1 Introduction

As we know the present generations are very much depend on applications of science and technologies which are based on industries. In aerodynamic, extrusion of polymers, hot rolling, cooling of metallic plates, glass-fiber production etc. The magneto-hydrodynamic flow has both liquid as well as magnetic properties it can exhibits particular characteristics in thermal conductivity. Raju et al. [1] considered a detail study of least square finite element on non-linear boundary layer problems analysed stagnation point flow. Gorla [2] has considered a non-Newtonian fluid of a stagnation point flow in the presence of the transverse magnetic field and resulted those shear stress co-efficient increases with increasing magnetic field strength for all the values of flow behavior index. Takhar et al. [3] studied MHD unsteady stagnation point boundary layer flow. Besser et al. [4] given a description of work of magnetic field annilation within the frame work of the MHD equations of an incompressible plasmas’ in the regions y<0 and y>0, along with the uniform resistivity and viscosity as a transport parameter. Massoudi and Ramezant [5] investigated heat transfer analysis of a viscoelastic fluid at a stagnation point. Donal [6] investigated the three dimensional stagnation point flow of a viscoelastic fluid. Mahapatra et al. [7] studied about a has a solution on Navier-Stokess’ equations with their study approached a stretching surface of the boundary layer structure and redoubted as velocity of the surface is very less than the free stream velocity along with this an opposite boundary layer can also be formed. Abel et al. [8] analysed effect of non-uniform heat source on magneto-hydrodynamic heat transfer in a liquid film over unsteady stretching sheet. Yazdi et al. [9] studied the MHD slip flow over nonlinear permeable stretching surface in the presence of a chemical reaction, the slip flow is about the characteristic size of the flow regime which is very small or at the very low pressure. Kai [10] has been studied the magnetic-hydrodynamic stagnation point flow of a non-Newtonian (viscoelastic fluid) flow and heat transfer due to a flow stretching sheet and they had a conclusion that the effect of non-Newtonian flow heat transfer is better than the Newtonian flow and heat transfer. Mahantesh et al. [11] focused on MHD flow and heat transfer with partial slip over a stretching surface and they presented analytical solutions of for two cases namely PST and PHF, in terms of Kummers’ function.

Natalia et al. [12] power law index flow and heat transfer with the non-linearly moving flat plate of the free stream slip velocity and they have given a particular attention on dual solutions (upper and lower branch solutions).Dessie and Kishan [13] considered MHD on boundary layer flow and heat transfer of a fluid with variable viscosity, porous medium, viscous dissipation and heat source/sink respectively. Mahantesh et al. [14] studied MHD flow and their importance in various fields of science and engineering applications. Hassan [15] considered the electrically conducting viscous boundary layer flow and heat transfer. Shen et al. [16] investigated magneto hydrodynamic mixed convection flow in the presence of stagnation point flow over a stretching sheet along with velocity slip. In the real world, we are depending upon various mechanical and non-mechanical things. Taufail and Ali [17] worked on properties of flow and heat transfer. Gireesha et al. [18] studied the magnetic heat transfer in dusty fluid on continuously stretching non-isothermal surface and studied effect of different parameters effects carried out. Zaidi and Din [19] considered convective heat transfer and importance of MHD effects on technology in various fields like automotive, aerospace and industry etc.

Abel et al. [20, 21, 22, 23, 24, 25] investigated MHD flow and heat transfer effect on thermal boundary layer, hydromagnetic flow of viscoelastic liquid, heat transfer in a viscoelastic boundary layer flow, boundary layer flow and heat transfer characteristics of a second grade fluid and non-newtonian fluid over a porous medium [24], MHD boundary layer flow and heat transfer characteristics of laminar liquid film [25] and continuously moving permeable stretching surface with non-uniform heat source/sink. Mahantesh et al. [26] worked on effects of thermal radiation [27], porous medium [28], heat transfer on non-linearly stretching sheet with non uniform heat source and variable wall temperature [28], liquid film flow due to unsteady stretching sheet [29], MHD stagnation point of flow and heat transfer [30] and second order slip effects on flow [31]. Ibrahim and Haq [32] examined the convective heat transfer and magneto hydrodynamic stagnation point flow and they showed the thermal boundary layer thickness increases as the thermophoresis parameter decreases with the prandlt number Pr.

