Z. angew. Math. Phys. 59 (2008) 100–123 0044-2275/08/010100-24 DOI 10.1007/s00033-006-6082-7 c 2006 Birkh¨ ° auser Verlag, Basel

Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux A. Ishak, J. H. Merkin, R. Nazar and I. Pop

Abstract. The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived. Mathematics Subject Classification (2000). 76R05, 76R10. Keywords. Boundary-layer, dual solutions, mixed convection, suction/injection.

1. Introduction Mixed convection flows, or combined forced and free convection flows, arise in many transport processes both naturally and in engineering applications. They play an important role, for example, in atmospheric boundary-layer flows, heat exchangers, solar collectors, nuclear reactors and in electronic equipment. Such processes occur when the effects of buoyancy forces in forced convection or the effects of forced flow in free convection become significant. The interaction of forced and free convection is especially pronounced in situations where the forced flow velocity is low and/or the temperature differences are large. Over the previous decades many analyses of mixed convection flow of a viscous and incompressible fluid over a vertical surface have been performed. Analytical and numerical solutions for the temperature and the velocity field have been obtained both for prescribed wall temperatures and for prescribed wall heat fluxes. A comprehensive description of the theoretical work prior to 1987 for both laminar and turbulent mixed convection boundary-layer flows for selected flow geometries has been given in a review paper by Chen and Corresponding author: [email protected] (I. Pop)

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Armaly [1] and recently in the book by Pop and Ingham [2]. In a large number of problems of fluid mechanics and heat transfer, the equations expressing the basic physical laws are partial differential equations, with the aim being to find exact solutions of these equations. Solutions of certain sets of partial differential equations occurring in applied fields can be found quite readily in spite of the failure of the more common classical methods to yield results. Notable among such solutions are those that have been obtained by employing transformations of the variables that reduce the system of partial differential equations to a system of ordinary differential equations. These solutions are generally designated as similarity solutions, see Hansen [3]. Such solutions for the problem under consideration here were recently studied in a series of papers by Merkin and Mahmood [4, 5, 6, 7], Rhida [8] and Merkin and Pop [9] for the steady mixed convection boundary-layer flow over an impermeable vertical flat plate and for impermeable vertical cylinders. The aim of this paper is to extend the paper by Merkin and Mahmood [4] to the case when the plate is permeable, i.e. when there is suction or injection (blowing) through the wall. Suction or injection of a fluid through the bounding surface, as, for example, in mass transfer cooling, can significantly change the flow field and, as a consequence, affect the heat transfer rate from the plate. In general, suction tends to increase the skin friction and heat transfer coefficients, whereas injection acts in the opposite manner [10]. Injection or withdrawal of fluid through a porous bounding heated or cooled wall is of general interest in practical problems involving film cooling, control of boundary layers etc. This can lead to enhanced heating (or cooling) of the system and can help to delay the transition from laminar flow (see Chaudhary and Merkin [11]).

2. Equations We consider a semi-infinite permeable vertical flat plate placed in a viscous and incompressible fluid of ambient temperature T∞ . It is assumed that the plate is subject to a variable heat flux qw (x) and that there is a transpiration velocity Vw (x) through the porous wall. It is also assumed that there is a free stream velocity U (x) flowing over the plate, and that the buoyancy force can act in the same direction as the flow (aiding flow) or can act in the opposite direction to the flow (opposing flow). Under these assumptions, along with the usual boundarylayer and Boussinesq approximations, the governing equations are

u

∂u ∂v + = 0, ∂x ∂y

(2.1)

∂u ∂u dU ∂2u +v =U + g ∗ β(T − T∞ ) + ν 2 , ∂x ∂y dx ∂y

(2.2)

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u

∂T ∂2T ∂T +v =α 2, ∂x ∂y ∂y

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(2.3)

together with the boundary conditions ∂T = −qw (x) on y = 0, ∂y u → U (x), T → T∞ as y → ∞,

v = Vw (x), u = 0, k

(2.4)

∗

where g is the acceleration due to gravity, β the coefficient of thermal expansion, ν and α are the kinematic viscosity and thermal diffusivity of the fluid, respectively. Following Merkin and Mahmood [4], we assume that U (x) and qw (x) are of the form ³ x ´m

