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MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate
Applied Mathematical Modelling 33 (2009) 4086–4096
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MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate Orhan Aydın *, Ahmet Kaya Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey
a r t i c l e
i n f o
Article history: Received 2 October 2007 Received in revised form 1 February 2009 Accepted 16 February 2009 Available online 23 February 2009
Keywords: MHD flow Mixed convection Vertical plate Viscous dissipation Suction/injection effect
a b s t r a c t The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, fw. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The study of magnetohydrodynamic (MHD) flow has received a great deal of research interest due to its importance in many engineering applications, such as plasma studies, petroleum industries, MHD power generators, cooling of nuclear reactors, the boundary layer control in aerodynamics, and crystal growth. Convection about a heated/cooled vertical plate is one of fundamental problems in heat and mass transfer studies. If the free convection existing for this case is accompanied by an external flow, the combined mode of free and forced convection exits, which is commonly called as mixed convection. On the effect of MHD flow, although there are many studies regarding the free convection regime [1,2], there are only a few regarding the mixed convection regime. Yıh [3] numerically analyzed the effect of heat source/sink on steady two-dimensional laminar MHD mixed convection in stagnation flows of incompressible, electrically conducting fluids about a vertical permeable flat plate with linear wall temperature proportional to the distance in a porous medium. Chamkha et al. [4] studied the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. Abo-Eldahab and Azzam [5,6] studied the MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow, past an isothermal
* Corresponding author. Tel.: +90 (462) 377 2974; fax: +90 (462) 325 5526. E-mail address: [email protected] (O. Aydın). 0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2009.02.002
O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
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Nomenclature cp Gr Ec F fw Pr Ri Re T u, v x,y
specific heat of the convective fluid Grashof number Eckert number dimensionless stream function suction/injection parameter Prandtl number Richardson number Reynolds number temperature, K velocities in x and y-directions, respectively, m s1 coordinates in horizontal and vertical directions, respectively, m
Greek symbols pseudo similarity variable, yRe1=2 x =x n magnetic interaction parameter, rB20 x=qu1 B0 magnetic flux density r electrical conductivity of the fluid q fluid density, kg m3 l dynamic viscosity, Pa s t kinematic viscosity, m2 s1 h dimensionless temperature profile in Eq. (5)
g
Subscripts w wall 1 free stream
semi-infinite inclined plate, for high temperature and concentration differences by the presence of radiation. In a recent study, Abdelkhalek [7] studied MHD mixed convection in stagnation point flow impinging on a heated vertical semi-infinite permeable surface. Aydin and Kaya [8] studied mixed convection of viscous dissipating fluid about a vertical plate. They observed four different flow situations according to the direction of the free stream flow and thermal boundary condition applied at the wall. The objective of this study is focused at extending that study including the suction/injection effect at the wall and the presence of a magnetic field. 2. Analysis Consider the steady, laminar, incompressible, two-dimensional, mixed convection boundary layer flow about a vertical flat plate shown in Fig. 1. The fluid is assumed to be electrically conducting. The coordinate system is chosen such that x measures the distance along the plate and y measures the distance normal to it. A uniform magnetic field is assumed to apply in the y-direction causing a flow resistive force in the x-direction. It is assumed that the induced magnetic field, the external
Tw
(Opposing-buoyancy) u∞, T∞ g
B0
x, u u∞, T∞ (Aiding-buoyancy) y, v Fig. 1. The schematic of the problem.
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or imposed electric field and the electric field due to the polarization of charges (i.e. Hall effect) are negligible. The plate is considered to be electrically non-conducting. Under foregoing assumptions and taking into account the Boussinesq approximation and the boundary layer approximation, the system of continuity, momentum and energy equations can be written:
ou ot þ ¼ 0; ox oy
ð1Þ
u
ou ou o2 u rB2 þ t ¼ v 2 gbðT T 1 Þ 0 ðu u1 Þ; ox oy oy q
ð2Þ
u
oT oT v o2 T v ou 2 rB20 2 þ þ ðu u1 uÞ: þt ¼ ox oy Pr oy2 cp oy qcp
ð3Þ
Here u and v are the velocity components parallel and perpendicular to the plate, T is the temperature, b is the coefficient of thermal expansion, v is the kinematic viscosity, q is the fluid density, g is the acceleration due to gravity, r is the electrical conductivity of the fluid, B0 is the magnetic flux density. The plus and minus signs of the buoyancy term denote the upward and downward flows of free stream, respectively. The appropriate boundary conditions for the velocity and temperature of this problem are:
x¼0
y > 0 T ¼ T1
u ¼ u1
x>0
y ¼ 0 T ¼ Tw
u¼0
y!1
T ! T1
v ¼ V w ðxÞ :
ð4Þ
u ! u1
Table 1 Comparison of the values h0 (0, 0) for various values Pr at Ri = 0.0, Ec = 0.0 and fw = 0.0. Pr
Lin and Lin [11]
Yih[12]
Chamkha et al. [13]
Nield and Kuznetsov [14,15]
Present study
0.01 0.1 1 10 100
0.051559 0.140032 0.332057 0.728148 1.571860
0.051589 0.140034 0.332057 0.728141 1.571831
0.051830 0.142003 0.332173 0.728310 1.572180
– 0.1580 0.3320 0.7300 1.5700
0.051437 0.148123 0.332000 0.727801 1.573141
Table 2 Comparison of the values h0 (0, 0) for various values Pr and Ri at Ec = 0.0 and fw = 0.0. Pr = 0.72
Pr = 7.0
Ri
Saeid [16]
Present study
Saeid [16]
Present study
0.0 0.2 0.4 0.6 0.8 1.0
0.309 0.332 0.361 0.382 0.402 0.416
0.297 0.332 0.356 0.376 0.392 0.406
0.628 0.698 0.752 0.791 0.822 0.851
0.646 0.698 0.740 0.772 0.802 0.827
Table 3 Comparison of the values h0 (n, 0) for various values Pr and Ec with Ri = 0.0 and fw = 0.0. Pr
n
Yih [12]
Chamkha et al. [13]
Watanabe and Pop [17]
Present study
Ec = 0.0
Ec = 1.0
Ec = 0.0
Ec = 1.0
Ec = 0.0
Ec = 1.0
Ec = 0.0
Ec = 1.0
0.733
0.0 0.5 1.0 1.5 2.0
0.297526 0.357022 0.382588 0.398264 0.409168
0.170272 0.210072 0.228813 0.240798 0.249316
0.29760 0.35704 0.38319 0.39998 0.40945
0.17018 0.20986 0.22900 0.24100 0.24931
0.29755 0.35699 0.38336 0.39959 0.41091
0.12395 0.20871 0.22857 0.24122 0.25022
0.29753 0.35709 0.38363 0.40012 0.41134
0.17031 0.21012 0.22873 0.24082 0.24963
1.0
0.0 0.5 1.0 1.5 2.0
0.332057 0.402864 0.433607 0.452634 0.465987
0.166029 0.201452 0.216814 0.226323 0.232998
0.33217 0.40310 0.43390 0.45280 0.46611
0.16630 0.20163 0.21672 0.22621 0.23310
0.33206 0.40280 0.43446 0.45413 0.46798
0.16603 0.20144 0.21727 0.22710 0.23401
0.33206 0.40259 0.43460 0.45302 0.46612
0.16599 0.20132 0.21710 0.22672 0.23296
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Here, u1 and T1 are the free stream velocity and temperature, respectively. To seek a solution, the following dimensionless variables are introduced:
nðxÞ ¼
u 1=2 rB20 x T T1 1 ; wðx; yÞ ¼ ðv u1 xÞ1=2 f ðn; gÞ; g ¼ y ; h¼ ; qu1 vx Tw T1
ð5Þ
where wðx; yÞ is the stream function that satisfies Eq. (1) with u = ow/oy and t ¼ ow=ox. In terms of these new variables, the velocity components can be expressed as,
u ¼ u1 f 0 ;
ð6Þ
( " #) 2 1 u1 v 1=2 g 0 1=2 rB0 of : t¼ f þ ðv u1 xÞ f 2 x qu1 on 2x
ð7Þ
The transformed momentum and energy equations together with the boundary conditions, Eqs. (2) (3) (4), can be written as:
1 00 of 0 of ; f 000 þ ff Rix h þ nð1 f 0 Þ ¼ n f 0 f 00 2 on on
ð8Þ
1 00 1 0 oh of h þ f h þ Ecðf 00 Þ2 Ecnf 0 þ Ecnðf 0 Þ2 ¼ nðf 0 h0 Þ; Pr 2 on on
ð9Þ
a
2.0 1.8
Ec=0.0 fw=0.0 ξ=0.0 ξ=0.1
1.6 1.4
f'(ξ,η)
1.2 1.0 0.8 0.6 Ri=5.0, 2.0, 1.0, 0.5, 0.0
0.4 0.2 0.0 0
1
2
3
4
5
6
5
6
η
b
1.0
0.8
Ec=0.0 fw=0.0 ξ=0.0 ξ=0.1
θ(ξ,η)
0.6
Ri=5.0, 2.0, 1.0, 0.5, 0.0 0.4
0.2
0.0 0
1
2
3
4
η Fig. 2. Dimensionless velocity (a) and temperature (b) profiles for different Ri (aiding-buoyancy) at Ec = 0.0, Pr = 1.0 and fw = 0.0.
