Arab J Sci Eng (2014) 39:3393–3401 DOI 10.1007/s13369-014-0991-0
RESEARCH ARTICLE - CHEMICAL ENGINEERING
Heat and Mass Transfer Effects on a Dissipative and Radiative Visco-Elastic MHD Flow over a Stretching Porous Sheet M. Kar · S. N. Sahoo · P. K. Rath · G. C. Dash
Received: 20 June 2012 / Accepted: 15 February 2013 / Published online: 8 March 2014 © King Fahd University of Petroleum and Minerals 2014
Abstract The objective of the present study is to consider a dissipative and radiative visco-elastic flow over a stretching porous sheet. The applicability of the present study is numerous in the polymer industry in stretching of sheets with dissipative environment. In view of its scope, the present study brings the work of Khan (Int J Heat Mass Transf 49, 1534– 1542, 2005) and Singh (Int Comm Heat Mass Transf 35, 637– 642, 2008) as particular cases. The solution of the momentum equation has been taken care of by similarity transformation and applying a particular solution satisfying boundary conditions. Further, the heat and mass transfer equations are solved by applying the confluent hypergeometric function (Kummer’s function). It is interesting to note that the presence of elastic element in the fluid, external magnetic field and suction at the plate reduce the skin friction favoring the stretching of sheets whereas heat loss has been minimized due to permeability of the medium, presence of heat source in case of low diffusive fluid. Keywords Visco-elastic · Heat source · Radiation · Viscous dissipation · Chemical reaction
M. Kar (B) Department of Mathematics, Christ College, Cuttack, 753007 Odisha, India e-mail:
[email protected] S. N. Sahoo · G. C. Dash Department of Mathematics, Institute of Technical Education and Research, Siksha ‘O’ Anusandhan University, Bhubaneswar, 751030 Odisha, India P. K. Rath Department of Mathematics, BRMIIT, Bhubaneswar, 751010 Odisha, India
1 Introduction The heat and mass transfer of an electrically conducting fluid with boundary layer behavior over a continuously moving flat wall has a lot of applications in various fields of electrochemistry, polymer technology, metallurgical operations and manufacturing process in industry. In polymer technology (where one deals with the stretching of plastic sheets) and specific metallurgical operations, the principles of magnetohydrodynamic techniques are applied to control the rate of cooling of continuous stretched strips or filaments by drawing them through a quiescent fluid. The application is well marked in case of drawing, annealing and thinning of copper wires. In all these cases, the properties of final product depend to a great extent on the rate of cooling. By drawing such strips in an electrically conducting fluid subject to a magnetic field, the rate of cooling can be controlled and the final products of desired characteristic might be achieved. Another interesting application of magnetohydrodynamics to metallurgy lies in the purification of molten metals from non-metallic inclusions by the application of magnetic field.
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In view of its application in various fields of science, the study of heat and mass transfer of an electrically conducting fluid has attracted the attentions of several authors. The momentum, heat and mass transfer in a viscous flow past a stretching sheet have been studied by Carragher and Crane [1]. Naseem and Khan [2] have investigated boundary layer flow past a stretching plate with suction, heat and mass transfer and with variable conductivity. Parida et al. [3] have studied the MHD heat and mass transfer in a rotating system with periodic suction. Anjali Devi et al. [4] have studied viscous dissipation effects on non-linear MHD flow in a porous medium over a stretching porous surface. Abdullah [5] found an analytical solution of heat and mass transfer over a permeable stretching plate affected by chemical reaction, internal heating, dufour-Soret effect and Hall effect using homotopy analysis method. Momentum, heat and mass transfer in a visco-elastic boundary layer over a linear stretching sheet have been studied extensively in the recent past because of its everincreasing usage in polymer processing industry. A new dimension is added to the study of visco-elastic boundary layer fluid flow, heat transfer by considering the effects of heat source, viscous dissipation and thermal radiation as well as mass transfer by considering the effect of chemical reaction. Khan et al. [6] have discussed visco-elastic MHD flow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work. Singh [7] has studied heat source and radiation effect on magneto convection flow of a visco-elastic fluid past a stretching sheet using a Kummer’s function. Sidheswar and Mahabalewar [8] have investigated MHD flow and heat transfer in a visco-elastic liquid over a stretching sheet with viscous dissipation, internal heat generation/absorption and radiation. Many other researchers like Datti et al. [9], Abel et al. [10], Cortell [11], Seddeek [12], Vajravelu [13] and Abel et al. [14] and Khan [15] have reported their works on visco-elastic boundary layer flow with heat transfer, but a few have considered mass transfer with chemical reaction. In the present analysis, we have considered a visco-elastic fluid of Walters B model. Further, we have incorporated heat source, radiation and viscous dissipation as well as mass transfer with chemical reaction aspect to the work of Singh et al. [7]. Our study is confined to convective MHD boundary layer flow of an incompressible visco-elastic fluid past a stretching porous wall embedded in a porous medium in the presence of transverse magnetic field. In the discussion part, we have studied the case of Sahoo and Dash [16] as a particular case by dropping the effect of elasticity, radiation, viscous dissipation and chemical reaction. Further, we have also discussed the case of Khan [15] as a particular case by omitting the effect of magnetic field, porosity and effect of mass transfer. The present study has relevance in its own right due to its method of solution and gen-
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eralized approach to study the flow and heat transfer phenomena of a visco-elastic fluid representing some common fluids.
