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1 

J. mt. area res., Vol. 4, 2019 

   

       Journal of Mountain Area Research 
            

 

 

MODELING AND ANALYSIS OF MAGNETIZED CHEMICALLY REACTIVE 

FLUID FLOW PAST OVER POROUS STRETCHED SHEET 

F. Haq1*, M. U. Rahman1 and S. Hussain1 

1 Department of Mathematical Sciences, Karakoram International University, Gilgit, Pakistan 

 

ABSTRACT 

The aim of this article to inspect the effect of nonlinear thermal radiation, heat joule, viscous 

dissipation and magnetic field on viscoelastic second grade fluid. Flow is generated due to 

stretching of sheet. Flow features are studied considering hydrodynamic boundary conditions. 

Chemical reaction on the surface is further accounted. The flow governing nonlinear partial system 

of differential equations is obtained incorporating boundary layer assumptions. The dimensional 

model is made dimensionless by taking suitable transformations and then tackled via HAM for 

convergent series solution. Effects of flow controlling parameters on velocity, concentration, 

temperature, local skin friction coefficient, Sherwood number and Nusselt numbers are discussed by 

plotting graphs. Main observations are listed at the end. 

 

KEYWORDS: Fluid flow, porous media, HAM 

 

*Corresponding author (email: fazal.haq@kiu.edu.pk) 

 

1. INTRODUCTION 

Magnetohydrodynamics(MHD) is a part of 

science which describes the relation between 

the magnetic fields and moving electrically 

conducting fluid. Hydromantic phenomenon 

becomes more important due to its practical 

applications in engineering, physics chemistry 

and industries, like MHD throttles, automatic fuel 

level indicator, nuclear reactor, 

magnetometer, electronic motors and 

transformer. The origin of hydromantic was first 

time introduced by Alfven[1]. Later on several 

researchers investigated MHD effects on the 

flow over different geometries [2-7]. Nadeem et 

al. [8] studied MHD liquid flow over shrinking 

sheet. Analytical and numerical solution of MHD 

boundary layer flow problem over an unsteady 

stretching sheet is reported by Sheikholeslami 

[9]. 

The flow field study near stretching sheet in 

boundary layer flow is an important process in 

fluid dynamics and engineering. The heat 

transfer process occurring in different 

engineering processes such as crystal growth, 

plastic sheets, glass fiber, polymer processing 

and metallurgy [10]. Ghosh et al. [11] applied 

Laplace transformation to study the behavior of 

flow of MHD viscoelastic incompressible fluid 

having small particles and moving between 

two parallel plates of infinite length. To 

investigate heat transfer in MHD viscoelastic 

fluid flow in presence of thermal radiation over 

a semi-infinite, non-isothermal stretching 

impermeable sheet with internal heat 

generation/ absorption, Datti et al. [12] applied 

fourth-order RK-4 (Runge-Kutta-4) method. 

Vol. 4, 2019 

http://journal.kiu.edu.pk/index.php/JMAR 

Full length article 



 

 

Haq et al., J. mt. area res. 04 (2019) 1-8 

2 

J. mt. area res., Vol. 4, 2019 

Analysis of Dufour effects on mass and heat 

transmission in micro polar liquid flow over an 

isothermal sphere Keller-box implicit technique 

is applied by Beg et al.[13]. Kamel et al.[14] 

used Laplace transformation technique to 

study MHD flow of vertical permeable sheet of 

infinite length. Chen et al. [15] used a 

central-difference scheme to deliberate the 

influence of involved physical parameters in 

governing equations power-law stretched 

sheet of non-Newtonian power-law fluids past 

with surface heat flux. 

Invent of modern high speed computers 

and development of new methods/techniques 

has a significant role in solving highly nonlinear 

problems. To solve nonlinear problems most of 

the researchers used analytical methods. These 

methods are reliable and have high 

convergence then other methods [16]. 

Homotopy Analysis Method (HAM) is one of 

reliable and efficient technique to get the 

analytical solution of nonlinear differential 

equations. HAM was introduced by Liao, as an 

analytical method for finding the solution of 

nonlinear problems [17]. Khan et al.[18] studied 

heat transfer in Magnetohydrodynamic Sisko 

fluid through a porous medium. Dufour and 

Soret’s effects over a vertical stretching sheet 

on mixed convection of a viscoelastic fluid flow 

is studied by Hayat et al.[19] using HAM. To 

study the behavior of unsteady flow in case of 

heat transfer over stretching sheet Rashdi et 

al.[20] used HAM. Maxwell fluids heaving mixed 

convection effects in a boundary layer over a 

vertical stretching surface is studied by Abbas 

et al. [21] via HAM. 

