Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction

Physics Letters A 372 (2008) 2400–2408 www.elsevier.com/locate/pla

Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction T. Hayat a , Z. Abbas a,∗ , M. Sajid b a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan

Received 10 August 2007; received in revised form 8 October 2007; accepted 27 October 2007 Available online 5 December 2007 Communicated by F. Porcelli

Abstract The laminar flow problem of convective heat transfer for a second grade fluid over a semi-infinite plate in the presence of species concentration and chemical reaction is investigated. The governing equations are transformed into a dimensionless system of three non-linear coupled partial differential equations. These equations have been solved analytically subject to the relevant boundary conditions by employing a homotopy analysis method (HAM). It is noted that for the arising system, the HAM performs extremely well in terms of efficiency and simplicity. The influence of dimensionless pertinent parameters on the velocity, temperature and concentration fields has been examined carefully. © 2007 Elsevier B.V. All rights reserved. PACS: 47.50.-d; 44.05.+e; 44.20.+b; 47.70.Fw Keywords: Second grade fluid; Heat and mass transfer; Chemical reaction; Analytic solution; Convergence

1. Introduction Due to the increasing importance in processing industries and elsewhere when materials whose flow behavior in shearing cannot be characterized by Newtonian relationships, a new stage in the evolution of fluid dynamic theory is in progress. An intensive research effort, both theoretical and experimental, has been devoted in the last few decades to problems of non-Newtonian fluids. This is mainly due to their several applications in polymer processing industries, biofluid dynamics, petroleum drilling and many other similar activities. Geophysical applications concerning ice and magna flows are also based on constitutive equations of nonNewtonian fluids. Because of practical and fundamental association of these fluids to industrial problems, quite a number of recent investigations [1–13] have been done with success in various geometrical configurations. Boundary layer theory has been successfully applied to non-Newtonian fluids of various models. The standard boundary layer equations play a central role in many aspects of fluid mechanics as they describe the motion of a slightly non-Newtonian fluid close to a surface. Especially the boundary layer concept of such fluids is of special importance owing to its applications in many engineering problems among which we cite the possibility of reducing frictional drag on the hulls of ships and submarines. Moreover the research on heat and mass transfer for non-Newtonian fluids is also very important in many engineering applications, such as oil recovery, food processing, paper making and slurry transporting. Furthermore, convective flow occurs in nuclear reactors cooled during emergency shutdown, electronics devices cooled by fans, heat exchangers placed in a low velocity environment and solar central receivers exposed to wind currents. The literature is replete on the topic dealing with the heat transfer in laminar flow of

* Corresponding author. Tel.: +92 51 2275341.

E-mail address: [email protected] (Z. Abbas). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.102

