Transp Porous Med (2012) 94:133–147 DOI 10.1007/s11242-012-9993-0
On the Temperature Slip Boundary Condition in a Mixed Convection Boundary-Layer Flow in a Porous Medium John H. Merkin · Azizah Mohd Rohni · Syakila Ahmad · Ioan Pop
Received: 22 December 2011 / Accepted: 14 March 2012 / Published online: 3 April 2012 © Springer Science+Business Media B.V. 2012
Abstract A problem derived previously (Rohni et al., Transp Porous Media 92:1–14, 2012) for unsteady mixed convection flow in a porous medium involving a ‘temperature slip’ boundary condition and fluid transfer through the boundary is considered. It is shown that the solution to this problem can be directly related to the solution of the corresponding problem for a prescribed surface temperature, involving a mixed convection parameter λ, an unsteadiness parameter A and transpiration parameter s. This latter problem is discussed in detail, particular attention being given to the steady analogue, A = 0, allowing for fluid transfer through the surface, and to the unsteady problem, A > 0, for an impermeable surface, s = 0. Asymptotic results are obtained for large fluid transfer rates, s 1 and s < 0, |s| 1 and for large A. Particular attention is given to deriving asymptotic results for the critical points which determine the range of existence of solutions. Keywords Porous medium · Unsteady flow · Mixed convection · Fluid transfer · Temperature slip condition
J. H. Merkin (B) Department of Applied Mathematics, University of Leeds, LS2 9JT, Leeds, UK e-mail:
[email protected] A. M. Rohni UUM College of Arts & Sciences, Physical Science Division, Building of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia S. Ahmad School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia I. Pop Faculty of Mathematics, University of Cluj, CP 253, 400082 Cluj, Romania
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1 Introduction The usual thermal boundary conditions that are applied on a body’s surface in the study of convective flows in porous media are either a prescribed wall temperature or a prescribed surface heat flux. Such flows have been extensively analysed under a wide variety of both geometrical and physical conditions, as described for example in the recent books by Nield and Bejan (2006), Ingham and Pop (2005) and Vafai (2005, 2010). A modification to this scenario arises when the surface heat flux is functionally related to the surface temperature, as for example in Newtonian or conjugate wall heating. However, Merkin and Pop (2010) have shown that this situation can usually be reduced to the corresponding problem for a prescribed surface heat flux by an appropriate transformation. A more recent alternative form for the wall condition is the ‘temperature slip’ condition proposed originally by Mukhopadhyay and Andersson (2009) and analysed in a little more detail by Rohni et al. (2012). The effects of both wall slip and temperature slip boundary conditions on the mixed convection boundary-layer flow on a vertical surface have been discussed by Cao and Baker (2009) who showed that this could have a significant influence on both the flow and heat transfer. The equivalent problem on a stretching surface has been considered by Mukhopadhyay (2010), again showing that slip effects can induce considerable differences in what would be seen otherwise. In these cases, the wall temperature and heat flux are related in a way analogous to the velocity slip condition. The idea of a wall slip (or partial slip) condition was suggested originally by Beavers and Joseph (1967) and has been applied recently in several contexts. For example, as well as Mukhopadhyay and Andersson (2009), Wang (2006) and Wang and Ng (2011) have considered the effects of wall slip on the flow on a stretching surface. Crane and McVeigh (2010, 2011) have treated the accelerated flow, or nonlinear shear flow, over cylinders with wall slip conditions. Fang et al. (2009) examined how MHD can affect this flow by obtaining an exact solution to their problem. The flow on a shrinking surface also including a wall slip has been discussed by Fang et al. (2010). Here we consider the problem derived in Rohni et al. (2012), namely the similarity system y f + f f + f (1 − f ) + A f − 1 + f = 0 (1) 2 on 0 ≤ y < ∞ subject to the boundary conditions that f (0) = s,
