In this study, we analyzed the induced magnetic field effect on the stagnationpoint flow of a nonNewtonian fluid over a stretching sheet with homogeneous–heterogeneous reactions and nonuniform heat source or sink. The transformed ordinary differential equations are solved numerically using Runge–Kutta and Newtons method. For physical relevance we analyzed the behavior of homogeneous and heterogeneous profiles individually in the presence of induced magnetic field. The effects of different nondimensional governing parameters on velocity, induced magnetic field, temperature and concentration profiles, along with the skin friction coefficient and local Nusselt number, are discussed and presented through graphs. The results of the present study are validated by comparing with the existed literature. Results indicate that induced magnetic field parameter and stretching ratio parameter have the tendency to enhance the heat transfer rate.
Induced magnetic field ; Homogeneous–heterogeneous reactions ; Nonuniform heat source/sink ; Casson fluid ; Stagnation flow
 Velocity components in directions
 Magnetic components in directions
 Magnetic component in free stream flow
 Distance along the surface
 Distance normal to the surface
 Specific heat capacity at constant pressure
 Dimensionless velocity
 Free stream velocity
 Temperature of the fluid
 Dimensionless nonuniform heat source/sink
, Nonuniform heat source/sink
 Skin friction coefficient
 Local Nusselt number
 Local Reynolds number
 Prandtl number
 Positive constants
 Thermal conductivity
 Homogeneous reaction
 Heterogeneous reaction
 Rate constants
 Similarity variable
 Electrical conductivity
 Dimensionless temperature
 Density
 Casson parameter
 Reciprocal magnetic Prandtl number
 Magnetic parameter
 Magnetic permeability
 Dynamic viscosity
 Kinematic viscosity
 Diffusion coefficients
 Thermal diffusivity
 stretching ratio parameter
 Fluid
 Condition at the wall
 Condition at the free stream
The study of convection flows over stretching sheet has extensive applications in the field of manufacturing production of plastic, polythene, paper, polymer extrusion, cooling of elastic sheets, and fiber technology, science and engineering technology. The applications of stretching surface were broadly discussed by Chaim [1] . The flow of a nonNewtonian fluid over a stretching sheet has potential applications in the biosciences, blood flows, jelly and engineering. Casson is a shear thinning liquid. It can exhibit yield stress. If applied yield stress is greater than the shear stress, then it acts as solid, whereas if yield stress is lesser than the applied shear stress then it starts to move. The application of Casson fluid was discussed by Hayat et al. [2] . The homogeneous–heterogeneous reactions on stagnationpoint flow past a stretching surface was discussed by Bachok et al. [3] . Peristaltic flow of a Carreau fluid flow with convective boundary conditions was analyzed by Hayat et al. [4] , and concluded that brinkman number and hall parameters help to enhance the thermal boundary layer thickness. Khan and Pop [5] investigated the viscoelastic fluid flow through a stretching surface with homogeneous–heterogeneous effects. Hayat et al. [6] illustrated the threedimensional MHD bidirectional flow of a nanofluid past a permeable stretching sheet in the presence of slip condition and homogeneous–heterogeneous reactions.
Heat transfer analysis of magneto hydrodynamic stagnationpoint flow toward a stretching surface in the presence of induced magnetic field was done by Ali et al. [7] . Abbas et al. [8] considered the slip conditions and homogeneous–heterogeneous effects on MHD stagnationpoint flow of viscous fluid past a stretching or shrinking surface. Sandeep and Sulochana [9] discussed the mixed convection flow of an unsteady micropolar fluid flow over a stretching or shrinking sheet in the presence of magnetic and nonuniform heat source/sink. Raju et al. [10] analyzed the nonuniform heat source or sink effects on ferrofluid flow through a flat plate with thermal radiation and aligned magnetic field effects. The stagnationpoint flow toward a vertical stretching surface was explained by Ishak et al. [11] . Mallikarjuna et al. [12] investigated the variable porosity regime on convection flow over a vertical cone with chemical reaction and magnetic field effect. An unsteady MHD nanofluid flow through a stretching surface in the presence of nonuniform heat source/sink and magnetic field effect was studied by Sandeep et al. [13] . Pal and Mandal [14] analyzed the induced magnetic field effect on stagnation point flow of a nanofluid past a nonisothermal stretching surface.
Heat transfer analysis of a stagnation point viscoelastic fluid flow toward a stretching sheet was investigated by Mahapatra and Gupta [15] . Gorla [16] depicted the stagnation point flow of nonNewtonian fluid flow in the presence of transverse magnetic field. Heat transfer characteristics of MHD Casson fluid flow past a permeable an exponentially stretching sheet was considered by Raju et al. [17] . Hayat et al. [18] analyzed the twodimensional flow of Eyring–Powell fluid flow toward a stretching sheet with homogeneous–heterogeneous reactions and concluded that the Eyring–Powell fluid material parameter enhances the boundary layer thickness. Heat transfer characteristics of Casson fluid flow past a permeable exponentially stretching sheet in the presence of thermal radiation was discussed by Pramanik [19] and concluded that an increasing value of Casson parameter suppresses the velocity profiles. Animasaun [20] discussed the MHD Casson fluid flow in the presence of viscous dissipation. Stagnation point flow of optimized exact solution for oblique flow of Cassonnanofluid with convective boundary conditions was discussed by Nadeem et al. [21] . Makinde and Aziz [22] and [23] examined the boundary layer analysis of a nanofluid flow through a stretching surface in the presence of convective boundary conditions and buoyancy effects. MHD heat transfer characteristics of an electrically conducting nanofluid flow through nonlinearly stretching surface in the presence of viscous dissipation was examined numerically by Mabood et al. [24] . Nadeem and Saleem [25] proposed an analytical solution for an unsteady mixed convection MHD flow on a rotating cone. Dissipation and magnetic field effects on the boundary layer flow of a powerlaw nanofluid induced by a permeable stretching/shrinking sheet were investigated numerically by Dhanai et al. [26] . Chaudhary and Merkin [27] proposed a simple isothermal model for homogeneous–heterogeneous reactions for boundary layer flow. Very recently, Hayat et al. [28] and Mahanta and Shaw [29] investigated the heat transfer characteristics of Casson fluid through different channels. Hayat et al. [30] studied the homogeneous and heterogeneous reactions on the flow over a nanotube with homogeneous heating. Further, Farooq et al. [31] discussed the homogeneous–heterogeneous characteristics on the flow of Jeffrey fluid.
In this study, we investigated the induced magnetic field and nonuniform heat source or sink effects on the stagnationpoint flow of a nonNewtonian fluid toward a stretching sheet in the presence of homogeneous–heterogeneous reactions. The emerging set of governing nonlinear partial differential equations is transformed into a set of ordinary differential equations using similarity transformation, which are then solved numerically using Runge–Kutta and Newtons method. The effects of different nondimensional governing parameters on velocity, induced magnetic field, temperature and concentration profiles, along with the friction factor and local Nusselt number, are discussed and presented through graphs and tables.
Consider a steady, incompressible, electrically conducting stagnation point flow of a Casson fluid toward a stretching sheet in the presence of induced magnetic field, nonuniform heat source/sink and homogeneous–heterogeneous reactions. The stretching sheet is considered along the axis and axis is normal to it as displayed in Fig. 1 .