T. Hayat et al. / Physics Letters A 372 (2008) 2400–2408

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viscous fluids. Anjalidevi and Kandasamy [14] have also made an attempt in this direction for the effects of chemical reaction, heat and mass transfer on laminar viscous flow along a semi-infinite horizontal plate. They gave the numerical solution. Although with the advent of computers approximate analytical solutions in fluid dynamics have lost some of their importance however even then such solutions still have their relevance for the following reasons. They give the solutions for each point within the domain of interest unlike the numerical solutions which are available for a particular run only for a set of discrete points in the domain. Such approximate solutions require minimal effort in comparison to the numerical solutions. Even with most of the scientific packages, some initial guess is required for the solution, as the algorithms, in general are not globally convergent. In such situations approximate solutions can provide an excellent starting guess, that can be readily refined to the exact numerical solution in a few iterations. Finally, aesthetically an approximate solution, if it is analytical, is more pleasing than a numerical solution. It is, therefore, not surprising that even after the numerical techniques for obtaining the solutions of the flow problems in fluid dynamics have peaked during the last 3–4 decades, there still have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. In recent years one such technique has drawn special status namely the HAM. Liao [15,16] has been the leading exponent of the HAM. Later, several authors [17–39] have applied HAM successfully for the solution of various complicated problems. In the present Letter our endeavor is to discuss the flow of a second grade fluid along a semi-infinite horizontal plate in the presence of species concentration and chemical reaction. The heat generated during chemical reaction has not been taken into account. HAM has been evolved for obtaining the solution of highly non-linear flow problem. The corresponding HAM solutions of viscous fluids which are not available in the literature yet can be recovered as a limiting case of the present analysis by choosing α1 = 0. The arrangement of the Letter is as follows. In Section 2 the governing equations and similarity transformations are presented. The boundary conditions are also defined here. In Section 3, HAM is used to derive the series solutions for velocity, temperature and concentration fields. The convergence of the HAM solution is briefly described in Section 4. The graphical results are discussed in Section 5 and finally, the conclusions of this study are summarized in Section 6. 2. Mathematical formulation We consider the steady, laminar convective flow of an incompressible second grade fluid bounded by a semi-infinite horizontal plate. The x-coordinate is measured from the leading edge of the plate and the y-coordinate is measured normal to the plate. The corresponding velocities in the x and y directions are u and v, respectively. The mass transfer is the flow along a flat plate that contains a species A slightly soluble in the fluid B. Let Cw be the concentration at the plate surface and the solubility of A in B and concentration of A far away from the plate is C∞ . Also the reaction of a species A with B be the first order homogeneous chemical reaction of rate constant k1 . The concentration of dissolved A is considered small enough. Under these assumptions, the governing equations can be written as: ∂u ∂v + = 0, ∂x ∂y   α1 ∂u ∂ 2 u ∂u ∂ 2 v ∂ 3u ∂u ∂ 2u ∂ 3u ∂u + + + v +v = ν 2 + gβT (T − T∞ ) + gβC (C − C∞ ) + u , u ∂x ∂y ρ ∂y ∂x∂y 2 ∂x ∂y 2 ∂y ∂y 2 ∂y 3       ν ∂u 2 α1 ∂u ∂ ∂T k ∂ 2T ∂u ∂u ∂T + + +v = u +v , u ∂x ∂y ρcp ∂y 2 cp ∂y ρcp ∂y ∂y ∂x ∂y

(1) (2) (3)

∂C ∂ 2C ∂C +v = D 2 − k1 C, (4) ∂x ∂y ∂y where ρ, ν, k, cp , T , C, D, g, βT , βC and α1 (> 0 [40]) are the density, kinematic viscosity, thermal conductivity, specific heat, temperature, the species concentration in the fluid, the mass diffusion, gravitational acceleration, the coefficient of thermal expansion, the coefficient of concentration expansion and material parameter of second grade fluid, respectively. For α1 = 0, Eqs. (2) and (3) reduces to that of a viscous fluid. The boundary conditions are given by u

u = 0, u → U∞ ,

v = 0,

T = Tw ,

T → T∞ ,

C = Cw

C → C∞

at y = 0,

as y → ∞.

To facilitate the solution of this problem, we now introduce similarity transformations as follows:   U∞ T − T∞ C − C∞ θ (x, η) = , φ(x, η) = . , ψ(x, y) = U∞ νxf (x, η), η(x, y) = y νx T w − T∞ C w − C∞ The stream function, ψ , satisfies the continuity equation given in Eq. (1) automatically with: u=

∂ψ , ∂y

v=−

∂ψ . ∂x

(5)

(6)

(7)

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According to Eqs. (6) and (7) we can deduce    U∞ ν 1 ∂f   dη u = U∞ f (x, η), v=− f +x + xf . x 2 ∂x dx

(8)

Substituting Eq. (6) into Eqs. (2)–(4) and then setting C∞ = 0 (see [14]) we have     1  x   ∂f  ∂f f + ff + x f −f + λ(θ + N φ) 2 ∂x ∂x L      ∂f  α  2   iv  ∂f  ∂f iv ∂f f − 2f f − ff + 2x f −f +f +f = 0, + 2x ∂x ∂x ∂x ∂x    ∂f ∂θ Pr f θ  + 2x θ  −f + Pr Ec f  2 θ  + 2 ∂x ∂x      α Ec Pr   2    ∂f  2 ∂f   ∂f + f f − ff f + 2x f −f −f f = 0, 2x ∂x ∂x ∂x    ∂f ∂φ Sc f φ  + 2x φ  −f + Sc γ xφ = 0, φ  + 2 ∂x ∂x

(9)

(10) (11)

where primes denote the differentiation with respect to η. In the above equations the chemical reaction parameter γ , the Schmidt number Sc, the Reynolds Re, the Prandtl number Pr, the Eckert number Ec, the Grashof number Gr, the buoyancy ratio parameter N , second grade fluid parameter α and mixed convection parameter λ are γ=

k1 L , U∞

Ec =

Sc =

ν , D

2 U∞ , cp (Tw − T∞ )

μcp U∞ L , Pr = , k ν2 βC (Cw − C∞ ) gβT (Tw − T∞ ) , N= Gr = , βT (Tw − T∞ ) ν2 Re =

