Physica A 389 (2010) 40–46

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Physica A

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MHDmixed convection flow near the stagnation-point on a verticalpermeable surfaceAnuar Ishak a,∗, Roslinda Nazar a, Norfifah Bachok b, Ioan Pop ca School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysiab Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysiac Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e i n f o

Article history:Received 17 June 2009Received in revised form 26 August 2009Available online 6 September 2009

PACS:47.15.Cb

Keywords:MagnetohydrodynamicMixed convectionSuction/injectionDual solutionsStagnation-point

a b s t r a c t

The steady magnetohydrodynamic (MHD) mixed convection boundary layer flow of aviscous and electrically conducting fluid near the stagnation-point on a vertical permeablesurface is investigated in this study. The velocity of the external flowand the temperature ofthe plate surface are assumed to vary linearly with the distance from the stagnation-point.The governing partial differential equations are first transformed into ordinary differentialequations, before being solved numerically by a finite-difference method. The features ofthe flow and heat transfer characteristics for different values of the governing parametersare analyzed and discussed. Both assisting and opposing flows are considered. It is foundthat dual solutions exist for both cases, and the range of the mixed convection parameterfor which the solution exists increases with suction.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The existence of dual solutions in mixed convection boundary layer flow has been pointed out by many researchers,for example, de Hoog et al. [1], Afzal and Hussain [2], Ramachandran et al. [3], Devi et al. [4], Ridha [5] and Lok et al. [6].Ramachandran et al. [3] studied the steady laminar mixed convection in two-dimensional stagnation flows around verticalsurfaces by considering both cases of an arbitrary wall temperature and arbitrary surface heat flux variations. They foundthat a reverse flow develops in the buoyancy opposing flow region, and dual solutions are found to exist for a certain rangeof the buoyancy parameter. This work was then extended by Devi et al. [4] to the unsteady case, and by Lok et al. [6]to a vertical surface immersed in a micropolar fluid. Dual solutions were found to exist by these authors only for theopposing flow case. The existence of dual solutions for both assisting and opposing flows was reported by Ridha [5] whenhe reconsidered the problems of mixed convection flow over a horizontal surface, mixed convection flow over a verticalsurface, and axisymmetric mixed convection flow, which were investigated previously by some authors. Ridha [5] pointedout that the failure of the previous investigations to report the existence of dual solutions for the assisting flow is perhapsdue to the misleading behavior of the non-dimensional temperature used in the similarity formulation.Motivated by the above investigations, the present paper investigates when the mixed convection boundary layer flow

with a constant magnetic field is introduced normal to the vertical plate. We also investigate the effects of suction andinjection on the surface shear stress and the heat transfer rate at the surface. Suction or injection of a fluid through thebounding surface, as, for example, in mass transfer cooling, can significantly change the flow field and, as a consequence,

∗ Corresponding author. Tel.: +603 8921 5756; fax: +603 8925 4519.E-mail addresses: anuarishak@yahoo.com, anuar_mi@ukm.my (A. Ishak).

0378-4371/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2009.09.008

A. Ishak et al. / Physica A 389 (2010) 40–46 41

u

g

y

xB0

Vw

Vw

B0

v

U (x)

T ∞

Tw (x) < T ∞

Tw (x) > T ∞

u

g

y

xB0

Vw

Vw

B0

v

U (x)

T ∞

Tw (x) > T ∞

Tw (x) < T ∞

(a) Assisting flow. (b) Opposing flow.

Fig. 1. Physical model and coordinate system.

affect the heat transfer rate at the plate. In general, suction tends to increase the skin friction and heat transfer coefficients,whereas injection acts in the opposite manner (Al-Sanea [7]). Injection or withdrawal of a fluid through a porous boundingheated or cooled wall is of general interest in practical problems involving boundary layer control applications such asfilm cooling, polymer fiber coating, coating of wires, etc. The process of suction and blowing has also its importance inmanyengineering activities such as in the design of thrust bearing and radial diffusers, and thermal oil recovery. Suction is appliedto chemical processes to remove reactants. Blowing is used to add reactants, cool the surfaces, prevent corrosion or scalingand reduce the drag (Labropulu et al. [8]).The governing partial differential equations are first transformed into ordinary differential equations using similarity

transformation, before being solved numerically by means of a finite-difference scheme known as the Keller-box method.The numerical results obtained are then compared with the data available in the literature for certain particular cases of thepresent problem, to support their validity.