,

qw (x) = q0

³ x ´(5m−3)/2

, ` ` where U0 , q0 and m are constants, and ` is a characteristic length scale. U (x) = U0

(2.5)

We look for a similarity solution of equations (2.1)–(2.3), subject to boundary conditions (2.4), by writing 1/2

ψ = (U0 ν`)

µ ¶1/2 ³ ´ q0 ν` x 2m−1 f (η), T − T∞ = θ(η), k U0 ` µ ¶1/2 ³ ´ x (m−1)/2 U0 . η=y ν` `

³ x ´(m+1)/2 `

From transformation (2.6), we obtain ³ x ´m f 0 (η), u = U0 ` ¶1/2 ³ ´ µ ¶ µ m−1 0 x (m−1)/2 m + 1 U0 ν f+ ηf , v=− ` ` 2 2

(2.6)

(2.7)

where prime denotes differentiation with respect to η. The existence of a similarity solution to equations (2.1)–(2.3), subject to boundary conditions (2.4), requires that we should take µ ¶1/2 ³ ´ m + 1 U0 ν x (m−1)/2 f0 , (2.8) Vw (x) = − 2 ` ` where f0 is a non-dimensional constant which determines the transpiration rate, with f0 > 0 for suction and f0 < 0 for blowing or injection. Using (2.6), (2.7) and (2.8), equations (2.2) and (2.3) become f 000 +

m + 1 00 2 f f + m(1 − f 0 ) + λθ = 0, 2

(2.9)

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1 00 m + 1 0 θ + f θ + (1 − 2m)f 0 θ = 0, (2.10) Pr 2 subject to boundary conditions (2.4) which become, on using (2.5) and (2.8), f (0) = f0 , f 0 (0) = 0, θ0 (0) = −1,

f 0 → 1, θ → 0 as η → ∞.

(2.11)

In equation (2.9), λ is the buoyancy parameter, defined as λ=

g ∗ βqw ν 1/2 `3/2 Gr = , 5/2 Re5/2 k U0

(2.12)

U0 ` g ∗ β(qw `/k)`3 are the Grashof number and Reynolds and Re = ν2 ν number respectively. We note that λ > 0 corresponds to assisting flow (free stream and buoyancy forces in the same direction) and that λ < 0 corresponds to opposing flow (free stream and buoyancy forces in the opposite direction). For the rest of this paper we consider only two values for m, namely m = 0.6 and m = 1, which correspond to constant heat flux and to a stagnation-point flow respectively. We also restrict attention to unit Prandtl number, taking P r = 1. We expect our findings to be qualitatively similar for other values of P r of O(1). The main physical quantities of interest are the values of f 00 (0), being a measure of the skin friction, and the non-dimensional wall temperature θ(0). Our main aim is to find how the values of f 00 (0) and θ(0) vary in terms of the parameters f0 and λ. where Gr =

3. Results Equations (2.9) and (2.10) subject to the boundary conditions (2.11) were integrated numerically for different values of the governing parameters f0 , m and λ using the finite-difference approximation known as the Keller-box method. This method is described fully by Cebeci and Bradshaw [12]. We compared our results for the skin friction coefficient f 00 (0) when λ = 0 and the critical values λc of the buoyancy parameter λ with those of Merkin and Mahmood [4]. These latter results were obtained by an essentially different method, using the boundary-value integrator D02AGF in the NAG library. The comparisons, which are given in tables 1 and 2, show excellent agreement between the two sets of results and so give confidence in our numerical approach. The variation of the skin friction parameter f 00 (0) with λ is shown in figures 1 and 2 for m = 1 and m = 0.6 respectively, for P r = 1 and for values of the suction/injection parameter f0 indicated on the figures. The corresponding wall temperatures θ(0) are shown in figures 3 and 4. These figures show that it is possible to obtain dual solutions of the similarity equations (2.9)–(2.11) for assisting flow, λ > 0, as well as for opposing flow, λ < 0, that have been reported previously. For λ > 0, there is a favourable pressure gradient due to the buoyancy forces. This results in the flow being accelerated and consequently there is a larger

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Figure 1. Variation of the skin friction coefficient f 00 (0) with λ for various values of f0 with m = 1 and P r = 1.

Figure 2. Variation of the skin friction coefficient f 00 (0) with λ for various values of f0 with m = 0.6 and P r = 1.