f (0) = 1 + λ + f (0),
f → 1 as y → ∞,
(2)
where primes denote differentiation with respect to the independent variable y and where we have changed the sign of the parameter A from Rohni et al. (2012). Equations (1, 2) arose in the unsteady mixed convection boundary–layer flow near the stagnation point on a permeable surface in a porous medium on assuming the ‘hyperbolic time variation’, Yang (1958). In (1, 2), the parameter A represents the time variation with A ≥ 0 (from the way this parameter is defined), s is the rate of fluid transfer through the surface, s > 0 for fluid withdrawal and s < 0 for fluid injection, λ is a mixed convection parameter that can be either positive (aiding flow) or negative (opposing flow), and is the temperature slip factor. Equations (1, 2) are derived fully in Rohni et al. (2012). The steady version of (1, 2) (A = 0) and without fluid transfer (s = 0) and temperature slip ( = 0) has already been considered by Nazar et al. (2004) and Merrill et al. (2006). We start by showing how we can relate the solution of (1, 2) for = 0 to the solution of (1, 2) with = 0 in a relatively simple way for given values of A and s. We then examine the solution of (1, 2) in the steady case, i.e., with A = 0, paying particular attention to
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the case when there is fluid transfer through the surface, extending the results given in Aly et al. (2003), Nazar et al. (2004) and Merrill et al. (2006). We complete our discussion by considering the unsteady case, A > 0, for an impermeable surface, i.e., taking s = 0.
2 Temperature Slip Effect Suppose ( f 0 (y), λ0 ) is a solution of Eqs. (1, 2) with = 0 for given values of s and A, i.e., ( f 0 (y), λ0 ) satisfies y (3) f 0 + f 0 f 0 + f 0 (1 − f 0 ) + A f 0 − 1 + f 0 = 0 2 now subject to the boundary conditions that f 0 (0) = s,
f 0 (0) = 1 + λ0 ,
f 0 → 1 as y → ∞
(4)
Now suppose Eqs. (3, 4) have a solution which has f 0 (0) = a0 = a0 (s, A). Then f 0 is also a solution to the problem (1, 2) with = 0 given by Eq. (3) and the boundary conditions f 0 (0) = s, f 0 → 1 as y → ∞ but now for the value λ1 where f 0 (0) = 1 + λ1 + f 0 (0)
or 1 + λ0 = 1 + λ1 + a0
(5)
giving λ1 = λ0 − a0
(6)
In particular, expression (6) expresses the critical values λ1,crit for problem (1, 2) in terms of the critical values λ0,crit for problem (3, 4) as λ1,crit = λ0,crit − a0,crit
(7)
f 0 (0) at the critical point λ0,crit . Values of a0 (s,
A) where a0,crit = a0,crit (s, A) is the value of are given in Rohni et al. (2012) plotted against λ for certain values of s and A as well as values of λ0,crit . As an example, when s = 0, A = 0, λ0,crit = −1.41743 and a0,critt = 0.20148 (Nazar et al. 2004; Merrill et al. 2006) giving λ1,crit = −1.41743 − 0.20148
(8)
We can see that the effect of the temperature slip, i.e., having > 0, is to increase the range of solutions in the opposing flow regime. However, if we allow to be negative, the effect is to decrease the range of solutions and, if || is sufficiently large, to limit solutions only to the aiding flow case. For A = s = 0, expression (8) gives < −7.0351 for this to occur. We next consider the solution to Eqs. (3, 4), dropping the suffix for convenience, starting with the steady case, A = 0. 3 The Problem with = 0 3.1 The Steady Case, A = 0 The case when s = 0 has been discussed by Nazar et al. (2004) who have shown that there is a critical value λcrit of λ, where λcrit −1.417, with a saddle–node bifurcation at λ = λcrit such that there are dual solutions to (3, 4) when λ > λcrit and no solutions when λ < λcrit , see Fig. 2 in their paper as well as Fig. 2 in Merrill et al. (2006).