α=

2 α U∞ 1 , νρ

λ=

Gr . Re2

Now define a new variable as   x ξ= . L

(12)

Therefore Eqs. (9)–(11) become   1    ∂f  ∂f f + ff + ξ f −f + ξ λ(θ + N φ) 2 ∂ξ ∂ξ      α ∂f   2   iv  ∂f  ∂f iv ∂f + f − 2f f − ff + 2ξ f −f +f +f = 0, 2ξ ∂ξ ∂ξ ∂ξ ∂ξ    ∂f ∂θ Pr f θ  + 2ξ θ  −f + Pr Ec f  2 θ  + 2 ∂ξ ∂ξ      α Ec Pr   2    ∂f  2 ∂f   ∂f + f f − ff f + 2ξ f −f −f f = 0, 2ξ ∂ξ ∂ξ ∂ξ    ∂f ∂φ Sc f φ  + 2ξ φ  −f + Sc γ ξ φ = 0, φ  + 2 ∂ξ ∂ξ

(13)

(14) (15)

with boundary conditions f (ξ, 0) = 0,

f  (ξ, 0) = 0,

θ(ξ, 0) = 1,

φ(ξ, 0) = 1,

f  (ξ, ∞) = 1,

θ(ξ, ∞) = 0,

φ(ξ, ∞) = 0.

(16) It should be noted that for N = 0 and α = 0, Eqs. (13)–(15) reduce to the corresponding equations of a viscous fluid [41]. The system of non-linear partial differential equations (13)–(16) will be solved analytically using HAM in the next section. 3. HAM solution The velocity, temperature and the concentration distributions f (η, ξ ), θ (η, ξ ) and φ(η, ξ ) can be expressed by the set of base functions of the form

 k j η ξ exp(−nη) k  0, j  0, n  0 (17)

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in the form of the following series 0 f (η, ξ ) = a0,0 +

∞  ∞ ∞  

j

an,k ηk ξ j exp(−nη),

θ(η, ξ ) =

n=0 k=0 j =0

φ(η, ξ ) =

∞  ∞ ∞  

∞  ∞ ∞  

j

bn,k ηk ξ j exp(−nη),

n=0 k=0 j =0

j

cn,k ηk ξ j exp(−nη),

(18)

n=0 k=0 j =0 j

j

j

in which an,k , bn,k and cn,k are the coefficients. Invoking the so-called rule of solution expressions for f (η, ξ ), θ (η, ξ ) and g(η, ξ ) and Eqs. (13)–(16) the initial guesses f0 (η), θ0 (η), φ0 (η) and linear operators L1 and L2 are f0 (η, ξ ) = η − 1 + exp(−η),

(19)

θ0 (η, ξ ) = exp(−η),

(20)

φ0 (η, ξ ) = exp(−η), L1 (f ) = f





−f ,

(21) 

L2 (f ) = f − f.

(22)

The operators L1 and L2 have the following properties:

L1 C1 + C2 e−η + C3 eη = 0,

L2 C4 e−η + C5 eη = 0

(23) (24)

in which Ci (i = 1, 2, . . . , 5) are arbitrary constants. Let p ∈ [0, 1] denote an embedding parameter and h¯ a non-zero auxiliary parameter. We then construct the zeroth order equations

ˆ ξ ; p) , (1 − p)L1 fˆ(η, ξ ; p) − f0 (η, ξ ) = p h¯ Nf fˆ(η, ξ ; p), θˆ (η, ξ ; p), φ(η, (25)

(1 − p)L2 θˆ (η, ξ ; p) − θ0 (η, ξ ) = p h¯ Nθ fˆ(η, ξ ; p), θˆ (η, ξ ; p) , (26)

