The effect of MHD on steady two-dimensional laminar mixed flow about a vertical porous surface is numerically analyzed. Also the effects of radiation ...

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MHD non-Darcian flow through a non-isothermal vertical surface embedded in a porous medium with radiation Alireza Taklifi · Cyrus Aghanajafi

Received: 26 January 2011 / Accepted: 19 September 2011 / Published online: 4 November 2011 © Springer Science+Business Media B.V. 2011

Abstract The effect of MHD on steady two-dimensional laminar mixed flow about a vertical porous surface is numerically analyzed. Also the effects of radiation and heat generation and absorption are considered. A power law variation of temperature along the vertical wall is assumed. The nonlinear boundarylayer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution and the temperature distribution are presented for various values of Prandtl number Pr, magnetic parameter, porous medium parameter and internal heat generation or absorption coefficient. Further validation with previous works is carried out. Keywords MHD · Porous medium · Mixed convection · Rosseland approximation · Power law temperature varying vertical wall Nomenclature a Constant b Constant B0 Magnetic field intensity (T) Cf Drag coefficient Cp Specific heat (J kg−1 K−1 ) A. Taklifi () · C. Aghanajafi Department of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran e-mail: [email protected]

fw g G K k M Nu Pr qr r Rd Re T u U∞ v vw x y

suction/blowing parameter Gravitational acceleration (m s−2 ) Mixed convection parameter Permeability (m2 ) Thermal conductivity (W m−1 K−1 ) Magnetic parameter Nusselt number Prandtle number Radiative heat transfer (W m−2 ) Dimensionless wall temperature Radiation parameter Reynolds number Temperature (K) Axial velocity (m s−1 ) Free stream velocity (m s−1 ) Normal velocity (m s−1 ) Suction/injection velocity (m s−1 ) Axial coordinate Transverse coordinate

Greek symbols βR Rosseland extinction coefficient βT Volumetric thermal expansion coefficient δ Inertia coefficient parameter ε Porosity η Pseudo similarity variable θ Dimensionless temperature ϑ Kinematic viscosity (m2 s−1 ) λ Constant ρ Density (kg m−3 ) σ Electrical conductivity (−1 s−1 )

930

σ∗ ψ

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Stefan–Boltzmann constant (W m2 K4 ) Stream function Internal heat generation/absorption coefficient porous medium parameter

Super/subscripts eff Effective properties f Fluid s Solid

1 Introduction The effect of MHD on convection flows in porous media has been the subject of many recent research papers, as pointed out in the review by Nield and Bejan [1]. The interest in this field is due to the wide range of applications either in engineering and in geophysics, such as the optimization of the solidification processes of metals and metal alloys, the study of geothermal source, the control of underground spreading of chemical wastes and pollutants, the design of MHD power generators, the design of heat exchangers and MHD accelerators. There is still a great deal of interest in this area, both from a theoretical and practical point of view. On the practical side, there is interest is problems connected to topical issues of MHD accelerators and porous fins [2–4] and on the theoretical side there is a great interest in analytical implementation of MHD to non-Newtonian fluids as works has been recently performed by Hayat and coworkers [5–7]. The first study to deal with the mixed convection flow over a horizontal surface embedded in a fluidsaturated porous medium was by Cheng [8]. It was found that similarity solutions exist for parallel flow (u∞ = constant) with non-isothermal wall temperature (Tw = T∞ + Ax 1/2 , where x is the distance along the surface, T is the temperature and A is a constant) and for stagnation flow (u∞ = Bx, where B is a constant) with surface wall temperature distribution Tw = T∞ + Ax 2 . The parallel flow case has important application to heat transfer about a hot bed rock where the parallel flow is due to the difference in hydrostatic heads resulting from recharge or discharge of meteoric water. The stagnation flow is applicable to cold water injection near a hot bed rock. Minkowycz et al. [9] extended Cheng’s work to the case of an arbitrary power-law variation of the wall