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Fig. 1 A plot of the critical values λcrit for s ≥ 0 for problem (3, 4) with A = 0 plotted against s. Asymptotic expression (13) is shown by the broken line
3.1.1 Fluid Withdrawal, s > 0 The case when there is fluid withdrawal through the surface, s > 0, has been discussed in Rohni et al. (2012). Though the situation when A = 0 was not specifically treated in Rohni et al. (2012) for s > 0, we can infer from their results that the values of |λcrit | increase as s is increased, thus giving a greater range of solutions in the opposing flow regime, λ < 0, when there is fluid withdrawal through the wall. To confirm this we calculated λcrit following the approach detailed in Merkin and Mahmood (1989) and Ling et al. (2007) and a graph of λcrit plotted against s is shown in Fig. 1 for s ≥ 0. Figure 1 shows that |λcrit | increases monotonically as s is increased becoming large for s large. The asymptotic behaviour of the solution to equations (3, 4) when s is large and λ is of O(1) has been given by Rohni et al. (2012). However, to determine how the critical values vary with s when s is large we need λ to be large, of O(s 2 ). To deal with this case, we put λ = s 2 μ, where μ is of O(1) for s large. We then make the scaling f = s F , η = s y. Equations (3, 4) then become F + F F − F 2 + s −2 F = 0,
F(0) = 1, F (0) = μ + s −2 , F → s −2 as η → ∞, (9)
where primes denote differentiation with respect to η. Equation (9) indicates looking for a solution by expanding F(η; s) = F0 (η) + s −2 F1 (η) + · · ·
(10)
The leading-order problem is, from (9), F0 + F0 F0 − F0 2 = 0,
F0 (0) = 1, F0 (0) = μ, F0 → 0 as η → ∞
(11)
Equation (11) has the solution μ F0 (η) = a − e−aη , a
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√
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(12)
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-1 -1.05 -1.1 -1.15 -1.2 -1.25 -1.3 -1.35 -1.4 -1.45
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Fig. 2 A plot of the critical values λcrit for s ≤ 0 for problem (3, 4) with A = 0 plotted against s
Expression (12) gives a critical value for μ as μcrit = −1/4. Hence λcrit ∼ −
s2 4
as s → ∞
(13)
Asymptotic expression (13) is shown in Fig. 1 by the broken line showing good agreement with the numerically determined values. The difference between the two sets of results indicates an O(1) correction to λcrit in (13), as is also suggested by expansion (10). Expression (12) also shows that there are two solutions for −1/4 < μ < 0 and only one solution for μ > 0 with, for s large, √ √ 1 + 1 + 4μ 1 − 1 + 4μ 3 3 f (0) ∼ −s μ + · · · and f (0) ∼ −s μ 2 2 + · · · ( for μ < 0) (14) We now consider the case when there is fluid injection into the flow through the surface, s < 0. 3.1.2 Fluid injection We calculated the values of λcrit in this case with these being plotted against s in Fig. 2. Here we see that there is a bound λb on s for λcrit , with λcrit → −1 as λ approach λb . Our numerical results suggest that λb is slightly less than s = −1.815. We note that the values of f (0) at the critical value λcrit become small as this bound is approached, at s = −1.8157 we found λcrit −1.00046, f (0) 0.000005. Figure 2 shows that, if the fluid injection through the wall is sufficiently strong, then this can inhibit the development of dual solutions. We illustrate this in Fig. 3 where we plot f (0) against λ for s = −0.5 (Fig. 3a) and for s = −2.0 (Fig. 3b) in the opposing flow, λ < 0, regime. These, and subsequent, numerical results were obtained using a standard boundary-value problem solver, D02AGF in the NAG library (www.nag.co.uk). In Fig. 3a, we see the existence of a critical value λcrit −1.214 consistent with Fig. 2 and dual solutions for λ > λcrit . This graph is similar in form to those shown in Nazar et al. (2004) and Merrill et al. (2006) for the s = 0 case. In Fig. 3b we do