Our main concern here is to scrutinize the 

magnetized flow of viscoelastic second grade 

fluid over stretched sheet. Thermal radiation, 

viscous dissipation and heat source are 

considered in energy relation. Concentration 

communication is modeled in view of chemical 

reaction. The impact of sundry variable on heat 

transfer, mass transfer, fluid velocity, 

temperature and concentration are analyzed 

through plots. 

2. PROBLEM FORMULATION 

In this study we consider 2-D boundary-layer 

flow of visco-elastic fluid over a stretching and 

electrically conducting sheet. The electrically 

conducting fluid through applied magnetic 

field  𝐵0  via thermal radiation, heat transfer 

characteristics is explored. Furthermore Joule 

heating and dissipation are also carried. Let us 

assume that 𝑢(𝑥) = 𝑏𝑥 is strains velocity in flow 

(see Fig. 1) direction and T and C are 

temperature and concentration of fluid. The 

governing equations in view of above 

assumptions are: 

𝜕𝑢

𝜕𝑥
+

𝜕𝑢

𝜕𝑦
= 0,                                   

(1) 

𝑢
𝜕𝑢

𝜕𝑥
+ 𝑣

𝜕𝑢

𝜕𝑦
= −𝑘0 {𝑢

𝜕3𝑢

𝜕𝑥𝜕𝑦2
+ 𝑣

𝜕3𝑣

𝜕𝑦3
+

𝜕𝑢

𝜕𝑥

𝜕2𝑢

𝜕𝑦2
−

  
𝜕𝑢

𝜕𝑦

𝜕2𝑣

𝜕𝑦2
} + 𝜐

𝜕2𝑢

𝜕𝑦2
−

𝜎

𝜌
𝐵0

2𝑢 −
𝜐

𝑘∗
 u ,                

(2) 

𝑢
𝜕𝑇

𝜕𝑥
+ 𝑣

𝜕𝑇

𝜕𝑦
=

𝑘

𝜌𝐶𝑝

𝜕2𝑇

𝜕𝑦2
+

𝜇

𝜌𝐶𝑝
(

𝜕𝑢

𝜕𝑦
)2 +

𝛼

𝜌𝐶𝑝
(𝑢

𝜕𝑢

𝜕𝑥

𝜕2𝑢

𝜕𝑦2
+

𝑣
𝜕𝑢

𝜕𝑦

𝜕2𝑢

𝜕𝑦2
) −  

1

𝜌𝐶𝑝

𝜕𝑞𝑟

𝜕𝑦
+

𝑄0

𝜌𝐶𝑝
(𝑇 − 𝑇∞) +  

𝜎

𝜌𝐶𝑝
𝐵0

2𝑢2    

(3) 

𝑢
𝜕𝐶

𝜕𝑥
+ 𝑣

𝜕𝐶

𝜕𝑦
= 𝐷

𝜕2𝐶

𝜕𝑦2
− 𝑘∗(𝐶 − 𝐶∞).               (4) 

With: 



 

 

Haq et al., J. mt. area res. 04 (2019) 1-8 

 

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J. mt. area res., Vol. 4, 2019 

𝑢(𝑥) = 𝑢𝑤(𝑥), 𝑣 = 0, 𝑇 = 𝑇𝑤 , 𝐶 = 𝐶𝑤  𝑎𝑡 𝑦 = ,0   

(5) 

𝑢(𝑥) = 0,  𝑢𝑦 = 0, 𝑇 = 𝑇∞, 𝐶 = 𝐶∞ 𝑎𝑠 𝑦 → ∞ .    

(6) 

Where u and v denotes component of velocity, 

  is dynamic viscosity,   is thermal diffusivity, 

T is temperature, 𝜌  is  density, 𝜎  is 

Stefan-Boltzmann constant, 𝑞𝑟  is  radiative 

heat flux, 𝐵0 is magnetic field strength, 𝑄0  is 

heat generation/absorption coefficient, 𝐶𝑝  is 

the specific heat, 𝑇∞  is ambient fluid 

temperature,  𝑘∗ mean absorption coefficient, 𝐶∞ 

is ambient fluid concentration, 𝑘0   is short 

memory coefficient, D is diffusion coefficient 

and 𝜐 is Kinematic viscosity. 

