J Eng Math (2011) 69:101–110 DOI 10.1007/s10665-010-9408-6

Forced-convection heat transfer over a circular cylinder with Newtonian heating M. Z. Salleh · R. Nazar · N. M. Arifin · I. Pop · J. H. Merkin

Received: 7 October 2009 / Accepted: 12 August 2010 / Published online: 2 September 2010 © Springer Science+Business Media B.V. 2010

Abstract A mathematical model for the forced convection boundary-layer flow over a circular cylinder is considered when there is Newtonian heating on the surface of the cylinder through which the heat transfer is proportional to the local surface temperature. The dimensionless version of the boundary-layer equations involve two parameters, the Prandtl number σ and γ measuring the strength of the surface heating. The solution near the stagnation point is considered first and this reveals that, to get a physically acceptable solution, γ must be less than some critical value γc , dependent on σ . Numerical solutions to the full boundary-layer problem are obtained which show that the surface temperature increases as the flow develops from the stagnation point. Keywords Boundary-layer flow · Circular cylinder · Forced convection · Newtonian heating · Stagnation point flow 1 Introduction The steady boundary-layer flow of a uniform stream over a circular cylinder is a classical problem in boundary-layer theory [1, Sect. 8.33], [2, pp. 260–264]. The flow near the forward stagnation point was described originally by Blasius [3] and Hiemenz [4], with the flow development from the stagnation point being determined through M. Z. Salleh Faculty of Industrial Science & Technology, Universiti Malaysia Pahang (UMP), Lebuhraya Tun Razak, 26300 Kuantan, Pahang, Malaysia R. Nazar School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor, Malaysia N. M. Arifin Department of Mathematics, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia I. Pop Faculty of Mathematics, University of Cluj, CP 253, 3400 Cluj, Romania J. H. Merkin (B) Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected]

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a series expansion [2, 3]. Later numerical solutions of the full boundary-layer problem [5] established that the solution terminated in a Goldstein–Stewartson singularity at the separation point [6, 7] before the rear stagnation point was reached. Thus a boundary-layer solution can exist on only a finite portion of the cylinder surface. The corresponding forced-convection problem was treated originally by Frössling [8] when the cylinder is maintained at a constant temperature different to the ambient conditions using a series-expansion technique. The analogous free-convection problem has been treated by [9–12] using series-expansion methods. Numerical solutions for the case when there is constant heat flux from the cylinder and for a range of different Prandtl numbers are given in [13]. Mixed convection from a horizontal cylinder held at a constant temperature above ambient has been considered by Merkin [14] who showed that the additional buoyancy forces delayed separation and could inhibit it altogether if sufficiently strong. Here we return to the forced-convection problem though now we consider the case of Newtonian heating where the rate of heat transfer from the boundary is proportional to the local surface temperature. Heat-transfer characteristics are dependent on the thermal boundary conditions. In general, there are three common heating processes, namely a prescribed surface-temperature distribution, a prescribed surface-heat-flux distribution, and conjugate conditions, whereby heat transfer through a bounding surface of finite thickness and finite heat capacity is specified. The interface temperature is not known a priori but depends on the intrinsic properties of the system, namely, the thermal conductivities of the fluid and solid. In Newtonian heating, the rate of heat transfer from the bounding surface with a finite heat capacity is proportional to the local surface temperature, and it is usually termed conjugate convective flow. This situation occurs in many important engineering devices, for example: (a) (b) (c)

in heat exchangers, where conduction in the solid tube wall is greatly influenced by convection in the fluid flowing past it; in conjugate heat transfer around fins, where conduction within the fin and convection in the fluid surrounding it must be simultaneously analyzed in order to obtain the vital design information; in a convective-flow set-up, where the bounding surfaces absorb heat from solar radiation.