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Figure 3. Variation of the wall temperature θ(0) with λ for various values of f0 with m = 1 and P r = 1.

Figure 4. Variation of the wall temperature θ(0) with λ for various values of f0 with m = 0.6 and P r = 1.

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Figure 5. Velocity profiles f 0 (η) for various f0 with m = 1, P r = 1 and λ = −0.5.

Figure 6. Velocity profiles f 0 (η) for various f0 with m = 1, P r = 1 and λ = 1.0.

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Table 1. Critical values λc of λ. f0

m = 0.6

Present (m = 1)

[4] (m = 1)

-0.1 0 0.1

-0.4066 -0.4924 -0.6025

-0.9942 -1.1936 -1.4310

-1.193568

Table 2. The values of f 00 (0) for various values of f0 when λ = 0 on the lower solution branch. f0

Present (m = 0.6)

[4] (m = 0.6)

Present (m = 1)

[4] (m = 1)

-0.1 0 0.1

-0.2657 -0.3058 -0.3494

-0.30579

-0.4867 -0.5626 -0.6461

-0.56258

skin friction coefficient than in the non-buoyant, λ = 0, case. For negative values of λ there is a critical value λc , with two solutions branches for λ > λc , a saddlenode bifurcation at λ = λc and no solutions for λ < λc . The values for λc are given in table 2, together with the values given in [4] for the impermeable surface. We identify the upper and lower branch solutions in the following discussion by how they appear in figures 1 and 2, i.e. the upper branch solution has a higher value of f 00 (0) for a given λ than the lower branch solution. For assisting flow, we found dual solutions to exist for all the (positive) values of λ considered, to much higher values of λ than shown in figures 1–4. These figures show that the critical value |λc | increases as f0 is increased, suggesting that suction (f0 > 0) increases the range of existence of solutions to equations (2.9)–(2.11), whereas injection (f0 < 0) reduces this range. The results for the wall temperature (figures 3, 4) show that, for the lower branch solution (as identified above), θ(0) becomes unbounded as λ → 0− and as λ → 0+ . Figures 5 and 6 present the velocity profiles for λ = −0.5 and λ = 1.0 respectively, with the corresponding temperature profiles being shown in figures 7 and 8, all for m = 1. In these figures the full lines are for the upper branch solution and the broken lines for the lower branch solution. As seen in figures 1 and 3, there are dual solutions both when λ = −0.5 and when λ = 1.0, with both these velocity and temperature profiles being solutions of the boundary-value problem (2.9)–(2.11). When λ = −0.5, the velocity profiles for the upper branch solution have a positive velocity gradient at the wall (as indicated in figure 1), with the opposite being the case for the lower branch solution. For λ = 1.0, the velocity gradient at the wall is positive for the solutions on both branches, in agreement with the curves of f 00 (0) shown in figure 1. However, the solution on the lower branch for λ = 1.0 has a region of reversed flow (has f 0 (η) < 0 for a finite range

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Figure 7. Temperature profiles θ(η) for various f0 with m = 1, P r = 1 and λ = −0.5.

of η) located away from the wall (η = 0). In figure 7 (λ = −0.5) we see that the temperature profiles are higher for the lower branch solutions compared to those for the upper branch solutions, consistent with figure 3. The opposite is the case in figure 8 (λ = 1.0) where we see that the temperature profiles are higher for the upper branch solutions, again consistent with figure 3. In figure 7, the temperature profiles for the solutions on both branches have θ(η) > 0 for all 0 ≤ η < ∞, whereas in figure 8, the solutions on the lower branch have regions of below ambient temperature, i.e. θ(η) < 0 for a range of η and θ → 0 from below as η → ∞. The dual nature of the present solutions is qualitatively similar with those reported by Ridha [8] and Steinr¨ uck [13] for mixed-convection flow along a vertical surface. It is worth commenting which dual solution is of physical relevance. Our similarity solutions should be regarded in the more general boundary-layer context as being possible asymptotic (large x) solutions. To which of our dual solutions the flow will approach as it develops from the leading edge (x = 0) depends essentially on the stability of the solution. A full stability analysis, along the lines described in [13, 18, 19, 20, 21, 22, 23], is beyond the scope of the present work. However, we suggest that it will be the upper branch solutions that are of the most physical relevance. The lower branch solutions have regions of reversed flow (see figures 5, 6) and this would seem to invalidate them as possible asymptotic solutions to which a fully forward flow developing near the leading edge could evolve. Also, the forced convection limit (λ = 0) is on the upper branch and we expect this solution to be stable, as it is the only solution for this case. The saddle-node bifurcation

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Figure 8. Temperature profiles θ(η) for various f0 with m = 1, P r = 1 and λ = 1.0.