temperature (Tw = T∞ + Ax λ ) for parallel and stagnation flows and obtained non-similar solutions by solving the partial differential equations by using the local non-similarity method [10, 11]. The analyses of Cheng [8] and Minkowycz et al. [9] were based on Darcian flow model which is valid only for relatively slow flows through the porous matrix. The effects of the fluid inertia and viscous diffusion at boundaries may become significant for materials with very high porosities such as fibrous media and foams. Lai [12] and Kumari et al. [13] have considered the non-Darcy mixed convection flow over a nonisothermal horizontal surface. They showed that the similarity solutions exist only for a parallel flow (u∞ = constant) with surface temperature distribution Tw = T∞ + Ax λ . Subsequently, Lai [12] extended their analysis to include the effects of thermal dispersion. Most of the published papers on convection and porous media under the action of a magnetic field deal with external flows and consider cases such that the magnetic field is uniform. Kumari et al. [14] studied the mixed convection in a porous medium around a vertical wedge. The boundary layer equations are solved by these authors considering the Brinkman model with inertia term for momentum transport and by taking into account both the effects of Joule heating and viscous dissipation in the energy balance. Chamkha and Quadri [15] consider hydromagnetic natural convection from a horizontal permeable cylinder and obtain a numerical solution of the non-similar boundary layer problem by using a finite difference method. El-Amin [16] investigates external free convection from either a horizontal plate or a vertical plate with uniform heat flux. The purpose of present work is to study the effect of MHD on non-Darcian mixed convection from a non-isothermal vertical surface embedded in a porous medium with presence of radiation and heat sources/ sinks effects. The non-uniform surface temperature treated here is of the form Tw = T∞ + Ax λ . A single mixed convection parameter has been used which covers the entire regime of mixed convection from pure forced convection limit to pure free convection limit. The results have been compared by works done by Abo-Eldahab and El-Gendy [17], Lai and Kulacki [18], Lin and Lin [19], Evans [20], Sparrow et al. [21] and Ariel [22]. A good agreement is seen between the results.

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(6) a uniform magnetic field is assumed to apply in y direction as depicted in Fig. 1, (7) it is assumed that the induced magnetic field imposed magnetic and electrical fields and induced electrical field due to polarization (including Hall Effect) are negligible, (8) following the works of Kamil Kahveci and Semiha Oztuna [24] and Abdelkhalek [25] the viscous dissipation and joule heating effects are neglected, (9) the fluid is electrically conducting and the fluid’s properties variation due to temperature changes are limited to density, (10) heat generation and absorptions are considered. The Brinkman-Forchheimer equation is used to model flow in the porous medium as a non-Darcian flow is under consideration. So according to the assumptions above and coordinates assigned in Fig. 1, if u, v and T are the fluid x-component velocity, y-component velocity and temperature, the governing equations that describe the case are as follows:

Fig. 1 Schematic view of geometry

2 Analysis Figure 1 depicts the geometry of the problem under consideration. In order to simplify the solution, the following assumptions are considered: (1) the problem is steady state, laminar, incompressible, two dimensional, non-Darcy, MHD mixed convection in stagnation flow over a vertical permeable surface embedded in a saturated porous medium. (2) the porous medium is homogeneous, isotropic, and saturated with a single-phase fluid, (3) both the fluid and the solid matrix have constant physical properties except the density in the buoyancy term where Boussenesq approximation is used, (3) the temperature along wall is varying in the form of power law, (4) following the work of Kiwan [23] and in order to reduce the complexity of the problem of radiative heat flux, the porous medium is assumed to behave as an optically thick gas,

∂u ∂v + =0 ∂x ∂y

(1)

2 ∂u 1 ∂p ∂u ∂ u u − +v =ϑ + βT g(T − T∞ ) 2 ∂x ∂y ρ ∂x ∂y σ B02 u ϑε u − Cε 2 u2 − (2) K ρ ∂T ∂T ∂ 2T ∂qr +v = keff 2 − + Q(T − T∞ ) ρCp u ∂x ∂y ∂y ∂y (3) −

where keff is the porous medium effective thermal conductivity and is defined as: keff = εkf + (1 − ε)ks

(4)

βT is the coefficient of thermal expansion and Q is the volumetric rate of heat generation/absorption. qr is the radiative heat transfer which is defined by Rosseland diffusion approximation (Hossain et al. [26]) as follows: qr =

4σ ∗ dT 4 3βR dy

(5)

