- Email: [email protected]

The effects of transpiration on the boundary layer ﬂow and heat transfer over a vertical slender cylinder Anuar Ishak a , Roslinda Nazar a,∗ , Ioan Pop b a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia b Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

Received 14 March 2006; received in revised form 18 May 2007; accepted 30 May 2007

Abstract This paper deals with a theoretical (numerical) analysis of the effects that blowing/injection and suction have on the steady mixed convection or combined forced and free convection boundary layer ﬂows over a vertical slender cylinder with a mainstream velocity and a wall surface temperature proportional to the axial distance along the surface of the cylinder. Both cases of buoyancy forces aid and oppose the development of the boundary layer are considered. Similarity equations are derived and their solutions are dependent upon the mixed convection parameter, the non-dimensional transpiration parameter and the curvature parameter, as well as of the Prandtl number. Dual solutions for the previously studied mixed convection boundary layer ﬂows over an impermeable surface of the cylinder are shown to exist also in the present problem for aiding and opposing ﬂow situations. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Combined forced and free convection ﬂows; Vertical slender cylinder; Boundary layer; Suction/injection; Numerical similarity solutions

1. Introduction Mixed convection ﬂows, or combined forced and free convection ﬂows arise in many transport processes in natural and engineering devices, such as, for example, atmospheric boundary layer ﬂow, heat exchangers, solar collectors, nuclear reactors, electronic equipments, etc. Such a process occurs when the effect of the buoyancy force in forced convection or the effect of forced ﬂow in free convection becomes signiﬁcant. The effect is especially pronounced in situations where the forcedﬂow velocity is low and the temperature difference is large (see [1]). Over the last several decades several analyses of mixed convection ﬂow of a viscous and incompressible ﬂuid over a vertical ﬂat plate have been performed. Analytical and numerical solutions for the temperature and the velocity ﬁelds have been obtained both for prescribed wall temperatures and for prescribed wall heat ﬂuxes. However, the problem of mixed convection in axisymmetric boundary layer ﬂow has received little attention so far. Mahmood and Merkin [2] have considered the ∗ Corresponding author. Tel.: +60 3 89213371; fax: +60 3 89254519.

E-mail address: [email protected] (R. Nazar). 0020-7462/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2007.05.004

similarity equations for the axisymmetric mixed convection boundary layer ﬂow along a vertical cylinder in the case of opposing ﬂow that is when the buoyancy force and the ﬂow are in opposite directions. Dual solutions were reported only for this opposing ﬂow. Ridha [3] has later shown that dual solutions exist also for the aiding ﬂow regime, that is when the buoyancy force acts in the same direction as the ﬂow. The interest in similarity solutions stems from the fact that they provide intermediate asymptotic solutions related to the more complex non-similar ones (see [4]). However, to our best knowledge, the similarity solutions for the mixed convection boundary layer ﬂow along a vertical permeable cylinder have not been studied before. It is worth mentioning that the similarity equations for the free convection boundary layer ﬂow on a vertical permeable (with blowing and suction) ﬂat plate with prescribed wall heat ﬂux and prescribed wall temperature proportional to x n , where x is the usual distance measured along the plate and n is a positive constant, have been considered by Chaudhary and Merkin [5], and Merkin [6]. The differences between blowing and suction were found. Such solutions for a vertical permeable surface embedded in a ﬂuid-saturated porous medium have also been studied by Chaudhary et al. [7,8].