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with the boundary conditions; @f f ðn; 0Þ þ 2n @n ¼ fw ; f 0 ðn; 0Þ ¼ 0; hðn; 0Þ ¼ 1;
f 0 ðn; 1Þ ¼ 1;
ð10Þ
hðn; 1Þ ¼ 0;
1=2 where fw ¼ 2x v V w Rex , the case fw > 0 designates suction while fw < 0 indicates injection or blowing, Pr and Ec are the main flow parameters and defined as:
Pr ¼
lc p k
¼
v a
;
Ec ¼
u21 ; cp ðT w T 1 Þ
Rix ¼
Grx Re1=2 x
;
Grx ¼
gbðT w T 1 Þx3
v2
and Rex ¼
u1 x
v
;
ð11Þ
where Pr is the Prandtl number, Ec is the Eckert number, Rix is the local Richardson number, Grx is the local Grashof number, Rex is the local Reynolds number. 3. Numerical solution The system of transformed equations together with the boundary conditions, Eqs. (8)–(10), have been solved numerically using the Keller box scheme, an efficient and accurate finite-difference scheme, similar to that described in Cebeci and Bradshaw [9]. For the sake of brevity, details of the numerical method are not described, referring the reader to Cebeci and Bradshaw [9]. This is a very popular implicit scheme, which demonstrates the ability to solve systems of differential equations of
a
1.2
1.0 Ri=1.0 Ri=1.0 ξ=0.1 ξ=0.1
fw=0.1, Ec=0.0 fw=0.1, Ec=0.25
f'(ξ,η)
0.8
fw= -0.1, Ec=0.0 fw= -0.1, Ec=0.25
0.6
0.4
0.2
0.0 0
1
2
3
4
5
6
5
6
η
b
1.0 fw=0.1, Ec=0.0 fw=0.1, Ec=0.25
0.8
Ri=1.0 ξ=0.1
fw= -0.1, Ec=0.0 fw= -0.1, Ec=0.25
θ(ξ,η)
0.6
0.4
0.2
0.0 0
1
2
3
4
η Fig. 3. Dimensionless velocity (a) and temperature (b) profiles for different Ec and fw at Pr = 1.0, Ri = 1.0 and n = 0.1.
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O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
any order as well as featuring second-order accuracy (which can be realized with arbitrary non-uniform spacing), allowing very rapid x or n variations [10]. A set of non-linear finite-difference algebraic equations derived are then solved by using the Newton quazi-linearization method. The same methodology as that followed by Takhar and Beg [10] is followed. Therefore, for the finite-difference forms of the equations, we refer the reader to Ref. [10] for the brevity of the article. In the calculations, a uniform grid of the step size 0.01 in the g-direction and a non-uniform grid in the n-direction with a starting step size 0.001 and an increase of 0.05 times the previous step size were found to be satisfactory in obtaining sufficient accuracy within a tolerance better than 106 in nearly all cases. The value of g1 = 16 is shown to satisfy the velocity to reach the relevant stream velocity. In order to verify the accuracy of the numerical results, the validity of the numerical code developed has been checked for a limiting case. For Pr = 1.0, Ri = 0.0, and Ec = 0.0, we compare our h0 (0, 0) results with those given by Lin and Lin [11], Yih [12], Chamkha et al. [13], Nield and Kuznetsov [14] and Kuznetsov and Nield [15] (Table 1). For different Pr and Ri, we compare our h0 (0, 0) results with those given by Saeid [16] (Table 2). For Pr = 1.0 and different Ec, our h0 (n, 0) results are compared with those given by Watanabe and Pop [17], Yih [12] and Chamkha et al. [13] (Table 3). As it is seen from Tables 1–3 excellent agreements have been observed.
4. Results and discussion In the mixed convection case, depending on the direction of external flow, two different mixed convection regimes occur: aiding and opposing [8]. In this study, the aiding and opposing regimes of mixed convection are examined including the viscous dissipation and magnetic field effects. The following ranges of the main parameters are considered: Ri = 0.15, 0.1, 0.075, 0.05, 0.025, 0.0, 0.25, 0.5, 0.75, 1.0, 2.0 and 5.0; Pr = 1.0; fw = 0.1, 0.0, 0.1; Ec = 0.0 and 0.25 and magnetic interaction parameter n, the range 0.0 6 n 6 1.0. The combined effects of Ri, Ec, fw and n on the momentum and heat transfer are analyzed. The Richardson number, Ri represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the heat and fluid flow. Outside the mixed convection region, either the pure forced convection or the free convection analysis can be used to describe accurately the flow or the temperature field. Forced convection is the dominant mode of transport when Ri ? 0, whereas free convection is the dominant mode when Ri ? 1. Buoyancy forces can enhance the surface heat transfer rate when they assist the forced convection, and vice versa [8]. The Eckert number, Ec represents the relative importance of viscous dissipation to thermal diffusion. Viscous dissipation plays a role like an energy source and therefore it affects the temperature profile. In the first case, an upward forced external flow is considered. For the heated wall case (Tw > T1), the upward free flow caused by the buoyancy is in the same direction with the external forced flow. This case is called the aiding mixed flow. For
2.0
2.0
2.0
1.5
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
1.0 fw= -0.1
0.5 0.0
0.2
0.4
0.6
0.8
f''(ξ,0)
c 2.5
f''(ξ,0)
b 2.5
f''(ξ,0)
a 2.5
1.5
fw=0.0
0.5
1.0
0.0
0.2
0.4
ξ
0.6
0.8
fw=0.1
0.5
1.0
0.0
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.2 0.1 0.0 -0.1 0.0
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
0.2
0.3 0.2 0.1
fw= -0.1
0.4
ξ
0.0 0.6
0.8
1.0
-θ'(ξ,0)
0.6
0.3
-0.1 0.0
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
0.2
0.0 0.6
0.8
1.0
0.4
ξ
0.6
0.8
1.0
0.2 0.1
ξ
0.2
0.3
fw=0.0
0.4
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
1.0
ξ
-θ'(ξ,0)
-θ'(ξ,0)
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
1.0
1.5
-0.1 0.0
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
0.2
fw=0.1
0.4
0.6
0.8
1.0
ξ
Fig. 4. Effects of Ri (aiding-buoyancy), Ec and n on the local skin friction parameter and local heat transfer parameter for fw = 0.1 (a) fw = 0.0 (b) and fw = 0.1 (c).
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the case without the viscous dissipation effect (Ec = 0) and impermeable wall, Fig. 2 shows velocity and temperature profiles for different values of the Richardson number, Ri and magnetic interaction parameter, n. An increase at Ri result in increasing velocities due to addition of buoyancy-induced flow onto the external forced flow. In order to understand the effect of the magnetic interaction parameter, we should examine Eqs. (2 and 3) closely. The rB2 sign of the last term in the right hand side of Eq. (2), q0 ðu u1 Þ, is directly related with the sign of (u u1). For the forced convection regime, when Ri ? 0, this term will always be negative since u1 > u. Therefore, it will generate a force in the main flow direction, which will aid the main flow. However, for the free convection regime, when Ri ? 1, buoyancy-driven flow will dominate the external flow and therefore this term will always be positive since u1 < u, which will slow down the rB20 u , represents the imposed pressure force in the inviscid main flow. In fact, this term has two components: The first one, 1 q rB2 region of the conducting fluid, while the second one, q0 u, represents the Lorentz force imposed by a transverse magnetic field to an electrically conducting, which slows down the fluid motion in the boundary layer region. When the imposed pressure force overcomes the Lorentz force, i.e. u1 > u, the effect of the magnetic interaction parameter is to increase velocity. Similarly, when the Lorentz force dominates over the imposed pressure force, i.e. u1 < u, the effect of the magnetic interaction parameter will decrease velocity. In the mixed convection regime, when Ri increases the Lorentz force and therefore opposes the flow and therefore velocity decreases. However, decreasing Ri strengthens the imposed pressure force, which aids the flow by increasing the velocity (Fig. 2a). The last term in the right hand side of Eq. (3), rB20 2 2 qcp ðu u1 uÞ, will behave like a 2heat sink or a heat source depending on the sign of (u u1u). This term consists of rB work, which is supposed two components. The first one, qcp0 u2 , represents the Ohmic heating due to the electromagnetic rB2 to behave like a heat source by increasing the fluid temperature, while the second one qcp0 u1 u is the stress work which
a 1.0 Ec=0.0 fw=0.0 ξ=0.1 ξ=0.0
0.8
0.6
f'(ξ,η)
Ri=0.0, -0.05, -0.1
0.4
0.2
0.0 0
2
4
6
8
ξ
b
1.0 Ec=0.0 fw=0.0 ξ=0.1 ξ=0.0
0.8
θ(ξ,η)
0.6
Ri=0.0, -0.05, -0.1
0.4
0.2
0.0 0
1
2
3
4
5
6
ξ Fig. 5. Dimensionless velocity (a) and temperature (b) profiles for different Ri (opposing-buoyancy) at Ec = 0.0, Pr = 1.0, fw = 0.0.