Fig. 1. Physical model and coordinate system.

The governing boundary layer equations for the flow and induced magnetic field profiles are given by [7] .

( 1) 

( 2) 

( 3) 

( 4) 
Subjected to the boundary conditions

( 5) 
where is the magnetic diffusivity of the fluid, which is given by , and are the velocity components along the and directions, respectively, is the density, and is the viscosity.
To convert the governing equations of the flow into a set of nonlinear ordinary differential equations, we now introduce the following similarity transformation:

( 6) 
Equation (6) identically satisfies the continuity equations (1) and (2) , and equations (3) and (4) become

( 7) 

( 8) 
Subjected to the transformed boundary conditions

( 9) 
where is the Casson parameter, is the magnetic parameter, and is the reciprocal magnetic Prandtl number, which are given by .
The governing equation for energy in the presence of nouniform heat source/sink is given by

( 10) 
With the boundary conditions

( 11) 
The space and temperaturedependent heat generation/absorption (nonuniform heat source/sink) is defined by Sandeep and Sulochana [9] :

( 12) 
where T is the fluid temperature, is the thermal diffusivity coefficient, is the kinematic viscosity of the base fluid, and and are parameters of the space and temperaturedependent internal heat generation/absorption. The positive and negative values of and represent heat generation and absorption parameters respectively. Now introducing the similarity transformation

( 13) 
By using equation (13) , equations (10) , (11) and (12) transformed to

( 14) 
With the transformed boundary conditions

( 15) 
where is the Prandtl number.
We considered a simple homogeneous–heterogeneous reaction model that exists as proposed by Chaudhary and Merkin [27] in the following form:

( 16) 

( 17) 
Subject to the boundary conditions

( 18) 
where are the diffusion coefficients. We now introduced the similarity transformations as

( 19) 
Using equation (19) , equations (16) , (17) and (18) transformed to

( 20) 

( 21) 
With the transformed boundary conditions

( 22) 
is the Schmidt number, is the measure of strength of homogeneous reaction, is the strength of heterogeneous reaction, is the Reynolds number and is the ratio of diffusion coefficient, which are represented below.