Therefore, we can conclude that the conventional assumption of the absence of an inter-relation between coupled conduction–convection effects is not always realistic, and this inter-relation must be considered when evaluating conjugate heat-transfer processes in many practical engineering applications; see [15–17] for example. Alternatively, this set-up can model the heat transfer when there is a weak exothermic catalytic reaction taking place on the surface generating heat at a rate proportional to the surface temperature. This is a reasonable assumption when the difference between the surface temperatures arising from the reaction and the ambient temperature are small, which is the situation envisaged here. The free-convection flow on a vertical surface resulting from Newtonian heating has been treated in [18] and more recently a stagnation-point flow has been considered by [19]. Convective boundary-layer flows in fluid-saturated porous media driven by Newtonian heating have also received some attention; see [20, 21] for example. In the present case the effect of applying a Newtonian-heating boundary condition is to introduce a dimensionless parameter γ (defined below) which gives a measure of the applied heating. Hence the solution depends on this parameter as well as the Prandtl number σ . The starting point for our discussion is a consideration of the solution near the leading stagnation point. The flow problem is classical [1], [2, pp. 231–233] and is here unaffected by the heat transfer from the boundary. An examination of the corresponding equations for the heat transfer reveals that there is a critical value γc of the parameter γ , dependent of the Prandtl number σ , so that physically acceptable solutions are possible only for γ < γc , with, for example, the surface temperature becoming unbounded when γ = γc . We start by describing the equations for our model after which we treat the solution near the stagnation point in detail. Guided by the nature of this solution we then obtain numerical solutions to the full problem for a range of σ and values of γ < γc .

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2 Model The boundary-layer equations for the steady flow and heat transfer on a circular cylinder are in dimensional form [1], [2, p. 202] ∂u ∂v + = 0, (1) ∂x ∂y ∂u dU ∂ 2u ∂u (2) +v =U +ν 2, u ∂x ∂y dx ∂y ∂T ∂T ν ∂2T , (3) u +v = ∂x ∂y σ ∂ y2 where x measures distance round the cylinder and y normal to it; u and v are the velocity components in the x- and y-directions, respectively, and T is the fluid temperature. Here ν is the kinematic viscosity with σ being the Prandtl number. The boundary conditions are ∂T = −h s T on y = 0, u → U (x), T → T∞ as y → ∞, (4) u = v = 0, ∂y where T∞ is the (constant) ambient temperature, h s is a heat-transfer parameter and U (x) = U0 sin(x/a) is the free stream corresponding to a uniform flow 21 U0 far from the cylinder with a being the radius of the circular cylinder. An assumption basic to our model is that the differences in temperature arising within the flow and the ambient temperature T∞ are sufficiently small for the effects of buoyancy forces to be ignored. Taking constant fluid properties is also consistent with this assumption. To make the equations of (1)–(4) dimensionless, we make the scalings x −1/2 1/2 y , T − T∞ = T∞ θ , v, x = , y = Re (5) u = U0 u, v = U0 Re a a where Re =

U0 a ν

is the Reynolds number. This results in the equations (on dropping bars for convenience)

∂u ∂v + = 0, ∂x ∂y ∂u dU ∂ 2u ∂u +v =U + 2, u ∂x ∂y dx ∂y ∂θ 1 ∂ 2θ ∂θ +v = . u ∂x ∂y σ ∂ y2 The boundary conditions become ∂θ = −γ (θ + 1) on y = 0, u → U (x), θ → 0 u = v = 0, ∂y

(6) (7) (8)

as y → ∞.

(9)

The free stream is now U (x) = sin x and the parameter γ = ah s Re−1/2 represents the strength of the Newtonian heating. To solve (6)–(9) numerically we make the transformation ψ(x, y) = sin x f (x, y)

(10)

and leave θ (x, y) unaltered. Here ψ is the stream function (u = ψ y , v = −ψx ). This results in the equations    2   ∂ f ∂2 f ∂2 f ∂f ∂3 f ∂ f ∂2 f , (11) − + cos x 1 + f − = sin x ∂ y3 ∂ y2 ∂y ∂ y ∂ x∂ y ∂ x ∂ y2   1 ∂ 2θ ∂θ ∂ f ∂θ ∂ f ∂θ , (12) + cos x f = sin x − σ ∂ y2 ∂y ∂y ∂x ∂x ∂y subject to the boundary conditions

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∂θ = −γ (θ + 1) ∂y

∂f = 0, ∂y

on y = 0,

∂f → 1, θ → 0 ∂y

as y → ∞.

(13)

Before considering the general solution to (11)–(13) we need to first consider the flow at the stagnation point x = 0.

