1 Introduction

The non-Newtonian fluids are now well suited in chemical,food,biological,and pharmaceu- tical industries^{[1, 2, 3, 4, 5]}. The Navier-Stokes equations are not able to explain the flow properties of such fluids. The non-Newtonian fluids in view of their diverse rheological characteristics are classified into three classes known as differential,integral,and rate types. The differential type subclasses,i.e.,second,third,and fourth grades,have been studied extensively by the scientists and engineers in the past. The Maxwell and Oldroyd-B fluids (subclasses of the rate type) have been also explored via reasonable attempts. However,the Burgers fluid (a subclass of the rate type) has not been given proper attention due to complex constitutive equation. Recently,four processes of heat transfer have been pointed by Merkin^[6]. These are due to the constant wall temperature (CWT),the constant heat flux at the surface,the flow of heat through wall with definite thickness and conductivity,and Newtonian heating (also called the conjugate heating process). Although Newtonian heating is important in various mechanical appliances including heat exchangers,heat fins,etc.,this process of heat transfer has scant information in the lit- erature. Lesnic et al.^[7] investigated Newtonian heating in the free convection flow saturating a porous medium. The free convective time-dependent flow subject to vertical surface with Newtonian heating was analyzed by Chaudhary and Jain^[8]. The forced convection flow with Newtonian heating was discussed by Salleh et al.^[9]. Salleh et al.^[10] also considered the circular cylinder and discussed the Newtonian heating phenomenon in detail for the mixed convective flow. Niu et al.^[11] performed the mathematical modeling for the flow of an Oldroyd-B fluid model subject to the Newtonian heating phenomenon. They discussed the stability analysis of the system in the presence of thermal convection.

Our interest here is to proceed further Newtonian heating analysis in the magnetohydrody- namic (MHD) stagnation point flow. The Burgers fluid model is considered. This viscoelastic model is able to predict the properties of asphalt,concrete,etc. The results of the Oldroyd- B and upper-convected Maxwell (UCM) fluid can be deduced from the present study as the special cases. The governing equation is derived by covariant differentiation rather than mate- rial differentiation. The homotopy analysis method (HAM) (see Refs. ^[12]-^[14] and references therein) is used in the computation of solutions. Graphical and numerical results are displayed to discuss the impact of various parameters. 2 Analysis

We consider the Burgers fluid flow by a stretching surface. An incompressible MHD fluid occupies the region y > 0. A constant magnetic field B0 is applied. The x-axis is taken along the sheet,while the y-axis is transverse to the sheet. The free stream velocity is

U_e(x) = ax,

whereas U_w(x) = cx is the stretching velocity. Notice that a and c are positive dimensional constants having the dimension of time. The equations representing the flow dynamics are given by

with Here,u and v are the velocities in the x- and y-directions,respectively,λ₁ and λ₃ are the relaxation time,whereas λ₂ is the retardation time,T is the temperature,h_s denotes the kinematic viscosity,α denotes the thermal diffusivity of the fluid,hs denotes the heat transfer parameter, and T∞ shows the ambient value of temperature.

Let

Then,Eq. (1) is satisfied identically,and Eqs. (2) and (3) become

Here,the Deborah numbers (β₁ and β₃) depend on the relaxation time,and the Deborah number (β₂) depends on the retardation time. Further,Pr represents the Prandtl number, is the conjugate parameter for Newtonian heating,M is the Hartman number,and A denotes the stagnation point parameter. The values of these variables are The local Nusselt number Nu_x is where the heat flux q_w is in which k is the thermal conductivity. In a dimensionless form,we get where Re_x = ax²/v is the local Reynolds number. Now,the HAM is used to evaluate the differential equations (7) and (8) with respect to the conditions (9). The parameters _f and _θ are important for convergence of functions f and θ, respectively. For this reason,the _f - and _θ-curves are prepared and displayed in Figs. 1 and 2. We observe that the ranges for suitable values of _f and _θ are

respectively.

Table 1 shows the required order of approximation for convergent series solution. From Table 1,we can see the convergence at 15th order of approximation. A comparison between the present and previous limiting solutions is presented. The comparison shows an excellent agreement (see Table 2). Our further interest is to address the influence of parameters of pref- erence on both the velocity and the temperature. Thus,Figs. 3-9 are shown for the velocity and temperature fields. Figure 3 elucidates that when we increase A,the corresponding boundary layer thickness and velocity increase (0≤A<1). However,when stretching is less than the free stream velocity,i.e.,A>1,the velocity is enhanced. In view of physical arguments,the larger values of A along with the greater free stream velocity result in an increase in the velocity of fluid. The effect of M on f′ is plotted in Fig. 4. This figure clearly indicates that the magni- tude of velocity component f′ decreases for larger M. We can conclude that the magnetic field retards and controls the velocity of the fluid flow. The effects of the viscoelastic parameters β₁,β₂,and β₃ on f′ are shown in Figs. 5-7. The velocity and boundary layer thickness reduce when β₁ and β₂ are increased. Moreover,the effect of β₃ is quite opposite to that of β₁ and β₂. Figures 8 and 9 analyze the behavior of the dimensionless temperature θ (η). Figure 8 shows the variation of on the temperature. It is observed that the temperature increases significantly near the wall with an increase in ,and the thermal slip is observed due to the Newtonian heating phenomenon. This deviation of temperature profiles is because of the fact that when we increase ,the heat transfer rate from the surface also increases and hence the temperature. The temperature via the Prandtl number Pr is observed in Fig. 9. It is observed that larger values of the Prandtl number decrease the thermal conductivity and consequently the thermal boundary layer decreases. It is found that the increase in the temperature profiles is more bulging at low Pr than larger values.

The Nusselt number is shown numerically in Table 3. Clearly,the Nusselt number reduces by increasing β₁,β₂,and M,whereas the increase in A,β₃,Pr,and enhances the numerical value of the Nusselt number.