On observing all above investigations, there are no investigations on stagnation point flow of viscous fluid and heat transfer with velocity and thermal jump in presence of non-uniform heat source due to moving surface. Hence in the present study we have considered these effects on flow and heat transfer.

2 Mathematical Formulation

We have considered two dimensional steady, laminar flow and heat transfer of a incompressible fluid in the presence of transverse magnetic field strength B(x), which is effected normally on flow and given in the unique form as

where n is constant and x is co-ordinate along the plate measured from the leading edge. The plate is moving inside or outside the origin with the velocity u_w(x) = axⁿ in an exterior (in viscid) flow of the velocityu_e(x) = axⁿ, where u and v are the corresponding velocity components in the x and y directions respectively. Here T_w(x) assumed as a temperature of the plate and ambient fluid is T_∞ which constant temperature is.

The governing equations of continuity, momentum and energy equations are as follows,

where ρ the electrical conductivity of the fluid, α is the thermal diffusivity, u and v are the velocity components x and y axis.

Associated with the initial and boundary conditions:

Here we guess the velocity slip factor N₁ and the temperature slip factor S₁ change with x in the form N1=Nx(1−n2) and S1=Sx(1−n2) respectively, where N and S are positive slip constants. Furthermore expected that surface temperature is T_w(x) = T_∞ + T₀x^p where T₀the characteristics temperature parameter and P is the wall temperature parameter. It is concluded from N1=Nx(1−n2) and S1=Sx(1−n2) against physical point of view n should be vary in the range 0 ≤ n ≤ 1. If n > 1 then N₁ and S₁ become singular at x chose to the leading edge of the plate. It is remembered that boundary layer does not start at x=0 but is starts in the vicinity of the leading edge of the plate [33]. Therefore the solution for n>1 is realizable from the mathematical point of view.

Cp is the specific heat at constant pressure, σ is the electrical conductivity, B₀ is the applied magnetic field, μ is the viscosity, T is the temperature, k is the thermal conductivity of the fluid and q‴ is the rate of non-uniform heat generation/absorption coefficient and defined as

Here A^* and B^* are the specifications of space and temperature dependent internal heat generation/absorption respectively. We can note that A^* > 0 & B^* > 0 for the internal thermal generation and A^* < 0 & B^* < 0 for the internal thermal absorption [28].

3 Tranformation of PDE of Flow and Heat transfer equations to ODEs

Using self-similar solution by means of the similarity function f defined by

To solve Eqs. (2) to (4) with the boundary conditions (5) we use the following similarity variables

where ψ is the stream function, which is defined as u=∂ψ∂y and v=−∂ψ∂x,υ=μρ is the kinematic viscosity.

The fundamental partial differential Eqs. (3) and (4) converted to the ordinary differential equations by substituting Eqs. (6) - (8) in Eqs. (2) – (4), we obtain the following ordinary differential equations,

where Pr=υα is the Prandtl number and the converted boundary conditions Eqs. (5) becomes,

where λ=ca is the moving parameter with λ > 0 corresponding to downstream movement of the plate from the origin while λ < 0 corresponding to the moving of the plate into the origin β=Naυ(n+1)2 is the velocity slip parameter and σ=Sa(n+1)2υ is the temperature slip parameter. It is worth mentioning that n = 1 (stagnation point flow), p = 0 (isothermal plate) when β = λ = σ = 0, Eqs. (9) and (10) along with the boundary conditions (11) become identical for m = 1 (stagnation point flow and heat transfer) Bejan [34].

The physical quantities of the local skin friction co-efficient C_f and local Nusselt number Nu_x in this problem are defined as

Where the skin friction (shear stress) along the plate is τ_w and wall heat q_w are given by

Using the Eqs. (8),(12) and (13), we get

where Rex=ue(x)xυ is the Reynolds number.