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Fig. 3 Plots of f (0) against λ for a s = −0.5 and b s = −2.0 obtained from the numerical solution of (3, 4) with A = 0
not see a critical value of λ. Here the solutions terminates as λ → −1 with a singularity appearing in the solution as this value is approached. We can get some further information about the nature of this singularity as λ → −1 from the profile plots of f against y in Fig. 4 for values of λ close to λ = −1. The structure of the solution appears to be an outer region where f takes the same form being increasingly further displaced from the wall, y = 0, as λ gets closer to −1. To obtain a solution valid for s < 0, |s| 1, we start in an inner region where we write f = |s|G, ζ = y |s|−1 , with equations (3, 4) then becoming G G + G − G 2 + |s|−2 G = 0, G(0) = −1, G (0) = 1 + λ
(15)
where the outer boundary condition is relaxed at this stage and where primes now denote differentiation with respect to ζ . An expansion in inverse powers of |s|−2 is suggested by (15), the leading–order term G 0 in this expansion is given by
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
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Fig. 4 Fluid injection, s < 0: profile plots of f against y for s = −2.0 and λ = −0.98, −0.985, −0.99, −0.995, −0.999, −0.9999 obtained from the numerical solution of (3, 4) with A = 0
1 1 − (1 + λ)e−λζ , G 0 = (1 + λ) e−λζ (16) λ From (16), G satisfies the outer boundary condition, at leading order, where ζ = ζ0 = 1 log(1 + λ). This leads to an outer region where we put λ |s| (17) log(1 + λ) + y and f = y + |s|−1 φ(y) y= λ The leading-order problem in the outer region is then, on matching with the solution in the inner region (16), G0 =
φ + y φ − φ = 0, From (18), we obtain
φ∼−
λ y2 + · · · as y → −∞, φ → 0 as y → ∞ 2 ⎛
1 ⎜ −y 2 /2 φ = √ −y ⎝e 2π
∞
(18)
⎞ e−s
2 /2
⎟ ds ⎠
(19)
y
From expression (17), we see that this asymptotic solution requires having λ > −1. This is consistent with Figs. 1 and 2 which show that there is a critical value scrit of s for all λ < −1 and none for λ > −1, as also seen in Fig. 5b, c. This then limits the range of λ for which there is a solution for all s to λ > −1 and for which this asymptotic solution is valid. Expression (16) gives 2 d f λ(1 + λ) + · · · as |s| → ∞, (λ > −1) (20) ∼− d 2 y y=0 |s| We can also assess the effect of fluid transfer through the boundary in both the aiding and opposing flow cases. In the aiding flow case, there are no critical points, as seen in Figs. 1 and 2, and a solution exists for all s. We illustrate this case with a plot of f (0) against s
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for λ = 1.0 in Fig. 5a. The asymptotic solutions for s 1 given in Rohni et al. (2012) and for |s| 1, s < 0, given by (20), are shown in Fig. 5a by broken lines. Expression (20) is in good agreement with the numerically determined values even at relatively small values of |s|. The asymptotic solution for s 1 in Rohni et al. (2012) also agrees well with the numerical values and suggests an O(1) correction again consistent with this theory. In the opposing flow case, there are two cases to consider, namely λ > −1 and λ < −1. In the former case there are no critical values, see Figs. 1 and 2, and there is a solution for all s. We illustrate this case in Fig. 5b with a plot of f (0) against s for λ = −0.5. Here we find two solution branches, compare this plot with Fig. 5a, diverging rapidly as s is increased. The upper solution branch approaches the