For radiation via Roseland approximation we 

have: 

𝑞𝑟 = −
4𝜎

3𝑘∗

𝜕𝑇4

𝜕𝑦
                               (7) 

Considering non dimensional forms of 

momentum, energy and concentration 

equations, the suitable dimensionless variables 

introduced are [15]: 

𝜂 = √
𝑢𝑤(𝑥)

𝜈𝑥
𝑦, 𝜓 = √𝜈𝑥𝑢𝑤 (𝑥) 𝑓(𝜂) , 𝜃(𝜂) =

𝑇−𝑇∞

𝑇𝑤−𝑇∞
,

∅(𝜂) =
𝐶 −𝐶∞

𝐶𝑤−𝐶∞
 .                              (8) 

 

Fig. 1: Schematic flow diagram. 

The continuity equation is satisfied identically 

and the remaining equations take the form: 

𝑓𝑓 𝑖𝑣 − (2𝑓 ′ +
1

𝑘1
) 𝑓 ′′′ −

1

𝑘1
(𝑀 − 𝑘2)𝑓

′ −
1

𝑘1
(𝑓 2 −

𝑓𝑓 ′′) = 0;                              (9) 

𝜃′′ + 𝑃𝑟𝐸𝑐𝑓 ′′
2

+ 𝑘1𝑃𝑟𝐸𝑐(𝑓
′𝑓 ′′

2
− 𝑓𝑓 ′′𝑓 ′′′)+𝑅[3(𝜃𝑤 −

1){1 + (𝜃𝑤 − 1)𝜃}
2]𝜃′

2
+ {1 + (𝜃𝑤 −

1)𝜃}3𝜃′′] +𝛿𝑃𝑟𝜃 + 𝑀𝑃𝑟𝐸𝑐𝑓 ′
2

+ 𝑃𝑟𝜃′𝑓 = 0,                                        

(10) 

∅′′ + 𝑆𝑐𝑓∅′ − 𝑆𝑐𝛾∅ = 0.                     (11) 

With: 

𝑓 ′(0) = 1, 𝑓(0) = 0, 𝑓 ′(∞) = 0 𝑎𝑛𝑑 𝑓 ′′(∞) = 0, (12) 

 𝜃(0) = 1 𝑎𝑛𝑑  𝜃(∞) = 0,                   

(13) 

∅(0) = 1 𝑎𝑛𝑑  ∅(∞) = 0.                   (14) 

Where 𝑃𝑟 =
𝜈

𝛼
  is Prandtl number, 𝑀 =

𝜎𝐵0
2

𝑏𝜌
 is 

Magnetic field parameter, 𝑘1 =
𝑘0𝑏

𝜈
 is the Visco–

elastic parameter , 𝑘2 =
𝜐

𝑏𝑘 ∗
 Permeability 

parameter, 𝛿 =
𝑄0

𝜌𝐶𝑝𝑏
 is Heat generation 

parameter, 𝑆𝑐 =
𝜈

𝐷
 is Schmidt number, 𝐸𝑐 =

𝑢𝑤
2(𝑥)

𝐶𝑝(𝑇𝑤−𝑇∞)
 is Eckert number 𝛾 =

𝑘∗

𝑏
 chemical 

reaction parameter and 𝑅𝑒𝑥 =
𝑥𝑢𝑤

𝜈
 is local 

Reynold number. 

Coefficients of Skin friction(𝐶𝑓𝑥 ), Nusselt (𝑁𝑢𝑥 ) 

and Sherwood (𝑆ℎ𝑥 ) numbers are: 

𝐶𝑓𝑥 =
−2𝜏𝑤

𝜌𝑢2𝑤
,                                 

(15) 



 

 

Haq et al., J. mt. area res. 04 (2019) 1-8 

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J. mt. area res., Vol. 4, 2019 

𝑁𝑢𝑥 = {−
𝑥

(𝑇𝑤−𝑇∞)

𝜕𝑇

𝜕𝑦
−

16𝜎∗𝑥

3𝑘𝑘∗(𝑇𝑤−𝑇∞)
𝑇3

𝜕𝑇

𝜕𝑦
}|

𝑦=0
,    (16) 

𝑆ℎ𝑥 = (
𝑥

(𝐶𝑤−𝐶∞)

𝜕𝐶

𝜕𝑦
)|

𝑦=0
.                     (17) 

Expression of shear stress 𝜏𝑤   at 𝑦 = 0  is: 

𝜏𝑤 = 𝜏𝑥𝑦 |𝑦=0
= 𝜇𝑏𝑥√

𝑏

𝜐
{𝑓 ′′(0) +

  
𝛼1𝑏

𝜇
(3𝑓 ′(0)𝑓 ′′(0) − 𝑓(0)𝑓 ′′′(0))}.                              