Figure 9. Variation of the skin friction coefficient f 00 (0) with f0 for λ = −1 and m = 1, 0.6 (P r = 1).

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Figure 10. Variation of the wall temperature θ(0) with f0 for λ = −1 and m = 1, 0.6 (P r = 1).

at λc corresponds to a change in the (temporal) stability of the solution and, unless there is a change in stability on the upper branch for λ 6= λc (which we cannot determine from our analysis), the saddle-node bifurcation gives a change in stability from stable (upper branch) to unstable (lower branch). As remarked in [14, 15], Spangenberg et al. [16] have reported in their experimental work on turbulent boundary layer under strong adverse pressure gradient that dual solutions were obtained as a function of how the pressure gradient was realized. Another example of non-unique flow is reported by Aidun et al. [17] where they have observed experimentally that the primary steady state flow in a through-flow lid-driven cavity was non-unique and only one of the multiple steadystate patterns can stabilize in the cavity. In figures 9 and 10 we show the variation of f 00 (0) and θ(0) with f0 for λ = −1.0 and for both m = 1 and m = 0.6. These figures support the dual nature of the solution to boundary-value problem (2.9)–(2.11) and that separation, f 00 (0) becoming zero, occurs for the opposing flow. For this value of λ, there is a critical value f0,crit of f0 , being negative and dependent on m, at which there is a saddle-node bifurcation, with two solutions for f0 > f0,crit and no solutions for f0 < f0,crit . This indicates that injection (having f0 < 0) limits the existence of solutions, whereas no such limit appears for suction (f0 > 0), with both branches of solutions continuing to large values of f0 .

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Figure 11. Graph of f 00 (0) plotted against f0 , obtained from the numerical solution of equations (4.13-4.15) for m = 1 and P r = 1.

Figure 12. Graph of g(0) plotted against f0 , obtained from the numerical solution of equations (4.13-4.15), with the - sign taken in (4.13), for m = 1 and P r = 1.

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4. Asymptotic limits 4.1. The forced convection limit, λ → 0 To discuss the behaviour as λ → 0 on the lower branch solution (as identified above) we follow [4] and put θ = g/|λ| leaving f and η unchanged. This results in, for P r = 1 and letting |λ| → 0, m + 1 00 2 f f + m(1 − f 0 ) ± g = 0, (4.13) f 000 + 2 m+1 0 f g + (1 − 2m)f 0 g = 0, (4.14) g 00 + 2 with f (0) = f0 , f 0 (0) = 0, g 0 (0) = 0,

f 0 → 1, g → 0 as η → ∞,

(4.15)

where, in equation (4.13), the + sign is for aiding flow (the limit λ → 0+ ) and the − sign is for opposing flow (the limit λ → 0− ). The case of opposing flow (the − sign in (4.13)) and f0 = 0 was considered in [4], which gave f 00 (0) = 1.62878 as λ → 0− (on the −0.56258, g(0) = 1.62878 for m = 1, so that θ(0) ∼ (−λ) lower branch solution). The result that f 00 (0) is of O(1) and θ(0) is of O(|λ|−1 ) as λ → 0− can be seen in figures 1-4. Graphs of f 00 (0) and g(0) against f0 , obtained from the numerical solution of equations (4.13)–(4.15) are shown in figures 11 and 12, respectively. If we now examine equations (4.13)–(4.15), we see that changing g to −g leaves the system unchanged apart from changing the − sign to a + sign before the final term in equation (4.13). This is, in effect, changing the situation from having 1.62878 λ → 0− (opposing flow) to having λ → 0+ (aiding flow). Thus θ(0) ∼ − λ as λ → 0+ (on the lower branch solution with m = 1) and this gives a basis for continuing the solution into λ > 0. This procedure for continuing the solution into λ > 0 to give dual solutions in aiding flow was not noticed in [4]. Our numerical investigation indicates that dual solutions continue well into λ > 0 and exist to large, positive values of λ. 4.2. Strong suction, f0 À 1 Here we obtain a solution to equations (2.9)–(2.11) valid for f0 large and positive. To do so we first make the transformation f = f0 (1 + f0−2 F ),

θ = f0−1 H,

ζ = f0 η.