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017

The fundamental governing equations for ﬂuid mechanics are the Navier–Stokes equations. This inherently non-linear set of partial differential equations has no general solution, and only a small number of exact solutions have been found (see [9]). Exact solutions are important for the following reasons: (i) the solutions represent fundamental ﬂuid-dynamic ﬂows. Also, owing to the uniform validity of exact solutions, the basic phenomena described by the Navier–Stokes equations can be more closely studied. (ii) The exact solutions serve as standards for checking the accuracies of the many approximate methods, whether they are numerical, asymptotic, or empirical. Explicit solutions are used as models for physical or numerical experiments, and often reﬂect the asymptotic behavior of more complicated solutions. All explicit solutions for the boundary layer equations are seemingly similarity solutions in the sense that the longitudinal velocity component displays the same shape of proﬁle across any transverse section of the layer, see Schlichting [10]. By an appropriate choice of the independent non-dimensional similarity variables, the boundary layer equations can therefore be reduced to ordinary differential equations. In the rare cases when these equations can be solved in closed form, the explicit solutions are obtained (see [11]). The aim of this paper is to study the effects of transpiration (suction or injection) on the mixed convection or combined forced and free convection ﬂows in boundary layer ﬂow along a vertical permeable cylinder. 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 [5]). We mention to this end that such a study has also been done by Massoudi [12], Weidman et al. [13,14] and Ishak et al. [15,16] for the classical problems of the boundary layers over a permeable wedge, moving ﬂat plates and permeable vertical ﬂat plates.

Consider the convective ﬂow and heat transfer along a vertical permeable slender cylinder of radius a placed in a viscous and incompressible ﬂuid of uniform ambient temperature T∞ and constant density ∞ . The equations of motion for such a ﬂuid, see Gebhart et al. [17] or Rajagopal et al. [18], are

(v · ∇)v = −

(1) 1 − ∞ ∇p + ∇ 2 v + g, ∞ ∞

(v · ∇)T = ∇ 2 T ,

approximation, see Gebhart et al. [17] or Pop and Ingham [19], = ∞ [1 − (T − T∞ )],

(4)

where is the thermal expansion coefﬁcient. Further, we shall write Eqs. (1)–(3) in cylindrical coordinates (x, r) with the x-axis is measured along the surface of the cylinder in the vertical direction and the r-axis is measured in the radial direction. We will then apply the following boundary layer approximations: x¯ = x/a,

r¯ = Re1/2 (r/a),

u¯ = u/U∞ ,

w¯ = Re1/2 (w/U∞ ), T¯ = (T − T∞ )/T , 2 p¯ = (p − p∞ )/U∞ ,

(5)

where u and w are the velocity components along the x and y axes, respectively, U∞ is the characteristic velocity, T is the characteristic temperature and Re = U∞ a/ is the Reynolds number with being the kinematic viscosity. We will consider that the ﬂow is symmetric relative to the transversal coordinate. Substituting variables (5) into Eqs. (1)–(3), using the boundary layer approximation that Re → ∞ and returning back to the dimensional (without bar) physical variables, we obtain the following boundary layer equations for the problem under consideration, see Mahmood and Merkin [2], j j (ru) + (rw) = 0, jx jr

(6)

2 ju ju dU j u 1 ju u +w + =U + + g(T −T∞ ), jx jr dx jr 2 r jr 2 jT 1 jT jT j T u + +w = , jx jr jr 2 r jr

(7)

(8)

where U (x) is the mainstream velocity. We assume that the appropriate boundary conditions are

2. Governing equations

∇ · v = 0,

1011

(2) (3)

where v is the velocity vector, p is the pressure, T is the temperature of the ﬂuid, g is the gravitation acceleration vector, is the kinematic viscosity, is the density, is the thermal diffusivity and ∇ 2 is the Laplacian operator. We assume that the density difference ( − ∞ ) in the buoyancy term of the momentum equation is given by the Oberbeck–Boussinesq

u = 0,

w = V,

u → U (x),

T = Tw (x)

T → T∞

at r = a,

as r → ∞,

(9)

where V is the constant velocity of injection (V >) or suction (V < 0). Further, we assume that the mainstream velocity U (x) and the temperature of the cylinder surface Tw (x) have the form x x , Tw (x) = T∞ + T , (10) U (x) = U∞ where is a characteristic length, U∞ is the characteristic velocity and T is the characteristic temperature with T > 0 for a heated surface and T < 0 for a cooled surface. 3. The solution It is worth mentioning that the partial differential equations (6)–(8) can be solved numerically using a ﬁnite-difference method or any other numerical methods, but in this paper we will solve these equations using a similarity transformation,

1012

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017

i.e. by reducing these equations to ordinary differential equations. Thus, following Mahmood and Merkin [2], we introduce the similarity variables r 2 − a 2 U 1/2 , = (U x)1/2 af (), = 2a x T − T∞ () = , Tw − T ∞

