c Pleiades Publishing, Ltd., 2016. ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2016, Vol. 57, No. 5, pp. 925–936. c A.A. Afify, Md.J. Uddin. Original Russian Text
LIE SYMMETRY ANALYSIS OF A DOUBLE-DIFFUSIVE FREE CONVECTIVE SLIP FLOW WITH A CONVECTIVE BOUNDARY CONDITION PAST A RADIATING VERTICAL SURFACE EMBEDDED IN A POROUS MEDIUM A. A. Afifya,b and Md. J. Uddinc
UDC 532.62
Abstract: A numerical study of a steady two-dimensional double-diffusive free convection boundary layer flow over a vertical surface embedded in a porous medium with slip flow and convective boundary conditions, heat generation/absorption, and solar radiation effects is performed. A scaling group of transformations is used to obtain the governing boundary layer equations and the boundary conditions. The transformed equations are then solved by the fourth- and fifth-order Runge–Kutta–Fehlberg numerical method with Maple 13. The results for the velocity, temperature, and concentration profiles, as well as the skin friction coefficient, the Nusselt number, and the Sherwood number are presented and discussed. Keywords: Lie group analysis, heat and mass transfer, porous medium, slip flow, convective boundary condition. DOI: 10.1134/S0021894416050217 INTRODUCTION The study of free convective heat and mass transfer in a porous medium has received noticeable attention from researchers during the last few decades due to its diverse applications, which include geothermal engineering, thermal insulation systems, packed bed chemical reactors, porous heat exchangers, oil separation from sand by steam, underground disposal of nuclear waste materials, food storage, and electronic device cooling. A comprehensive review on this area was presented by Nield and Bejan [1] and Ingham and Pop [2]. Double-diffusive convection is formed due to the combination of temperature and concentration gradients in the fluid, in which the thermal and mass diffusivities are different. Thus, the heat and mass transfer occurs simultaneously [3–8]. The cases of cooperating thermal and concentration buoyancy forces where both forces act in the same direction and opposing thermal and concentration buoyancy forces where both forces act in the opposite directions were considered in the literature. Additionally, Chamkha and Khaled [9] presented nonsimilar solutions for hydromagnetic simultaneous heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium. They found that the porous medium inertia effect decreases the local Nusselt number. Yih [10] examined the coupled heat and mass transfer in mixed convection over a variable heat flux/variable mass flux wedge in a porous medium for the entire regime by using the finite difference method. They showed that the local Nusselt number and the local Sherwood number increase with increasing buoyancy ratio parameter (ratio of the mass to thermal forces). Recently, Bachok et al. [11] studied a mixed convection boundary layer flow near the stagnation point on a vertical surface embedded in an anisotropic porous medium. They adopted the existing similarity transformation to convert the a
Department of Mathematics, Deanship of Educational Services, Qassim University, Buraidah, 51452 Saudi Arabia; afi
[email protected]. b Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt. c American International University-Bangladesh, Banani, Dhaka 1213, Bangladesh; jashim
[email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 57, No. 5, pp. 186– 198, September–October, 2016. Original article submitted July 25, 2014; revision submitted October 17, 2014. c 2016 by Pleiades Publishing, Ltd. 0021-8944/16/5705-0925
925
governing equations into a system of ordinary differential equations. They showed that dual solutions exist for both assisting and opposing flows. The method of the Lie group transformations was used to derive all group-invariant similarity solutions of the boundary layer equations. On the other hand, it is now well known that the classical Lie symmetry method can be used to find similarity solutions (see, e.g., [12–14]). Many authors applied the group method to solve various transport problems [15–18]. Recently, Jalil and Asghar [19] investigated similarity solutions for a boundary layer flow of a power-law fluid over a permeable stretching surface. They found that the surface wall gradient −f (0) decreases as the flow index parameter n increases both in shear thinning and shear thickening fluids. It may be remarked that earlier studies did not include the slip flow and convective surface boundary condition effects despite their frequent occurrence in many real boundary layer flows. This fact motivated researchers to investigate the combined effects of thermal and mass diffusion behaviors on mixed convection boundary layer flows with slip flow and convective surface boundary conditions. The aim of the present study is to investigate a double-diffusive free convection boundary layer flow over a vertical surface embedded in a porous medium under slip flow and convective surface boundary conditions, heat generation/absorption, and solar radiation effects.