3 Stagnation-point flow To discuss the flow near the stagnation point at x = 0 we assume that f = f (y), θ = θ (y). This results in the ordinary differential equations f  + f f  + 1 − f  2 = 0 with f (0) = f  (0) = 0,

f → 1

as y → ∞,

(14)

where primes denote differentiation with respect to y. Equation 14 is a standard boundary-layer problem [1], [2, pp. 231–233] and has f  (0) = 1.23259 f ∼ y − δ as y → ∞, δ = 0.6479.

(15)

The problem for the heat transfer is 1  θ + f θ  = 0 with θ  = −γ (θ + 1) on y = 0, θ → 0 as y → ∞. (16) σ From problem (16) we can calculate θ0 ≡ θ (0), where θ0 = θ0 (σ, γ ), using standard numerical techniques. However, we can formally express the solution to (16) in terms of the solution f (y) to the stagnation-point flow (14). To do so, we introduce the function y q(y) =

f (s) ds.

(17)

0

From (14), q(y) is defined for all y ≥ 0 with q ∼ for y large. Then ∞ θ (y) = B0

e

−σ q(s)

∞ ds, with θ (0) = B0

y

a0 3 6 y +· · · for

y small (where a0 = f  (0)) and q ∼ 21 y 2 −δy +· · ·

e−σ q(y) dy

(18)

0

for some constant B0 . The constant B0 is determined from the condition on y = 0, which gives γ B0 = γ (B0 I0 + 1) or B0 = where I0 = I0 (σ ) = 1 − γ I0

∞

e−σ q(y) dy,

(19)

0

with (18) and (19) then giving θ0 =

γ I0 (σ ) . 1 − γ I0 (σ )

(20)

We note that (19) gives θ ≡ 0 when γ = 0, corresponding to having h s = 0 and hence no heating, and that, from(18), θ  (y) and hence θ (y) is of one sign on 0 ≤ y < ∞. Expressions (19) and (20) show that there is a critical value γc of γ , given by γc (σ ) = I0 (σ )−1 , where the solution becomes unbounded. Now, from boundary condition (4), we must have





∂T ∂ y y=0 <

(21) 0, as the

applied heating condition is given in terms of the actual fluid temperature T not a temperature difference. Hence we can only have physically acceptable solutions to (16) which have θ  (0) < 0. From (18) and (19) this means

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Fig. 1 Plots of θ0 for the stagnation-point flow (14) and (16) plotted against a σ for γ = 0.5, asymptotic expression (25) for σ large is shown by the broken line, and b γ for σ = 1. The positions where the solution becomes unbounded at (21) are shown 1.6

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Fig. 2 Temperature profiles at the stagnation point shown as plots of θ against y for γ = 0.2 and for the values of σ labelled on the figure

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Fig. 3 A plot of γc , given by (21), against σ , the region where θ0 > 0 and solutions for the stagnation-point flow (14) and (16) exist is indicated. The asymptotic expressions (24) for small and large σ are shown by broken lines

that we can have solutions only when γ < γc and this puts a bound (dependent on the Prandtl number σ ) for the existence of forced-convection solutions with the Newtonian heating given in (4). We illustrate this in Fig. 1 where we plot θ0 against σ for γ = 0.5 (Fig. 1a), obtained from the numerical solution of (14), (16), and against γ for σ = 1 (Fig. 1b). Figure 1a shows that θ0 becomes large as σ approaches the value σ = 0.7148 which satisfies (21) when γ = 0.5. For values of σ less than this, our numerical integrations gave θ0 < 0, in line with (20). Figure 1b also shows that θ0 becomes large as γ approaches γc (for σ = 1, γc = 0.5705), starting from small values when γ is small. Again our numerical integrations gave θ0 < 0 for values of γ > γc . In Fig. 2 we show representative temperature profiles, here for γ = 0.2 and for the values of σ labelled on the figure. This figure shows that both the temperatures within the profile and their spread increase as σ is decreased, consistent with the results shown in Fig. 1a. We note that, for this value of γ , the critical value σc of σ from (21) is σc 0.028.

