Transp Porous Med DOI 10.1007/s11242-014-0333-4
Mixed Convection in a Darcy–Brinkman Porous Medium with a Constant Convective Thermal Boundary Condition Asterios Pantokratoras
Received: 7 April 2014 / Accepted: 8 May 2014 © Springer Science+Business Media Dordrecht 2014
Abstract The characteristics of the boundary layer flow past a plane surface adjacent to a saturated Darcy–Brinkman porous medium are investigated in this paper. The flow is driven by an external free stream moving with constant velocity. The surface is heated with a convective boundary condition with constant heat transfer coefficient. The problem is non-similar and is investigated numerically by a finite difference method. The problem is governed by four non-dimensional parameters, that is, the convective Darcy number, the convective Grashof number, the Prandtl number, and the axial distance along the plate. The influence of these parameters on the results is investigated, and the results are presented in tables and figures. The Darcy term and the Grashof term in the momentum equation contradict each other and this contradiction makes the problem complicated. However, the wall shear stress and the wall temperature increase continuously along the plate and the wall temperature always tends to 1. Keywords Darcy-Brinkman porous medium · Mixed convection · Heat transfer · Convective parameter · Non-similar
1 Introduction The idea of using a convective thermal boundary condition was first introduced by Merkin (1994) for the problem of free convection past a vertical flat plate. More recently, Aziz (2009), used the convective boundary condition to study the classical problem of forced convection boundary layer flow over a flat plate. The appearance of the above paper stimulated a large number of subsequent papers concerning different boundary layer problems. See for example, Ishak (2010), Makinde and Aziz (2010), Merkin and Pop (2011), Yao et al. (2011) and Magyari (2011) to mention just a few of them.
A. Pantokratoras (B) School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece e-mail:
[email protected]
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A. Pantokratoras
In a recent paper, Merkin et al. (2013) investigated the mixed convection on a vertical surface in a Darcy porous medium with a constant convective boundary condition. The present paper is an extension of the work of Merkin et al. (2013) to a Darcy–Brinkman porous medium.
2 Problem Definition and Solution Procedure Consider the flow along a vertical semi-infinite plane surface adjacent to a saturated Darcy– Brinkman porous medium with u and v denoting, respectively, the velocity components in the x and y directions, where x is the coordinate along the plate and y is the coordinate perpendicular to x. For a steady, two-dimensional flow, the boundary layer equations are ∂u ∂v + =0 ∂x ∂y ∂ 2u υϕ 2 ∂u ∂u Momentum equation: u +v = υe f f ϕ 2 2 − (u − u ∞ ) ∂x ∂y ∂y K
Continuity equation:
(1)
+ ϕ 2 gβ(T − T∞ ) ∂T ∂T ∂2T Energy equation: u +v =a 2 ∂x ∂y ∂y
(2) (3)
subject to boundary conditions u = v = 0, −k
∂T = h f (T f − T ) at y = 0, u → u ∞ , T → T∞ as y → ∞, ∂y
(4)
where ϕ is the porous medium porosity, υeff is the effective viscosity, υ is the normal kinematic viscosity, K is the porous medium permeability, a is the fluid thermal diffusivity, k is the fluid thermal conductivity, and T is the fluid temperature. It is assumed that the plate is heated by convection from another fluid with constant temperature T f with a constant heat transfer coefficient h f . Following Merkin and Pop (2011) the following dimensionless quantities have been introduced X = Y = U = V = Pr = = η=
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ϕ 2 υe f f h 2f u∞k2 hf y k u u∞ kv ϕ 2 υe f f h f υe f f ϕ2 a T − T∞ T f − T∞ Y X 1/2
x
(5) (6) (7) (8) (9) (10) (11)
Mixed Convection in a Darcy–Brinkman Porous Medium
Da = Gr c =
υk 2 υe f f K h 2f
(12)
k 2 1 gβ(T f − T∞ ) u∞ h 2f υe f f
(13)