1. MATHEMATICAL FORMULATION The problem geometry, the rectangular coordinates x and y, the corresponding velocity components u and v, and the flow configuration are depicted in Fig. 1. It is assumed that the uniform temperature of the ambient fluid is T∞ , the unknown temperature of the plate is Tw , and the left surface of the plate is heated from a hot fluid having a temperature Tf > T∞ by the process of convection. This then yields a variable heat transfer coefficient hf (x/L). It is assumed that the thermal radiation is present in the form of a unidirectional flux applied transversely to the plate surface and obeys the Rosseland diffusion approximation. This model is valid for optically thick media in which thermal radiation propagates within a limited distance prior to experiencing scattering or absorption. It is further assumed that the ambient fluid has a uniform concentration C∞ , and the unknown concentration of the plate is Cw . The fluid properties are assumed to be invariant, except for density, which is assumed to vary only in some particular cases (i.e., in the Boussinesq approximation). It is considered that there is internal heat generation or absorption. In the boundary layer approximation, the governing boundary layer equations relevant to our problem are written in the dimensional form as follows [20]: ∂u ∂v + = 0; ∂x ∂y
(1)
∂u ∂u ∂2u μ ρ u +v =μ u; + ρgβT (T − T∞ ) + ρgβC (C − C∞ ) − 2 ∂x ∂y ∂y k(¯ x/L)
(2)
u
∂T ∂T k2 ∂ 2 T Q(¯ x/L) 1 ∂q r +v = + − (T − T∞ ); ∂x ∂y ρCp ∂y 2 ρCp ∂y ρCp
(3)
∂C ∂C ∂2C +v =D . ∂x ∂y ∂y 2
(4)
u The boundary conditions are y = 0:
v = 0,
u = N1 (x/L)ν y → ∞:
∂u , ∂y
u → 0,
−k2
∂T = hf (x/L)[Tf − Tw ], ∂y
T → T∞ ,
C = Cw , (5)
C → C∞ .
Here T is the temperature, C is the concentration, ν is the kinematic viscosity, k2 is the thermal conductivity, D is the mass diffusivity of the medium, βT is the volumetric thermal coefficient, βC is the volumetric concentration coefficient, g is the acceleration due to gravity, α = k2 /(ρcp ) is the thermal diffusivity of the fluid, ρ is the density of the base fluid, μ is the dynamic viscosity, k(x/L) is the permeability of the porous medium, N1 (x/L) is the variable velocity slip factor, and Q(x/L) is the volumetric heat generation or absorption. 926
I
II
T, C Tf >T1
u
2
g 1 3
x, u y, v
Fig. 1. Flow configuration and coordinate system in a porous medium: (1) hot fluid; (2) quiescent fluid; (3) porous medium; the arrows show the flow directions; the boundaries of the hydrodynamic and thermal boundary layers are indicated by I and II, respectively.
The radiative heat flux term q r is written by employing the Rosseland diffusion approximation [21] as 4σ1 ∂T 4 , 3k1 ∂y where σ1 and k1 are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. Expanding T 4 into the Taylor series about T∞ and neglecting higher-order terms, we can write qr = −
3 4 − 3T∞ . T 4 ≈ 4T T∞
Therefore, Eq. (3) becomes u
3 ∂T k2 ∂ 2 T ∂T ∂2T 16σ1 T∞ Q(x/L) +v = + + (T − T∞ ). ∂x ∂y ρCp ∂y 2 3ρCp k1 ∂y 2 ρCp
(6)
1.1. Normalization Let us introduce the dimensionless variables x=
x , L
y=
yGr1/4 , L
u=
u , Ur
v=
vL νGr1/4
,
θ=
T − T∞ , Tf − T∞
ϕ=
C − C∞ . Cw − C∞
(7)
Here Gr = gβT (Tf − T∞ )L3 /ν 2 is the Grashof number based on the characteristic length L and Ur = gβT (Tf − T∞ )L is the reference velocity [22]. Let us also introduce the stream function ψ: u=
∂ψ , ∂y
v=−
∂ψ . ∂x
The continuity equation (1) is satisfied identically, and Eqs. (2), (4), and (6) yield Δ1 ≡
∂ψ ∂ 2 ψ ∂ 3 ψ ∂ψ ∂ 2 ψ νL ∂ψ − = 0; − − λ(θ + N ϕ) + ∂y ∂x ∂y ∂x ∂y 2 ∂y 3 k(x)Ur ∂y
Δ2 ≡ Pr
(8)
∂ψ ∂θ ∂ψ ∂θ ∂ 2θ Q(x)L − − (1 + R) 2 − Pr θ = 0; ∂y ∂x ∂x ∂y ∂y ρCp Ur
(9)
∂ψ ∂ϕ ∂ψ ∂ϕ ∂ 2 ϕ − − = 0. ∂y ∂x ∂x ∂y ∂y 2
(10)
Δ3 ≡ Sc
Here Pr = ρCp ν/k2 is the Prandtl number, Sc = ν/D is the Schmidt number, N = βC (Cw − C∞ )/(βT (Tf − T∞ )) is 3 /(3k1 k2 ) is the radiation parameter. the buoyancy ratio parameter, λ = gβT L(Tf − T∞ )/Ur2 = 1, and R = 16σ1 T∞ 927
The boundary conditions (5) take the form y = 0:
Gr1/4 N1 (x)ν ∂ 2 ψ ∂ψ = , ∂y L ∂y 2
∂ψ = 0, ∂x
∂ψ → 0, ∂y
y → ∞:
∂θ Lhf (x) =− (1 − θ), ∂y k2 Gr1/4
θ → 0,
ϕ = 1,
ϕ → 0.