2 Flow Analysis In this paper, we have considered a two-dimensional convective steady laminar flow of an incompressible electrically conducting visco-elastic fluid past a vertical stretching porous wall in the presence of heat source and chemical reaction. The flow is generated due to linear stretching of the boundary sheet, caused by simultaneous application of equal and opposite forces along x-axis while keeping the origin fixed (Fig. 1). A uniform magnetic field of strength B0 is applied to the flow. Let u and v be the components of the velocity along x- and y-axes, respectively. The stretching sheet is supposed to start from a thin slit at the origin and the speed of a point on the plate is proportional to its distance from the plate, but the boundary layer approximations still hold. We have neglected the induced magnetic field. Under these assumptions, the steady state boundary layer equations of visco-elastic (Walters B model) fluid are given by ∂v ∂u + =0 ∂x ∂y u
(2.1)
σ B02 ∂u ∂ 2u ∂u ν +v = ν 2 − ∗u − u ∂x ∂y ∂y Kp ρ ∂ 3u ∂u ∂ 2 u K0 ∂ 3u ∂u ∂ 2 u (2.2) u + . − + v − ρ ∂ x∂ y 2 ∂ y3 ∂ y ∂ x∂ y ∂ x ∂ y2
where v is the kinematic coefficient of viscosity, K 0 is the elasticity of the fluid and K ∗p is the permeability of the porous medium. The appropriate boundary conditions for the problem are y = 0 : u = Ax, v = −V0 and y → ∞ : u → 0
(2.3)
where V0 is the suction velocity across the stretching sheet and A, being a positive constant, is the stretching rate. x
B0
Boundary layer
Permeable stretching sheet
v= -V0 Slit
Fig. 1 Flow diagram
y
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Following Rajagopal [17], we have introduced following similarity transformations satisfying the Eq. (2.1) √ A u = Ax f (η), v = − Aν f (η), η = y (2.4) ν where prime denotes the differentiation with respect to η, the similarity variable, and f is the dimensionless stream function. Using Eq. (2.4) in Eq. (2.2), we get f 2 − f f = f − Rc (2 f f − f f iv − f 2 ) 1 f − M+ Kp
(2.5)
general method and obtained all the non-unique solutions of the equation of (2.7) with transverse magnetic field. Recently, Khan [15] obtained a closed form solution of the Eq. (2.5) subject to the boundary condition (2.6) in the absence of magnetic field and porosity. Among all these solutions, the solution (2.8) is realistic one, as one can recover the Navier–Stokes solution only in its limiting case Rc = 0. In the present study with magnetic field, porosity and suction, the most appropriate solution of the Eq. (2.5) with boundary condition (2.6) is given by 1 1 (2.11) f (η) = + N0 − e−αη α α
(Porosity parameter) and M = Aρ0 (magnetic parameter). The corresponding boundary conditions are
where α is a real positive root of the cubic equation 1 N0 Rc α 3 − (1 − Rc )α 2 + N0 α + M + +1 =0 Kp (2.12)
f (0) = 1, f (0) = N0 , f (∞) = 0
The skin friction in non-dimensional form is given by
where Rc =
K0 A ρν
(visco-elastic parameter), K p = σ B2
AK ∗p ν
(2.6)
where N0 = √V0 is the suction parameter. Aν Making use of the boundary conditions (2.6) with V0 = 0, Rajagopal et al. [17] obtained the corresponding solution of the Eq. (2.5) without magnetic field and porosity. Subsequently, Mcleod and Rajagopal [18] and Troy et al. [19] obtained unique solution of Eq. (2.5) in the form f (η) = 1 − e−η
(2.7)
when Rc = 0, M = 0 and K p → ∞. Without considering the effect of magnetic field and porosity, Troy et al. [19] also found a solution of the Eq. (2.5) in the form −√ η (2.8) f (η) = 1 − Rc 1 − e 1−Rc Later on Chang [20] showed that the solution of Eq. (2.5) satisfying the boundary conditions (2.6) along with V0 = 0, M = 0 and K p → ∞ is not unique. Taking Rc = 1/2, he presented a solution of the form √ √ 3η − √η f (η) = 2 1 − e 2 cos (2.9) 2 Further, Rao [21] derived another closed form solution of the Eq. (2.5) with boundary condition (2.6) of the form 1 f (η) = √ Rc ×