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The appropriate boundary conditions for the velocity and temperature of this problem are: y = 0: u = 0,

v = −vw ,

T = Tw (x) = T∞ ± bx λ y → ∞: u = U∞ = ax,

(6) T = T∞

U∞ and T∞ are the free stream velocity and temperature, respectively. a and b are constants and vw is the suction or injection velocity. In free stream condition (2) becomes: U∞

σ B02 U∞ 1 ∂p ϑε dU∞ 2 = − U∞ −Cε 2 U∞ (7) − dx ρ ∂x K ρ

By eliminating u

∂p ∂x

∂u ∂u +v ∂x ∂y 2 ∂ u dU∞ =ϑ + U∞ + βT g(T − T∞ ) 2 dx ∂y ϑε 2 (u − U∞ ) − Cε 2 (u2 − U∞ ) K

−

σ B02 (u − U∞ ) ρ

(8)

η = y(a/ϑ)1/2

θ=

(9)

T − T∞ Tw − T∞

where ψ(x, y) is the stream function. Finally the momentum and energy equations (see (8) and (3)) with the boundary conditions can be written as: f + ff − (f )2 − (M + )f + M + + 1 + δ(1 − (f )2 ) + Gθ = 0

σ B02 , ρa

G=

gβT (Tw − T∞ ), aU∞

=

ϑε , Ka

3 σ∗ 16aT∞ Rd = − , 3ρCp ϑβR

Tw − T∞ , T∞

δ = Cε 2 x,

=

Q , ρcp a

(13)

vw fw = − , (aϑ)1/2 Pr =

ρCp ϑ keff a

3 Numerical solution

In order to investigate heat transfer in porous medium the following dimensionless variables are introduced:

ψ = x(aϑ)1/2 f (η)

M=

r=

between (2) and (7),

−

where the magnetic parameter (M), porous medium parameter (), inertia coefficient parameter (δ), mixed convection parameter (G), internal heat generation or absorption coefficient ( ), radiation parameter (Rd), suction/blowing parameter (fw ) and r and Pr are:

(10)

The system of equations and boundary conditions, (10)–(12), have been solved numerically using the Keller box scheme, a finite-difference scheme, similar to that described in Cebeci and Bradshaw [27]. This is an implicit scheme, which demonstrates the ability to solve systems of differential equations of any order as well as featuring second-order accuracy. The resulting nonlinear algebraic system is solved by Newton’s method with step control. The linearized system of equations is solved by Gauss elimination. The same methodology as that used by Kiwan [23] is followed. In the calculations, a uniform grid of the step size 0.001 in the η-direction and a non-uniform grid in the ψ -direction with step size 0.001 is used. The iterative solution is considered to have converged when the maximum values of the normalized absolute residuals across all nodes are less than 10−6 . Further validation is carried out by the other works (Abo-Eldahab and ElGendy [17], Sparrow et al. [21], Evans [20], Lin and Lin [19], Ariel [22]) as they are shown in Tables 1, 2, 3, 4.

θ − Prλf θ + Prf θ + P r θ − RdPr(θ )2 [3r(θ )2 (1 + rθ )2 + (1 + rθ )3 θ ] = 0 (11) and boundary conditions are: f (η, 0) = fw ,

θ (η, 0) = 1

f (η, ∞) = 1,

θ (η, ∞) = 0

(12)

4 Results and discussion Table 5 illustrates the effect of magnetic field parameter (M), porous medium parameter (), internal heat generation or absorption parameter ( ), Prandtl number Pr, λ, ε, r on f (0) and heat flux θ (0) at the surface. This table shows that, the shear stress on the

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Table 1 Validation of the values of f (0), θ (0) at the plate with different M, G, Pr, λ, δ/ and ε values m

0

G

0.2

δ/

ε

0.35

0.45

λ

1

f (0)

Pr

3

θ (0)

Present method

Ab-Eldahab and El-Gendy [17]

Present method

Abo-Eldahab and El-Gendy [17]