This leads to the equations, for P r = 1, µ ¶ m + 1 00 m+1 −2 000 00 02 F + f0 F F + m(1 − F ) + λf0−3 H = 0, F + 2 2

(4.16)

(4.17)

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Figure 13. Variation of the skin friction coefficient f 00 (0) with f0 for various λ and m = 1 (P r = 1).

H 00 +

m+1 0 H + f0−2 2

µ

¶ m+1 F H 0 + (1 − 2m)F 0 H) = 0, 2

(4.18)

now subject to the boundary conditions F (0) = 0, F 0 (0) = 0, H 0 (0) = −1,

F 0 → 1, H → 0 as ζ → ∞,

(4.19)

and where now primes denote differentiation with respect to ζ. An expansion of the form H = H0 + f0−2 H1 + · · · , (4.20) F = F0 + f0−2 F1 + · · · , is suggested by equations (4.17), (4.18). The leading order terms are easily found to be ´ 2 ³ 2 1 − e−(m+1)ζ/2 , e−(m+1)ζ/2 . H0 = (4.21) F0 = ζ − m+1 m+1 The higher order terms can be derived in a straightforward way. From (4.16), (4.21) it follows that f 00 (0) ∼

m+1 f0 + · · · , 2

θ(0) ∼

2 f −1 + · · · , m+1 0

(4.22)

for f0 large. In figures 13 and 14 we plot f 00 (0) and θ(0) against f0 obtained from the numerical solution of equations (2.9)–(2.11) for λ = 1, 2, 5 and m = 1. The figures show the linear increase in f 00 (0) with f0 and that θ(0) becomes small for f0 large, as predicted in (4.22). The figures also show that the forms for f 00 (0)

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Figure 14. Variation of the wall temperature θ(0) with f0 for various λ and m = 1 (P r = 1).

Figure 15. The critical value (bifurcation point) λc plotted against f0 for m = 0.6, 1 (P r = 1).

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and θ(0) are independent of λ for f0 large, as is also seen in (4.22). From equation (4.17), the effect of the buoyancy parameter λ will appear at leading order only when λ is large, of O(f03 ). To examine this case we put λ = f03 µ, where µ is of O(1). Equation (4.18) is unaltered as is the leading order solution H0 given in (4.21). Equation (4.17) now gives, at leading order, 2µ −(m+1)ζ/2 e , m+1 F0 (0) = 0, F00 (0) = 0, F00 → 1 as ζ → ∞. F0000 + F000 = −

The required solution is F00

" µ =1+ µ

2 m+1

¶2

(4.23)

# ζ − 1 e−(m+1)ζ/2 .

(4.24)

From (4.24), it follows that ¶" µ ¶3 # µ 2 m + 1 f 00 (0) ∼ 1+ µ f0 + · · · , 2 m+1

(4.25)

for f0 À 1, with θ(0) still given by (4.22). ¶3 m+1 . From (4.25) we can see that, for f0 large, f (0) = 0 when µ = − 2 00 Figures 1, 2 and 9 suggest that the solution terminates at λ = λc when f (0) = 0, ¶3 µ m+1 f03 for f0 large. In figure 15 its critical value. This suggests that λc ∼ − 2 we plot the critical value λc against f0 for m = 0.6 and m = 1. The figure shows that |λc | increases rapidly with f0 (for f0 > 0) consistent with this result, though the values of f0 plotted in figure 15 are, perhaps, too small to give quantitative agreement. µ

00

4.3. Strong injection, f0 < 0, |f0 | À 1 We start by considering the situation when λ is small, following the scalings for strong blowing given in [24] for the case without buoyancy effects. Here there is an inner region of thickness O(|f0 |) made up of fluid blown through the permeable wall and a thinner outer region (shear layer), of thickness O(1), through which the outer conditions are attained. For simplicity we treat only the case m = 1 (with P r = 1) and write δ = −f0 , where δ > 0 and δ À 1. For the inner region we put f = δφ,

θ = δ 3 H,

Y = η δ −1 .