asymptotic form given in Rohni et al. (2012) that f (0) ∼ −s λ as s → ∞. The two solution branches merge when s < 0, both approaching the asymptotic limit (20) as |s| increases, shown by the broken line. When λ < −1, there are critical values and hence there is a lower bound on s for the existence of a solution. We consider two cases in Fig. 5c where we plot f (0) against s for λ = −1.2 and −2.0. For λ = −1.2, there is a critical value scrit −0.542 of s in the opposing flow region and for λ = −2.0, scrit 0.919 in the aiding flow region. In both cases, there are dual solutions for s > scrit which continue further into the fluid withdrawal, s > 0, region. The upper solution branch has the asymptotic form given in Rohni et al. (2012) whereas on the lower solution branch f (0) < 0 and | f (0)| increases very rapidly as s is increased. 3.2 The Unsteady Case, A > 0 Here we assume that the surface is impermeable, s = 0. We calculated the values of λcrit for a given value of A > 0 following the approach described in Merkin and Mahmood (1989) and Ling et al. (2007) and a plot of λcrit against A is shown in Fig. 6. We again find that λcrit < 0 and that |λcrit | decreases as A is increased, thus reducing the range of solutions possible in the opposing flow case. We also note that the values of f (0) at λcrit are negative for A 0.45 and increase in size as A is increased. In Fig. 7, we plot f (0) against λ for A = 1.0, 5.0. We see critical values of λ, dependent on A, consistent with Fig. 6, with dual solutions for λ > λcrit . These dual solutions continue into the aiding flow, λ > 0, regime. For A = 5.0, f (0) ≤ 0 for opposing flow, however for A = 1.0 there is a range −1.1 < λ < 0 where f (0) > 0. In a recent paper, Magyari (2001) has shown that, for a given value of A > 1 (and for given values of the other parameters), it is possible to have a range of values of f 0 (0) all of which give a solution to (3) having f 0 → 1 as y → ∞. In effect this arises from the fact that f 0 ∼ 1 + O(y 2(1−A)/2+A ) for y large with this range of solutions for A > 1 then having algebraic decay, rather than the exponential decay when A < 1, as y → ∞. The numerical solutions presented here in Figs. 7 and 8 (below) for A > 0 are obtained by continuing the corresponding solution for A = 0 into the parameter range A > 0 by successively increasing the value of A. Thus we continue the numerical solution from a parameter region where there is a single solution (A < 1) into a region (A > 1) where there is a range of possible solutions Magyari (2001). This continuation process across A = 1 appears to be smooth in our numerical computations and gives a single, continued numerical solution. To determine the nature of the critical values for A large we need λ to be small, of O(A−1 ). We put λ = ν A−1 and then make the transformation f = A−1/2 g, Y = A1/2 y
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Fig. 5 Plots of f (0) against s for a λ = 1.0, asymptotic expression (20) for s < 0, |s| 1 and for s 1 given in Rohni et al. (2012) are shown by broken lines, b λ = −0.5 and c for λ = −1.2, −2.0 obtained from the numerical solution of (3, 4) with A = 0
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Fig. 6 A plot of the critical values λcrit for problem (3, 4) with s = 0 plotted against A. Asymptotic expression (29) is shown by the broken line 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9
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Fig. 7 Plots of f (0) against λ for A = 1.0, 5.0 obtained from the numerical solution of (3, 4) with s = 0
Equations (3, 4) become g +
Y g + g − 1 + A−1 (gg + g − g 2 ) = 0, 2 (22)
g(0) = 0, g (0) = 1 + ν A−1 , g → 1
as Y → ∞