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O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
decreases the fluid temperature. When the latter component suppresses the former one, this term will serve as a heat sink decreasing the fluid temperature (Fig. 2b). Fig. 3 illustrates the effects of Ec and fw at Ri = 1 and n = 0.1. As expected, when compared to an impermeable plate, the suction (fw > 0) decreases velocity and temperature while injection (fw > 0) increases them. Therefore, suction of fluid at wall reduces both the hydrodynamic and thermal boundary layers, which result in increasing fluid velocity and decreasing fluid temperature. Opposite is true for the case of injecting fluid at wall. The viscous dissipation, as a heat generation inside the fluid, increases the bulk fluid temperature. As seen from Fig. 3, an increase in Ec increases fluid temperature. Fig. 4 shows the effect of the magnetic interaction parameter, n on the momentum and heat transfer for various values of Ec and fw. As it is stated above, the effect of this parameter will vary according to the value of Ri. In the forced regime (Ri ? 0), the magnetic field parameter, increasing n will increase the local skin friction parameter, f 00 ðn; 0Þ and the heat transfer parameter, h0 (n, 0) as a result of increased velocity and temperature gradients at wall. In the free convection regime (Ri ? 1), an increase in n will increase f 00 ðn; 0Þ and will decrease h0 (n, 0). In the mixed convection regime, the level of Ri will determine the effect of n. As expected, increasing Ec increases local skin friction and decreasing local heat transfer parameters. As seen, effect of Ec becomes more considerable with increasing Ri. Fig. 4 also aims to explore the effects of the suction/injection parameter on the temperature profile. As expected, when compared to an impermeable wall case (fw = 0, Fig. 4b) increasing fw increases the local skin friction and the local heat transfer parameter since injecting fluid into the boundary layer (fw = 0.1, Fig. 4a) broadens the velocity distribution and increases the hydrodynamic boundary layer thicknesses, while sucking of fluid from the boundary layer through wall (fw = 0.1, Fig. 4c) will results in an opposing effect.
a
1.0 Ri=-0.1 ξ=0.1 0.8 fw=0.1, Ec=0.0 fw=0.1, Ec=0.25
0.6
f'(ξ,η)
fw=-0.1, Ec=0.0 fw=-0.1, Ec=0.25
0.4
0.2
0.0 0
2
4
6
8
ξ
b 1.0
Ri=-0.1 ξ=0.1
0.8 fw=0.1, Ec=0.0 fw=0.1, Ec=0.25 fw=-0.1, Ec=0.0
θ(ξ,η)
0.6
fw=-0.1, Ec=0.25 0.4
0.2
0.0 0
1
2
3
4
5
6
ξ Fig. 6. Dimensionless velocity (a) and temperature (b) profiles for different Ec and fw at Pr = 1.0, Ri = 0.1 and n = 0.1.
4094
b 1.0
c 1.0
0.8
0.8
0.8
0.6
0.6
0.4
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.0
0.0
0.2
0.4
0.45
ξ
0.6
0.8
0.0 0.0
1.0
0.35
0.35
0.20 0.0
fw= -0.1 Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.2
0.4
0.6
ξ
0.2
0.4
0.45 0.40
0.25
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.2
0.40
0.30
fw=0.0
0.4
0.8
ξ
0.0 0.0
1.0
0.2
0.4
ξ
0.6
0.8
1.0
0.40
fw=0.0
0.30
0.20 0.0
0.8
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.45
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.25
1.0
0.6
fw=0.1
0.4 0.2
-θ'(ξ,η)
fw= -0.1
0.2
-θ'(ξ,0)
f''(ξ,0)
0.6
f''(ξ,0)
1.0
-θ'(ξ,0)
f''(ξ,0)
a
O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
0.2
0.4
0.6
ξ
0.8
0.35 fw=0.1 Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.30 0.25
1.0
0.20 0.0
0.2
0.4
ξ
0.6
0.8
1.0
Fig. 7. Effects of Ri (opposing-buoyancy), Ec and n on the local skin friction parameter and local heat transfer parameter for fw = 0.1 (a) fw = 0.0 (b) and fw = 0.1 (c).
In the second case, a downward forced external flow is assumed. The upward flow caused by the buoyancy for the heated wall case (Tw > T1) has a retarding effect on the external forced flow, which is called as opposing mixed flow. For the case without the viscous dissipation effect (Ec = 0) and the impermeable wall (fw = 0.1), the velocity and temperature profiles for different values of the Richardson number, Ri are shown in Fig. 5. An increase at Ri results in decreasing velocities because of opposing effect of the upward buoyancy-induced flow on the downward external forced flow. For this opposing mixed convection regime, Fig. 6 illustrates the effects of Ec and fw at Ri = 1 and n = 0.1. As expected, the suction (fw > 0) decreases velocity and temperature while injection (fw > 0) increases them. The viscous dissipation increases the bulk fluid temperature. Therefore, in addition to opposing effect of the buoyancy, viscous dissipation will have an opposing effect on the heat transfer, too. For the opposing mixed convection case, Fig. 7 shows the effect of the magnetic interaction parameter, n on the momentum and heat transfer for various values of Ec and fw. For this case, increasing n increases the local skin friction, f 00 ðn; 0Þ and the local heat transfer parameter, h0 (n, 0). Similar to the case of the aiding mixed convection, when compared to the impermeable wall case, the heat transfer parameter increases at the suction case, while it is decreased at the injection case. When injection is applied at the wall, hydrodynamic and thermal boundary layers will thicken, thereby resulting in a decrease in the heat transfer parameter. Results of the values of the local skin friction parameter, f 00 ðn; 0Þ and the local heat transfer parameter h0 (n, 0) for various values of the suction/injection parameter, fw, the mixed convection parameter, Ri, the viscous dissipation parameter, Ec and the magnetic parameter, n are given in Table 4. 5. Conclusions Steady MHD mixed convection about a heated vertical plate has been analyzed including the effects of suction/injection at the wall and viscous dissipation. A transformed set of non-similar equations have been solved using the Keller box scheme, an efficient and accurate finite-difference scheme. Both the aiding- and opposing-buoyancy modes of the mixed convection have been studied. The conclusions drawn from the study can be summarized as follows: (i) As expected, the mixed convection parameter (i.e. the Richardson number, Ri) increases the momentum and heat transfer for the aiding-buoyancy mode, while the opposite is true for the opposing-buoyancy mode. (ii) Suction at the wall increases the local skin friction parameter and the local heat transfer parameter due to decreased thermal boundary layer thickness, while injection has an opposite effect. (iii) The viscous dissipation increases the fluid temperature and therefore decreases the temperature gradient at the wall, which results in decreased heat transfer values. (iv) The presence of the magnetic field influences flow and temperature fields. The effect of the magnetic interaction parameter is found to vary according to the value of the Richardson number, Ri.
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Table 4 Results of the values of the local skin friction parameter, f 00 ðn; 0Þ and the local heat transfer parameter h0 (n, 0) for various values of the suction/injection parameter, fw, the mixed convection parameter, Ri, the viscous dissipation parameter, Ec and the magnetic parameter, n. n
fw = 0.1
fw = 0.0
Ec = 0.0 f 00 ðn; 0Þ
Ec = 0.25
fw = 0.1
Ec = 0.0
Ec = 0.25
Ec = 0.0
Ec = 0.25
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
Ri = 0.1 0.0 0.1285 0.05 0.2138 0.10 0.2772 0.15 0.3343 0.20 0.3846 0.25 0.4314 0.30 0.4746 0.35 0.5154 0.40 0.5538 0.45 0.5905 0.50 0.6255 0.55 0.6591 0.60 0.6913 0.65 0.7225 0.70 0.7527 0.75 0.7819 0.80 0.8102 0.85 0.8378 0.90 0.8646 0.95 0.8908 1.0 0.9163
0.2599 0.2879 0.3037 0.3169 0.3272 0.3362 0.3438 0.3507 0.3567 0.3623 0.3673 0.3719 0.3762 0.3801 0.3838 0.3873 0.3905 0.3936 0.3965 0.3992 0.4018
0.1194 0.2079 0.2725 0.3304 0.3814 0.4287 0.4722 0.5133 0.5520 0.5889 0.6240 0.6578 0.6903 0.7216 0.7518 0.7812 0.8096 0.8373 0.8642 0.8904 0.9160
0.2322 0.2595 0.2749 0.2877 0.2979 0.3067 0.3144 0.3212 0.3274 0.3331 0.3383 0.3431 0.3476 0.3518 0.3558 0.3596 0.3632 0.3665 0.3698 0.3729 0.3758