(18) 

In non-dimensional form Coefficient of skin 

friction(𝐶𝑓𝑥 ), Nusselt (𝑁𝑢𝑥 ) and Sherwood (𝑆ℎ𝑥 ) 

numbers are: 

𝐶𝑓𝑥 𝑅𝑒𝑥
0.5 = −2{𝑓 ′′(0) + 𝑘1(3𝑓

′(0)𝑓 ′′(0) −

𝑓 ′(0)𝑓 ′′(0))},                               

(19) 

𝑅𝑒𝑥
−0.5𝑁𝑢𝑥 =  − (1 +

4

5
𝑅{1 + (𝜃𝑤 −

   1)  𝜃(0)}3)  𝜃′(0),      (20) 

𝑅𝑒𝑥
−0.5𝑆ℎ𝑥 = −∅

′(0).                        

(21) 

 

3. SOLUTION PROCEDURE 

Suitable initial approximations are chosen 

which satisfy the boundary conditions. 

Following initial guesses, linear operators are 

taken and homotopic concept is applied to 

obtain solutions of nonlinear expressions. 

𝑓0(𝜂) = 1 − 𝑒
−𝜂 , 𝜃0(𝜂) = 𝑒

−𝜂 , 𝜙0(𝜂) = 𝑒
−𝜂,     (22) 

£𝑓 = 𝑓
′′′ − 𝑓 ′, £𝜃 = 𝜃

′′ − 𝜃,  £𝜙 = ϕ
′′ − 𝜙.      (23) 

With: 

£𝑓 (𝐶1 + 𝐶2𝑒
𝜂 + 𝐶3𝑒

−𝜂 ), 

£𝜃 (𝐶4𝑒
𝜂 + 𝐶5𝑒

−𝜂 ), 

£𝜙 (𝐶6𝑒
−𝜂 + 𝐶7𝑒

𝜂 ). 

Where 𝐶𝑖, (𝑖 = 1 − 7)  are arbitrary constants, 

the auxiliary variables ℏ𝑓 , ℏ𝜃  𝑎𝑛𝑑 ℏ𝜙
 
have key 

role in regulating and controlling the 

convergence region of homotopic expressions. 

By plotting ℏ -curves suitable ranges of these 

variables are obtained. 

3.1 Convergence analysis 

Homotopy analysis method consists of auxiliary 

parameters, which control and regulate the 

region of convergence for homotopic 

expressions. By plotting  ℏ -curves (see Fig. 2). 

Appropriate values of ℏ𝑓 , ℏ𝜃 , 𝑎𝑛𝑑 ℏ𝜙 are found 

in the ranges 1.6 0.3
f

    , 1.4 0.4


     

and 1.6 0.3


     

Table 1 shows convergence numerically. 

Table 1: Convergence analysis in case of HAM when 

𝑘1 = 𝑘2 = 𝑀 = 0.1, 𝐸𝑐 = 0.2, 𝑃𝑟 = 𝜃𝑤 = 0.01, 𝑅 =

0.03, 𝑆𝑐 = 0.7, 𝛿 = 0.02, 𝛾 = 1.0. 

Order of 

approximations 
-

//
(0)f  -

/
(0)  -

/
(0)  

1 1.086 0.801 0.986 

10 1.244 0.294 0.965 

20 1.248 0.204 0.965 

25 1.248 0.182 0.965 

30 1.248 0.158 0.965 

35 1.248 0.158 0.965 

 

After 15th order of approximations we get 

convergence of Eq. (9), after 30th order of 

approximations convergence of Eq. (10) is 

obtained and 10th order approximation is 

enough for convergence of Eq. (11). 