1.2. Symmetry Analysis A one-parameter scaling group of transformations that is a simplified form of the Lie group transformation is selected [23, 24]: Γ:
x∗ = x eεα1 ,
y ∗ = y eεα2 ,
ψ ∗ = ψ eεα3 ,
N1∗ = N1 eεα7 ,
θ∗ = θ eεα4 ,
k ∗ = k eεα8 ,
ϕ∗ = ϕ eεα5 ,
h∗f = hf eεα6 , (11)
Q∗ = Q eεα9 .
Here ε is the parameter of the group and αi (i = 1, 2, . . . , 7) are arbitrary real numbers. Transformation (11) is treated as a point transformation that transforms the coordinates (x, y, ψ, θ, ϕ, hf , N1 , k, Q) to (x∗ , y ∗ , ψ ∗ , θ∗ , ϕ∗ , h∗f , N1∗ , k ∗ , Q∗ ). We now investigate the relationship among the exponents αi (i = 1, 2, . . . , 7): ∂ 3 ψ∗ Δj x∗ , y ∗ , u∗ , v ∗ , θ∗ , ϕ∗ , . . . , ∂y ∗3 ∂3ψ ∂3ψ = H1 x, y, u, v, θ, ϕ, . . . , 3 ; α Δj x, y, u, v, θ, ϕ, . . . , 3 , ∂y ∂y
j = 1, 2, 3.
According to this relationship, the differential forms Δ1 ,Δ2 , and Δ3 should be conformal invariants of transformations of the group Γ [23]. The system remains invariant under transformations of the group Γ if the following relations are satisfied: 2α3 − 2α2 − α1 = α3 − 3α2 = α4 = α5 = α3 − α2 − α8 , α4 + α3 − α2 − α1 = α4 − 2α2 = α4 + α9 ,
(12)
α5 + α3 − α2 − α1 = α5 − 2α2 . The boundary conditions are invariant under transformations of the group Γ if the following equations hold: α3 − α2 = α7 + α3 − 2α2 , α4 − α2 = α6 = α6 + α4 ,
(13)
α5 = 0. Solving Eqs. (12) and (13), we obtain α4 = α5 = 0,
α2 = α7 =
1 α1 , 4
α3 =
3 α1 , 4
1 α6 = − α1 , 4
α8 =
1 α1 , 2
1 α9 = − α1 . 2
To find the absolute invariants, we expand transformations (11) into the Taylor series, keeping terms up to the first order of ε inclusive. Thus, we have dx dy dψ dθ dϕ dN1 dQ dhf dk = = = = = = . = = x (1/4)y (3/4)ψ 0 0 (1/4)N1 (−1/4)hf (1/2)k (−1/2)Q 928
(14)
1.3. Similarity Transformations Solving Eq. (14), we obtain the similarity transformations (absolute invariants) η = yx−1/4 ,
ψ = x3/4 f (η),
hf = x−1/4 (hf )0 ,
N1 = x1/4 (N1 )0 ,
θ = θ(η),
ϕ = ϕ(η),
k = k0 x1/2 ,
Q = Q0 x−1/2 ,
(15)
where η is an independent similarity variable, f (η), θ(η), and ϕ(η) are the dimensionless velocity, temperature, and concentration function, respectively, (hf )0 is the constant heat transfer coefficient, (N1 )0 is the constant hydrodynamic slip factor, k0 is the constant permeability of the porous media, and Q0 is the constant heat generation or absorption. 1.4. Similarity Equations Substituting Eqs. (15) into Eqs. (8)–(10), we obtain the equations f + (3f f − 2f 2 − 4Kf ) + λ(θ + N ϕ) = 0,
(16)
(1 + R)θ + (3/4)Pr f θ + Pr Gθ = 0,
(17)
ϕ + (3/4)Scf ϕ = 0
(18)
subject to the boundary conditions f (0) = af (0),
f (0) = 0,
θ (0) = − Bi [1 − θ(0)],
ϕ(0) = 1,
f (∞) = θ(∞) = ϕ(∞) = 0,
(19)
where primes denote differentiation with respect to η, Bi = (hf )0 L/(k2 Gr1/4 ) is the Biot number, a = (N1 )0 νGr1/4 L is dimensionless hydrodynamic slip parameter, K = νL/(Ur k0 ) is the dimensionless permeability parameter, and G = Q0 L/(ρCp Ur ) is the dimensionless heat generation/absorption parameter. The physical parameters of interest in the present problem are the skin friction factor Cf x , the local Nusselt number Nux , and the local Sherwood number Shx , which can be found from the relations ∂T ∂C μ ∂u x x Cf x = − − , Nu = , Sh = . x x ρUr2 (x) ∂y y=0 Tf − T∞ ∂y y=0 Cw − C∞ ∂y y=0 By virtue of Eqs. (7) and (15), the dimensionless skin friction coefficient, local Nusselt number, and local Sherwood number can be presented as Gr1/4 x Cf x = f (0),