(4.26)

To determine the scaling for θ in (4.26), we note that, using the scaling for η in (4.26), the viscous/heat conduction terms are of O(δ −2 ) compared to the other terms in equations (2.9), (2.10) and hence (with m = 1) θ = −k0 δφ at leading

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order, for some constant k0 . This might suggest scaling θ with δ. However, this would mean that the boundary condition θ0 (0) = −1 is not satisfied, since φ0 (0) = 0. To obtain a scaling which allows this form for θ at leading order and the boundary condition on θ to be satisfied we are led to the scaling for θ given in (4.26). Following from these scalings is the restriction that λ must be small, of O(δ −3 ) and we write λ = δ −3 υ, where υ is of O(1). Using this and (4.26), equations (2.9), (2.10) become 2 (4.27) δ −2 φ000 + φφ00 + 1 − φ0 + υH = 0, δ −2 H 00 + φH 0 − φ0 H = 0, subject to φ(0) = −1,

φ0 (0) = 0,

(4.28)

H 0 (0) = −δ −2 .

(4.29)

The outer boundary conditions are relaxed at this stage and primes denote differentiation with respect to Y . Equations (4.27), (4.29) suggest an expansion of the form φ(Y ; δ) = φ0 (Y ) + δ −2 φ1 (Y ) + · · · ,

H(Y ; δ) = H0 (Y ) + δ −2 H1 (Y ) + · · · . (4.30)

At leading order we have H0 = −k0 φ0 and then 2

φ0 φ000 + 1 − φ00 − υk0 φ0 = 0,

φ0 (0) = −1, φ00 (0) = 0.

Equation (4.31) has the solution ¶ µ p υk0 π 1 + υk0 sin( 1 + 2υk0 Y − ) − . φ0 = 1 + 2υk0 2 1 + 2υk0

(4.31)

(4.32)

This solution holds for 0 ≤ Y ≤ Y0 , where φ0 (Y0 ) = 0, so that Y0 is given by p υk0 π sin( 1 + 2υk0 Y0 − ) = (4.33) 2 1 + υk0 and φ0 ∼ (Y − Y0 ) + · · ·

for |Y − Y0 | small.

(4.34)

To determine the constant k0 we need to consider the equation for H1 at O(δ −2 ) in expansion (4.30). This is φ0 H10 + φ1 H00 − φ00 H1 − φ01 H0 + H000 = 0,

H10 (0) = −1.

(4.35)

Applying the boundary conditions on Y = 0 and using (4.32), we have 1 − k0 φ000 (0) = 0 or k0 (1 + υk0 ) = 1, giving √ ± 1 + 4υ − 1 . (4.36) k0 = 2υ The + sign in (4.36) corresponds to solutions on the upper branch and the − sign to solutions on the lower branch, as defined by figures 1, 2.

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To complete the solution we require an outer (shear) layer for which we put η = η¯ + Y0 δ, with Y0 defined in (4.33), and θ = δ 2 h. This leads to, at leading order, 2 (4.37) f 000 + f f 00 + 1 − f 0 = 0, h00 + f h0 − f 0 h = 0,

(4.38)

subject to, on using (4.34) to match with the inner solution, f ∼ η¯ + · · · , h ∼ −k0 η¯ + · · · as η¯ → −∞,

f 0 → 1, h → 0 as η¯ → ∞, (4.39)

where k0 is given in (4.36) and primes denote differentiation with respect to η¯. This gives f = η¯ and then ¶ µ Z η¯ 2 2 1 e−s /2 ds . e−¯η /2 + η¯ h = η¯ − √ π −∞ From (4.26), (4.32) and (4.36), we have ¶ µ√ 1 + 4υ + 1 00 δ −1 + · · · , upper branch: f (0) ∼ 2 ¶ µ√ 1 + 4υ − 1 δ3 + · · · , θ(0) ∼ 2υ ¶ µ√ 1 + 4υ − 1 δ −1 + · · · , 2 ¶ µ√ 1 + 4υ + 1 δ3 + · · · , θ(0) ∼ − 2υ