( 23) 
In most of the applications, we expect the diffusion coefficient of chemical species A and B to be of comparable size. This leads us to make an assumption that the diffusion coefficients D_{A} and D_{B} are equal. That is the case to .
For engineering interest, the shear stress coefficient (friction factor) and the local Nusselt number are given by

( 24) 
Here, is the wall shear stress and is the wall heat flux, which are given by

( 25) 
Using (6) and (13) we get

( 26) 

( 27) 
where is the local Reynolds number.
The set of nonlinear ordinary differential equations (7) , (8) , (10) , (20) and (21) subjected to the boundary conditions (9), (11) and (22) are solved numerically using Runge–Kutta and Newtons method (Mallikarjuna et al. [12] ). The influence of nondimensional governing parameters on velocity, induced magnetic field, temperature and concentration profiles, along with the friction factor coefficient and local Nusselt number, are discussed and presented through graphs and tables. For numerical calculations, we considered , . In the entire study we kept these values as common except the variations in the corresponding graphs and tables. In this paper green color indicates the Newtonian fluid case and red color indicates the nonNewtonian fluid case.
Fig. 2 , Fig. 3 , Fig. 4 and Fig. 5 depict the influence of stretching ratio parameter on velocity, induced magnetic field and concentration profiles for both Newtonian and nonNewtonian fluid cases. It is evident that the velocity and homogeneous concentration profiles are encouraged with increasing values of stretching ratio parameter, and reverse profiles are observed in induced magnetic field and heterogeneous concentration profiles. Basically, an increase in the stretching ratio parameter initially develops the more pressure on the flow; due to this reason, we have seen an enhancement in the velocity profiles.

Fig. 2. Velocity profiles for different values of stretching ratio parameter.


Fig. 3. Induced magnetic field profiles for different values of stretching ratio parameter.


Fig. 4. Concentration profiles for different values of stretching ratio parameter.


Fig. 5. Concentration profiles for different values of stretching ratio parameter.

The effects of induced magnetic field parameter on velocity, induced magnetic field and concentration profiles for both Newtonian and nonNewtonian cases are plotted in Fig. 6 , Fig. 7 , Fig. 8 and Fig. 9 . It is noticed that the momentum, thermal and concentration boundary layer thicknesses are enhanced with an increase in the induced magnetic field parameter. Usually an increase in the induced magnetic field develops the electric current. This electric force can help to enhance the momentum and thermal boundary layer thickness. This leads to an increase in the momentum and thermal boundary layer thickness.

Fig. 6. Velocity profiles for different values of induced magnetic field parameter.


Fig. 7. Induced velocity profiles for different values of induced magnetic field parameter.


Fig. 8. Concentration profiles for different values of induced magnetic field parameter.


Fig. 9. Concentration profiles for different values of induced magnetic field parameter.

The influence of homogeneous reaction parameter on concentration profiles for both Newtonian and nonNewtonian cases is depicted in Fig. 10 and Fig. 11 . It displays that an increasing value of enhances the concentration profiles and depreciates the concentration profiles. It may be due to the fact that the reaction rates dominate diffusion coefficients. Quite similar results are displayed in Fig. 12 and Fig. 13 , with an increase in the heterogeneous parameter. This agrees with the general physical behavior of homogeneous and heterogeneous reactions.

Fig. 10. Concentration profiles for different values of strength of homogeneous reaction parameter.


Fig. 11. Concentration profiles for different values of strength of homogeneous reaction parameter.


Fig. 12. Concentration profiles for different values of heterogeneous reaction parameter.


Fig. 13. Concentration profiles for different values of strength of heterogeneous reaction parameter.

The effect of the ratio of the diffusion coefficient on the concentration profiles is depicted in Fig. 14 . Mixed profiles in the concentration are observed; initially, less diffused particles are developed in the concentration. Due to this reason, initially we have noticed depreciation in the concentration profiles. Fig. 15 and Fig. 16 illustrate the effect of nonuniform heat source or sink parameter on temperature profiles. Generally, higher values of nonuniform heat source or sink parameter enhance temperature profiles. Because it generates an internal energy in the flow, we have seen an improvement in temperature profiles.