0.1776 0.2536 0.3138 0.3685 0.4174 0.4631 0.5054 0.5456 0.5834 0.6197 0.6543 0.6875 0.7195 0.7505 0.7804 0.8094 0.8376 0.8650 0.8917 0.9177 0.9431
0.3011 0.3235 0.3377 0.3498 0.3595 0.3679 0.3752 0.3818 0.3876 0.3930 0.3978 0.4023 0.4065 0.4103 0.4139 0.4173 0.4205 0.4235 0.4263 0.4290 0.4316
0.1697 0.2479 0.3091 0.3646 0.4141 0.4602 0.5029 0.5434 0.5815 0.6180 0.6527 0.6862 0.7184 0.7494 0.7795 0.8086 0.8369 0.8644 0.8912 0.9173 0.9428
0.2697 0.2913 0.3053 0.3170 0.3266 0.3349 0.3423 0.3489 0.3549 0.3604 0.3655 0.3702 0.3746 0.3787 0.3826 0.3863 0.3899 0.3932 0.3964 0.3995 0.4024
0.2247 0.2940 0.3513 0.4038 0.4513 0.4959 0.5374 0.5768 0.6141 0.6498 0.6840 0.7169 0.7486 0.7793 0.8090 0.8378 0.8657 0.8930 0.9195 0.9454 0.9707
0.3410 0.3597 0.3726 0.3836 0.3926 0.4006 0.4075 0.4138 0.4194 0.4245 0.4292 0.4336 0.4376 0.4413 0.4448 0.4481 0.4512 0.4542 0.4569 0.4596 0.4621
0.2175 0.2885 0.3467 0.3999 0.4480 0.4929 0.5348 0.5745 0.6121 0.6481 0.6824 0.7155 0.7474 0.7782 0.8080 0.8369 0.8650 0.8923 0.9189 0.9449 0.9702
0.3057 0.3237 0.3363 0.3471 0.3561 0.3640 0.3710 0.3774 0.3831 0.3885 0.3934 0.3980 0.4023 0.4064 0.4102 0.4138 0.4173 0.4206 0.4237 0.4267 0.4296
Ri = 0.0 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
0.2963 0.3536 0.4036 0.4503 0.4934 0.5341 0.5724 0.6090 0.6439 0.6774 0.7096 0.7407 0.7708 0.7999 0.8282 0.8557 0.8824 0.9085 0.9340 0.9589 0.9832
0.2975 0.3112 0.3217 0.3309 0.3387 0.3457 0.3519 0.3576 0.3627 0.3675 0.3718 0.3759 0.3797 0.3832 0.3866 0.3897 0.3927 0.3955 0.3981 0.4007 0.4031
0.2963 0.3536 0.4036 0.4503 0.4934 0.5341 0.5724 0.6090 0.6439 0.6774 0.7096 0.7407 0.7708 0.7999 0.8282 0.8557 0.8824 0.9085 0.9340 0.9589 0.9832
0.2621 0.2747 0.2846 0.2933 0.3009 0.3077 0.3138 0.3194 0.3246 0.3294 0.3339 0.3381 0.3421 0.3459 0.3494 0.3528 0.3561 0.3592 0.3621 0.3650 0.3677
0.3320 0.3871 0.4357 0.4813 0.5236 0.5636 0.6015 0.6376 0.6722 0.7054 0.7373 0.7682 0.7981 0.8270 0.8551 0.8825 0.9091 0.9351 0.9605 0.9852 1.0090
0.3320 0.3445 0.3543 0.3630 0.3705 0.3772 0.3832 0.3886 0.3936 0.3982 0.4024 0.4064 0.4101 0.4136 0.4168 0.4199 0.4228 0.4255 0.4282 0.4306 0.4330
0.3320 0.3871 0.4357 0.4813 0.5236 0.5636 0.6015 0.6376 0.6722 0.7054 0.7373 0.7682 0.7981 0.8270 0.8551 0.8825 0.9091 0.9351 0.9605 0.9852 1.0090
0.2927 0.3043 0.3136 0.3219 0.3292 0.3357 0.3417 0.3471 0.3522 0.3569 0.3613 0.3655 0.3694 0.3731 0.3766 0.3799 0.3831 0.3862 0.3891 0.3919 0.3946
0.3686 0.4216 0.4689 0.5134 0.5549 0.5942 0.6315 0.6672 0.7014 0.7343 0.7659 0.7966 0.8262 0.8550 0.8829 0.9101 0.9366 0.9624 0.9876 1.0120 1.0360
0.3671 0.3786 0.3877 0.3959 0.4030 0.4094 0.4151 0.4204 0.4252 0.4297 0.4338 0.4376 0.4412 0.4446 0.4478 0.4508 0.4536 0.4563 0.4589 0.4613 0.4636
0.3686 0.4216 0.4689 0.5134 0.5549 0.5942 0.6315 0.6672 0.7014 0.7343 0.7659 0.7966 0.8262 0.8550 0.8829 0.9101 0.9366 0.9624 0.9876 1.0120 1.0360
0.3239 0.3346 0.3434 0.3513 0.3582 0.3645 0.3703 0.3756 0.3805 0.3851 0.3894 0.3934 0.3973 0.4009 0.4044 0.4077 0.4108 0.4138 0.4167 0.4195 0.4221
Ri = 0.1 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
0.4307 0.4767 0.5188 0.5589 0.5966 0.6327 0.6671 0.7003 0.7321 0.7629 0.7927 0.8216 0.8496 0.8768 0.9034 0.9293 0.9545 0.9792 1.0030 1.0270 1.0500
0.3215 0.3304 0.3379 0.3447 0.3507 0.3563 0.3613 0.3660 0.3702 0.3743 0.3780 0.3815 0.3848 0.3879 0.3909 0.3936 0.3963 0.3988 0.4012 0.4035 0.4057
0.4356 0.4810 0.5227 0.5624 0.5998 0.6356 0.6698 0.7027 0.7343 0.7650 0.7946 0.8233 0.8512 0.8783 0.9047 0.9305 0.9557 0.9803 1.0040 1.0280 1.0510
0.2761 0.2841 0.2911 0.2975 0.3033 0.3086 0.3135 0.3181 0.3224 0.3265 0.3303 0.3339 0.3374 0.3407 0.3438 0.3468 0.3497 0.3525 0.3552 0.3578 0.3602
0.4614 0.5065 0.5479 0.5874 0.6247 0.6605 0.6946 0.7275 0.7591 0.7897 0.8193 0.8481 0.8759 0.9031 0.9295 0.9553 0.9805 1.0050 1.0290 1.0530 1.0760
0.3535 0.3620 0.3692 0.3758 0.3816 0.3870 0.3919 0.3965 0.4007 0.4046 0.4083 0.4117 0.4149 0.4180 0.4209 0.4236 0.4262 0.4287 0.4311 0.4334 0.4355
0.4663 0.5108 0.5518 0.5910 0.6280 0.6634 0.6973 0.7300 0.7614 0.7918 0.8213 0.8499 0.8776 0.9046 0.9309 0.9566 0.9817 1.0060 1.0300 1.0540 1.0770
0.3046 0.3123 0.3191 0.3253 0.3310 0.3362 0.3410 0.3456 0.3498 0.3538 0.3576 0.3612 0.3646 0.3679 0.3710 0.3740 0.3768 0.3796 0.3822 0.3848 0.3873
0.4932 0.5373 0.5781 0.6170 0.6539 0.6892 0.7230 0.7556 0.7870 0.8174 0.8468 0.8753 0.9031 0.9301 0.9564 0.9821 1.0070 1.0320 1.0560 1.0790 1.1020
0.3864 0.3945 0.4014 0.4077 0.4134 0.4186 0.4233 0.4278 0.4319 0.4357 0.4393 0.4426 0.4458 0.4488 0.4516 0.4543 0.4569 0.4593 0.4617 0.4639 0.4660
0.4980 0.5417 0.5820 0.6206 0.6571 0.6922 0.7258 0.7582 0.7893 0.8196 0.8488 0.8772 0.9048 0.9317 0.9579 0.9834 1.008 1.0330 1.0570 1.0800 1.1030
0.3339 0.3414 0.3479 0.3540 0.3595 0.3646 0.3693 0.3737 0.3779 0.3819 0.3856 0.3891 0.3925 0.3957 0.3988 0.4018 0.4046 0.4073 0.4099 0.4125 0.4149
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Acknowledgement The first author of this article is indebted to the Turkish Academy of Sciences (TUBA) for the financial support provided under the Programme to Reward Success Young Scientists (TUBA-GEBIT). References [1] C.H. Chen, Combined heat and mass transfer MHD free convection from a vertical surface with ohmic heating and viscous dissipation, Int. J. Eng. Sci. 42 (2004) 699–713. [2] C.H. Chen, Heat and mass transfer in MHD flow by natural convection from a permeable, inclined surface with variable wall temperature and concentration, Acta Mech. 172 (2004) 219–235. [3] K.A. Yıh, Heat source/sink effect on mhd mixed convection in stagnation flow on a vertical permeable plate in porous media, Int. Commun. Heat Mass 25 (3) (1998) 427–442. [4] A.J. Chamkha, H.S. Takhar, G. Nat, Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection), Heat Mass Transfer 40 (2004) 835–841. [5] E.M. Abo-Eldahab, G.E.A. Azzam, Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion, Heat Mass Transfer 41 (12) (2005) 1056–1065. [6] E.M. Abo-Eldahab, G.E.A. Azzam, Thermal radiation effects on magnetohydrodynamic flow past a semi-infinite vertical plate in the presence of mass diffusion, Can. J. Phys. 83 (3) (2005) 243–256. [7] M.M. Abdelkhalek, The skin friction in the MHD mixed convection stagnation point with mass transfer, Int. Commun. Heat Mass 33 (2006) 249–258. [8] O. Aydin, A. Kaya, Mixed convection of a viscous dissipating fluid about a vertical flat plate, Appl. Math. Model 31 (2007) 843–853. [9] T. Cebeci, P. Bradshaw, Momentum Transfer in Boundary layers, Hemisphere, Washington D.C., USA, 1977. [10] H.S. Takhar, O.A. Beg, Effects of transverse magnetic field, Prandtl number and Reynolds number on non-darcy mixed convective flow of an incompressible viscous fluid past a porous vertical flat plate in a saturated porous medium, Int. J. Energy Res. 21 (1997) 87–100. [11] H.T. Lin, L.K. Lin, Similarity solutions for laminar forced convection heat transfer from wedges to fluids of any Prandtl number, Int. J. Heat Mass Transfer 30 (1987) 1111–1118. [12] K.A. Yih, MHD forced convection flow adjacent to a non-isothermal wedge, Int. Commun. Heat Mass 26 (1999) 819–827. [13] A.J. Chamkha, M. Mujtaba, A. Quadri, C. Issa, Thermal radiation effects on MHD forced convection flow adjacent to a non-isothermal wedge in the presence of heat source or sink, Heat Mass Transfer 39 (2003) 305–312. [14] D.A. Nield, A.V. Kuznetsov, Boundary layer analysis of forced convection with a plate and porous substrate, Acta Mech. 166 (2003) 141–148. [15] A.V. Kuznetsov, D.A. Nield, Boundary layer treatment of forced convection over a wedge with an attached porous substrate, J. Porous Media 9 (7) (2006) 683–694. [16] N.W. Saeid, Mixed convection flow along a vertical plate subjected to time-periodic surface temperature oscillations, Int. J. Therm. Sci. 44 (2005) 531– 539. [17] T. Watanabe, I. Pop, Thermal boundary layers in magnetohydrodynamic flow over a flat plate in the presence of a transverse magnetic field, Acta Mech. 105 (1994) 233–238.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate Orhan Aydın *, Ahmet Kaya Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey
a r t i c l e
i n f o
Article history: Received 2 October 2007 Received in revised form 1 February 2009 Accepted 16 February 2009 Available online 23 February 2009
Keywords: MHD flow Mixed convection Vertical plate Viscous dissipation Suction/injection effect
a b s t r a c t The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, fw. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The study of magnetohydrodynamic (MHD) flow has received a great deal of research interest due to its importance in many engineering applications, such as plasma studies, petroleum industries, MHD power generators, cooling of nuclear reactors, the boundary layer control in aerodynamics, and crystal growth. Convection about a heated/cooled vertical plate is one of fundamental problems in heat and mass transfer studies. If the free convection existing for this case is accompanied by an external flow, the combined mode of free and forced convection exits, which is commonly called as mixed convection. On the effect of MHD flow, although there are many studies regarding the free convection regime [1,2], there are only a few regarding the mixed convection regime. Yıh [3] numerically analyzed the effect of heat source/sink on steady two-dimensional laminar MHD mixed convection in stagnation flows of incompressible, electrically conducting fluids about a vertical permeable flat plate with linear wall temperature proportional to the distance in a porous medium. Chamkha et al. [4] studied the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. Abo-Eldahab and Azzam [5,6] studied the MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow, past an isothermal
* Corresponding author. Tel.: +90 (462) 377 2974; fax: +90 (462) 325 5526. E-mail address: [email protected] (O. Aydın). 0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2009.02.002
O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
4087
Nomenclature cp Gr Ec F fw Pr Ri Re T u, v x,y
specific heat of the convective fluid Grashof number Eckert number dimensionless stream function suction/injection parameter Prandtl number Richardson number Reynolds number temperature, K velocities in x and y-directions, respectively, m s1 coordinates in horizontal and vertical directions, respectively, m
Greek symbols pseudo similarity variable, yRe1=2 x =x n magnetic interaction parameter, rB20 x=qu1 B0 magnetic flux density r electrical conductivity of the fluid q fluid density, kg m3 l dynamic viscosity, Pa s t kinematic viscosity, m2 s1 h dimensionless temperature profile in Eq. (5)
g
Subscripts w wall 1 free stream
semi-infinite inclined plate, for high temperature and concentration differences by the presence of radiation. In a recent study, Abdelkhalek [7] studied MHD mixed convection in stagnation point flow impinging on a heated vertical semi-infinite permeable surface. Aydin and Kaya [8] studied mixed convection of viscous dissipating fluid about a vertical plate. They observed four different flow situations according to the direction of the free stream flow and thermal boundary condition applied at the wall. The objective of this study is focused at extending that study including the suction/injection effect at the wall and the presence of a magnetic field. 2. Analysis Consider the steady, laminar, incompressible, two-dimensional, mixed convection boundary layer flow about a vertical flat plate shown in Fig. 1. The fluid is assumed to be electrically conducting. The coordinate system is chosen such that x measures the distance along the plate and y measures the distance normal to it. A uniform magnetic field is assumed to apply in the y-direction causing a flow resistive force in the x-direction. It is assumed that the induced magnetic field, the external
Tw
(Opposing-buoyancy) u∞, T∞ g
B0
x, u u∞, T∞ (Aiding-buoyancy) y, v Fig. 1. The schematic of the problem.
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or imposed electric field and the electric field due to the polarization of charges (i.e. Hall effect) are negligible. The plate is considered to be electrically non-conducting. Under foregoing assumptions and taking into account the Boussinesq approximation and the boundary layer approximation, the system of continuity, momentum and energy equations can be written:
ou ot þ ¼ 0; ox oy
ð1Þ
u
ou ou o2 u rB2 þ t ¼ v 2 gbðT T 1 Þ 0 ðu u1 Þ; ox oy oy q
ð2Þ
u
oT oT v o2 T v ou 2 rB20 2 þ þ ðu u1 uÞ: þt ¼ ox oy Pr oy2 cp oy qcp
ð3Þ
Here u and v are the velocity components parallel and perpendicular to the plate, T is the temperature, b is the coefficient of thermal expansion, v is the kinematic viscosity, q is the fluid density, g is the acceleration due to gravity, r is the electrical conductivity of the fluid, B0 is the magnetic flux density. The plus and minus signs of the buoyancy term denote the upward and downward flows of free stream, respectively. The appropriate boundary conditions for the velocity and temperature of this problem are:
x¼0
y > 0 T ¼ T1
u ¼ u1
x>0
y ¼ 0 T ¼ Tw
u¼0
y!1
T ! T1
v ¼ V w ðxÞ :
ð4Þ
u ! u1
Table 1 Comparison of the values h0 (0, 0) for various values Pr at Ri = 0.0, Ec = 0.0 and fw = 0.0. Pr
Lin and Lin [11]
Yih[12]
Chamkha et al. [13]
Nield and Kuznetsov [14,15]
Present study
0.01 0.1 1 10 100
0.051559 0.140032 0.332057 0.728148 1.571860
0.051589 0.140034 0.332057 0.728141 1.571831
0.051830 0.142003 0.332173 0.728310 1.572180
– 0.1580 0.3320 0.7300 1.5700
0.051437 0.148123 0.332000 0.727801 1.573141
Table 2 Comparison of the values h0 (0, 0) for various values Pr and Ri at Ec = 0.0 and fw = 0.0. Pr = 0.72
Pr = 7.0
Ri
Saeid [16]
Present study
Saeid [16]
Present study
0.0 0.2 0.4 0.6 0.8 1.0
0.309 0.332 0.361 0.382 0.402 0.416
0.297 0.332 0.356 0.376 0.392 0.406
0.628 0.698 0.752 0.791 0.822 0.851
0.646 0.698 0.740 0.772 0.802 0.827
Table 3 Comparison of the values h0 (n, 0) for various values Pr and Ec with Ri = 0.0 and fw = 0.0. Pr
n
Yih [12]
Chamkha et al. [13]
Watanabe and Pop [17]
Present study
Ec = 0.0
Ec = 1.0
Ec = 0.0
Ec = 1.0
Ec = 0.0
Ec = 1.0
Ec = 0.0
Ec = 1.0
0.733
0.0 0.5 1.0 1.5 2.0
0.297526 0.357022 0.382588 0.398264 0.409168
0.170272 0.210072 0.228813 0.240798 0.249316
0.29760 0.35704 0.38319 0.39998 0.40945
0.17018 0.20986 0.22900 0.24100 0.24931
0.29755 0.35699 0.38336 0.39959 0.41091
0.12395 0.20871 0.22857 0.24122 0.25022
0.29753 0.35709 0.38363 0.40012 0.41134
0.17031 0.21012 0.22873 0.24082 0.24963
1.0
0.0 0.5 1.0 1.5 2.0
0.332057 0.402864 0.433607 0.452634 0.465987
0.166029 0.201452 0.216814 0.226323 0.232998
0.33217 0.40310 0.43390 0.45280 0.46611
0.16630 0.20163 0.21672 0.22621 0.23310
0.33206 0.40280 0.43446 0.45413 0.46798
0.16603 0.20144 0.21727 0.22710 0.23401
0.33206 0.40259 0.43460 0.45302 0.46612
0.16599 0.20132 0.21710 0.22672 0.23296
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Here, u1 and T1 are the free stream velocity and temperature, respectively. To seek a solution, the following dimensionless variables are introduced:
nðxÞ ¼
u 1=2 rB20 x T T1 1 ; wðx; yÞ ¼ ðv u1 xÞ1=2 f ðn; gÞ; g ¼ y ; h¼ ; qu1 vx Tw T1
ð5Þ
where wðx; yÞ is the stream function that satisfies Eq. (1) with u = ow/oy and t ¼ ow=ox. In terms of these new variables, the velocity components can be expressed as,
u ¼ u1 f 0 ;
ð6Þ
( " #) 2 1 u1 v 1=2 g 0 1=2 rB0 of : t¼ f þ ðv u1 xÞ f 2 x qu1 on 2x
ð7Þ
The transformed momentum and energy equations together with the boundary conditions, Eqs. (2) (3) (4), can be written as:
1 00 of 0 of ; f 000 þ ff Rix h þ nð1 f 0 Þ ¼ n f 0 f 00 2 on on
ð8Þ
1 00 1 0 oh of h þ f h þ Ecðf 00 Þ2 Ecnf 0 þ Ecnðf 0 Þ2 ¼ nðf 0 h0 Þ; Pr 2 on on
ð9Þ
a
2.0 1.8
Ec=0.0 fw=0.0 ξ=0.0 ξ=0.1
1.6 1.4
f'(ξ,η)
1.2 1.0 0.8 0.6 Ri=5.0, 2.0, 1.0, 0.5, 0.0
0.4 0.2 0.0 0
1
2
3
4
5
6
5
6
η
b
1.0
0.8
Ec=0.0 fw=0.0 ξ=0.0 ξ=0.1
θ(ξ,η)
0.6
Ri=5.0, 2.0, 1.0, 0.5, 0.0 0.4
0.2
0.0 0
1
2
3
4
η Fig. 2. Dimensionless velocity (a) and temperature (b) profiles for different Ri (aiding-buoyancy) at Ec = 0.0, Pr = 1.0 and fw = 0.0.