(4.40)

lower branch: f 00 (0) ∼ −

(4.41)

for δ large and λ of O(δ −3 ). Expression (4.36) puts a bound on the existence of 1 solutions of υ ≥ υc = − , or that 4 1 δ −3 = − |f0 |−3 for |f0 | large, (4.42) 4 4 is required for the existence of a solution. The emergence of the two solution branches at the saddle-node bifurcation at υc is readily identified in (4.40), (4.41). The lower branch solutions also have θ(0) → ∞ as υ → 0 (λ → 0) in line with the results shown in figures 3, 4 and discussed above, with the sign of θ(0) changing from positive to negative as υ passes through zero, again in line with figures 3, 4. The result that λc → 0 as |f0 | increases can clearly be seen in figure 15. To obtain a solution for δ large (|f0 | large) and λ of O(1), the above discussion suggests that we introduce the scalings λ ≥ λc = −

f = δφ,

θ = δ 3/2 G,

ξ = δ −1/4 η.

(4.43)

We then apply (4.43) in equations (2.9), (2.10) and look for a solution by expanding in inverse powers of δ, the first perturbation being of O(δ −5/4 ). For the leading

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order terms φ0 , G0 we have G0 = −K0 φ0 (as before) for some constant K0 to be determined and then 2

φ0 φ000 − φ00 − λK0 φ0 = 0,

φ0 (0) = −1, φ00 (0) = 0.

Equation (4.44) has the solution ´ p 1³ π sin( 2K0 λ ξ − ) − 1 , φ0 = 2 2 π for 0 ≤ ξ < ξ0 , where ξ0 = √ and 2K0 λ

(4.44)

(4.45)

K0 λ (ξ − ξ0 )2 for |ξ − ξ0 | small. (4.46) 2 To determine K0 we have, as above, to consider the equation for the next term in the series for G1 . This is, in effect, equation (4.35) and gives, as before, 1 − K0 φ000 (0) = 0, which in this case leads to, from (4.45), K0 = ±λ−1/2 , with the + sign corresponding to solutions on the upper branch and the − sign to solutions on the lower branch. To complete the solution, we need an outer (shear) layer in which we put φ0 ∼ −

f = δ 1/6 F,

θ = δ 2/3 g,

ζ = (η − δ 1/4 ξ0 )δ 1/6 .

(4.47)

This gives, at leading order, 2

F 000 + F F 00 − F 0 + λg = 0, 00

0

0

g + F g − F g = 0,

(4.48) (4.49)

subject to, from (4.46), F ∼∓

1 λ1/2 2 ζ + · · · , g ∼ ± ζ 2 + · · · as ζ → −∞, 2 2 F 0 → 0, g → 0 as ζ → ∞.

(4.50)

From (4.44) we have f 00 (0) ∼ ±λ1/2 δ 1/2 + · · · ,

θ(0) ∼ ±λ−1/2 δ 3/2 + · · · ,

(4.51)

for δ large and λ of O(1). Expressions (4.40), (4.41) give f 00 (0) ∼ ±υ 1/2 δ −1 and θ(0) ∼ ±υ −1/2 δ 3 for υ large, which agree with (4.51) on writing υ = λδ 3 . The forms for f 00 (0) and θ(0) given by (4.51) are in good agreement with the results shown in figures 13, 14 for f0 large and negative. 4.4. The free convection limit, λ → ∞ To obtain a solution for λ large we follow [4] and put f = λ1/5 f¯,

¯ θ = λ−1/5 θ,

η¯ = λ1/5 η.

(4.52)

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Figure 16. Graph of f¯00 (0) against ² (where ² = λ−2/5 ) obtained from equations (4.53-4.55) for m = 1, f0 = 0 (P r = 1).

This results in the equations, for P r = 1, m + 1 ¯ ¯00 f¯000 + θ¯ + f f − mf¯02 + m²2 = 0, 2