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We plot γc , obtained by solving the stagnation-point flow (14), calculating q(y) defined in (17) and the integral I0 defined in (19), against σ in Fig. 3. The region where θ0 > 0 and physically acceptable solutions to (16) exist is indicated on the figure. The figure shows the γc increases monotonically with σ becoming large as σ increases. For σ large, the integral I0 is dominated by the behaviour of q(y) for y small, giving  1/3   1 6 ! σ −1/3 + · · · for σ 1. I0 (σ ) ∼ (22) a0 3 For σ small, q ∼ 21 y 2 in the integral I0 , giving π −1/2 + · · · for σ  1. (23) I0 (σ ) ∼ σ 2 Hence  a 1/3 1 2 1/2 0 1/3 σ γc ∼ (σ 1), γc ∼ (σ  1). (24)

1 σ 6 π 3 ! Asymptotic expressions (24) are shown in Fig. 3 by broken lines. From (22),  1/3   1 6 θ0 ∼ γ ! σ −1/3 + · · · for σ large. (25) a0 3 Expression (25) is shown in Fig. 1a by a broken line with the numerical solution approaching this value only slowly as σ is increased, as might be expected from the O(σ −1/3 ) decrease in θ0 . We now describe numerical solutions to (11)–(13) for different values of γ and σ guided by the results from the stagnation-point flow. 4 Numerical results The problem given by (11)–(13) was solved numerically using a scheme based on the Crank–Nicolson method and described in [14]. The flow problem is unaffected by the heat transfer and its numerical solution terminates in a Goldstein–Stewartson singularity [6, 7] at the separation point x = xs , where we estimated xs 1.838, a value slightly higher than the value given originally by Terrill [5]. As a consequence, our numerical solutions for the temperature θ are restricted to 0 ≤ x < xs . In Fig. 4 we plot θ0 against x for σ = 1.0 and a range of values of γ . Here θ0 ≡ θ (x, 0) and, as noted above, for this value of σ, γc = 0.5705. These results show that θ0 increases as x increases, with this increase becoming more marked as γ increases. For γ = 0.4 and more so for γ = 0.45 there is a large increase in θ0 as the separation point at x = xs (shown in the figure by a vertical axis) is approached. For even higher values of γ , though still less than γc (not shown in the figure), even higher increases in θ0 are seen in the numerical solution as x gets close to xs . In Fig. 5 we plot the temperature profiles as the solution develops from the stagnation point. Here we took γ = 0.4, σ = 1.0 and plotted the profiles at x = 0, 0.5, 1.0, 1.5 and 1.82. The temperatures within the boundary layer increase as x increases, becoming much larger as the separation point is approached, consistent with Fig. 4. The temperature profiles spread further from the wall as x increases, again becoming large near separation, consequent on the boundary thickness increasing rapidly as separation is approached. In Fig. 6 we plot θ0 against x for γ = 0.25 and a range of values of the Prandtl number σ . This value of γ gives a critical value σc for the stagnation-point flow of σc = 0.0811. Again we see that θ0 increases with x and that there is a more noticeable increase in θ0 as the separation point at xs is approached, with this being greater as σ decreases towards its critical value. 5 Discussion and conclusions We have considered the forced-convection heat transfer from a circular cylinder in a uniform stream when there is Newtonian heat transfer from the cylinder. This introduces a dimensionless parameter γ giving a measure of the

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Fig. 4 Plots of the wall temperature θ0 against x for σ = 1.0 and for the values of γ labelled on the figure obtained from the numerical solution of (11)–(13)

Fig. 5 Temperature profiles as the flow develops from the stagnation-point shown by plots of θ against y for σ = 1.0, γ = 0.4 at x = 0.0, 0.5, 1.0, 1.5, 1.82

Fig. 6 Plots of the wall temperature θ0 against x for γ = 0.2 and for the values of σ labelled on the figure obtained from the numerical solution of (11)–(13)

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surface heating. We started by considering the solution near the forward stagnation point, where the flow problem is standard; see [1, 2] for example. The solution for the dimensionless temperature can be expressed in terms of this solution; see expression (18) with this solution showing that there is a critical value γc = γc (σ ) of γ requiring that γ < γc to obtain the (physically acceptable) solutions which have a dimensionless surface temperature θ0 > 0; see expression (21) and Fig. 3. The boundary condition (4) is given in terms of the absolute temperature T and of necessity this must be positive. However, we can still obtain numerical solutions to (16) when γ > γc without encountering any difficulties, though these solutions have θ0 < 0. The surface temperature becomes unbounded at γ = γc (see Fig. 1a, b). For values of γ close to γc the fluid temperatures become large; see Fig. 1. In this situation the basic assumption underlying our model of small temperature differences breaks down with other effects, e.g. buoyancy-driven flows and variable fluid properties, become much more significant. This result restricts our numerical solution to the full boundary-layer problem (11)–(13) to values of γ < γc . Also the solution to the flow problem encounters a Goldstein–Stewartson singularity [6, 7] at x = xs (xs 1.84) and this further restricts our attention to values of x in 0 ≤ x < xs . Our numerical results show that the wall temperature θ0 (x) increases with x for all the parameter values tried. The values of θ0 increase as γ is increased