4 Procedure of Numerical computation

The governing non-linear partial differential equations Eq. (3) and (4) are converted into non-linear ordinary differential equations Eq. (9) and (10). To find the solutions of Eqs. (9) and (10) we required five boundary conditions, three on equation of motion and two on equations of temperature respectively. But here f″(η ) and θ′ (η ) are missing boundary conditions, hence solving the boundary value problem of Eq. (9) and (10) is difficulty, therefore, in the boundary conditions Eq. (11) we replace infinity to a finite value. The Eq. (9) and (10) with boundary conditions (11) are solved a numerical BVP method with efficient shooting technique and for the practical we considered the relative tolerance to 10⁻⁶. Finally, the converged result of BVPs’ are obtained. These all the above said procedure is implemented through the MAPLE13 computing software.

5 Results and Discussion

We consider an analysis to study MHD slip flow and heat transfer in presence of stagnation point of a stretching sheet with non-linearly moving flat plat in a parallel free stream. The boundary layer equations of momentum and heat transfer are solved analytically and the different analytical expressions are obtained for non-dimensional temperature profile for different parameters. Numerical computations of results are demonstrated in the following figures. The characteristics of the parameters are studied in the present problem are power law index parameter n, wall temperature parameter p, magnetic field Mn, prandlt number Pr, space-dependent heat source/sink parameter A^* and temperature dependent heat source/sink parameterB^*. We now proceed with the discussion of results.

Fig. 1 Shows the physical phenomena of the considered problem.

Fig. 1

Physical sketch of the stagnation point flow

Fig. 2 and 3 show the flow velocity profile respectively for the different values of λ and β. It is observed that as increasing the value of λ and β which reduces the thickness of the boundary layer flow.

Fig. 2

Flow profile for different values of λ when β = σ = 0.5

Fig. 3

Flow profile for different values of β when λ = σ = 0.5

Fig. 4 & 5 are depicted for flow and velocity profile of different values of index law parameter n (remaining parameters are remains same), as increasing power law index parameter n both the flow and velocity increases, which intern increases the thickness of the momentum boundary layer.

Fig. 4

Velocity profile for different values of n and different parameter p = 1, pr = 1, M = 0, A = −0.05, B = −0.05

Fig. 5

Velocity profile for different values of n and different parameter p = 1, pr = 1, M = 0.5, A = −0.05, B = −0.05

Fig. 6 is demonstrated the velocity profile for different values of M, due the application of Lorentz force, the velocity of the boundary layer decreases as increases the M.

Fig. 6

Velocity profile for different values of M and n = 1.5, Pr = 1, A = 0.5, B = 0.5, p = 1

Fig. 7 and 8 plotted for the velocity profile for the different values of λ and β. In this figures we observed that as increasing the value of λ it increases the thickness of the velocity but in the case of velocity slip parameter β found that the opposite result.

Fig. 7

Velocity profile for different values of λ when β = σ = 0.5

Fig. 8

Velocity profile for different values of β when λ = σ = 0.5

Fig. 9 and 10 demonstrated the combined profile for different values of n and M and other physical parameters respectively. In both the profile, as increasing the values of n and M which increases the thickness of the velocity of the boundary layer flows respectively.

Fig. 9

Combined profile of power law index for flow and velocity for different values of n and p = 1, M = 1.5, Pr = 1, A = 0.5, B = 0.5

Fig. 10

Combined profile of flow and velocity for different values of M and p = 1, M = 1.5, Pr = 1, A = 0.5, B = 0.5, n = 1

The Fig. 11 and 12 directed for the skin friction profile for various values of moving parameter λ and velocity slip parameter β. Here the effects are found as, as increasing the values of moving parameter λ which increases the thickness of the skin-friction of the boundary layer but decreases as increasing the velocity slip parameter β.

Fig. 11

Skin-friction profile for different values of λ when β = σ = 0.5

Fig. 12

Skin-friction profile for different values of β when λ = σ = 0.5

Fig. 13 & 14. Presented the effect of physical parameter power law index n is depicted for temperature profile, which shows the result as increasing the non-linear stretching parameter n decreasing the temperature i.e., decreasing the thickness of the thermal boundary layer. Due to the Lorentz force, increases the thermal boundary layer thickness as increasing the magnetic field effects, hence the Fig. 15 plotted for the effect of magnetic field on temperature.