2. Problem formulation

Consider the steady, two-dimensional flow of a viscous and incompressible electrically conducting fluid near thestagnation-point on a vertical permeable flat plate as shown in Fig. 1. It is assumed that the x-component velocity of theflow external to the boundary layer U(x) and the temperature Tw(x) of the plate are proportional to the distance from thestagnation-point, i.e. U(x) = ax and Tw(x) = T∞ + bx, where a and b are constants with a > 0. The assisting flow (b > 0)occurs if the upper half of the plate is heated while the lower half of the plated is cooled. In this case, the flow near theheated plate tends to move upward and the flow near the cooled plate tends to move downward, therefore this behavioracts to assist the flow field. The opposing flow (b < 0) occurs if the upper part of the plate is cooled while the lower partof the plate is heated (see Lok et al. [9]). A uniform magnetic field of strength B0 is assumed to be applied in the positivey-direction normal to the plate. The induced magnetic field is assumed to be small compared to the applied magnetic field,and is neglected. Also the viscous dissipation effect is neglected. Under these assumptions along with the Boussinesq andboundary layer approximations, the system of equations, which models the problem under consideration is given by:

∂u∂x+∂v

∂y= 0, (1)

u∂u∂x+ v

∂u∂y= ν

∂2u∂y2−1ρ

dpdx−σB20ρu+ gβ(T − T∞), (2)

u∂T∂x+ v

∂T∂y= α

∂2T∂y2

, (3)

subject to the boundary conditions

u = 0, v = Vw, T = Tw(x) at y = 0,u→ U(x), T → T∞ as y→∞, (4)

where u and v are the velocity components in the x and y directions, respectively, ρ is fluid density, ν is the kinematicviscosity, β is the thermal expansion coefficient, α is the thermal diffusivity, p is the pressure, T is the fluid temperature,B0 is the uniform magnetic field, σ is the electrical conductivity and g is the acceleration due to gravity. Further, Vw is theuniform surface mass flux, where Vw < 0 corresponds to suction and Vw > 0 corresponds to injection. By employing the

42 A. Ishak et al. / Physica A 389 (2010) 40–46

generalized Bernoulli’s equation, in free-stream, Eq. (2) becomes

UdUdx= −

1ρ

dpdx−σB20ρU . (5)

Using (5), Eq. (2) can be written as

u∂u∂x+ v

∂u∂y= ν

∂2u∂y2+ U

dUdx+σB20ρ(U − u)+ gβ(T − T∞). (6)

We introduce now the following similarity variables:

η = (U/νx)1/2y, ψ = (Uνx)1/2f (η), θ(η) = (T − T∞)/(Tw − T∞), (7)

where ψ is the stream function defined as u = ∂ψ/∂y and v = −∂ψ/∂x so as to identically satisfy Eq. (1). Substituting (7)into Eqs. (3) and (6), we obtain the following ordinary differential equations:

f ′′′ + ff ′′ + 1− f ′2 +M(1− f ′)+ λθ = 0, (8)1Prθ ′′ + f θ ′ − f ′θ = 0, (9)

where primes denote differentiation with respect to η,M = σB20/(ρa) is the magnetic parameter, λ = Grx/Re2x(= gβb/a

2)

is the buoyancy or mixed convection parameter and Pr = ν/α is the Prandtl number. Further, Grx = gβ(Tw − T∞)x3/ν2and Rex = Ux/ν are respectively the Grashof number and the Reynolds number. We notice that λ is a constant, with λ > 0and λ < 0 corresponding to assisting and opposing flows, respectively, while λ = 0 represents the case when the buoyancyforce is absent. The boundary conditions (4) now become

f (0) = f0, f ′(0) = 0, θ(0) = 1,

f ′(∞)→ 1, θ(∞)→ 0, (10)

where f0 = f (0) = −Vw/(νa)1/2 is a constant with f0 > 0 and f0 < 0 corresponding to mass suction and mass injection, re-spectively. It is worth mentioning that whenM = 0 (magnetic field is absent) and f0 = 0 (impermeable plate), Eqs. (8)–(10)reduce to those found by Ramachandran et al. [3] 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 Nux, which are defined

by

Cf =τw

ρU2/2, Nux =

xqwk(Tw − T∞)

, (11)

where the wall shear stress τw and the wall heat flux qw are given by

τw = µ

(∂u∂y

)y=0

, qw = −k(∂T∂y

)y=0

, (12)

withµ and k being the dynamic viscosity and thermal conductivity, respectively. Using the similarity variables (7), we obtain

12Cf Re1/2x = f

′′(0), Nux/Re1/2x = −θ′(0). (13)