The quantities given by Eqs. (12) and (13) are new non-dimensional parameters, which are introduced here and called convective Darcy number and convective Grashof numbers respectively. Using the above quantities, the Eqs. (1)–(3) take the following dimensionless form ∂U ∂V + =0 ∂X ∂Y ∂U ∂U ∂ 2U − Da(U − 1) + Gr c U +V = ∂X ∂Y ∂Y 2 ∂ ∂ 1 ∂ 2 U +V = ∂X ∂Y Pr ∂Y 2
(14) (15) (16)
The new boundary conditions are U (X, 0) = 0, V (X, 0) = 0, U (X, ∞) = 1 ∂ = −(1 − ) on Y = 0, = 0 as Y → ∞ ∂Y
(17) (18)
Equations (14)–(16) represent a two-dimensional parabolic problem. Such a flow has a predominant velocity in the streamwise coordinate which is the direction along the plate. In this type of flow, convection always dominates the diffusion in the streamwise direction. Furthermore, no reverse flow is acceptable in the predominant direction. The solution of this problem in the present work is obtained using a finite difference algorithm as described by Patankar (1980). In order to obtain complete form of both the temperature and velocity profiles at the same cross-section, a nonuniform lateral grid has been used. Y is small near the surface (dense grid points near the surface) and increases with Y. A total of 500 lateral grid cells were used. It is known that the boundary layer thickness changes along X. For that reason, the calculation domain must always be at least equal to or wider than the boundary layer thickness. In each case, the goal was to have a calculation domain wider than the real boundary layer thickness. This has been done by trial and error. If the calculation domain was thin, the velocity and temperature profiles were truncated. In this case, a wider calculation domain was used in order to capture the entire velocity and temperature profiles. The parabolic (space marching) solution procedure is described analytically in the textbook of Patankar (1980) which “remains to this day a model of simplicity and clarity and one of the most coherent explications of the finite volume technique ever written” (Acharya and Murthy 2007). That solution procedure is implicit and unconditionally stable (White 2006, p. 276), has been used extensively in the literature and has been included in fluid mechanics and heat transfer textbooks (see Anderson et al. (1984), p. 364; White (2006), p. 271; and Oosthuizen and Naylor (1999), p. 124). The method has been used successfully in a series of papers by the present author (Pantokratoras 2009a,b, 2010, 2014).
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A. Pantokratoras Table 1 Values of dimensionless wall temperature (0) for the non-similar case in forced convection flow with constant heat transfer coefficient in a clear fluid (no porous medium) for Pr = 1 (validation test) X
(0) Merkin and Pop (2011)
(0) Present method
0.001
0.062
0.062
0.124
0.457
0.446
1.867
0.784
0.780
8.335
0.895
0.892
110.335
1.0
0.970
3 Results and Discussion The problem is non-similar and is governed by four non-dimensional parameters, that is, the Prandtl number, the convective Darcy number, the convective Grashof number and the non-dimensional distance along the plate X. The most important parameters for this problem are the non-dimensional wall temperature, the non-dimensional wall shear stress, and non-dimensional wall heat transfer defined as ∂U ∂U U (0) = = X 1/2 (19) ∂η η=0 ∂Y Y =0 ∂ 1/2 ∂ (0) = =X (20) ∂η η=0 ∂Y Y =0 Before applying the current numerical solution procedure to the present problem, it was applied to a case treated by Merkin and Pop (2011) in order to check its accuracy. The results are shown in Table 1. Table 1 is a validation test of the present numerical solution procedure. The results of Merkin and Pop (2011) have been taken from their Fig. 4 due to a lack of numerical data in their work. Taking into account this fact that the agreement between the results of Merkin and Pop (2011) and the results obtained by the present solution procedure are satisfactory. Next results are presented for the mixed convection in a Darcy–Brinkman porous medium. Table 2 includes some of the results of the present work. More results are presented in the following figures. The influence of the convective