ˆ ξ ; p) , ˆ ξ ; p) − φ0 (η, ξ ) = p h¯ Nφ fˆ(η, ξ ; p), φ(η, (1 − p)L2 φ(η, (27) subject to the conditions fˆ(0, ξ ; p) = 0, θˆ (0, ξ ; p) = 1,

fˆ (0, ξ ; p) = 0, θˆ (∞, ξ ; p) = 0,

fˆ (∞, ξ ; p) = 1, ˆ ξ ; p) = 1, φ(0,

(28) ˆ φ(∞, ξ ; p) = 0,

(29)

where the non-linear operators are defined as:   1 ∂ 3 fˆ(η, ξ ; p) ∂ 2 fˆ(η, ξ ; p) ˆ (η, ξ ; p) + N φ(η, ˆ ξ ; p) + fˆ(η, ξ ; p) + ξ λ θ 2 ∂η3 ∂η2  2 ˆ  ∂ f (η, ξ ; p) ∂ fˆ(η, ξ ; p) ∂ fˆ(η, ξ ; p) ∂ fˆ (η, ξ ; p) +ξ − ∂ξ ∂η ∂ξ ∂η2  2 ˆ 2 3 4 ˆ α ∂ fˆ(η, ξ ; p) ∂ fˆ(η, ξ ; p) ∂ f (η, ξ ; p) ˆ(η, ξ ; p) ∂ f (η, ξ ; p) + − 2 − f 2ξ ∂η ∂η2 ∂η3 ∂η4  ∂ fˆ (η, ξ ; p) ∂ fˆ (η, ξ ; p) ∂ fˆ (η, ξ ; p) + 2ξ fˆ(η, ξ ; p) − fˆ (η, ξ ; p) + fˆ (η, ξ ; pp) ∂ξ ∂ξ ∂ξ   ˆ ∂ f (η, ξ ; p) , + fˆiv (η, ξ ; p) ∂ξ ∂ 2 θˆ (η, ξ ; p) Pr ∂ θˆ (η, ξ ; p) ˆ + f (η, ξ ; p) Nθ = 2 ∂η ∂η2   ˆ ∂ fˆ(η, ξ ; p) ∂ 2 fˆ(η, ξ ; p) ˆ(η, ξ ; p) ∂ θ (η, ξ ; p) ˆ  (η, ξ ; p) + Pr ξ θ − f + Pr Ec ∂ξ ∂ξ ∂η2  ˆ  2 ˆ 2 2 3 α Pr Ec ∂ f (η, ξ ; p) ∂ f (η, ξ ; p) ∂ fˆ(η, ξ ; p) ∂ fˆ(η, ξ ; p) + − fˆ(η, ξ ; p) 2 2ξ ∂η ∂η ∂η2 ∂η3   ∂ fˆ (η, ξ ; p) ∂ fˆ (η, ξ ; p) ∂ fˆ(η, ξ ; p)   2   ˆ ˆ ˆ ˆ + 2ξ f (η, ξ ; p) − f (η, ξ ; p) − f (η, ξ ; p)f (η, ξ ; p) , ∂ξ ∂ξ ∂ξ

Nf =

(30)

(31)

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T. Hayat et al. / Physics Letters A 372 (2008) 2400–2408

Nφ =

ˆ ξ ; p) ˆ ξ ; p) Sc ∂ φ(η, ∂ 2 φ(η, ˆ ξ ; p) fˆ(η, ξ ; p) + γ ξ Sc φ(η, + 2 2 ∂η ∂η   ˆ ξ ; p) ∂ fˆ(η, ξ ; p) ∂ φ(η, + Sc ξ φˆ  (η, ξ ; p) − fˆ (η, ξ ; pp) . ∂ξ ∂ξ

(32)

For p = 0 and p = 1 we have fˆ(η, ξ ; 0) = f0 (η, ξ ),

fˆ(η, ξ ; 1) = f (η, ξ ),

θˆ (η, ξ ; 0) = θ0 (η, ξ ),

θˆ (η, ξ ; 1) = θ (η, ξ ),

ˆ ξ ; 0) = φ0 (η, ξ ), φ(η,

ˆ ξ ; 1) = φ(η, ξ ). φ(η,

(33)

ˆ ξ ; p) vary from f0 (η, ξ ), θ0 (η, ξ ) and φ0 (η, ξ ) to the exact solutions As p increases from 0 to 1, fˆ(η, ξ ; p), θˆ (η, ξ ; p) and φ(η, f (η, ξ ), θ (η, ξ ) and φ(η, ξ ). By Taylor’s theorem and Eq. (33), we can write fˆ(η, ξ ; p) = f0 (η, ξ ) +