(4.53)

m + 1 ¯ ¯0 θ¯00 + f θ + (1 − 2m)f¯0 θ¯ = 0, 2

(4.54)

now subject to the boundary conditions f¯(0) = f0 ²1/2 , f¯0 (0) = 0, θ¯0 (0) = −1,

f¯0 → ², θ¯ → 0 as η¯ → ∞,

(4.55)

where ² = λ−2/5 (or λ = ²−5/2 ) and primes denote differentiation with respect to η¯. The free convection limit is obtained by letting ² → 0 in equations (4.53)– (4.55). We note that the effects of suction/injection do not appear in this limit when f0 is of O(1), for these effects to appear at leading order in this limit requires |f0 | to be large, of O(²−1/2 ) (or of O(λ1/5 )). This is consistent with the results shown in figures 1–4, particularly with the upper branch solutions given in figures 1, 2, even at the relatively small values of λ used for these figures. In figures 16 ¯ obtained from the numerical solution and 17 we respectively plot f¯00 (0) and θ(0) of equations (4.53)–(4.55) for m = 1 and f0 = 0, starting with the solution of equations (2.9)–(2.11) at λ = ² = 1. This figure shows that the upper branch solution (full lines) continues back to ² = 0, giving the free convection limiting ¯ = 1.51472 at ² = 0. The behaviour of the upper values of f¯00 (0) = 1.00969, θ(0)

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¯ Figure 17. Graph of θ(0) against ² (where ² = λ−2/5 ) obtained from equations (4.53-4.55) for m = 1, f0 = 0 (P r = 1).

branch solution for f0 of O(1) and ² small is readily obtained, we find that f¯00 (0) ∼

1.00969 − 0.37189f0 ²1/2 − (0.05721f02 + 0.01280)² + · · · , (4.56)

¯ θ(0) ∼ 1.51472 − 0.86791f0 ²1/2 + (0.22088f02 − 0.06974)² + · · · , as ² → 0. We were unable to continue the lower branch solutions back to ² = 0, our numerical integrations were stopped at ² = 0.03. The extent of the computational domain η¯∞ for these lower branch solutions had to be increased considerably as ² was decreased, at ² = 0.03 we needed η¯∞ = 90. This suggests that there are no dual solutions in the free convection limit.

5. Conclusions We have studied the similarity solutions for mixed convection boundary-layer flow along a vertical permeable surface arising from a prescribed wall heat flux. The governing similarity equations were solved numerically using the Keller-box method. We discussed in detail the effects of the suction/injection parameter f0 and the buoyancy parameter λ on the skin friction, given through f 00 (0), and the wall temperature θ(0) for the specific cases of stagnation point flow (m = 1) and

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uniform wall heat flux (m = 3/5). A new feature to emerge from our results is the existence of dual solutions in the aiding (λ > 0) case for both permeable and impermeable walls, see figures 1–4. Previous work for impermeable walls [4] had failed to notice that the lower branch solution could be continued through λ = 0 with f 00 (0) remaining continuous and g(0) for λ small, where g(0) θ(0) having a discontinuity of the form θ(0) ∼ − λ is given by equations (4.13)–(4.15). These dual solutions could be extended to large (positive) values of λ, though do not appear to arise in the free convection (λ → ∞) limit. We found solutions for all positive values of λ (aiding case) with both suction and injection. In the opposing case (λ < 0) the solution terminated with a saddlenode bifurcation at λ = λc (λc < 0), see figures 1–4, 9, 10. The value of |λc | increased (for given values of m and P r) for suction (f0 > 0) over its value for the impermeable wall, see for example table 2. The value of |λc | increases rapidly with f0 , being of O(f03 ) for f0 large. Thus suction through the wall can greatly increase the range of similarity solutions, see figures 9, 10 and figures 13, 14. The effect of injection (f0 < 0) is just the opposite, it decreases the range of the similarity solutions, with λc of O(|f0 |−3 ) for |f0 | large. This was seen in the solution for strong injection, which also clearly revealed the start of the dual solutions at λc = −|f0 |−3 /4, in expression (4.42).

Acknowledgements The authors would like to thank the reviewers for their valuable time spent reading this paper and for their valuable comments and suggestions. This work is supported by a research grant (IPRA project code: 09-02-02-10038-EAR) from the Ministry of Science, Technology and Innovation (MOSTI), Malaysia.

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Vol. 59 (2008)

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J.H. Merkin Department of Applied Mathematics University of Leeds Leeds LS2 9JT, UK e-mail: [email protected] R. Nazar School of Mathematical Sciences Faculty of Science and Technology National University of Malaysia 43600 UKM Bangi Selangor Malaysia e-mail: [email protected] I. Pop Faculty of Mathematics University of Cluj R-3400 Cluj CP 253 Romania e-mail:[email protected] (Received: June 26, 2006; revised: January 9, 2007) Published Online First: June 6, 2007

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