(11)

where is the stream function deﬁned as u = r −1 j/jr and w = −r −1 j/jx, which identically satisfy Eq. (6). By using this deﬁnition, we obtain a U∞ 1/2 f (), (12) u = Uf (), w = − r where primes denote differentiation with respect to . In order that similarity solutions exist, V has to be of the form a U∞ 1/2 f0 , (13) V =− r where f0 = f (0) and f0 < 0 is for mass injection and f0 > 0 is for mass suction. Substituting (11) into Eqs. (7) and (8), we get the following ordinary differential equations: (1 + 2 )f + 2 f + ff + 1 − f + = 0,

(14)

(1 + 2 ) + 2 + P r(f − f ) = 0,

(15)

2

subject to the boundary conditions (9) which become f (0) = f0 ,

f (0) = 0,

f → 1, → 0

(0) = 1,

as → ∞,

(16)

where is the curvature parameter and is the buoyancy or mixed convection parameter deﬁned as 1/2 gT , = , (17) = 2 U∞ a 2 U∞ respectively. In Eq. (17), > 0 and < 0 correspond to the aiding ﬂow (heated cylinder) and to the opposing ﬂow (cooled cylinder), respectively, while = 0 represents the pure forced convection ﬂow (buoyancy force is absent). It is worth mentioning that, the similarity solution of Eqs. (14) and (15) is not necessarily the only solution to the problem as the governing equations are non-linear. We notice that when = 0 (i.e. a → ∞), the problem under consideration reduces to the ﬂat plate case considered by Ishak et al. [16], while when f0 = 0 it reduces to the impermeable cylinder considered by Mahmood and Merkin [2]. Furthermore, when both and f0 are zero, the present problem reduces to the problem considered by Ramachandran et al. [20] for the case of an arbitrary surface temperature with n = 1 in their paper. The physical quantities of interest are the skin friction coefﬁcient Cf and the local Nusselt number N ux , which are deﬁned by w xq w Cf = , Nux = , (18) 2 U /2 k(Tw − T∞ )

where the skin friction w and the heat transfer from the plate qw are given by ju jT , qw = −k , (19) w = jr r=a jr r=a with and k being the dynamic viscosity and thermal conductivity, respectively. Using the similarity variables (11), we get 1 1/2 Cf Rex = f (0), 2

1/2

N ux /Rex

= − (0),

(20)

where Rex = U x/ is the local Reynolds number. 4. Results and discussion Eqs. (14) and (15) subject to the boundary conditions (16) have been solved numerically for some values of the governing parameters , f0 and using a very efﬁcient ﬁnite-difference scheme known as the Keller-box method, which is described in the book by Cebeci and Bradshaw [21]. The solution is obtained in the following four steps: • Reduce Eqs. (14) and (15) to a ﬁrst-order system. • Write the difference equations using central differences. • Linearize the resulting algebraic equations by Newton’s method, and write them in matrix–vector form. • Solve the linear system by the block-tridiagonal-elimination technique. To conserve space, the details of the solution procedure are not presented here. Following Mahmood and Merkin [2], we considered only Prandtl number unity throughout the paper, except for comparison with previously reported cases, by Ramachandran et al. [20], Hassanien and Gorla [22] and Lok et al. [23]. This comparison is shown in Tables 1 and 2, and is found to be in a very good agreement. The variations of the skin friction coefﬁcient f (0) with

together with their velocity proﬁles are shown in Figs. 1–7 for = 1, f0 = 0.5 and f0 = −0.5, respectively, while the respective local Nusselt number (0) together with their temperature proﬁles are shown in Figs. 8–14, to support the validity of the numerical results obtained. It is worth mentioning that all the velocity and temperature proﬁles satisfy the boundary conditions (16). The results for the skin friction coefﬁcient f (0) and the local Nusselt number (0) as a function of show that it is possible to get dual solutions of the similarity equations (14) and (15) subject to the boundary conditions (16) for the aiding ﬂow ( > 0) as well, besides that usually reported in the literature for the opposing ﬂow ( < 0). Also for > 0, there is a favorable pressure gradient due to the buoyancy effects, which results in the ﬂow being accelerated in a larger skin friction coefﬁcient than in the non-buoyant case ( = 0). For negative values of , dual solutions ( c < < 0), unique solution ( = c ) or no solution ( < c ) is obtained, where c is the critical value of for which the solution exists. At = c , both solution branches are connected, thus a unique solution is obtained. For the aiding ﬂow, dual solutions exist for all values of