∞ 

m

fm (η, ξ )p ,

m=1 ∞ 

θˆ (η, ξ ; p) = θ0 (η, ξ ) +

θm (η, ξ )p m ,

m=1

ˆ ξ ; p) = φ0 (η, ξ ) + φ(η,

∞ 

m

φm (η, ξ )p ,

m=1

1 ∂ m fˆ(η, ξ ; p) fm (η, ξ ) = , m! ∂p m p=0 θm (η, ξ ) =

(34)

1 ∂ m θˆ (η, ξ ; p) , m! ∂p m p=0

(35)

ˆ ξ ; p) 1 ∂ m φ(η, φm (η, ξ ) = . m! ∂p m p=0

(36)

The convergence of the series given in Eqs. (34)–(36) strongly depends upon parameter h¯ . Therefore h¯ is properly chosen so that the series (34)–(36) are convergent at p = 1 and thus by using Eq. (33) one obtains f (η, ξ ) = f0 (η, ξ ) +

∞ 

fm (η, ξ ),

(37)

θm (η, ξ ),

(38)

m=1 ∞ 

θ (η, ξ ) = θ0 (η, ξ ) +

m=1

φ(η, ξ ) = φ0 (η, ξ ) +

∞ 

(39)

φm (η, ξ ).

m=1

In order to obtain the mth-order deformation problems, we differentiate Eqs. (25)–(28) m times with respect to p, dividing it by m! and then set p = 0. The resulting mth-order problem satisfy

L1 fm (η, ξ ) − χm fm−1 (η, ξ ) = h¯ Rf (η, ξ ),

L2 θm (η, ξ ) − χm θm−1 (η, ξ ) = h¯ Rθ (η, ξ ),

L2 φm (η, ξ ) − χm φm−1 (η, ξ ) = hR ¯ φ (η, ξ ),

(40) (41) (42)

fm (0, ξ ) = fm (0, ξ ) = fm (∞, ξ ) = 0, θm (0, ξ ) = θm (∞, ξ ) = 0,

(43)

φm (0, ξ ) = φm (∞, ξ ) = 0,

 (η, ξ ) + ξ λ(θm−1 + N φm−1 ) + Rf (η, ξ ) = fm−1

1 2

m−1  k=0

fm−1−k fk + ξ

(44) m−1 

 fm−1−k

k=0

∂f ∂fk − fm−1−k k ∂ξ ∂ξ

m−1 

   α  fk − fm−1−k fkiv fm−1−k fk − 2fm−1−k 2ξ k=0   ∂fk ∂fk ∂fk ∂fk   iv + 2ξ fm−1−k − fm−1−k + fm−1−l + fm−1−k , ∂ξ ∂ξ ∂ξ ∂ξ +



(45)

T. Hayat et al. / Physics Letters A 372 (2008) 2400–2408

Fig. 1. h¯ -curves at 12th-order of approximations.  Rθ (η, ξ ) = θm−1 (η, ξ ) +

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Fig. 2. Influence of α on f  at h¯ = −1.

  m−1 ∂fk ∂θm−1−k Pr    − fk θm−1−k fk + 2ξ θm−1−k 2 ∂ξ ∂ξ k=0

k  ∂fk α   Pr Ec     fk + 2αfm−1−k fl + fm−1−k fk−l fl + 2fm−1−k fm−1−k fk−l 2 ∂ξ ξ k=0 l=0      ∂fl   ∂fl − 2ξ fm−1−k fk−l fk−l − fm−1−1k , ∂ξ ∂ξ   m−1 ∂fk ∂φm−1−k 1     Rφ (η, ξ ) = φm−1 (η, ξ ) + ξ Sc γ φm−1 + Sc − fk , φm−1−k fk + 2ξ φm−1−k 2 ∂ξ ∂ξ

+

m−1 

(46)

(47)

k=0

where  χm =

0, m  1, 1, m > 1.

(48)

The general solutions of Eqs. (40)–(48) can be written as fm (η, ξ ) = fm (η, ξ ) + C1 + C2 exp(−η) + C3 exp(η), θm (η, ξ ) = θm (η, ξ ) + C4 exp(−η) + C5 exp(η),  φm (η, ξ ) = φm (η, ξ ) + C6 exp(−η) + C7 exp(η),

(49)

 (η, ξ ) are the particular solutions and the constants are determined by the boundary conditions (43) where fm (η, ξ ), θm (η, ξ ) and φm and (44) which are given by ∂fm (η, ξ ) C2 = C1 = −C2 − fm (0, ξ ), , ∂η η=0

C 3 = C5 = C7 = 0

C4 = −θm (0, ξ ),

 C6 = −φm (0, ξ ).