Gr1/4 x Nux = −θ (0),
Gr1/4 x Shx = −ϕ (0),
where Grx is the local Grashof number.
2. NUMERICAL SOLUTIONS The system of nonlinear ordinary differential equations (16)–(18) with the boundary conditions (19) form a two-point boundary-value problem, which is solved numerically by using an efficient fourth- and fifth-order Runge– Kutta–Fehlberg numerical method with Maple 13 for various values of flow parameters. The accuracy of the method is tested in various transport problems [18, 25]. The asymptotic boundary condition (19) is replaced at infinity by a finite value of ηmax : ηmax = 15,
f (15) = θ(15) = ϕ(15) = 0.
The choice ηmax = 15 ensures that all numerical solutions obey the far-field asymptotic values correctly. Pantokratoras [26] noticed that the erroneous result was found by many researchers in the field of convective heat and mass transfer because of taking small far-field asymptotic values of η during their numerical computations. The present numerical computations are carried out for 0 Bi 2, 0 R 2, 0 K 2, 0 a 1, 0 G 0.3, −1 N 1, Pr = 0.72, and Sc = 0.24. 929
Table 1. Heat transfer rates for Bi → ∞ and different values of the Prandtl number −θ (0) Pr
Present data
Data of Bejan [20]
0.01 0.72 1.00 2.00 10.00 100.00 1000.00
0.180 0.387 0.401 0.426 0.464 0.489 0.497
0.162 0.387 0.401 0.426 0.465 0.490 0.499
Table 2. Skin friction coefficients and heat transfer rates for different values of the Prandtl number −f (0)
−θ (0)
Pr
Data of Ostrach [22]
Present data
Data of Ostrach [22]
Present data
0,72 1.00 10.00
0.6760 0.6421 0.4192
0.67602 0.64219 0.41919
0.5046 0.5671 1.1694
0.50463 0.56715 1.16933
3. RESULTS AND DISCUSSION In order to assess the accuracy of the numerical method, we compared our results with those of Bejan [20] (Table 1) and found excellent agreement. We also compared the present results for f (0) and θ (0) with the previously published data of Ostrach [22] for Bi → ∞ and a = R = K = 0 (Table 2), which are also found to be in excellent agreement. Finally, the numerical results for different values of flow parameters are summarized in Table 3. As is seen from Table 3, owing to the uniform distributions of the Biot number, dimensionless radiation parameter, generation/absorption parameter, and slip parameter, the shear stress and the rates of heat and mass transfer decrease as the dimensionless permeability parameter increases. On the other hand, it is observed that the shear stress and the rates of heat and mass transfer increase with increasing Biot number and buoyancy ratio along the plate. All physical quantities decrease with increasing radiation parameter. It is also observed that the shear stress and the mass transfer rate increase whereas the heat transfer rate decreases with increasing generation/absorption parameter. Finally, the shear stress decreases whereas the rates of heat and mass transfer increase due to an increase in the dimensionless slip parameter. Furthermore, as the dimensionless slip parameter increases, permitting more fluid to