considered in this study, whereas for the opposing ﬂow, the

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017

1013

Table 1 Values of f (0) for different values of Pr when = 1, f0 = 0 and = 0 (ﬂat plate) Pr

Ramachandran et al. [20]

0.7 1 7 10 20 40 50 60 80 100

Hassanien and Gorla [22]

1.7063 – 1.5179 – 1.4485 1.4101 – 1.3903 1.3774 1.3680

Lok et al. [23]

1.70632 – – 1.49284 – – 1.40686 – – 1.38471

Present results

1.7064 – 1.5180 – 1.4486 1.4102 – 1.3903 1.3773 1.3677

Upper branch

Lower branch

1.7063 1.6754 1.5179 1.4928 1.4485 1.4101 1.3989 1.3903 1.3774 1.3680

1.2387 1.1332 0.5824 0.4958 0.3436 0.2111 0.1720 0.1413 0.0947 0.0601

Table 2 Values of − (0) for different values of Pr when = 1, f0 = 0 and = 0 (ﬂat plate) Pr

Ramachandran et al. [20]

0.7 1 7 10 20 40 50 60 80 100

Hassanien and Gorla [22]

0.7641 – 1.7224 – 2.4576 3.1011 – 3.5514 3.9095 4.2116

Lok et al. [23]

0.76406 – – 1.94461 – – 3.34882 – – 4.23372

0.7641 – 1.7226 – 2.4577 3.1023 – 3.5560 3.9195 4.2289

Upper branch

Lower branch

0.7641 0.8708 1.7224 1.9446 2.4576 3.1011 3.3415 3.5514 3.9095 4.2116

1.0226 1.1691 2.2192 2.4940 3.1646 4.1080 4.4976 4.8572 5.5166 6.1230

1

4

Pr = 1 γ=1

3

0.8

f0 = -0.5, 0, 0.5

f = − 0.5, 0, 0.5 0

0.6

2

0.4 f ′(η)

f" (0)

Present results

1

0.2

Pr = 1

0

0

γ=1 λ = −2

− 0.2

-1 -2

upper branch lower branch

−0.4

-5

-4

-3

-2

-1

0

1

2

3

4

λ Fig. 1. Skin friction coefﬁcient f (0) as a function of for various values of f0 when Pr = 1 and = 1.

solutions exist up to certain values of , i.e. c . Beyond these critical values, the boundary layer separates from the surface, thus no solution is obtained using the boundary layer approximations. Moreover, from Figs. 1 and 8, we found that the values of | | for which the solution exists increase as f0 increases.

− 0.6

0

2

4

6

8

10

12

14

16

18

η Fig. 2. Velocity proﬁles f () for various values of f0 when Pr = 1, = 1 and = −2.

Hence, suction delays the boundary layer separation. It is noticed that, for the opposing ﬂow ( < 0), the lower branch solutions in Figs. 1–7 correspond to the upper branch solutions in Figs. 8–14, and vice versa. Numerical results for the local Nusselt number as presented in Figs. 8, 11 and 14 show that (0) approaches +∞ as → 0− , and −∞ as → 0+ .

1014

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017

1.2

1.2

1

1

0.8

0.8

f0 = -0.5, 0, 0.5

0.6

0.6 Pr = 1 γ=1 λ=1

0.2 f0 = -0.5, 0, 0.5

0 -0.2

f' (η)

f' (η)

0.4

5

10

15

20

25

30

35

Pr = 1 f0 = 0.5 λ=2

0.2

-0.2

upper branch γ = 0, 1, 2

-0.4

lower branch 0

0.4

0

upper branch

-0.4 -0.6

γ = 0, 1, 2

-0.6

40

0

10

20

30

40

50

60

70

η

η Fig. 3. Velocity proﬁles f () for various values of f0 when Pr = 1, = 1 and = 1.