Fig. 13

Temperature profile for different values of n and different parameters p = 1, M = 0, A = −0.05, B = −0.05, pr = 1

Fig. 14

Temperature profile for different values of n and different parameters p = 1, pr = 1, M = 0.5, A = 0.5, B = 0.5

Fig. 15

Temperature profile for different values of M and n = 1.5, p = 1, A = 0.5, B = 0.5, pr = 1

Fig. 16 & 17 shown for temperature profile for various values of prandlt number Pr, the characteristics of the parameter prandlt number shows that decreasing the thickness of the thermal boundary layer as increasing the Pr, hence decreasing the temperature due to the thermal layer.

Fig. 16

Temperature profile for different values of Pr and M = 0, p = 1, n = 1.5, A = 0.5, B = 0.5

Fig. 17

Temperature profile for different values of Pr and M = 0.5, p = 1, n = 1.5, A = 0.5, B = 0.5

Fig. 18 & 19 presented temperature profile for different values of A^* and B^* which are called space-dependent heat source/sink parameter and temperature dependent heat source/sink parameter. And Fig. 20 shows the profile for wall temperature parameter as increasing the power law index p which reduces the temperature in the boundary layer.

Fig. 18

Temperature profile for different values of A = −0.5, 0.5, 1 and p = 1, pr = 1, M = 0.5, B = 0.5, n 1.5

Fig. 19

Temperature profile for different values of B = −0.5, 0, 0.5, 1 and p = 1, pr = 1, M = 0.5, B = 0.5, n = 1.5

Fig. 20

Temperature profile for different values of power law index p = 0, 1, 2, 3 and pr = 1, M = 0.5, A = 0.5, B = 0.5, n = 1.5

Fig. 21 and 22 plotted for temperature profile different values of moving parameter λ and velocity slip parameter β. As increasing the moving parameter λ which increases the thickness of the thermal boundary layer and velocity slip parameter β also shows that the same effects of thermal boundary layer.

Fig. 21

Temperature profile for different values of λ when β = σ = 0.5

Fig. 22

Temperature profile for different values of β when λ = σ = 0.5

Fig. 23, 24 and 25 directed for the temperature gradient profile for various values of the physical parameters such as moving parameter, velocity slip and temperature slip parameters respectively. Here some points of effects of these physical parameters are found as, as increasing the values of moving parameter and velocity slip parameter the temperature gradient increases and temperature gradient decreases as increasing the temperature velocity parameter.

Fig. 23

Temperature profile for different values of σ when β = λ = 0.5

Fig. 24

Temperature gradient profile for different values of λ when β = σ 0.5

Fig. 25

Temperature gradient profile for different values of β when λ = σ = 0.5

6 Conclusions

The problem of the stagnation point of MHD fluid flow and heat transfer has been studied in the presence of slip velocity, thermal jump and non-uniform heat source/sink. The numerical results are presented to analyze the various parameter characteristics. Among these results we found some important notices as follows

1. As increasing the power law index parameter which increases the velocity where as opposite result observed in the case of magnetic field

2. The velocity increases as increasing the power law index in both the cases i.e., the magnetic field presents and absent.

3. The temperature increases as increasing the power law index when magnetic field is absent and decreases when magnetic field is present.

4. Whether the magnetic field is present or absent, the temperature decreases as increasing the Prandtl number.

5. The temperature decreases as increasing the parameters A, B and p in presence of magnetic field but increases in case of variation of magnetic field (see Fig. 15).

6. The moving parameter λ reduces the flow, velocity and temperature gradient and decreases the skin-friction as well as thermal boundary layer.

7. The velocity slip parameter reduces the flow, velocity and skin-friction of the boundary layer but increases the thermal boundary layer and temperature gradient.

8. The temperatures slip parameter effects only on thermal boundary layers and which reduces the thermal boundary layer where as decreases the temperature gradient.