3. Results and discussion

The system of nonlinear ordinary differential equations (8)–(10) has been solved numerically for some values of thebuoyancy parameter λ, magnetic parameter M and suction/injection parameter f0, while the Prandtl number Pr is fixed tobe unity (Pr = 1), except for comparisons with previously reported cases. These equations have been solved using theKeller-boxmethod by integrating forwards in η until a predetermined large value of η is reached, say η∞, where we assumethe infinity boundary condition may be enforced. The Keller-box method is very well described in the book by Cebeci andBradshaw [10].The values of the dimensionless skin friction coefficient f ′′(0) and local Nusselt number −θ ′(0) are obtained and

compared with previously reported cases. These comparisons are shown in Tables 1 and 2, which showed a very goodagreement with the results reported by Ramachandran et al. [3], Devi et al. [4], Lok et al. [6] and Hassanien and Gorla [11],for some particular cases of the present study.The variations of the skin friction coefficient f ′′(0) and the local Nusselt number−θ ′(0)with buoyancy parameter λ for

M = 1 and some values of f0 are shown in Figs. 2 and 3 respectively, all for Pr = 1. These figures show that it is possibleto obtain dual solutions of the similarity equations (8)–(10) also for assisting flow (λ > 0), apart of those for opposing flow

A. Ishak et al. / Physica A 389 (2010) 40–46 43

Table 1Values of f ′′(0) for various values of Pr whenM = 0, f0 = 0 and λ = 1 (assisting flow)

Pr Ramachandran Devi et al. Lok et al. Hassanien and Present resultset al. [3] [4] [6] Gorla [11] Upper branch Lower branch

0.7 1.7063 1.7064 1.706376 1.70632 1.7063 1.23871 – – – – 1.6754 1.13327 1.5179 1.5180 1.517952 – 1.5179 0.582410 – – – 1.49284 1.4928 0.495820 1.4485 1.4485 1.448520 – 1.4485 0.343640 1.4101 – 1.410094 – 1.4101 0.211150 – – – 1.40686 1.3989 0.172060 1.3903 1.3903 1.390311 – 1.3903 0.141380 1.3774 – 1.377429 – 1.3774 0.0947100 1.3680 1.3680 1.368070 1.38471 1.3680 0.0601

Table 2Values of−θ ′(0) for various values of Pr whenM = 0, f0 = 0 and λ = 1 (assisting flow).

Pr Ramachandran Devi et al. Lok et al. Hassanien and Present resultset al. [3] [4] [6] Gorla [11] Upper branch Lower branch

0.7 0.7641 0.7641 0.764087 0.76406 0.7641 1.02261 – – – – 0.8708 1.16917 1.7224 1.7223 1.722775 – 1.7224 2.219210 – – – 1.94461 1.9446 2.494020 2.4576 2.4574 2.458836 – 2.4576 3.164640 3.1011 – 3.103703 – 3.1011 4.108050 – – – 3.34882 3.3415 4.497660 3.5514 3.5517 3.555404 – 3.5514 4.857280 3.9095 – 3.914882 – 3.9095 5.5166100 4.2116 4.2113 4.218462 4.23372 4.2116 6.1230

Fig. 2. Skin friction coefficient f ′′(0) as a function of λ for different values of f0 when Pr = 1 andM = 1.

(λ < 0), that have been reported by Ramachandran et al. [3], Devi et al. [4] and Lok et al. [6]. For λ > 0, there is a favorablepressure gradient due to the buoyancy forces, which results in the flow being accelerated and consequently there is a largerskin friction coefficient than in the non-buoyant case (λ = 0) as well as the opposing flow case (λ < 0). For negative valuesof λ, there is a critical value λc(< 0), with two solution branches for λ > λc , a saddle-node bifurcation at λ = λc and nosolutions for λ < λc . Based on our computations, λc = −3.38,−3.85 and−5.51 for f0 = −0.2, 0 and 0.5, respectively.We identify the upper and lower branch solutions in the following discussion by how they appear in Fig. 2, i.e. the upper

branch solution has a higher value of f ′′(0) for a given λ than the lower branch solution. For an assisting flow, dual solutionsare found to exist for all positive values of λ considered, to much higher values than shown in Fig. 2. This figure also showsthat the critical value |λc | increases as the suction/injection parameter f0 is increased, suggesting that suction increasesthe range of existence of the solutions to Eqs. (8)–(10). The results shown in Fig. 3 for the heat transfer rate at the surface−θ ′(0) suggest that for the lower branch solution,−θ ′(0) becomes unbounded as λ→ 0. The small gap in the dual solutionbranch of f ′′(0) around λ = 0, for each value of f0 (see Fig. 2), is due to the lack of a converged numerical solution, since thecorresponding behavior of−θ ′(0) (see Fig. 3) is that it tends to large negative and positive values as λ→ 0− and λ→ 0+,respectively.Figs. 4 and 5 illustrate the samples of velocity and temperature profiles for assisting flow, λ = 1, while the corresponding

opposing flow, λ = −1, is shown in Figs. 6 and 7, all for Pr = 1 andM = 1. In these figures the solid lines are for the upper

44 A. Ishak et al. / Physica A 389 (2010) 40–46

Fig. 3. Local Nusselt number−θ ′(0) as a function of λ for different values of f0 when Pr = 1 andM = 1.