Fig. 6. Velocity proﬁles f () for various values of when Pr = 1, f0 = 0.5 and = 2.

4

5

Pr = 1 f0 = 0.5

4

Pr = 1 f0 = -0.5

3

3

γ = 0, 1, 2

γ = 0, 1, 2

2 f" (0)

2 f" (0)

lower branch

1

1 0

0 -1

-1

-2 -3

-2 -6

-4

-2

0

2

4

λ Fig. 4. Skin friction coefﬁcient f (0) as a function of for various values of when Pr = 1 and f0 = 0.5.

1 0.8

γ = 0, 1, 2

0.6

′

f (η)

0.4 0.2

Pr = 1 f0 = 0.5

0

λ = −2

γ = 0, 1, 2

− 0.2

upper branch lower branch

− 0.4 − 0.6

0

5

10

15

η

20

25

30

Fig. 5. Velocity proﬁles f () for various values of when Pr = 1, f0 = 0.5 and = −2.

-4

-3

-2

-1

0

1

2

3

4

λ Fig. 7. Skin friction coefﬁcient f (0) as a function of for various values of when Pr = 1 and f0 = −0.5.

In Figs. 1, 4 and 7, following the upper branch solution for a particular value of f0 or , one may expect that the solution suddenly disappears at the separation point = c , but this is not the case. The solution makes an U −turn at this point and form the lower branch solution. It is worth mentioning that the separation occurs here at a point where f (0) = 0, that is a different feature from the classical boundary layer theory, where the separation takes place at f (0) = 0. Wilks and Bramley [24] stopped the lower branch solutions when the wall heat transfer goes to zero. Although physically it is a realistic thing to do, it was shown in [2] that the lower branch solutions could be continued further to the point where the buoyancy parameter goes to zero and terminated at this point. It seems that Ridha [3,4] was the ﬁrst to show the existence of dual (non-uniqueness) solutions for both aiding and opposing ﬂow situations. In the present paper, we show that the lower branch solutions exist in the opposing ﬂow regime ( < 0) and they continue to the aiding ﬂow regime ( > 0), which is in

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017

6

2

Pr = 1 γ=1

4

1

θ' (0)

θ' (0)

2 0

γ = 0, 1, 2

0 γ=0

-1

-2 f0 = -0.5, 0, 0.5

-4 6

Pr = 1 f0 = 0.5

1015

γ=1 γ=2

-2 f0 = -0.5, 0, 0.5

-5

-4

-3

-2

-1

0

1

2

3

-3

4

γ = 0, 1, 2 -6

-4

-2

0

2

λ Fig. 8. Variation of (0) with for various values of f0 when Pr = 1 and = 1.

Fig. 11. Variation of (0) with for various values of when Pr = 1 and f0 = 0.5.

Pr = 1 f = 0.5

Pr = 1

1

γ=1

1

0

λ = −2

λ = −2 upper branch

lower branch

lower branch

0.6

upper branch

0.8

θ (η)

θ (η)

0.8

0.6

γ = 0, 1, 2

0.4

0.4 f = − 0.5, 0, 0.5

0.2 0

4

λ

0.2

0

0

5

10

15

η

Fig. 9. Temperature proﬁles () for various values of f0 when Pr = 1, = 1 and = −2.

1 0.5

0

20

0

5

10

15

20

η

1

0.6

upper branch lower branch

0.4 θ (η)

θ (η)

f0 = -0.5, 0, 0.5

30

Pr = 1 f0 = 0.5 λ=2

0.8

0 -0.5

25

Fig. 12. Temperature proﬁles () for various values of when Pr = 1, f0 = 0.5 and = −2.

Pr = 1 γ=1 λ=1

f0 = -0.5, 0, 0.5

γ = 0, 1, 2

γ = 0, 1, 2

0.2 0 -0.2

-1 -1.5

0

5

10

15

20

25

upper branch

-0.4

lower branch

-0.6

30

35

40

η

Fig. 10. Temperature proﬁles () for various values of f0 when Pr = 1, = 1 and = 1.