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(for a fixed value of the Prandtl number σ ); see Fig. 4, and as σ is decreased (for a fixed value of γ ); see Fig. 6. There is also a steeper increase in θ0 as the separation point xs is approached, more noticeable for the larger values of γ and smaller values of σ . For small values of γ the scaling θ = γ θ is suggested by (13). This still results in Eq. 12 though now subject to the boundary condition at leading order that ∂θ ∂ y = −1 on y = 0. For the stagnation-point solution this gives θ0 ∼ γ I0 (σ ) for γ  1. When σ is large, the heat transfer is confined to a narrow region next to the body indicated in Fig. 2). There  (as 2 = A0 (x). In the inner is an outer region in which y is unscaled and θ = 0, with the flow solution giving ∂∂ y 2f y=0

region Y = σ 1/3 y, θ = σ −1/3 h and f ∼

1 A0 (x)σ −2/3 Y 2 + · · · 2

(26)

with (12) then giving, at leading order, Y2 d ∂h ∂h ∂ 2h (A0 (x) sin x) − sin x A0 (x) Y =0 (27) + 2 ∂Y 2 dx ∂Y ∂x subject to, still at leading order, ∂h = −1 on Y = 0, h → 0 as Y → ∞. (28) ∂Y For x small (26)–(28) gives the result (25) for θ0 on noting that A0 (x) = a0 + O(x 2 ). The scaling for θ in (26) indicates that the wall temperature θ0 is small, of O(σ −1/3 ) for σ large, accounting for the low values seen in Fig. 6 for σ = 4.0. Since A0 (x) is of O((xs − x)1/2 ) close to the separation point at xs [6, 7], a singularity in the solution to (27) should be expected as x → xs . When σ is small, expression (24) suggests assuming that γ = γ0 σ 1/2 with γ0 of O(1). In this case the inner region has y of O(1) and is effectively the boundary-layer flow with θ = C0 − γ0 (C0 + 1) yσ 1/2 + · · ·

(29)

for some C0 (x). In the outer region ζ = σ 1/2 y,

f ∼ ζ σ −1/2 + · · ·

(30)

with the leading-order problem becoming ∂ 2θ ∂θ ∂θ − sin x = 0, + cos x ζ ∂ζ 2 ∂ζ ∂x θ ∼ C0 − γ0 (C0 + 1)ζ + · · · as ζ → 0, θ → 0 as ζ → ∞. For x small we have ∞ γ0 2 θ (ζ ) = e−s /2 ds, 2 π − γ0 ζ

(31) (32)

(33)

giving the critical value in (24) for σ small. Expression (29) and (31) suggest that θ will remain of O(1) for σ small, provided that γ ∼ σ 1/2 and that the temperature is, at least to leading order, unaffected by the boundary-layer flow and is determined entirely by the outer flow. The Goldstein–Stewartson singularity at separation [6, 7] has an inner region where the singularity develops in which 1/4 1/4 y , (34) ψ = ξ 3/4 b0 f (ξ, η), η = b0 ξ 1/4 where b0 = sin xs cos xs , ξ = xs − x with a solution being sought for ξ small. The function f (ξ, η) is expanded in powers of ξ , the first two terms being f (ξ, η) =

123

η2 1/4 η3 + b1 ξ + ···, 6 2

(35)

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so that the skin friction τw ∼ (b0 b1 )(xs − x)1/2 + · · · as x → xs . If we apply transformation (34) in (8) for the temperature and then look for a solution as a series expansion in powers of ξ , we find that θ ∼ D0 − γ (D0 + 1)η ξ 1/4 − b1 γ (D0 + 1) ξ 1/2 + · · · for ξ small,

(36)

where D0 is a constant that depends on both γ and σ and is indeterminate from this expansion reflecting the asymptotic nature of the solution [22]. Expression (36) shows that θ0 ∼ D0 − b1 γ (D0 + 1) (xs − x)1/2 + · · ·