where primes now denote differentiation with respect to Y . Eq. (22) suggests an expansion of the form g(Y ; A) = g0 (Y ) + A−1 g1 (Y ) + · · ·
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Fig. 8 Plots of f (0) against A obtained from the numerical solution of (3, 4) with s = 0 for a λ = 0.5, 1.0 (representative of the aiding flow case) and b λ = −0.75, −1.0 (representative of the opposing flow case)
The leading–order problem has the solution g0 = Y + 2B0 (1 − e−Y
2 /4
)
giving g0 = 1 + B0 Y e−Y
2 /4
(24)
for some constant B0 to be determined. At O(A−1 ), we obtain g1 +
Y 2 2 g + g1 = 2B02 e−Y /2 − B0 e−Y /4 2 1
2B0 − B0 Y 2 −
Y3 2
(25)
subject to g1 (0) = 0, g1 (0) = ν,
g1 → 0 as Y → ∞
(26)
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Since g1 = Y e−Y /4 is a complementary function for Eq. (25), we find that, if we multiply Eq. (25) by Y and integrate, we obtain the condition, on applying boundary conditions (26), that ∞ Y3 2 2 ν= Y 2B02 e−Y /2 − B0 e−Y /4 2B0 − B0 Y 2 − dY (27) 2 2
0
On evaluating the integral, we obtain √ 6B02 + 6 π B0 − ν = 0
giving B0 = √
Expression (28) gives a critical value at B0 = − f (0) ∼ B0 A1/2 , we have λcrit ∼ −
3π −1 A + · · · and 2
d2 f dy 2
√ √ −3 π ± 9π + 6ν 6
3π π , with νcrit = − . Noting that 2 2
∼−
(y=0, crit)
(28)
√ π 1/2 A + · · · as A → ∞ (29) 2
consistent with our numerical results. Expression (29) is shown in Fig. 6 by a broken line and is approaching the numerical values relatively slowly as A is increased. To derive a solution for the general case when A large and λ of O(1) we follow the above discussion for λ small, starting by putting f = y + h(Y ), with Y defined above. Eqs. (3, 4) become Y h + h + h + A−1/2 (h h − h 2 ) + A−1 (Y h − h ) = 0, 2 (30) h(0) = 0, h (0) = λ A−1/2 , h → 0 as Y → ∞ We now expand h(Y ; A) = h 0 (Y ) + A−1/2 h 1 (Y ) + · · ·
(31)
The leading-order problem has the solution h 0 = 2C0 (1 − e−Y
2 /4
),
h 0 = C0 Y e−Y
2 /4
(32)
for some constant C0 to be determined. At O(A−1/2 ), we then have h 1
Y Y 2 −Y 2 /4 2 −Y 2 /2 + h 1 + h 1 = 2C0 e − (1 − )e , 2 2 (33)
h 1 (0) = 0, h 1 (0) = λ, h 1 → 0 as Y → ∞ As before we obtain an integral relation, namely ∞ λ=
2C02
Y e
−Y 2 /2
Y 2 −Y 2 /4 )e dY − (1 − 2
(34)
0
Evaluating the integral in (34) gives λ = 6C02
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(35)
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giving λ1/2 f (0) ∼ ± √ A + · · · as A → ∞ (for λ > 0) 6
(36)
Expression (36) agrees with the solution for λ small on letting ν → ∞ in (28). In Fig. 8a, we plot f (0) against A for λ = 0.5, 1.0 representative of the aiding flow case. We see that, for each λ, there are two disjoint solution branches in A ≥ 0 with f (0) starting at the values for A = 0 given in Nazar et al. (2004), Merrill et al. (2006) and increasing on the upper branch and decreasing on the lower branch. The solutions on the lower branches are approaching, though relatively slowly, the asymptotic form (36) as A increases. However, the upper branch solutions are well away from this asymptotic limit and require very much larger values of A than are used to plot this figure. In Fig. 8b, we plot f (0) against A for λ = −0.75, −1.0 representative of the opposing flow case. Here, from Fig. 6, we see that there are critical values Acrit of A (approximately 2.880 and 1.338 respectively for λ = −0.75, −1.0). Now in each case the solution exists only for 0 ≤ A ≤ Acrit , and so there is an upper limit on the value of A for the existence of a solution in the opposing flow regime. This is consistent with asymptotic expression (36) which clearly requires λ > 0. Finally we note that, if λ < λcrit −1.417 (for A = 0), there are no solutions to (3, 4) in the opposing flow case for any A ≥ 0.