Darcy number is shown in Figs. 1, 2, 3 and 4. It is seen that an increase of the Darcy number causes a reduction of the maximum velocity and an increase of the velocity boundary layer thickness. At a fixed value of the distance X, the wall shear stress is high at low Darcy numbers, then decreases reaching a minimum and increases again as Da increases. From Figs. 2 and 4, it is seen that as the Da increases the wall temperature and the temperature boundary layer thickness both increase. The influence of the convective Grashof number is presented in Figs. 5, 6, 7 and 8. As was expected, an increase of the Grashof number produces a higher velocity which is accompanied by higher wall shear stress. The influence on temperature is opposite, that is, as Grc increases the wall temperature and the temperature boundary layer thickness both get lower values. As the problem is non-similar the results change along the non-dimensional distance X. The influence of X is shown in Figs. 9, 10, 11, 12, 13, 14, 15 and 16. As X increases the velocity and wall shear stress both increase. The same behavior exists for wall temperate, that is, it increases with increasing X and always tends to 1 as X → ∞. The influence of the Prandtl number is presented in Figs. 17 and 18 where the usual behavior appears, that is, as the Pr number increases the velocity and temperature profiles become thinner.
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Mixed Convection in a Darcy–Brinkman Porous Medium Table 2 Values of wall shear stress U (0), wall heat transfer (0), and wall temperature (0) for different parameter values for Pr = 0.72 Da = 10 X
Gr c = 1 U (0)
Gr c = 10 − (0)
(0)
U (0)
Gr c = 100 − (0)
(0)
U (0)
− (0)
(0)
0.01
0.4391
0.0800
0.1912
0.4541
0.0790
0.1903
0.5903
0.0817
0.1830
0.1
1.0334
0.1897
0.3977
1.2276
0.1933
0.3879
2.6833
0.2070
0.3403
1
3.3324
0.3528
0.6460
4.7476
0.3796
0.6194
14.9964
0.4787
0.5203
10
10.8477
0.4646
0.8531
17.7400
0.5451
0.8277
73.2891
0.8801
0.7218
100
34.5683
0.4899
0.9509
60.7214
0.6011
0.9399
298.6614
1.1791
0.8820
1000
108.5379
0.4947
0.9844
0.6109
0.9807
1.2700
0.9598
0.0838
0.1644
195.194
1035.030
Da = 100 Gr c = 1 0.01 0.1 1 10
1.0095
Gr c = 10 0.0840
0.1670
1.0179
Gr c = 100 0.0823
0.1666
1.1000
3.1719
0.2057
0.3500
3.2529
0.2059
0.3484
4.0159
0.2101
0.3346
10.0612
0.3730
0.6269
10.5875
0.3762
0.6227
15.4895
0.4098
0.5894
0.4623
0.8538
0.4720
0.8505
0.5603
0.8226
100
100.249
31.8793
0.4803
0.9521
108.701
34.2554
0.4917
0.9508
192.008
57.0599
0.6049
0.9396
1000
301.222
0.4806
0.9848
327.603
0.4947
0.9844
590.207
0.6105
0.9807
Da = 1000 Gr c = 1 0.01 0.1
3.1554 10.000
Gr c = 10
Gr c = 100
0.0853
0.1400
3.1586
0.0846
0.1400
3.1908
0.0854
0.1396
0.2119
0.3252
10.0274
0.2121
0.3250
10.2989
0.2126
0.3232
1
31.6312
0.3796
0.6194
0.3801
0.6189
0.3840
0.6143
10
99.4705
0.4619
0.8535
100.229
0.4629
0.8531
107.784
0.4740
0.8497
0.4773
0.9522
299.373
0.4791
0.9521
324.658
0.4920
0.9508
100
296.797
31.8038
33.5156
Although most of the above results are clear, the present problem is complicated for the following reasons: In momentum Eq. (15) there are two source terms, the Darcy term and the Grashof term. In most cases U 1 and the Darcy term is negative, whereas the Grashof term is always positive. The two terms compete each other. From the classical convection theory, in a clear fluid, it is known that when the Grashof number increases, the fluid velocity increases. However, in a porous medium, the increase of velocity causes an increase of the term Da(U −1) which is negative, and this negative term tends to reduce the velocity increase due to increase of the Grashof number. There is continuous competition between the Darcy term and the Grashof term and it is difficult to find the ranges where each term is dominant taking into account that the temperature changes continuously along the plate. Probably for this reason some results show strange behavior. For example in Figs. 11 and 12, the increase of the wall shear stress is monotonic with Da (higher Da, higher U (0)), but in Fig. 13 this trend is destroyed.