(37)

as the separation point at x = xs is approached. This singular nature of θ0 near separation can be seen in the numerical results shown in Figs. 4 and 6. Finally we note that, if we apply the heat-flux boundary condition in (4) in terms of a temperature difference, i.e., we apply instead ∂∂Ty = −h s (T − T∞ ) on y = 0 with the corresponding dimensionless version ∂θ ∂ y = −γ θ , we still have expression (18) for the stagnation-point solution. However, the constant B0 occurring in expression (18) is now given by B0 = γ I0 (σ )B0 . This means that, in this case, B0 is arbitrary and a nontrivial solution is possible only if γ = γc = I0 (σ )−1 . In this case θ cannot be determined uniquely only to within an arbitrary multiple as is also the case for the full problem (12). Acknowledgments This work was supported by the university research grant UKM-GUP-BTT-07-25-174 from Universiti Kebangsaan Malaysia. RN wishes to thank the financial support given by the Ministry of Science, Technology and Innovation (Malaysia) and the Academy of Sciences (Malaysia) under the Brian Gain (BGM) Programme.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Schlichting H, Gersten K (2000) Boundary layer theory, 8th edn. Springer, Berlin Rosenhead L (ed) (1963) Laminar boundary layers. Clarendon Press, Oxford Blasius H (1908) Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z Math Phys 56:1–37 Hiemenz K (1911) Die Grenzschicht an einem in den gleichfömigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers J 326:321–324 Terrill RM (1960) Laminar boundary-layer flow near separation with and without suction. Philos Trans Roy Soc Lond A 253:55–100 Goldstein S (1948) On laminar boundary-layer flow near a position of separation. Q J Mech Appl Math 1:43–69 Stewartson K (1958) On the Goldstein theory of laminar separation. Q J Mech Appl Math 11:399–410 Frössling N (1940) Verdungstung Wärmeübergang und Geschwindigkeitsverteilung bei zweidimensionaler und rotations symmetrischer laminarer Grenzschichtströmung. Lunds Univ Arsskr NF Avd 2 36(4) Koh JCY (1964) Laminar free convection from a horizontal cylinder. In: Paper 76-HT-16 ASME-AIChE heat transfer conference, vol 7, pp 811–813 Saville DA, Churchill SW (1967) Laminar free convection in boundary layers near horizontal cylinders and vertical axisymmetric bodies. J Fluid Mech 29:391–399 Elliott L (1970) Free convection on a two-dimensional or axisymmetric body. Q J Mech Appl Math 23:153–162 Wilks G (1972) External natural convection about two-dimensional bodies with constant heat flux. Int J Heat Mass Transf 15:351– 354 Merkin JH, Pop I (1988) A note on the free convection boundary layer on a horizontal circular cylinder with constant heat flux. Wärme-und-Stoffübertragung 22:79–81 Merkin JH (1977) Mixed convection from a horizontal circular cylinder. Int J Heat Mass Transf 20:73–77 Pop I, Sunada JK, Cheng P, Minkowycz WJ (1985) Conjugate free convection from long vertical plate fins embedded in a porous medium. Int J Heat Mass Transf 28:1629–1636 Pozzi A, Lupo M (1988) The coupling of conduction with laminar natural convection along a flat plate. Int J Heat Mass Transf 31:1807–1814 Chaudhary RC, Jain P (2007) An exact solution to the unsteady free convection boundary-layer flow past an impulsively started vertical surface with Newtonian heating. J Eng Phys Thermophys 80:954–960 Merkin JH (1994) Natural convection boundary-layer flow on a vertical surface with Newtonian heating. Int J Heat Fluid Flow 15:392–398 Salleh MZ, Nazar R, Pop I (2009) Forced convection boundary layer flow at a forward stagnation point with Newtonian heating. Chem Eng Commun 196:987–996 Lesnic D, Ingham DB, Pop I (1999) Free convection boundary layer flow along a vertical surface in a porous medium with Newtonian heating. Int J Heat Mass Transf 42:2621–2627

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21. Lesnic D, Ingham DB, Pop I, Storr C (2004) Free convection boundary-layer flow above a nearly horizontal surface in a porous medium with Newtonian heating. Heat Mass Transf 40:665–672 22. Stewartson K (1955) On asymptotic expansions in the theory of boundary layers. J Math Phys 13:113–122

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