-0.8

γ = 0, 1, 2 0

10

20

30

40

50

60

70

η

Fig. 13. Temperature proﬁles () for various values of when Pr = 1, f0 = 0.5 and = 2.

1016

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017

Pr = 1 f0 = -0.5

2

vertical slender cylinder in an incompressible viscous ﬂuid. The governing boundary layer equations were solved numerically for both aiding and opposing ﬂow regimes using the Keller-box method. Discussions for the effects of the curvature parameter , suction or blowing parameter f0 and the buoyancy parameter on the skin friction coefﬁcient f (0) and the local Nusselt number (0) for Pr = 1 have been done. From the present investigation, it may be concluded that:

γ = 0, 1, 2

θ' (0)

1 0

γ=0 γ=1 γ=2

-1 -2

γ = 0, 1, 2 -4

-3

-2

-1

0

1

2

3

4

λ Fig. 14. Variation of (0) with for various values of when Pr = 1 and f0 = −0.5.

agreement with Ridha [3,4]. However, as discussed by Ridha [3,4] and Ishak et al. [16], the lower branch solutions have no physical meaning. Although such solutions are deprived of physical signiﬁcance, they are nevertheless of mathematical interest as well as of physical terms in so far as the differential equations are concerned. Besides, similar equations may arise in other situations where the corresponding solutions could have more realistic meaning. The effects of the curvature parameter on the skin friction coefﬁcient f (0) are shown in Figs. 4 and 7, for f0 = 0.5 (suction) and f0 = −0.5 (injection), respectively, while the respective local Nusselt number (0) are presented in Figs. 11 and 14. The results for f0 = 0 (impermeable plate) were reported by Mahmood and Merkin [2], but the lower branch solutions terminate as → 0− , i.e. dual solutions were only reported for situations of opposing ﬂow. Our further investigation showed that these dual solutions extend into the aiding ﬂow regime that is when the buoyancy force and the ﬂow are in the same direction. As mentioned in [3], the reason of this omission is perhaps due to the misleading behavior of the non-dimensional temperature function used in the similarity formulation, which shows the existence of singularity at = 0 (i.e. T = 0). However, it does not reﬂect the solution of the similarity stream function f terminates in a singularity as → 0− . As can be seen from Figs. 4 and 7, f (0) remains regular and ﬁnite in the neighborhood of = 0, even (0) undergoes a discontinuity at = 0 (see Figs. 11 and 14). The results for the case of suction and injection showed similar features as the case of impermeable surface. From the numerical results shown in Figs. 4 and 7, as well as Figs. 11 and 14, it can be concluded that larger values of (smaller values of the cylinder diameter) delay the boundary layer separation, which is in agreement with the results reported by Mahmood and Merkin [2] for an impermeable surface. 5. Conclusions We have theoretically studied the existence of dual similarity solutions in mixed convection boundary layer ﬂow about a