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A. Pantokratoras 4 Da=0(clear fluid) Pr=0.72 Grc=100 X=1
3
U (η )
Da=10
2
Da=100
1
Da=1000
0 0
1
2
3
4
η
Fig. 1 Velocity profiles for Pr = 0.72, Grc = 100, X = 1, and different values of Darcy number
Pr=0.72 Grc=100 X=1
0.6
Θ (η)
0.4
Da=1000
0.2
100
Da=10 clear fluid Da=0
0 0
1
2
3
4
η
Fig. 2 Temperature profiles for Pr = 0.72, Grc = 100, X = 1, and different values of Darcy number
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Mixed Convection in a Darcy–Brinkman Porous Medium 1000 Pr=0.72 Grc=100
U'(0)
X=100
100 X=10
X=1
10 1
10
100
1000
Da
Fig. 3 Variation of wall shear stress as a function of Da for Pr = 0.72, Grc = 100, and different values of the distance X along the plate
1 Pr=0.72 Grc=100
X=100
0.9 X=10
Θ (0)
0.8
0.7
0.6
X=1
0.5
0.4 1
10
100
1000
Da
Fig. 4 Variation of wall temperature as a function of Da for Pr = 0.72, Grc = 100, and different values of the distance X along the plate
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A. Pantokratoras 4 Pr=0.72 Da=100 X=1
Grc=1000
U(η )
3
2
Grc=100 Grc=10
1
Grc=1
0 0
1
2 η
3
4
Fig. 5 Velocity profiles for Pr = 0.72, Da = 100, X = 1, and different values of the convective Grashof number
Pr=0.72 Da=100 X=1
0.6
Θ (η)
0.4 Grc=1
100
0.2 Grc=1000
0 0
1
2
η
3
4
5
Fig. 6 Temperature profiles for Pr = 0.72, Da = 100, X = 1, and different values of convective Grashof number
123
Mixed Convection in a Darcy–Brinkman Porous Medium 1000 Pr=0.72 Da=100
U'(0)
X=100
100
X=10
X=1
10 1
10
100
1000
Grc
Fig. 7 Variation of wall shear stress as a function of Grc for Pr = 0.72, Da = 100, and different values of the distance X along the plate
Pr=0.72 Da=100
1
X=100
0.9
X=10
Θ (0)
0.8
0.7
0.6
X=1
0.5
0.4 1
10
100
1000
Grc
Fig. 8 Variation of wall temperature as a function of Grc for Pr = 0.72, Da = 100, and different values of the distance Xalong the plate
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A. Pantokratoras 2 X=100
1.6
Pr=0.72 Grc=100 Da=100
X=10
X=1
U (η )
1.2 X=0.1
0.8 X=0.01
0.4
0 0
1
2
η
3
4
5
Fig. 9 Velocity profiles for Pr = 0.72, Da = 100, Grc = 100, and different values of the distance X
1
Pr=0.72 Grc=100 Da=100
X=100
0.8
0.6
Θ (η )
X=10
0.4 X=1
0.2
X=0.1
X=0.01
0 0
1
2
η
3
4
5
Fig. 10 Temperature profiles for Pr = 0.72, Da = 100, Grc = 100, and different values of the distance X
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Mixed Convection in a Darcy–Brinkman Porous Medium 1000 Pr=0.72 Grc=1
100 Da=1000
U '(0 )
Da=100
10 Da=10
1
0.1 0.01
0.1
1
10
100
X
Fig. 11 Variation of wall shear stress along the plate for Pr = 0.72, Grc = 1, and different values of the convective Darcy number
1000 Pr=0.72 Grc=10
100 Da=1000
U'(0)
Da=100
10 Da=10
1
0.1 0.01
0.1
1
10
100
X
Fig. 12 Variation of wall shear stress along the plate for Pr = 0.72, Grc = 10, and different values of the convective Darcy number
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A. Pantokratoras 1000 Pr=0.72 Grc=100