4 Conclusions We have established an equivalence between the solution to a similarity problem involving a slip temperature condition and the corresponding one with a prescribed surface temperature. In detail, we have shown that the solution to problem (1, 2) can be expressed in terms of the solution to problem (3, 4). In particular, we identified a relation (6) between the mixed convection parameter λ for each problem in terms of the surface heat flux derived from (3, 4). This led to a relation (7) between the critical values of these two problems. Although we considered only a specific problem, our approach has a wider applicability to problems of a similar nature. We then examined the problem (3, 4) in more detail, extending the results of Rohni et al. (2012) and of Nazar et al. (2004), Merrill et al. (2006) for the steady case. Our results for the steady, A = 0, case indicate that f (0) > 0 at the critical point λcrit for all appropriate values of s. On physical grounds, Cao and Baker (2009) we can expect ≥ 0 and hence, from (7), the effect of including a slip temperature boundary condition is to reduce the value of λcrit , thus increasing the range for a solution to exist in the opposing flow regime. When there is fluid withdrawal through the surface, our asymptotic analysis shows that the solution continues to large values of s in the opposing flow region. At the critical point f (0) ∼ s 3 /8 from (14) with (7) and (13) giving λ1,crit ∼ −
s2 s3 + ··· − + · · · for s large 4 8
(37)
where λ1,crit is the critical value for the original problem (1, 2). Expression (37) shows that, as the value of s is increased for opposing flow, the temperature slip effects become increasingly more dominant. There are critical points only for λ > −1 when there is fluid injection through the wall, see Fig. 2. In this case, the values of f (0) decrease to zero as λ → −1 so that expression (7) shows that the effect of temperature slip on the
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boundary become less significant in modifying the range of solutions for opposing flow as |s| is increased. The unsteady, A > 0, case is different in that now f (0) < 0 for A 0.45 with the effect that the range of λ for the existence of a solution is decreased. Applying our asymptotic result (29) in (7) gives λ1,crit ∼ −
√ 3 −1 π 1/2 A + ··· + A + ··· 2π 2
(38)
Expression (38) suggests that, for sufficiently large values of A and , the critical value will be in the aiding, λ > 0, flow region thus restricting solutions only to aiding flows. Our analysis of the unsteady problem assumed that s = 0. We can include effects of fluid transfer through the wall directly into this analysis if we assume that s is small, of O(A−1/2 ) 2 for A large. If we put s = γ A−1/2 , expression (24) becomes G 0 = γ +Y +2B0 (1−e−Y /4 ). This then modifies the right-hand side of Eq. 25 and the integral condition (27) which now gives 6B02
√
+ 2B0 (3 π + γ ) − ν = 0 giving νcrit
√ (3 π + γ ) =− 6
(39)
Expression (39) indicates that having a small amount of fluid withdrawal through the wall, γ > 0, can √ increase the range of solutions for opposing flow. Small amounts of fluid injection, −3√ π < γ < 0 is to decrease the range of solutions for opposing flow, to zero at γ = −3 π , before increasing again as γ is decreased further. Finally we comment on how the lower branch solutions continue through λ = 0, as seen in Figs. 3a, 5 and 7. Our numerical integrations of Eqs. (3, 4) indicate that the numerical solution passes smoothly through λ = 0 without any suggestion of a singularity developing, as was the case in Merkin (1985) for example. In fact we can find numerical solutions to Eqs. (3, 4) on the lower solution branch for λ = 0 which are not the ‘forced convection’ solution f = y + s. For example, with s = A = 0, we find f 0 (0) = −0.43815 on the lower solution branch for λ = 0. However, these are spurious solutions to the original problem, as commented on in Rohni et al. (2012). The reason for this is that, in the original problem motivating this and the previous paper by Rohni et al. (2012), the (dimensionless) temperature difference θ is related to the flow through f = 1 + λ θ . Then θ is eliminated using θ = ( f − 1)/λ, clearly requiring λ = 0 unless f = 1. However, regarded purely as solutions to the problem given by Eqs. (3, 4), in effect disregarding the original physical set up, these are solutions for λ = 0 in boundary conditions (4) which are not the forced convection solution f = y + s. The existence of these lower branch solutions for λ = 0 may be perhaps be regarded as a mathematical artefact, though they do allow the lower branch solutions to continue smoothly into the aiding flow, λ > 0, region. We expect these lower branch solutions to be temporally unstable from the change in stability at the saddle–node bifurcation at λcrit . Acknowledgments AMR and SA gratefully acknowledge the financial support received in the form of research grant: Research University Grant (1001/PMATHS/811166) from Universiti Sains Malaysia (USM). I.P. wishes to express his thanks to USM for the opportunity to visit USM. We also wish to thank the referees for their helpful comments and suggestions.
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