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with the boundary conditions; @f f ðn; 0Þ þ 2n @n ¼ fw ; f 0 ðn; 0Þ ¼ 0; hðn; 0Þ ¼ 1;
f 0 ðn; 1Þ ¼ 1;
ð10Þ
hðn; 1Þ ¼ 0;
1=2 where fw ¼ 2x v V w Rex , the case fw > 0 designates suction while fw < 0 indicates injection or blowing, Pr and Ec are the main flow parameters and defined as:
Pr ¼
lc p k
¼
v a
;
Ec ¼
u21 ; cp ðT w T 1 Þ
Rix ¼
Grx Re1=2 x
;
Grx ¼
gbðT w T 1 Þx3
v2
and Rex ¼
u1 x
v
;
ð11Þ
where Pr is the Prandtl number, Ec is the Eckert number, Rix is the local Richardson number, Grx is the local Grashof number, Rex is the local Reynolds number. 3. Numerical solution The system of transformed equations together with the boundary conditions, Eqs. (8)–(10), have been solved numerically using the Keller box scheme, an efficient and accurate finite-difference scheme, similar to that described in Cebeci and Bradshaw [9]. For the sake of brevity, details of the numerical method are not described, referring the reader to Cebeci and Bradshaw [9]. This is a very popular implicit scheme, which demonstrates the ability to solve systems of differential equations of
a
1.2
1.0 Ri=1.0 Ri=1.0 ξ=0.1 ξ=0.1
fw=0.1, Ec=0.0 fw=0.1, Ec=0.25
f'(ξ,η)
0.8
fw= -0.1, Ec=0.0 fw= -0.1, Ec=0.25
0.6
0.4
0.2
0.0 0
1
2
3
4
5
6
5
6
η
b
1.0 fw=0.1, Ec=0.0 fw=0.1, Ec=0.25
0.8
Ri=1.0 ξ=0.1
fw= -0.1, Ec=0.0 fw= -0.1, Ec=0.25
θ(ξ,η)
0.6
0.4
0.2
0.0 0
1
2
3
4
η Fig. 3. Dimensionless velocity (a) and temperature (b) profiles for different Ec and fw at Pr = 1.0, Ri = 1.0 and n = 0.1.
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O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
any order as well as featuring second-order accuracy (which can be realized with arbitrary non-uniform spacing), allowing very rapid x or n variations [10]. A set of non-linear finite-difference algebraic equations derived are then solved by using the Newton quazi-linearization method. The same methodology as that followed by Takhar and Beg [10] is followed. Therefore, for the finite-difference forms of the equations, we refer the reader to Ref. [10] for the brevity of the article. In the calculations, a uniform grid of the step size 0.01 in the g-direction and a non-uniform grid in the n-direction with a starting step size 0.001 and an increase of 0.05 times the previous step size were found to be satisfactory in obtaining sufficient accuracy within a tolerance better than 106 in nearly all cases. The value of g1 = 16 is shown to satisfy the velocity to reach the relevant stream velocity. In order to verify the accuracy of the numerical results, the validity of the numerical code developed has been checked for a limiting case. For Pr = 1.0, Ri = 0.0, and Ec = 0.0, we compare our h0 (0, 0) results with those given by Lin and Lin [11], Yih [12], Chamkha et al. [13], Nield and Kuznetsov [14] and Kuznetsov and Nield [15] (Table 1). For different Pr and Ri, we compare our h0 (0, 0) results with those given by Saeid [16] (Table 2). For Pr = 1.0 and different Ec, our h0 (n, 0) results are compared with those given by Watanabe and Pop [17], Yih [12] and Chamkha et al. [13] (Table 3). As it is seen from Tables 1–3 excellent agreements have been observed.
4. Results and discussion In the mixed convection case, depending on the direction of external flow, two different mixed convection regimes occur: aiding and opposing [8]. In this study, the aiding and opposing regimes of mixed convection are examined including the viscous dissipation and magnetic field effects. The following ranges of the main parameters are considered: Ri = 0.15, 0.1, 0.075, 0.05, 0.025, 0.0, 0.25, 0.5, 0.75, 1.0, 2.0 and 5.0; Pr = 1.0; fw = 0.1, 0.0, 0.1; Ec = 0.0 and 0.25 and magnetic interaction parameter n, the range 0.0 6 n 6 1.0. The combined effects of Ri, Ec, fw and n on the momentum and heat transfer are analyzed. The Richardson number, Ri represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the heat and fluid flow. Outside the mixed convection region, either the pure forced convection or the free convection analysis can be used to describe accurately the flow or the temperature field. Forced convection is the dominant mode of transport when Ri ? 0, whereas free convection is the dominant mode when Ri ? 1. Buoyancy forces can enhance the surface heat transfer rate when they assist the forced convection, and vice versa [8]. The Eckert number, Ec represents the relative importance of viscous dissipation to thermal diffusion. Viscous dissipation plays a role like an energy source and therefore it affects the temperature profile. In the first case, an upward forced external flow is considered. For the heated wall case (Tw > T1), the upward free flow caused by the buoyancy is in the same direction with the external forced flow. This case is called the aiding mixed flow. For
2.0
2.0
2.0
1.5
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
1.0 fw= -0.1
0.5 0.0
0.2
0.4
0.6
0.8
f''(ξ,0)
c 2.5
f''(ξ,0)
b 2.5
f''(ξ,0)
a 2.5
1.5
fw=0.0
0.5
1.0
0.0
0.2
0.4
ξ
0.6
0.8
fw=0.1
0.5
1.0
0.0
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.2 0.1 0.0 -0.1 0.0
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
0.2
0.3 0.2 0.1
fw= -0.1
0.4
ξ
0.0 0.6
0.8
1.0
-θ'(ξ,0)
0.6
0.3
-0.1 0.0
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
0.2
0.0 0.6
0.8
1.0
0.4
ξ
0.6
0.8
1.0
0.2 0.1
ξ
0.2
0.3
fw=0.0
0.4
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
1.0
ξ
-θ'(ξ,0)
-θ'(ξ,0)
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
1.0
1.5
-0.1 0.0
Ec=0.0, Ri=0.5 Ec=0.0, Ri=1.0 Ec=0.0, Ri=2.0 Ec=0.25, Ri=0.5 Ec=0.25, Ri=1.0 Ec=0.25, Ri=2.0
0.2
fw=0.1
0.4
0.6
0.8
1.0
ξ
Fig. 4. Effects of Ri (aiding-buoyancy), Ec and n on the local skin friction parameter and local heat transfer parameter for fw = 0.1 (a) fw = 0.0 (b) and fw = 0.1 (c).
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the case without the viscous dissipation effect (Ec = 0) and impermeable wall, Fig. 2 shows velocity and temperature profiles for different values of the Richardson number, Ri and magnetic interaction parameter, n. An increase at Ri result in increasing velocities due to addition of buoyancy-induced flow onto the external forced flow. In order to understand the effect of the magnetic interaction parameter, we should examine Eqs. (2 and 3) closely. The rB2 sign of the last term in the right hand side of Eq. (2), q0 ðu u1 Þ, is directly related with the sign of (u u1). For the forced convection regime, when Ri ? 0, this term will always be negative since u1 > u. Therefore, it will generate a force in the main flow direction, which will aid the main flow. However, for the free convection regime, when Ri ? 1, buoyancy-driven flow will dominate the external flow and therefore this term will always be positive since u1 < u, which will slow down the rB20 u , represents the imposed pressure force in the inviscid main flow. In fact, this term has two components: The first one, 1 q rB2 region of the conducting fluid, while the second one, q0 u, represents the Lorentz force imposed by a transverse magnetic field to an electrically conducting, which slows down the fluid motion in the boundary layer region. When the imposed pressure force overcomes the Lorentz force, i.e. u1 > u, the effect of the magnetic interaction parameter is to increase velocity. Similarly, when the Lorentz force dominates over the imposed pressure force, i.e. u1 < u, the effect of the magnetic interaction parameter will decrease velocity. In the mixed convection regime, when Ri increases the Lorentz force and therefore opposes the flow and therefore velocity decreases. However, decreasing Ri strengthens the imposed pressure force, which aids the flow by increasing the velocity (Fig. 2a). The last term in the right hand side of Eq. (3), rB20 2 2 qcp ðu u1 uÞ, will behave like a 2heat sink or a heat source depending on the sign of (u u1u). This term consists of rB work, which is supposed two components. The first one, qcp0 u2 , represents the Ohmic heating due to the electromagnetic rB2 to behave like a heat source by increasing the fluid temperature, while the second one qcp0 u1 u is the stress work which
a 1.0 Ec=0.0 fw=0.0 ξ=0.1 ξ=0.0
0.8
0.6
f'(ξ,η)
Ri=0.0, -0.05, -0.1
0.4
0.2
0.0 0
2
4
6
8
ξ
b
1.0 Ec=0.0 fw=0.0 ξ=0.1 ξ=0.0
0.8
θ(ξ,η)
0.6
Ri=0.0, -0.05, -0.1
0.4
0.2
0.0 0
1
2
3
4
5
6
ξ Fig. 5. Dimensionless velocity (a) and temperature (b) profiles for different Ri (opposing-buoyancy) at Ec = 0.0, Pr = 1.0, fw = 0.0.