slip past the plate, the skin friction coefficient decreases in magnitude and approaches zero. Figures 2–7 are drawn in order to see the influence of the Biot number, slip parameter a, radiation parameter R, permeability parameter K, generation/absorption parameter G, and buoyancy ratio parameter N on the velocity, temperature, and concentration fields at Pr = 0.72 and Sc = 0.24. It is seen from Fig. 2 that the velocity distribution at any point near the plate increases with increasing Bi, whereas the opposite result is observed further away from the plate. Simultaneously, the temperature increases, whereas the concentration decreases owing to an increase in the Biot number. At Bi = 0, the left side of the plate is thermally insulated from the medium on the right from the plate (the internal thermal resistance of the plate is extremely high), and no convective heat transfer between the left and right surfaces of the plate takes place. It is interesting to note that the peak velocity is low at Bi = 0. As the Biot number increases, the thermal resistance of the plate decreases; consequently, the peak velocity and the velocities in the neighborhood of the peak increase significantly. Moreover, it is interesting to note that the fluid temperature on the right side of the plate increases with an increase in the Biot number, the thermal resistance of the plate decreases, and the rates of heat and mass transfer on the right side of the plate increase (see Table 3). It is seen from Fig. 3 that the velocity distribution at any point near the plate slightly increases with increasing a, whereas the opposite result is observed further away from the plate. Furthermore, an increase in the slip parameter leads to a decrease in the flow velocity, because not all the pulling force of the plate can be transmitted to the fluid under the slip condition. On the other hand, the fluid temperature decreases with 930
(a)
f0 1.4
5 4 3 2 1
1.2 1.0
(b)
o 4
5 4
3
3
0.8
2
0.6 0.4
1
0.2
2
0
1
0 0
2
4
6
8
12 n
10
f 1.0
0
2
4
6
8
10
12 n
8
10
12 n
(c)
0.8 0.6 1 2
0.4 3
0.2
4 5
0 0
2
4
6
Fig. 2. Velocity (a), temperature (b), and concentration (c) profiles for N = λ = 1, a = G = K = 0.1, Pr = 0.72, Sc = 0.24, R = 0.5, and Bi = 0.4 (1), 0.8 (2), 1.2 (3), 1.6 (4), and 1.8 (5).
f0 1.2 5 4 3
o
(a)
(b)
0.6 2
1.0
0.5 1
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
1 2
0
2
4
6
8
10
12 n
3 4 5
0
2
4
6
8
10
12 n
Fig. 3. Velocity (a) and temperature (b) profiles for N = λ = 1, G = K = 0.1, Pr = 0.72, Sc = 0.24, Bi = R = 0.5, and a = 0 (1), 0.1 (2), 0.3 (3), 0.5 (4), and 0.8 (5).
931
(a)
f0 0.9 5
4
0.8
(b) o
3 2
0.6
1
0.7
0.5
0.6
0.4
0.5
5
0.4
0.3
0.3
0.2
4
0.2
3
0.1
0.1
2 1
0
0 0
2
4
6
8
12 n
10
0
2
4
6
8
10
12 n
Fig. 4. Velocity (a) and temperature (b) profiles for N = λ = 1, Bi = 0.5, a = G = K = 0.1, Pr = 0.72, Sc = 0.24, and R = 0 (1), 0.4 (2), 0.8 (3), 1.2 (4), and 2.0 (5).
(a)
f0 1 2 3
0.8 0.7
0.6
0.6
0.5
0.5
0.4
0.4
5 4 3
0.3
4
0.3
(b)
o 0.7
5
0.2
0.2
2
0.1
0.1
1
0
0 0
5
15 n
10 f 1.0
0
5
10
15 n
10
15 n
(c)
0.8 0.6 5 4
0.4 0.2
3 2 1
0 0
5
Fig. 5. Velocity (a) and temperature (b) profiles for N = λ = 1, Bi = R = 0.5, a = G = 0.1, Pr = 0.72, Sc = 0.24, and K = 0 (1), 0.5 (2), 1.0 (3), 1.5 (4), and 2.0 (5).