U'(0)
100
Da=1000
10 Da=100
Da=10
1
0.1 0.01
0.1
1
10
100
X
Fig. 13 Variation of wall shear stress along the plate for Pr = 0.72, Grc = 100, and different values of the convective Darcy number
1 Pr=0.72 Grc=1
Θ (0)
Da=10
100
Da=1000
0.1 0.01
0.1
1
10
100
X
Fig. 14 Variation of wall temperature along the plate for Pr = 0.72, Grc = 1, and different values of the convective Darcy number
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Mixed Convection in a Darcy–Brinkman Porous Medium 1 Pr=0.72 Grc=10
Θ (0)
Da=10
100
Da=1000
0.1 0.01
0.1
1
10
100
X
Fig. 15 Variation of wall temperature along the plate for Pr = 0.72, Grc = 10, and different values of the convective Darcy number
1 Da=1000 Pr=0.72 Grc=100
100
Θ (0)
Da=10
Da=10
Da=1000
0.1 0.01
0.1
1
10
100
X
Fig. 16 Variation of wall temperature along the plate for Pr = 0.72, Grc = 100, and different values of the convective Darcy number
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A. Pantokratoras Grc=100 Da=100 X=1
Pr=0.72
1.2
Pr=1 Pr=10
Pr=100
U(η)
0.8
0.4
0 0
1
2
3
4
η
Fig. 17 Velocity profiles for Da = 100, Grc = 100, X = 1, and different values of the Prandtl number
0.6 Grc=100 Da=100 X=1
0.4
Θ (η )
Pr=0.72
Pr=1
0.2 Pr=10
Pr=100
0 0
1
2
3
4
η
Fig. 18 Temperature profiles for Da = 100, Grc = 100, X = 1, and different values of the Prandtl number
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Mixed Convection in a Darcy–Brinkman Porous Medium
4 Conclusions The mixed convection problem along a vertical plate with constant convective thermal boundary condition in a Darcy-Brinkman porous medium has been investigated in this paper. The main conclusions can be summarized as follows: 1. In contrast to the corresponding problem of a Darcy porous medium which is governed by two non-dimensional parameters the Darcy-Brinkman problem is governed by four nondimensional parameters, that is, the convective Darcy number, the convective Grashof number, the Prandtl number, and the axial distance along the plate. 2. When the convective Darcy number increases the velocity decreases in contrast to temperature which increases and vice versa. A minimum appears in wall shear stress as Da varies at a fixed distance X. 3. An increase of the convective Grashof number causes an increase in velocity and a reduction in temperature. 4. The velocity, wall shear stress, and wall temperature all increase along the plate. In addition, the wall temperature always tends to 1 as X → ∞. 5. As the Pr number increases the velocity and temperature profiles become thinner. 6. There is a continuous competition between the Darcy term and the Grashof term in the momentum equation and this competition makes the problem complicated. Probably this competition is responsible for the appearance of a minimum in Fig.3 and the destruction of the monotonic behavior of of wall shear stress in Fig. 13.
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