Fig. 4. Velocity profiles f ′(η) for different values of f0 when Pr = 1,M = 1 and λ = 1 (assisting flow).

Fig. 5. Temperature profiles θ(η) for different values of f0 when Pr = 1,M = 1 and λ = 1 (assisting flow).

branch solutions and the dash lines for the lower branch solutions. As seen in Figs. 4–7, there are dual solutions both whenλ = 1 and λ = −1. When λ = 1, the velocity gradient at the surface is positive for the solutions on both branches (seeFig. 4), in agreement with the curves of f ′′(0) shown in Fig. 2. For λ = −1 (see Fig. 6), the velocity profiles for the upperbranch solutions show a positive velocity gradient at the surface, while for the lower branch solutions, the velocity gradientat the surface is positive when f0 = −0.2 and f0 = 0, but negative when f0 = 0.5. Again, this observation is in agreementwith the curves shown in Fig. 2. Further, the solution on the lower branch for both cases λ = ±1 has a region of reversedflow (has f ′(η) < 0). We expect that the upper branch solution is stable, while the lower branch solution is not, since theupper branch solution is the only solution for the case λ = 0, and the existence of reverse flow region for the lower branchsolution. The saddle-node bifurcation at λ = 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 , the saddle-node bifurcation gives a change in stabilityfrom stable (upper branch) to unstable (lower branch). Although the lower branch solutions seem to be deprived of physical

A. Ishak et al. / Physica A 389 (2010) 40–46 45

Fig. 6. Velocity profiles f ′(η) for different values of f0 when Pr = 1,M = 1 and λ = −1 (opposing flow).

Fig. 7. Temperature profiles θ(η) for different values of f0 when Pr = 1,M = 1 and λ = −1 (opposing flow).

Fig. 8. Skin friction coefficient f ′′(0) as a function of λ for different values ofM when Pr = 1 and f0 = 0.

significance, they are nevertheless of interest so far as the differential equations are concerned. Similar results may arise inother situations where the corresponding solutions have more realistic meaning [5].The sample of velocity and temperature profiles presented in Figs. 4–7 show that the far field boundary conditions (10)

are approached asymptotically, and thus support the numerical results obtained, besides supporting the dual nature of thesolutions to the boundary-value problem (8)–(10).Finally, Figs. 8 and 9 show the variations of the skin friction coefficient f ′′(0) and the local Nusselt number−θ ′(0) with

buoyancy parameter λ for different values ofM when Pr = 1 and f0 = 0 (impermeable plate). It is seen from these figuresthat the range of λ for which the solution exists increases with M . For the upper branch solution, which we expect to bethe physically relevant solution, the skin friction coefficient f ′′(0) increases asM increases, and in consequence increase thelocal Nusselt number−θ ′(0), as presented in Fig. 9.

46 A. Ishak et al. / Physica A 389 (2010) 40–46

Fig. 9. Local Nusselt number−θ ′(0) as a function of λ for different values ofM when Pr = 1 and f0 = 0.

4. Conclusions

We have theoretically studied the similarity solutions of the steady mixed convection boundary layer flow near thestagnation-point on a vertical permeable surface embedded in a viscous and electrically conducting fluid. We discussed theeffects of the governing parameters λ,M and f0 on the fluid flow and heat transfer characteristics, while the Prandtl numberPr is fixed to be unity except for comparisons with previously reported cases available in the literature, which showed avery good agreement. We found that dual solutions exist for both assisting and opposing flows. For the assisting flow case, asolution could be obtained for all positive values of λ, while for the opposing case, the solution terminated in a saddle-nodebifurcation at λ = λc(λc < 0). The value of |λc | increases with an increase in f0 and M , thus suction as well as a magneticfield increases the range of λ for which the solution exists.

Acknowledgements

The authors wish to express their very sincere thanks to the anonymous referee for his valuable comments andsuggestions. This work is supported by research grants, i.e. Science Fund (Project Code: 06-01-02-SF0610) fromMinistry ofScience, Technology and Innovation (MOSTI), Malaysia and GUP (Project Code: UKM-GUP-BTT-07-25-174) from UniversitiKebangsaan Malaysia.

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