1−e
− 2√ηR
c
√
√ 3η 3η 1−2Rc cos √ + √ sin √ Rc 2 Rc 3 (2.10)
in which the effects of magnetic field and porosity were not considered. Moreover, Lawrence and Rao [22] presented a
τ = f (0) = −α
(2.13)
3 Heat Transfer Analysis We consider that the whole flow field is exposed under thermal radiation. To get the effect of temperature difference between the surface and the ambient fluid, we consider temperature-dependent heat source in the flow region. Since, the fluid considered for analysis is visco-elastic, the energy will be stored in the fluid by means of frictional heating due to viscous dissipation. So we take into account of this. However, we assume that the fluid possesses strong viscous property in comparison with the elastic property. Also, the effect of elastic deformation terms might not be significant as the momentum boundary layer equation is valid at low shear rate and small values of elastic parameter [17]. Numerous works are also available in the literature of visco-elastic boundary layer flow which recognizes this fact while studying heat transfer [8–14]. Taking all above discussions into account, the governing boundary layer equation for heat transfer as presented by Cortell [11] and Vajravelu and Soewono [13] is given by 2 ∂T K ∂2T ∂u ν ∂T +v = + u ∂x ∂y ρC P ∂ y 2 CP ∂y 1 ∂qr − S (T − T∞ ) − (3.1) ρC P ∂ y where K , S , ρ and C p are, respectively, the thermal conductivity of the fluid, the dimensional heat source parameter, the density of the fluid and the specific heat of the fluid at constant pressure. Using Rosseland approximation for radiation [23], we can write
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qr = −
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4σ ∂ T 4 3K ∗ ∂ y
(3.2)
where σ , the Stefan–Boltzmann constant and K ∗ is the absorption coefficient. Further, we assume that the temperature difference within the flow is such that T 4 may be expanded in the Taylor series. Hence, expanding T 4 about T∞ and neglecting higher order terms, we get 3 4 T 4 ≡ 4T∞ T − 3T∞
(3.3)
Therefore, the Eq. (10) is simplified to ∂T ∂T K ∂2T ν ∂u 2 u +v = + ∂x ∂y ρC p ∂ y 2 Cp ∂y +
3 ∂2T 16σ T∞ − S (T − T∞ ) 3K ∗ ρC p ∂ y 2
(3.4)
The appropriate boundary conditions on the temperature are y = 0 : T → Tw (= T∞ + A1 x r ), where A1 is a constant. y → ∞ : T → T∞
(3.5) To solve the Eq. (3.4), we now introduce the following non-dimensional temperature variable. θ (η) =
T − T∞ Tw − T∞
(3.6)
the Eq. (3.8) becomes Pr S 3Rd dθ d 2θ + 2− 2 θ ξ 2 + (1 − a0 − ξ ) dξ dξ d ξ 3Rd + 4 E c α 4 3Rd + 4 =− ξ (3.11) Pr 3Rd subject to the boundary conditions Pr 3Rd θ ξ =− 2 = 1, θ (ξ = 0) = 0 α 3Rd + 4
r Rd (1 + α N0 ). Equations (3.11) and where a0 = α 23P (3Rd +4) (3.12) constitute a non-homogeneous boundary value problem. Denoting the solution of the homogenous part of Eq. (3.11) by θc and further introducing the transformation θc = ξ δ1 w(ξ ), we get
d 2w dw a0 + b0 − 4 − w=0 (3.13) + (1 + b0 − ξ ) 2 dξ dξ 2 d 0 where δ1 = a0 +b and b = a02 + 4Pαr2 S ( 3R3Rd +4 ). 0 2 Now Eq. (3.13) is a Kummer’s equation (Abramowitz and Stegum [25]). Hence, the complementary function of Eq. (3.13) is ξ
θc (ξ ) = ξ δ1 [C1 F(δ1 − 1, 1 + b0 , ξ ) + C2 ξ −b0 F(δ1 − b0 − 1, 1 − b0 , ξ )]
T (y) = T∞ + A1 x r θ (η)
(3.7)
In the present case, we have assigned r = 2. Using Eqs. (2.4), (2.11), (3.6) and (3.7) in the Eq. (3.4), we obtain
(3.14)
To obtain the particular integral of Eq. (3.11), we take θ p (ξ ) = b0 + b1 ξ + b2 ξ 2 + 0(ξ 3 )