−1.85133

−1.85095

−1.85571

−1.87514

0.2

0.2

0.35

0.45

1

3

−1.11024

−1.09669

−1.88237

−1.89762

0.5

0.2

0.35

0.45

1

3

−0.95915

−0.95196

−1.92259

−1.93376

0.5

0.2

0

0.45

1

3

−0.94978

−0.94005

−1.93001

−1.93611

0.5

0.2

0.35

0.45

1

3

−0.95915

−0.95196

−1.92998

−1.93376

0.5

0.2

2.01

0.45

1

3

−1.0506

−1.0068

−1.92005

−1.92294

0.5

0

0.35

0.45

1

3

−0.90553

−0.89102

−1.93994

−1.94484

0.5

0.2

0.35

0.45

1

3

−0.95915

−0.95196

−1.92259

−1.93376

0.5

0.5

0.35

0.45

1

3

−1.10021

−1.04556

−1.90979

−1.91606

0.5

0.7

0.35

0.45

1

3

−1.11098

−1.10946

−1.90058

−1.90342

0.5

0.2

0.35

0.45

1

0.3

−1.08042

−1.01712

−0.53924

−0.55894

0.5

0.2

0.35

0.45

1

1

−0.99471

−0.98736

−1.00473

−1.01941

0.5

0.2

0.35

0.45

1

3

−0.95915

−0.95196

−1.92259

−1.93376

Table 2 Validation of the values of 0.5Cf M

√ Rex for different M values when G = δ = Rd = = 0 and Pr = 1

fw = −1 Sparrow et al. [21]

fw = 0 Present method

Sparrow et al. [21]

Ariel [22]

Present method

0

0.7605

0.7598

1.231

1.232588

1.232588

1

1.124

1.1171

1.584

1.585331

1.585331

2

1.892

1.8894

2.345

2.346663

2.346663

√ Table 3 Validation of the values of Nux / Rex for different values of Pr and λ when G = δ = Rd = M = = 0 Pr

λ=0 Sparrow et al. [21]

Evans [20]

Lin and Lin [19]

Present method

0.01

0.07596

0.075927

0.075973

0.075973

0.1

02194

0.219503

0.219505

0.219503

1

0.5705

0.570466

0.570466

0.570464

10

1.349

1.3388

1.3388

1.338799

surface by increases as the magnetic parameter (M) increases, also increasing the porous medium parameter () has the same effect on shear stress. Further more by increasing the internal heat generation or absorption parameter ( ) an increase could be resulted in shear stress on the surface. On the other hand as Prandtle number increases the shear stress at the surface decreases.

Table 5 also shows that the magnitude of the wall temperature gradient increases as the magnetic parameter (M), porous medium (), Prandtle number Pr increase and the internal heat generation or absorption parameter ( ) decreases. Figures 2 and 3 depict the dimensionless x-directional velocity f (η) and dimensionless temperature

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Meccanica (2012) 47:929–937 √ Table 4 Validation of the values of Nux / Rex = −θ (0) for different values of fw when = δ = 10 Rd = M = 0, Pr = 1 and = 10 fw

Lai and Kulacki [18]

Present method

−1

0.7766

0.776614

−0.5

0.9909

0.990905

0

1.2533

1.253298

0.5

1.5599

1.559852

1

1.9043

1.904287

Table 5 Variation of the values of f (0), θ (0) at the plate surface with M, G, Pr, λ, δ, , , ε, Rd, r parameters M

R

G

δ

λ

Pr

ε

r

f (0)

θ (0)