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O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
decreases the fluid temperature. When the latter component suppresses the former one, this term will serve as a heat sink decreasing the fluid temperature (Fig. 2b). Fig. 3 illustrates the effects of Ec and fw at Ri = 1 and n = 0.1. As expected, when compared to an impermeable plate, the suction (fw > 0) decreases velocity and temperature while injection (fw > 0) increases them. Therefore, suction of fluid at wall reduces both the hydrodynamic and thermal boundary layers, which result in increasing fluid velocity and decreasing fluid temperature. Opposite is true for the case of injecting fluid at wall. The viscous dissipation, as a heat generation inside the fluid, increases the bulk fluid temperature. As seen from Fig. 3, an increase in Ec increases fluid temperature. Fig. 4 shows the effect of the magnetic interaction parameter, n on the momentum and heat transfer for various values of Ec and fw. As it is stated above, the effect of this parameter will vary according to the value of Ri. In the forced regime (Ri ? 0), the magnetic field parameter, increasing n will increase the local skin friction parameter, f 00 ðn; 0Þ and the heat transfer parameter, h0 (n, 0) as a result of increased velocity and temperature gradients at wall. In the free convection regime (Ri ? 1), an increase in n will increase f 00 ðn; 0Þ and will decrease h0 (n, 0). In the mixed convection regime, the level of Ri will determine the effect of n. As expected, increasing Ec increases local skin friction and decreasing local heat transfer parameters. As seen, effect of Ec becomes more considerable with increasing Ri. Fig. 4 also aims to explore the effects of the suction/injection parameter on the temperature profile. As expected, when compared to an impermeable wall case (fw = 0, Fig. 4b) increasing fw increases the local skin friction and the local heat transfer parameter since injecting fluid into the boundary layer (fw = 0.1, Fig. 4a) broadens the velocity distribution and increases the hydrodynamic boundary layer thicknesses, while sucking of fluid from the boundary layer through wall (fw = 0.1, Fig. 4c) will results in an opposing effect.
a
1.0 Ri=-0.1 ξ=0.1 0.8 fw=0.1, Ec=0.0 fw=0.1, Ec=0.25
0.6
f'(ξ,η)
fw=-0.1, Ec=0.0 fw=-0.1, Ec=0.25
0.4
0.2
0.0 0
2
4
6
8
ξ
b 1.0
Ri=-0.1 ξ=0.1
0.8 fw=0.1, Ec=0.0 fw=0.1, Ec=0.25 fw=-0.1, Ec=0.0
θ(ξ,η)
0.6
fw=-0.1, Ec=0.25 0.4
0.2
0.0 0
1
2
3
4
5
6
ξ Fig. 6. Dimensionless velocity (a) and temperature (b) profiles for different Ec and fw at Pr = 1.0, Ri = 0.1 and n = 0.1.
4094
b 1.0
c 1.0
0.8
0.8
0.8
0.6
0.6
0.4
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.0
0.0
0.2
0.4
0.45
ξ
0.6
0.8
0.0 0.0
1.0
0.35
0.35
0.20 0.0
fw= -0.1 Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.2
0.4
0.6
ξ
0.2
0.4
0.45 0.40
0.25
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.2
0.40
0.30
fw=0.0
0.4
0.8
ξ
0.0 0.0
1.0
0.2
0.4
ξ
0.6
0.8
1.0
0.40
fw=0.0
0.30
0.20 0.0
0.8
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.45
Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.25
1.0
0.6
fw=0.1
0.4 0.2
-θ'(ξ,η)
fw= -0.1
0.2
-θ'(ξ,0)
f''(ξ,0)
0.6
f''(ξ,0)
1.0
-θ'(ξ,0)
f''(ξ,0)
a
O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
0.2
0.4
0.6
ξ
0.8
0.35 fw=0.1 Ec=0.0, Ri= -0.05 Ec=0.0, Ri= -0.1 Ec=0.0, Ri= -0.15 Ec=0.25, Ri= -0.05 Ec=0.25, Ri= -0.1 Ec=0.25, Ri= -0.15
0.30 0.25
1.0
0.20 0.0
0.2
0.4
ξ
0.6
0.8
1.0
Fig. 7. Effects of Ri (opposing-buoyancy), Ec and n on the local skin friction parameter and local heat transfer parameter for fw = 0.1 (a) fw = 0.0 (b) and fw = 0.1 (c).
In the second case, a downward forced external flow is assumed. The upward flow caused by the buoyancy for the heated wall case (Tw > T1) has a retarding effect on the external forced flow, which is called as opposing mixed flow. For the case without the viscous dissipation effect (Ec = 0) and the impermeable wall (fw = 0.1), the velocity and temperature profiles for different values of the Richardson number, Ri are shown in Fig. 5. An increase at Ri results in decreasing velocities because of opposing effect of the upward buoyancy-induced flow on the downward external forced flow. For this opposing mixed convection regime, Fig. 6 illustrates the effects of Ec and fw at Ri = 1 and n = 0.1. As expected, the suction (fw > 0) decreases velocity and temperature while injection (fw > 0) increases them. The viscous dissipation increases the bulk fluid temperature. Therefore, in addition to opposing effect of the buoyancy, viscous dissipation will have an opposing effect on the heat transfer, too. For the opposing mixed convection case, Fig. 7 shows the effect of the magnetic interaction parameter, n on the momentum and heat transfer for various values of Ec and fw. For this case, increasing n increases the local skin friction, f 00 ðn; 0Þ and the local heat transfer parameter, h0 (n, 0). Similar to the case of the aiding mixed convection, when compared to the impermeable wall case, the heat transfer parameter increases at the suction case, while it is decreased at the injection case. When injection is applied at the wall, hydrodynamic and thermal boundary layers will thicken, thereby resulting in a decrease in the heat transfer parameter. Results of the values of the local skin friction parameter, f 00 ðn; 0Þ and the local heat transfer parameter h0 (n, 0) for various values of the suction/injection parameter, fw, the mixed convection parameter, Ri, the viscous dissipation parameter, Ec and the magnetic parameter, n are given in Table 4. 5. Conclusions Steady MHD mixed convection about a heated vertical plate has been analyzed including the effects of suction/injection at the wall and viscous dissipation. A transformed set of non-similar equations have been solved using the Keller box scheme, an efficient and accurate finite-difference scheme. Both the aiding- and opposing-buoyancy modes of the mixed convection have been studied. The conclusions drawn from the study can be summarized as follows: (i) As expected, the mixed convection parameter (i.e. the Richardson number, Ri) increases the momentum and heat transfer for the aiding-buoyancy mode, while the opposite is true for the opposing-buoyancy mode. (ii) Suction at the wall increases the local skin friction parameter and the local heat transfer parameter due to decreased thermal boundary layer thickness, while injection has an opposite effect. (iii) The viscous dissipation increases the fluid temperature and therefore decreases the temperature gradient at the wall, which results in decreased heat transfer values. (iv) The presence of the magnetic field influences flow and temperature fields. The effect of the magnetic interaction parameter is found to vary according to the value of the Richardson number, Ri.