(50)

In the next section, the linear non-homogeneous Eqs. (40)–(48) are solved using Mathematica in the order m = 1, 2, 3, . . . . 4. Convergence of the HAM solution The series in Eqs. (37)–(39) are the solutions of the considered problem if one guarantee the convergence of these series. As pointed out by Liao [15] the convergence and rate of approximation for the HAM solution strongly depends upon h¯ . In order to obtain the admissible value of h¯ for the present problem, the h¯ -curves are plotted for 12th-order of approximations. It can be easily seen from Fig. 1 that the range for the admissible value of h¯ for f , θ and φ are −1.9  h¯ < −0.1, −4.35  h¯ < −0.3 and −1.8  h¯ < −0.1, respectively. The presented calculations clearly indicate that the series (37)–(39) converge for the whole region of η when h¯ = −1.

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Fig. 3. Influence of Sc on f  at h¯ = −1.

Fig. 4. Influence of γ on f  at h¯ = −1.

Fig. 5. Influence of α on θ at h¯ = −1.

Fig. 6. Influence of Pr on θ at h¯ = −1.

Fig. 7. Influence of Ec on θ at h¯ = −1.

Fig. 8. Influence of α on φ at h¯ = −1.

5. Results and discussion In this section our interest lies seeing the effects of second grade parameter α, the Schmidt number Sc, the chemical reaction parameter γ , the Prandtl number Pr and the Eckert number Ec on the velocity, temperature and concentration fields. For this purpose, Figs. 2–10 have been plotted and their initial behaviors are analyzed. Figs. 2–4 have been prepared for the effects of α, Sc and γ on the velocity component f  . It is evident from Fig. 2 that f  is an increasing function of α. The boundary layer thickness decreases as α increases. Fig. 4 displays that f  is a decreasing function of

T. Hayat et al. / Physics Letters A 372 (2008) 2400–2408

Fig. 9. Influence of Sc on φ at h¯ = −1.

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Fig. 10. Influence of γ on φ at h¯ = −1.

Sc and the boundary layer thickness increases in this case. Fig. 5 indicates the effects of γ on f  . It is noted that the velocity f  and the boundary layer thickness are increased when γ increases. Figs. 5–8 are plotted just to explore the effects of α, Pr and Ec on the temperature field θ . Fig. 5 illustrates that θ decreases for large values of α. The thermal boundary layer thickness also decreases as α increases. Fig. 6 gives the variation of Pr on θ . It is observed that θ is a decreasing function of Pr. Moreover the thermal boundary layer thickness also decreases for large values of Pr. Fig. 7 clearly indicates that the effect of Ec on θ is quite opposite to that of Pr. Here the thermal boundary layer thickness is increased when compared with Fig. 6. Figs. 8–10 are drawn just to point out the variations of α, Sc and γ on the concentration field φ. Fig. 8 elucidates the influence of α on concentration field φ. It is found that φ is the decreasing function of α. The boundary layer thickness also decreases. Fig. 9 shows the variation of Sc on φ. As expected, it is found that φ decreases for increasing Sc. The boundary layer thickness is also decreased as Sc is increased. Fig. 10 indicates the effects of γ on φ. Fig. 10 depicts that the concentration field φ and the boundary layer thickness are the increasing function of γ . It shows the opposite results when compared with Fig. 9. 6. Concluding remarks The effects of heat and mass transfer on the flow of a second grade fluid with chemical reaction is investigated in this Letter. The reduced system of coupled non-linear ordinary differential equations was analytically solved using HAM. The effects of emerging parameters have been seen and discussed through graphs. The following observations have been made. • • • •

The boundary layer thickness for f  is decreased when α and Sc increases. This layer thickness increases when γ increases. The thermal boundary layer thickness decreases when the values of α and Pr increases. However it increases by increasing Ec. By increasing α and Sc, the boundary layer thickness for the concentration field decreases. The role of γ on the boundary layer thickness in the concentration field is opposite to that of α and Sc.

Acknowledgements The authors are thankful to the referees for their useful suggestions. The financial support from Higher Education Commission (HEC) is also gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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