.7 Velocity,pressure and temperature profiles for M=10 on y,see Fig.2(b).Our asymptotic results showed perature condition on the boundary-layer convection that the skin friction was large,of (M )and the on a horizontal surface.These types of flows.some wall temperature became small,of O(M- )Our times referred to as'indirect co ction'.have feature asymptotic theory for M large showed that,for the nt to those seen in temperature field,a more appropriate parameter was tion on 6=Co()2.This asymptotic limit was from studies of these latter also considered,see expression(33). over to horiz We have limited our attention to a relatively sim- ple set up,namely we have attempted to assess the ef- nding the cal in fects of an outer flow and a convective surface tem- est.For example,convection of nanoparticles on hori Springer
Meccanica (2013) 48:2149–2158 2157 Fig. 7 Velocity, pressure and temperature profiles for M = 10 on γ , see Fig. 2(b). Our asymptotic results showed that the skin friction was large, of O(M3/2) and the wall temperature became small, of O(M−1/2). Our asymptotic theory for M large showed that, for the temperature field, a more appropriate parameter was δ = γ M1/2 = C1/5 0 ( ν U0 )1/2. This asymptotic limit was also considered, see expression (33). We have limited our attention to a relatively simple set up, namely we have attempted to assess the effects of an outer flow and a convective surface temFig. 8 Velocity, pressure and temperature profiles for M = 100 perature condition on the boundary-layer convection on a horizontal surface. These types of flows, sometimes referred to as ‘indirect convection’, have features which can be quite different to those seen in convection on a vertical surface. As a consequence, conclusions drawn from studies of these latter flows cannot always be taken over to horizontal convective flows. Future studies are suggested extending the present results to problems with, perhaps, a more practical interest. For example, convection of nanoparticles on hori-
2158 Meccanica(2013)48:2149-2158 zontal surfaces,of interest here would be to see what tures seen nd what a of the 16 ntial An altemative extension to our work could be to hor. 227-158 izontal convection flows driven by combustion.Here 12ro the main feature is to include a combustion reaction into the model consistent with a boundary-layer ap- rity solution for laminar the p( C10010 at p. UEFISCDI,project number PN-11-RU-TE-2011-3-0013. 20 References 97842 21. $艺0g出e6ho0ioa1 y Math Ph ous m ith Fluid Mech39:173-192 23. Naju PO (2010)Bu ffec es DR(1973)Free 24. 26: kin JH,Pop I(2011)The force 25. 752-760 1984)Mixed 26. conditionActa PolytechHung 8131-140 G.Merkin ( aniels PG(1902)A sin n th y-l 28. Uddn M.Khan WA.Ismail AIM (2)Free ed ho. 6 3 0 Kuzn 14. iting Prand Math Mech (ZAMM) Springer
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