932
Table 3. Skin friction coefficients and rates of heat and mass transfer for Pr = 0.72, Sc = 0.24, and different values of K, N , G, a, Bi, R, and λ K
N
G
a
Bi
R
λ
f (0)
−θ (0)
−ϕ (0)
0.1 0.5 1.0 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.10 0.10 0.10 0.10 0.10 0.15 0.18 0.10 0.10 0.10 0.10 0.10 0.10
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5 1.0 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5 1.0 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 2.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.497 612 0.442875 0.402692 0.779532 1.131216 0.517824 0.557065 0.371884 0.279671 0.745404 0.830363 0.573449 0.546043
0.065 543 0.060536 0.054901 0.072974 0.077262 0.061849 0.058093 0.068087 0.069639 0.165996 0.209475 0.062723 0.061016
0.183 437 0.174352 0.167407 0.206939 0.231749 0.226268 0.228785 0.188984 0.192817 0.199246 0.204038 0.194673 0.190542
(à)
f0 4
3
0.7
(b)
o 1.0
2
0.8
0.6 1
0.5 0.6
4
0.4
3
0.3
0.4
0.2
2
0.2
1
0.1 0
0 0
2
4
6 f 1.0
10 n
8
0
2
4
6
8
10 n
(c)
0.8 0.6
1 2
0.4 3 4
0.2 0 0
2
4
6
8
10 n
Fig. 6. Velocity (a), temperature (b), and concentration (c) profiles for N = 0.5, λ = R = 1.0, Bi = 10000, a = 0, K = 0.5, Pr = 0.72, Sc = 0.24, and G = 0 (1), 0.1 (2), 0.2 (3), and 0.3 (4).
933
(a)
f0 5
0.7
4
0.7
3
0.6
(b)
o 0.8
0.6
0.5
0.5
0.4
1 2
0.4
3
2
0.3
0.3
1
0.2
0.2
0.1
0.1
0
0 0
5
15 n
10 f 1.0
4 5
0
5
10
15 n
10
15 n
(c)
0.8 0.6 1 2
0.4
3
0.2 4 5
0 0
5
Fig. 7. Velocity (a), temperature (b), and concentration (c) profiles for Bi = 0.5, λ = R = 1.0, Sc = 0.24, G = K = a = 0.1, Pr = 0.72, and N = −0.5 (1), −0.3 (2), 0 (3), 0.3 (4), and 0.5 (5).
increasing slip parameter a in the boundary layer region; as a consequence, the thickness of the thermal boundary layer decreases. It is seen from Fig. 4 that the velocity and temperature distributions increase with increasing radiation parameter R. As the Rosseland mean absorption coefficient k1 decreases, the heat transfer rate decreases, whereas the divergence ∂qr /∂y increases; as a consequence, the temperature increases. It is seen from Fig. 5 that the velocity decreases, whereas the temperature and concentration distributions increase with increasing permeability parameter K. That is because the presence of a porous medium increases the resistance to the flow, resulting in a decrease in the flow velocity and an increase in both the temperature and concentration distributions. It is seen from Fig. 6 that the velocity and temperature distributions increase, whereas the concentration distribution decreases with increasing heat generation/absorption parameter G. The buoyancy effects tend to increase the velocity of fluid motion along the plate surface (the concentration and temperature gradients have the same direction N > 0). Both the hydrodynamic and thermal boundary layers tend to increase as G increases, whereas the reverse trend is seen for the concentration boundary layer. The velocity, temperature, and concentration profiles are plotted in Fig. 7 for different values of the buoyancy ratio. At N > 0, the velocity increases, whereas the temperature and concentration distributions decrease. At N < 0, vice versa, the velocity decreases, whereas the temperature and concentration distributions increase. 934
CONCLUSIONS In the present paper, we performed a numerical study of a steady two-dimensional double-diffusive free convection boundary layer flow over a vertical surface embedded in a porous medium. With the use of the Lie group analysis, we first found the symmetries of the governing equations and then reduced the equations to ordinary differential equations. The main conclusions emerging from this study are listed below. The results presented in this paper can be helpful for possible technological applications in liquid-based systems involving stretchable materials. An increase in the permeability parameter leads to a decrease in the shear stress and the rates of heat and mass transfer. An increase in the Biot number and buoyancy ratio parameter leads to an increase in the shear stress and the rates of heat and mass transfer, whereas the radiation parameter reduces all of these physical quantities. An increase in the slip parameter leads to a decrease in the shear stress, whereas the rates of heat and mass transfer increase thereby. An increase in the Biot number leads to an increase in the velocity and temperature distributions, whereas the concentration distribution decreases. An increase in the slip parameter leads to an increase in the velocity distribution and to a decrease in the temperature distribution.
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