Following Nield and Bejan [24], the temperature T is now expressed as
(3.12)
(3.15)
Using (3.15) in (3.11) we get 3Rd +4 4 ξ2 3Rd α Ec θ p (ξ ) = − Pr P 4 − 2a − S Pr r
0
α2
3Rd 3Rd +4
(3.16)
Hence, the complete solution of the Eq. (3.11) is 3Pr Rd 3Rd Pr θ + f θ − (2 f + S)θ 3Rd + 4 3Rd + 4 3Rd Pr =− E c f 2 3Rd + 4
where Pr =
ρνC p K
θ (ξ ) = θc (ξ ) + θ p (ξ ) (3.8)
(Prandtl number), S =
dimensional heat source parameter), E c =
A2 A1 C p
S A
(non-
θ (0) = 1, θ (∞) = 0 By introducing the new variable Pr 3Rd ξ =− 2 e−αn α 3Rd + 4
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Now making use of the boundary condition (3.12) and changing the variable ξ to η, the complete solution of Eq. (3.11) is given by
⎡
(Eckert
K K∗ number) and Rd = 4σ 3 (Radiation parameter). T∞ The boundary conditions (3.5) now reduce to
θ (η) = ⎣1 +
(3.9)
F
a0 +b0 2
E c Pr 4 − 2a0 −
3Rd 3Rd +4
S Pr α2
3Rd 3Rd +4
⎤ ⎦
Pr 3Rd e−αη α 2 3Rd +4 d 0 F( a0 +b − 2, 1+b0, − αP2r 3R3Rd +4 ) 2
⎡
(3.10)
(3.17)
−⎣
− 2, 1+b0, −
E c Pr 4−2a0 −
3Rd 3Rd +4 S Pr 3Rd α 2 3Rd +4
a0 +b0 αη exp − 2
⎤
⎦ e−2αη
(3.18)
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The heat transfer in terms of Nusselt number (Nu = θ (0)) at the wall is calculated with the help of Eq. (3.18) as d ) E c Pr ( 3R3Rd +4 Nu = 1 + d 4 − 2a0 − SαP2r 3R3Rd +4 Pr 3Rd a0 +b0 −4 a0 +b0 −2 d , 2 + b0, , − αP2r 3R3Rd +4 ) 2α 3Rd +4 1+b0 F( 2 × a0 +b0 Pr 3Rd F( 2 − 2, 1 + b0, − α 2 3Rd +4 ) d E c Pr 3R3Rd +4 α(a0 + b0 ) (3.19) − + 2α 2 4 − 2a0 − S P2r 3Rd α
3Rd +4
4 Mass Transfer Analysis The equation for species concentration is given by u
∂C ∂C ∂ 2C +v = D 2 − K c (C − C∞ ) ∂x ∂y ∂y
(4.1)
where D is the diffusivity of the medium and K c is the chemical reaction parameter. The appropriate boundary conditions are y = 0 : C → Cw (= C∞ + A2 x s ), where A2 is a constant y → ∞ : C → C∞
(4.2) To solve Eq. (4.1), we now introduce the non-dimensional concentration variable φ(η) defined by φ(η) =
C − C∞ Cw − C∞
(4.3)
Again, the concentration variable C can be expressed as C(y) = C∞ + A2 x φ(η) s
(4.4)
The boundary value problem (4.1) and (4.2), in view of equations (2.19), (2.26), (4.3) and (4.4) reduces to φ + Sc f φ − s Sc f φ − Sc K c φ = 0
(4.5)
and φ(0) = 1, φ(∞) = 0 where Sc = Dν (Schmidt number) and K c = reaction parameter) To solve (4.5), we introduce ζ =−
Sc −αη e α2
(4.6) K c A
(Chemical
(4.7)
Using (4.7) in the boundary value problem (4.5) and (4.6), we get Sc K c d 2φ dφ φ=0 (4.8) ζ 2 + (1 − c0 − ζ ) 2 + s − 2 dζ d ζ α ζ Sc (4.9) φ ζ = − 2 = 1, φ (ζ = 0) = 0 α
where c0 = αSc2 (1 + α N0 ) Here, we note that the Eqs. (4.8) and (4.9) constitute a homogeneous boundary value problem. To solve it, we put φ = ζ δ2 h(ζ )
(4.10)
Using (4.10) in (4.8) we obtain c0 + d0 − 2s dh d2h − h=0 (4.11) + (1 − d0 − ζ ) dζ 2 dζ 2 0 where δ2 = c0 +d and d = c0 − 4Sαc2K c 0 2 As Eq. (4.11) is a Kummer’s equation, so its solution subjects to the boundary condition (4.9) is c0 + d0 αη φ(η) = exp − 2 Sc −αη 0 − s, 1 + d , − e F c0 +d 0 2 2 α (4.12) × Sc c0 +d0 F − s, 1 + d , − 0 2 α2
ζ
The non-dimensional mass transfer in terms of Sherwood number (Sh ) is given by Sh = φ(0)
Sc c0 +d0 −s F , 2 + b , − 0, 2 2 Sc c0 + d0 − 2s α = c0 +d0 2α 1 + d0 F − s, 1 + d , − Sc 2
−
α (c0 + d0 ) 2
0
α2
(4.13)
