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 SpringerMeccanica (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 veloc￾ity, 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 boundary￾layer flow on an upward facing horizontal surface heated convectively by being in contact with a fluid held a constant temperature Tf . We reduced the prob￾lem 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 in￾volved, apart from the Prandtl number σ , two dimen￾sionless parameters, namely M, a measure of the rela-