• Dual solutions exist also for the aiding/assisting ﬂow ( > 0). • For the opposing ﬂow ( < 0), there are dual solutions, unique solution or no solution. The solution curves bifurcate at the critical point c (< 0). • Suction (f0 > 0) delays the boundary layer separation, while injection (f0 < 0) accelerates it. • Larger values of (smaller values of the cylinder diameter) delay the boundary layer separation. Acknowledgments The authors wish to express their very sincere thanks to the reviewers for their valuable comments and suggestions. This work is supported by a research grant (IRPA project code: 0902-02-10038-EAR) from the Ministry of Science, Technology and Innovation (MOSTI), Malaysia. References [1] T.S. Chen, B.F. Armaly, Mixed convection in external ﬂow, in: S. Kakaç, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987, pp. 14.1–14.35. [2] T. Mahmood, J.H. Merkin, Similarity solutions in axisymmetric mixedconvection boundary-layer ﬂow, J. Eng. Math. 22 (1988) 73–92. [3] A. Ridha, Aiding ﬂows non-unique similarity solutions of mixedconvection boundary-layer equations, J. Appl. Math. Phys. 47 (1996) 341–352. [4] A. Ridha, Three-dimensional mixed convection laminar boundary-layer near a plane of symmetry, Int. J. Eng. Sci. 34 (1996) 659–675. [5] M.A. Chaudhary, J.H. Merkin, The effect of blowing and suction on free convection boundary layers on vertical surfaces with prescribed heat ﬂux, J. Eng. Math. 27 (1993) 265–292. [6] J.H. Merkin, A note on the similarity equations arising in free convection boundary layers with blowing and suction, J. Appl. Math. Phys. 45 (1994) 258–274. [7] M.A. Chaudhary, J.H. Merkin, I. Pop, Similarity solutions in free convection boundary-layer ﬂows adjacent to vertical permeable surfaces in porous media. I. Prescribed surface temperature, Eur. J. Mech. B/Fluids 14 (1995) 217–237. [8] M.A. Chaudhary, J.H. Merkin, I. Pop, Similarity solutions in free convection boundary-layer ﬂows adjacent to vertical permeable surfaces in porous media: II. Prescribed surface heat ﬂux, Heat Mass Transfer 30 (1995) 341–347. [9] C.Y. Wang, Exact solutions of the steady-state Navier–Stokes equations, Annu. Rev. Fluid Mech. 23 (1991) 159–177. [10] H. Schlichting, Boundary Layer Theory, sixth ed., McGraw-Hill, New York, 1968. [11] G.I. Burde, The construction of special explicit solutions of the boundarylayer equations. Steady Flows, Q. J. Mech. Appl. Math. 47 (1994) 247–260. [12] M. Massoudi, Local non-similarity solutions for the ﬂow of a nonNewtonian ﬂuid over a wedge, Int. J. Non-Linear Mech. 36 (2001) 961–976.

A. Ishak et al. / International Journal of Non-Linear Mechanics 42 (2007) 1010 – 1017 [13] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on self-similar boundary layer ﬂow over moving surfaces, Int. J. Eng. Sci. 44 (2006) 730–737. [14] P.D. Weidman, A.M. Davis, D.G. Kubitschek, Crocco variable formulation for uniform shear ﬂow over a stretching surface with transpiration: multiple solutions and stability. J. Appl. Math. Phys., 2007, doi: 10.1007/s00033-006-6018-2. [15] A. Ishak, J.H. Merkin, R. Nazar, I. Pop, Mixed convection boundary layer ﬂow over a permeable vertical surface with prescribed wall heat ﬂux, J. Appl. Math. Phys., 2007, doi: 10.1007/s00033-006-6082-7. [16] A. Ishak, R. Nazar, I. Pop, Dual solutions in mixed convection ﬂow near a stagnation point on a vertical porous plate, Int. J. Therm. Sci., 2007, doi: 10.1016/j.ijthermalsci.2007.03.005. [17] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Samakia, Buoyancy Induced Flows and Transport, Hemisphere, New York, 1988. [18] K.R. Rajagopal, M. Ruzicka, A.R. Srinivasa, On the Oberbeck– Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996) 1157–1167.

1017

[19] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001. [20] N. Ramachandran, T.S. Chen, B.F. Armaly, Mixed convection in stagnation ﬂows adjacent to a vertical surfaces, ASME J. Heat Transfer 110 (1988) 373–377. [21] T. Cebeci, P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, Springer, New York, 1988. [22] I.A. Hassanien, R.S.R. Gorla, Combined forced and free convection in stagnation ﬂows of micropolar ﬂuids over vertical non-isothermal surfaces, Int. J. Eng. Sci. 28 (1990) 783–792. [23] Y.Y. Lok, N. Amin, I. Pop, Unsteady mixed convection ﬂow of a micropolar ﬂuid near the stagnation point on a vertical surface, Int. J. Therm. Sci. 45 (2006) 1149–1157. [24] G. Wilks, J.S. Bramley, Dual solutions in mixed convection, Proc. R. Soc. Edinburgh A 87 (1981) 349–358.

Copyright © 2021 COEK.INFO. All rights reserved.