0.001

0.185

0.0305

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

2.8656

−2.2185

0.001

0.185

5

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

3.7212

−2.36

0.001

0.185

5

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

3.7618

−2.002

0.001

0.185

5

0.0056

0.5

5

1

3.3

0.45

0.0016

3.8137

−1.575

0.001

0.185

5

0.0056

1

5

1

3.3

0.45

0.0016

3.9454

−0.58

0.001

0.185

5

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

3.7212

−2.36

0.001

0.185

5

0.0056

2.41e–9

5

1

4

0.45

0.0016

3.6164

−2.9265

0.001

0.185

5

0.0056

2.41e–9

5

1

4.8

0.45

0.0016

3.448

−4.2355

0.001

0.185

0.0305

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

2.8656

−2.2185

0.001

0.185

0.0305

0.5

2.41e–9

5

1

3.3

0.45

0.0016

2.9506

−2.234

0.001

0.185

0.0305

4

2.41e–9

5

1

3.3

0.45

0.0016

3.4938

−2.32

0.001

0.185

0.0305

10

2.41e–9

5

1

3.3

0.45

0.0016

4.267

−2.427

0.001

0.185

0.0305

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

2.8656

−2.2185

4

0.185

0.0305

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

3.4946

−2.321

10

0.185

0.0305

0.0056

2.41e–9

5

1

3.3

0.45

0.0016

4.26768

−2.427

λ = 1,

θ (η) profiles for the case δ = 5,

= 2.41e–9,

M = 0.001, λ = 1,

Pr = 3.3,

r = 0.0016,

G = 0.0305, R = 0.185, ε = 0.45

for change in the values of the porous medium parameter . These figures show that both of the dimensionless temperature and dimensionless velocity increase as increases. Figure 4 represents the dimensionless x-directional velocity variations of the fluid f (η) profiles for the case δ = 5, G = 5,

= 2.41e–9, Pr = 3.3,

= 0.0056, R = 0.185,

r = 0.0016,

ε = 0.45

for different values of magnetic field parameter M. This figure shows that by increasing of M the velocity of fluid in x-direction increases. Figure 5 depicts the dimensionless temperature θ (η) profiles for the case δ = 5,

= 2.41e–9,

= 0.0056,

G = 5,

Pr = 3.3,

R = 0.185,

λ = 1,

r = 0.0016,

ε = 0.45

for different values of M which shows that the dimensionless temperature increases as Pr number decreases.

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Fig. 2 Dimensionless velocity profiles f (η) for different values of

Fig. 4 Dimensionless velocity profiles f (η) for different values of M

Fig. 3 Dimensionless temperature profiles θ(η) for different values of

Fig. 5 Dimensionless temperature profiles θ(η) for different values of M

Figure 6 illustrates the variations of dimensionless x-directional velocity f (η) for the case δ = 5,

= 2.41e–9,

M = 0.001, λ = 1,

G = 5,

r = 0.0016,

= 0.0056, Rd = 0.185, ε = 0.45

for different values of Prandtle number which shows that the velocity increases as Pr number decreases. Also Fig. 7 depicts the dimensionless temperature

θ (η) variations for the same conditions which shows that by increasing of the Prandtle number dimensionless temperature θ (η) decreases. Figure 8 depicts the variations of dimensionless xdirectional velocity f (η) for the case δ = 5,

Pr = 3.3,

M = 0.001, λ = 1,

G = 5,

r = 0.0016,

= 0.0056, Rd = 0.185, ε = 0.45

936

Fig. 6 Dimensionless velocity profiles f (η) for different values of Pr

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Fig. 8 Dimensionless velocity profiles f (η) for different values of

Fig. 9 Dimensionless temperature profiles θ(η) for different values of Fig. 7 Dimensionless temperature profiles θ(η) for different values of Pr

5 Conclusion

for different values of internal heat generation or absorption coefficient and also Fig. 9 shows the dimensionless temperature θ (η) variations for the same conditions. These figures show that both of the dimensionless velocity f (η) and dimensionless temperature θ (η) increase by increasing of the values of .

The problem of hydromagnetic boundary layer flow and heat transfer of an electrically conducting fluid on a vertical surface embedded in a non-Darcian porous medium in the presence of radiation and heat source/sink generation was investigated. The governing equations were simplified to partial differential equations and they were reduced to ordinary differential equations by using similarity solutions. Finally the set of ordinary differential equations

Meccanica (2012) 47:929–937

were solved. Further validations were carried out by works of other researchers. These validations show excellent agreement between the current results and previous works’ results. Following results were achieved by this work: By increasing the magnetic field parameter M, the dimensionless velocity and shear stress at the wall increases and the dimensionless temperature decreases. By increasing the Prandtle number, the dimensionless velocity, the shear stress at the wall and the dimensionless temperature decrease slightly but the wall temperature gradient increases. By increasing the porous medium parameter , the dimensionless velocity, shear stress at the wall, dimensionless temperature and the wall temperature gradient increase. By increasing the internal heat generation or absorption, the dimensionless velocity, shear stress at the wall, dimensionless temperature increase and the wall temperature gradient decreases. So, by regarding the results above it is possible to enhance the heat transfer of an electrically conducting fluid on a vertical surface embedded in a non-Darcian porous medium, by controlling the magnetic field and porosity of the medium which could has a great range of interest by the industrial engineers.

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