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Table 4 Results of the values of the local skin friction parameter, f 00 ðn; 0Þ and the local heat transfer parameter h0 (n, 0) for various values of the suction/injection parameter, fw, the mixed convection parameter, Ri, the viscous dissipation parameter, Ec and the magnetic parameter, n. n
fw = 0.1
fw = 0.0
Ec = 0.0 f 00 ðn; 0Þ
Ec = 0.25
fw = 0.1
Ec = 0.0
Ec = 0.25
Ec = 0.0
Ec = 0.25
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
f 00 ðn; 0Þ
h0 (n, 0)
Ri = 0.1 0.0 0.1285 0.05 0.2138 0.10 0.2772 0.15 0.3343 0.20 0.3846 0.25 0.4314 0.30 0.4746 0.35 0.5154 0.40 0.5538 0.45 0.5905 0.50 0.6255 0.55 0.6591 0.60 0.6913 0.65 0.7225 0.70 0.7527 0.75 0.7819 0.80 0.8102 0.85 0.8378 0.90 0.8646 0.95 0.8908 1.0 0.9163
0.2599 0.2879 0.3037 0.3169 0.3272 0.3362 0.3438 0.3507 0.3567 0.3623 0.3673 0.3719 0.3762 0.3801 0.3838 0.3873 0.3905 0.3936 0.3965 0.3992 0.4018
0.1194 0.2079 0.2725 0.3304 0.3814 0.4287 0.4722 0.5133 0.5520 0.5889 0.6240 0.6578 0.6903 0.7216 0.7518 0.7812 0.8096 0.8373 0.8642 0.8904 0.9160
0.2322 0.2595 0.2749 0.2877 0.2979 0.3067 0.3144 0.3212 0.3274 0.3331 0.3383 0.3431 0.3476 0.3518 0.3558 0.3596 0.3632 0.3665 0.3698 0.3729 0.3758
0.1776 0.2536 0.3138 0.3685 0.4174 0.4631 0.5054 0.5456 0.5834 0.6197 0.6543 0.6875 0.7195 0.7505 0.7804 0.8094 0.8376 0.8650 0.8917 0.9177 0.9431
0.3011 0.3235 0.3377 0.3498 0.3595 0.3679 0.3752 0.3818 0.3876 0.3930 0.3978 0.4023 0.4065 0.4103 0.4139 0.4173 0.4205 0.4235 0.4263 0.4290 0.4316
0.1697 0.2479 0.3091 0.3646 0.4141 0.4602 0.5029 0.5434 0.5815 0.6180 0.6527 0.6862 0.7184 0.7494 0.7795 0.8086 0.8369 0.8644 0.8912 0.9173 0.9428
0.2697 0.2913 0.3053 0.3170 0.3266 0.3349 0.3423 0.3489 0.3549 0.3604 0.3655 0.3702 0.3746 0.3787 0.3826 0.3863 0.3899 0.3932 0.3964 0.3995 0.4024
0.2247 0.2940 0.3513 0.4038 0.4513 0.4959 0.5374 0.5768 0.6141 0.6498 0.6840 0.7169 0.7486 0.7793 0.8090 0.8378 0.8657 0.8930 0.9195 0.9454 0.9707
0.3410 0.3597 0.3726 0.3836 0.3926 0.4006 0.4075 0.4138 0.4194 0.4245 0.4292 0.4336 0.4376 0.4413 0.4448 0.4481 0.4512 0.4542 0.4569 0.4596 0.4621
0.2175 0.2885 0.3467 0.3999 0.4480 0.4929 0.5348 0.5745 0.6121 0.6481 0.6824 0.7155 0.7474 0.7782 0.8080 0.8369 0.8650 0.8923 0.9189 0.9449 0.9702
0.3057 0.3237 0.3363 0.3471 0.3561 0.3640 0.3710 0.3774 0.3831 0.3885 0.3934 0.3980 0.4023 0.4064 0.4102 0.4138 0.4173 0.4206 0.4237 0.4267 0.4296
Ri = 0.0 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
0.2963 0.3536 0.4036 0.4503 0.4934 0.5341 0.5724 0.6090 0.6439 0.6774 0.7096 0.7407 0.7708 0.7999 0.8282 0.8557 0.8824 0.9085 0.9340 0.9589 0.9832
0.2975 0.3112 0.3217 0.3309 0.3387 0.3457 0.3519 0.3576 0.3627 0.3675 0.3718 0.3759 0.3797 0.3832 0.3866 0.3897 0.3927 0.3955 0.3981 0.4007 0.4031
0.2963 0.3536 0.4036 0.4503 0.4934 0.5341 0.5724 0.6090 0.6439 0.6774 0.7096 0.7407 0.7708 0.7999 0.8282 0.8557 0.8824 0.9085 0.9340 0.9589 0.9832
0.2621 0.2747 0.2846 0.2933 0.3009 0.3077 0.3138 0.3194 0.3246 0.3294 0.3339 0.3381 0.3421 0.3459 0.3494 0.3528 0.3561 0.3592 0.3621 0.3650 0.3677
0.3320 0.3871 0.4357 0.4813 0.5236 0.5636 0.6015 0.6376 0.6722 0.7054 0.7373 0.7682 0.7981 0.8270 0.8551 0.8825 0.9091 0.9351 0.9605 0.9852 1.0090
0.3320 0.3445 0.3543 0.3630 0.3705 0.3772 0.3832 0.3886 0.3936 0.3982 0.4024 0.4064 0.4101 0.4136 0.4168 0.4199 0.4228 0.4255 0.4282 0.4306 0.4330
0.3320 0.3871 0.4357 0.4813 0.5236 0.5636 0.6015 0.6376 0.6722 0.7054 0.7373 0.7682 0.7981 0.8270 0.8551 0.8825 0.9091 0.9351 0.9605 0.9852 1.0090
0.2927 0.3043 0.3136 0.3219 0.3292 0.3357 0.3417 0.3471 0.3522 0.3569 0.3613 0.3655 0.3694 0.3731 0.3766 0.3799 0.3831 0.3862 0.3891 0.3919 0.3946
0.3686 0.4216 0.4689 0.5134 0.5549 0.5942 0.6315 0.6672 0.7014 0.7343 0.7659 0.7966 0.8262 0.8550 0.8829 0.9101 0.9366 0.9624 0.9876 1.0120 1.0360
0.3671 0.3786 0.3877 0.3959 0.4030 0.4094 0.4151 0.4204 0.4252 0.4297 0.4338 0.4376 0.4412 0.4446 0.4478 0.4508 0.4536 0.4563 0.4589 0.4613 0.4636
0.3686 0.4216 0.4689 0.5134 0.5549 0.5942 0.6315 0.6672 0.7014 0.7343 0.7659 0.7966 0.8262 0.8550 0.8829 0.9101 0.9366 0.9624 0.9876 1.0120 1.0360
0.3239 0.3346 0.3434 0.3513 0.3582 0.3645 0.3703 0.3756 0.3805 0.3851 0.3894 0.3934 0.3973 0.4009 0.4044 0.4077 0.4108 0.4138 0.4167 0.4195 0.4221
Ri = 0.1 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
0.4307 0.4767 0.5188 0.5589 0.5966 0.6327 0.6671 0.7003 0.7321 0.7629 0.7927 0.8216 0.8496 0.8768 0.9034 0.9293 0.9545 0.9792 1.0030 1.0270 1.0500
0.3215 0.3304 0.3379 0.3447 0.3507 0.3563 0.3613 0.3660 0.3702 0.3743 0.3780 0.3815 0.3848 0.3879 0.3909 0.3936 0.3963 0.3988 0.4012 0.4035 0.4057
0.4356 0.4810 0.5227 0.5624 0.5998 0.6356 0.6698 0.7027 0.7343 0.7650 0.7946 0.8233 0.8512 0.8783 0.9047 0.9305 0.9557 0.9803 1.0040 1.0280 1.0510
0.2761 0.2841 0.2911 0.2975 0.3033 0.3086 0.3135 0.3181 0.3224 0.3265 0.3303 0.3339 0.3374 0.3407 0.3438 0.3468 0.3497 0.3525 0.3552 0.3578 0.3602
0.4614 0.5065 0.5479 0.5874 0.6247 0.6605 0.6946 0.7275 0.7591 0.7897 0.8193 0.8481 0.8759 0.9031 0.9295 0.9553 0.9805 1.0050 1.0290 1.0530 1.0760
0.3535 0.3620 0.3692 0.3758 0.3816 0.3870 0.3919 0.3965 0.4007 0.4046 0.4083 0.4117 0.4149 0.4180 0.4209 0.4236 0.4262 0.4287 0.4311 0.4334 0.4355
0.4663 0.5108 0.5518 0.5910 0.6280 0.6634 0.6973 0.7300 0.7614 0.7918 0.8213 0.8499 0.8776 0.9046 0.9309 0.9566 0.9817 1.0060 1.0300 1.0540 1.0770
0.3046 0.3123 0.3191 0.3253 0.3310 0.3362 0.3410 0.3456 0.3498 0.3538 0.3576 0.3612 0.3646 0.3679 0.3710 0.3740 0.3768 0.3796 0.3822 0.3848 0.3873
0.4932 0.5373 0.5781 0.6170 0.6539 0.6892 0.7230 0.7556 0.7870 0.8174 0.8468 0.8753 0.9031 0.9301 0.9564 0.9821 1.0070 1.0320 1.0560 1.0790 1.1020
0.3864 0.3945 0.4014 0.4077 0.4134 0.4186 0.4233 0.4278 0.4319 0.4357 0.4393 0.4426 0.4458 0.4488 0.4516 0.4543 0.4569 0.4593 0.4617 0.4639 0.4660
0.4980 0.5417 0.5820 0.6206 0.6571 0.6922 0.7258 0.7582 0.7893 0.8196 0.8488 0.8772 0.9048 0.9317 0.9579 0.9834 1.008 1.0330 1.0570 1.0800 1.1030
0.3339 0.3414 0.3479 0.3540 0.3595 0.3646 0.3693 0.3737 0.3779 0.3819 0.3856 0.3891 0.3925 0.3957 0.3988 0.4018 0.4046 0.4073 0.4099 0.4125 0.4149
4096
O. Aydın, A. Kaya / Applied Mathematical Modelling 33 (2009) 4086–4096
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