 

 

Haq et al., J. mt. area res. 04 (2019) 1-8 

 

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J. mt. area res., Vol. 4, 2019 

 

Fig. 2: ℏ  -curves for velocity, temperature and 

concentration 

4. RESULTS AND DISCUSSION 

Effect of involved variables versus 

temperature (𝜃) , concentration  (𝜙) , skin 

friction  (𝐶𝑓𝑥 𝑅𝑒𝑥
0.5) , Nusselt and Sherwood 

numbers (𝑅𝑒𝑥
−0.5𝑁𝑢𝑥 , 𝑅𝑒𝑥

−0.5𝑆ℎ𝑥 )  are 

highlighted in this section. The values of 

involved variables taken for plotting graphs are: 

 𝑘1 = 𝑘2 = 𝑀 = 0.1, 𝐸𝑐 = 0.2, 𝑃𝑟 = 𝜃𝑤 = 0.01, 𝑅 =

0.03, 𝛿 = 0.02, 𝑆𝑐 = 0.7 𝑎𝑛𝑑 𝛾 = 1. 

Fig. 3 explains the curves of 𝑓 ′ for 𝑘1. Here for 

greater values of 𝑘1, 𝑓
′ and its related layer is 

thicker .Velocity 𝑓 ′  increases when 𝑘1  is 

increased. In fact liquid viscosity diminishes via 

an increment in 𝑘1 which yields higher 𝑓
′. Fig. 4 

shows the variation in 𝑓 ′  against 𝑘2  . The 

thermal layer and 𝑓 ′ increase with the increase 

in 𝑘2. Ultimately there is an increment in 𝑓
′  for 

larger values of  𝑘2 . Fig. 5 represents the effect 

of M on 𝑓 ′. Larger values of M improves both 

velocity and related layer thickness. Here we 

observe an improvement in 𝑓 ′. Fig. 6 shows that  

𝜃  and thickness of related layer diminish for 

greater values of  𝑘1.The effect of Pr number 

on temperature distribution is depicted in Fig. 7. 

The related layer and temperature distribution 

decreases as Pr increases. Importance of Sc is 

shown in Fig. 8. Here larger Sc increases the 

concentration and thickness of related layer. 

Fig 9 witnesses that 𝜙  and thickness of 

corresponding layer decreases with the 

increase of 𝛾. 

 

Fig. 3: Influence of 𝑘1 on velocity distribution. 

Fig. 4: Influence of 𝑘2  on velocity distribution. 



 

 

Haq et al., J. mt. area res. 04 (2019) 1-8 

6 

J. mt. area res., Vol. 4, 2019 

 

Fig. 5: Influence of M on velocity distribution. 

Fig. 6: Influence of 𝑘2 on temperature distribution. 

 

Fig. 7: Influence of Pr on temperature distribution.  

Fig. 8: Influence of Sc on concentration distribution. 

Fig. 9: Influence of 𝛾 on concentration distribution. 

Fig. 10: Effects of 𝑘1 and M on 𝑅𝑒𝑥
0.5𝐶𝑓𝑥. 

Fig. 11: Effects of R and 𝜃𝑤 on 𝑅𝑒𝑥
−0.5𝑁𝑢𝑥 . 

Fig. 12: Effects of Sc and   on 𝑅𝑒𝑥
−0.5𝑆ℎ𝑥. 



 

 

Haq et al., J. mt. area res. 04 (2019) 1-8 

 

7 

J. mt. area res., Vol. 4, 2019 

 Fig.10 Show that larger 
1

k  and M corresponds 

to higher 𝑅𝑒𝑥
0.5𝐶𝑓𝑥. and associated layer. Fig. 11 

Show that for larger  and R corresponds to 

higher 𝑅𝑒𝑥
−0.5𝑁𝑢𝑥. Fig. 12 divulges the variation 

of   and Sc against 0.5Re
x x

Sh


. Higher values of 

  and Sc reported higher 𝑅𝑒𝑥
−0.5𝑆ℎ𝑥. 

CONCLUSIONS 

Here we study nonlinear radiation and mixed 

convection aspects in MHD boundary layer 

visco-elastic 2nd grade fluid flow over a moving 

stretching sheet in a pours medium. The 

formulated equations are transformed to 

ordinary differential equations by taking 

suitable transformations. The solution of 

governing equations is obtained by using HAM. 

 It is observed that visco-elastic parameter 

effects velocity and temperature distribution, 

the velocity increases and temperature 

decreases in boundary-layer. This shows the 

effect of visco-elastic parameter. The magnetic 

field parameter increases velocity distribution it 

has no any effect on temperature and 

concentration. The prandtl number has no 

effect on velocity and concentration but it 

decreases the temperature distribution. The 

permeability parameter increases the velocity 

distribution. The Renold number, Ecrect number 

and 𝜃𝑤  have no any effect on velocity, 

temperature and concentration distributions. 

The Schmidt number and radiation parameter 

𝛾  decreases concentration distribution.  

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