Fig. 14. Concentration profiles for different values of ratio of diffusion parameter.


Fig. 15. Temperature profiles for different values of nonuniform heat source/sink parameter.


Fig. 16. Temperature profiles for different values of nonuniform heat source/sink parameter.

Table 1 and Table 2 depict the comparison of the present results with previously published work under some special limited cases. We found an excellent agreement of the present results with existing results. This proves the validity of the present results along with the accuracy of the numerical technique we used in this study. Table 3 and Table 4 presented the effect of various emerging thermo physical parameters for skin friction coefficients and local Nusselt number for both Newtonian and nonNewtonian fluid cases. It is evident that the friction factor coefficients and heat transfer rate are enhanced with an increase in the induced magnetic field parameter. The quite similar types of results are displayed in stretching ratio parameter. It is also observed that an increase in the nonuniform heat source or sink parameter declines the rate of heat transfer.

Ishak et al. [11]  Mahapatra and Gupta [15]  Ali et al. [7]  Present  

0.1  −0.9694  −0.9694  −0.9694  −0.96941  
0.2  −0.9181  −0.9181  −0.9181  −0.91810  
0.5  −0.6673  −0.6673  −0.6673  −0.66731  
2  2.0175  2.0175  2.0175  2.01750  
3  4.7294  4.7293  4.7293  4.72931 

 

Mahapatra and Gupta [15]  Present results  
0.05  0.5  1  1.5  0.05  0.5  1  1.5  
0.1  −0.081  −0.383  −0.603  −0.777  −0.0811  −0.3833  −0.6031  −0.7772  
0.2  −0.099  −0.408  −0.625  −0.797  −0.0991  −0.4082  −0.6252  −0.7971  
0.5  −0.136  −0.473  −0.692  −0.863  −0.1360  −0.4731  −0.6921  −0.8632  
1  −0.178  −0.563  −0.796  −0.974  −0.1782  −0.5632  −0.7962  −0.9741  
2  −0.241  −0.709  −0.974  −1.171  −0.2413  −0.7094  −0.9743  −1.1713  
3  −0.289  −0.829  −1.124  −1.341  −0.2891  −0.8290  −1.1241  −1.3411 








 

0.5  −0.510405  1.731575  
0.6  −0.458589  1.739244  
0.7  −0.371571  1.753433  
0.1  −0.516500  1.723825  
0.15  −0.435305  1.739196  
0.2  −0.321719  1.760058  
1  −0.516500  1.723825  
1.4  −0.516500  1.723825  
1.8  −0.516500  1.723825  
0.4  −0.516500  1.723825  
0.6  −0.516500  1.723825  
0.8  −0.516500  1.723825  
0.5  −0.520685  1.268812  
1  −0.520685  0.707954  
1.5  −0.520685  0.147097  
1  −0.475514  1.521289  
2  −0.475514  1.216227  
3  −0.475514  0.860203  
1.5  −0.516500  1.723825  
2  −0.516500  1.723825  
2.5  −0.516500  1.723825 








 

0.5  −0.305742  1.766651  
0.6  −0.277062  1.769630  
0.7  −0.225497  1.777041  
0.1  −0.305683  1.760185  
0.15  −0.253464  1.769873  
0.2  −0.183574  1.782195  
1  −0.305683  1.760185  
1.4  −0.305683  1.760185  
1.8  −0.305683  1.760185  
0.4  −0.305683  1.760186  
0.6  −0.305683  1.760185  
0.8  −0.305683  1.760185  
0.5  −0.305679  1.297576  
1  −0.305679  0.726128  
1.5  −0.305679  0.154680  
1  −0.328580  1.551278  
2  −0.328580  1.255117  
3  −0.328580  0.913042  
1.5  −0.305683  1.760185  
2  −0.305683  1.760185  
2.5  −0.305683  1.760185 
This study deals with the stagnationpoint flow of a Casson fluid past a stretching sheet in the presence of induced magnetic field and homogeneous–heterogeneous reactions. The nonuniform heat source/sink is also taken into account. The arising sets of governing nonlinear coupled partial differential equations are transformed into a set of nonlinear coupled ordinary differential equations using selfsimilarity transformations, which are then solved numerically. The conclusions are as follows:
Published on 10/04/17
Licence: Other
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