5 Results and Discussion The problem of steady laminar flow and mass transfer of an incompressible electrically conducting radiative and dissipative visco-elastic fluid past a stretching porous wall well embedded in a porous medium in the presence of a uniform transverse magnetic field and heat source has been considered. The solutions for velocity, heat and mass transfer are given in (2.11), (3.18) and (4.12), respectively. The Eq. (2.5) characterizing the flow field, is a fourthorder differential equation. The elastic property of the fluid has increased the order of the equation by one. Many closed form solutions are obtained without magnetic field, suction and elasticity or with elasticity and constant suction. In the present study, it is possible to derive the viscous case by putting Rc = 0.0 in the momentum equation, but it is not possible to derive the case of without stretching, i.e., A = 0 (because A appears in the denominator of the solution). The solution of heat transfer Eq. (3.8) without radiation (Rd = 0.0) is a linear one which is also exhibited in the Fig. 4 (curve XV). Figures 2 and 3 show the effect of suction parameter (N0 ), magnetic parameter (M), porosity parameter (K p ) and
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Fig. 2 Non-dimensional transverse velocity profile
the elasticity parameter (Rc ) on transverse and longitudinal velocity components. From Fig. 3, it is observed that the longitudinal component of the velocity decreases in the presence of elastic elements with an increasing magnetic field and stronger suction whereas an increase in K p , increases it. On comparison with transverse velocity it is concluded that an opposite effect is observed in case of suction whereas the effects of other parameters remain same. Therefore, it is remarked that suction is found to be counterproductive for enhancing transverse velocity. The reduction is caused due to Lorentz force which acts in an opposite direction of the fluid flow and also fluid experiences tensile stress due to elastic property of the fluid. This is an established result reported by Acharya et al. [26].
Fig. 3 Non-dimensional longitudinal velocity profile
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Figure 4 shows the effects of suction, magnetic parameter, Prandtl number, porosity parameter, radiation parameter, Eckert number and source parameter on temperature distribution of viscous and visco-elastic fluids, respectively. Now, it is observed that elastic parameter, magnetic parameter, and Eckert number increase the temperature whereas the presence of suction, porous matrix, radiation and heat source decrease it at all points. Moreover, higher Prandtl number fluid reduces the temperature at all points. The parameters characterizing heat transfer phenomena have the same effects on both viscous and visco-elastic fluids. Further, it is pointed out that negative value of S, i.e., presence of sink is not admissible in the present problem. Thus, it may be concluded that decrease of temperature can be attributed to release of heat energy from the flow region by way of radiation and on the other hand, increase in temperature is due to frictional heating and shearing stress experienced by visco-elastic property of the fluid leading to thickening of thermal boundary layer. The same result in case of radiation and elastic parameter has been observed by Khan [15] (curve XII and X). Figure 5 shows the concentration variation for various parameters in case of viscous and visco-elastic fluids. It is to note that concentration level attains maximum value in both the cases, i.e., in the absence of magnetic field and chemical reaction. It is quite interesting to note that the concentration decreases due to heavier species, i.e., for higher value of Sc , higher rate of chemical reaction, stronger suction and higher value permeability parameter whereas it increases in the presence of elasticity of the fluid and magnetic field. It is further to note that negative value of K c is not admissible for computation as present formulation of the problem is not suitable for the negative value of K c (exothermic reaction).
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Fig. 4 Temperature profile
Fig. 5 Concentration profile for s = 2.0
Table 1 presents the values of skin friction for different values of Rc , M, K p and N0 . All entries are found to be negative. It is to note that elasticity, magnetic field and suction reduce the skin friction providing a favorable situation for reducing drag at the solid surface, whereas presence of porous medium acts adversely.
Table 2 presents the values of Nusselt number measuring the rate of heat transfer at the solid surface. The Nusselt number decreases due to increase in permeability of the medium K p , radiation parameter Rd , heat source parameter S for the low diffusive fluid, but it increases in the presence of magnetic field, elastic elements and stronger suction at the plate.
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6 Conclusion
Table 1 Skin friction N0
τ
0.5
0.4
–1.943560
0.5
0.5
0.4
–2.081489
0.5
100.0
0.8
–1.692285
0.05
0.0
0.5
0.4
–2.049978
0.05
0.5
0.5
0.4
–2.198555
0.20
0.5
0.5
0.4
–2.847254
0.05
1.0
0.5
0.4
–2.337426
0.05
0.5
1.0
0.4
–1.889170
0.05
0.5
0.5
0.8
–2.557915
0.05
0.5
100.0
0.8
–1.846477
Rc
M
Kp
0.00
0.0
0.00 0.00
Table 2 Nusselt number Rc
M
Kp
0.00
0.5
0.5
N0
Rd
Ec
S
Pr
Nu
0.4
1
1
0.5
0.71
–0.27737
0.05
0.0
0.5
0.4
1
1
0.5
0.71
–0.29308
0.05
0.5
0.5
0.4
1
1
0.5
0.71
–0.23267
0.20
0.5
0.5
0.4
1
1
0.5
0.71
–0.16623
0.05
1.0
0.5
0.4
1
1
0.5
0.71
–0.19938
0.05
0.5
1.0
0.4
1
1
0.5
0.71
–0.41157
0.05
0.5
0.5
0.8
1
1
0.5
0.71
–0.15056
0.05
0.5
0.5
0.4
2
1
0.5
0.71
–0.68459
0.05
0.5
0.5
0.4
1
2
0.5
0.71
–0.57699
0.05
0.5
0.5
0.4
1
1
1.0
0.71
–0.61835
0.05
0.5
100.0
0.8
1
1
0.5
0.71
–0.51064
0.05
0.5
0.5
0.4
1
1
0.5
7.00
–1.56653
Table 3 Sherwood number Rc
M
Kp
N0
Kc
Sc
Sh
0.00
0.5
0.5
0.4
1.00
0.22
–0.58871
0.05
0.5
100.0
0.8
1.00
0.22
–0.70326
0.05
0.0
0.5
0.4
0.00
0.22
–0.12380
0.05
0.5
0.5
0.4
1.00
0.22
–0.58675 –0.58056
0.20
0.5
0.5
0.4
1.00
0.22
0.05
1.0
0.5
0.4
1.00
0.22
–0.57670
0.05
0.5
1.0
0.4
1.00
0.22
–0.60391
0.05
0.5
0.5
0.8
1.00
0.22
–0.64231
0.05
0.5
0.5
0.4
2.00
0.22
–0.82995
0.05
0.5
0.5
0.4
1.00
0.78
–1.33514
Sherwood number given in Table 3 measures the rate of mass transfer at the solid surface. From numerical values, it is observed that heavier diffusing species, higher rate of chemical reaction, the increasing permeability and stronger suction decrease the Sherwood number, but magnetic field and elasticity increase it.
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• Presence of elastic element, magnetic field and suction is counterproductive for primary velocity, whereas porous matrix favors it. • Suction parameter increases transverse velocity. • Visco-elastic property of the fluid vis-à-vis frictional heating and shearing stress experienced by the fluid causes the rise in temperature, whereas radiation reduces it. • Heavier species, higher rate of chemical reaction, stronger suction and permeability decrease the level of concentration, whereas magnetic field and elasticity enhance it. • Elasticity, magnetic parameter and suction reduce the skin friction favoring the stretching of the surface. • Presence of porous matrix, radiation, heat source reduces the rate of heat transfer in case of low diffusive fluid, whereas heavier diffusing species, higher rate of chemical reaction, stronger suction and presence of porous matrix reduce the rate of mass transfer.
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