**1. Introduction**

The existence of the closed-form solution, the simplicity of the mathematical expression, and the numerous applications, such as plastic sheet extrusion, drawing of plastic films, metallic plate cooling, and the glass blowing of the boundary layer flow past a stretching sheet, have received much attention in recent years. In the process of manufacturing the sheets (metal and plastic), it is required that the melt materials from a slit be stretched continuously until the required thickness is achieved. However, the desired final product of the production process largely depends on the rate at which the sheet cools. The rate of cooling is fundamentally influenced by the type of fluid adjacent to the boundary layer surface [1,2]. The rate at which the sheet is stretched and the rate at which it cools are the two major mechanisms that influence the mechanical properties of the desired product. Similarly, the behaviors of the fluid flows and heat transfer induced by elongating or a moving sheet play a vital role in an industrial process [3].

Many researchers have shown keen interest in the study of flow past a stretching sheet, since the work of Crane [4] on the flow past a flat plate. However, Gupta and Gupta [5] observed that the sheet stretching is not always continuous and, hence, may not necessarily conform to the linear speed. For instance, the stretching of the plastic sheet is inextensible and therefore nonlinear. In view of this, Kumaran and Ramanaiah [6], for the first time, presented a note on two-dimensional boundary layer flow past a stretching sheet. They considered the stretching velocity to be quadratic polynomial and obtained a closed-form solution for the problem. Later, the work of Gupta and Gupta [5] was extended by Vajravelu [7] for a nonlinear stretching sheet. He observed that shear stress is an increasing function of the nonlinear stretching sheet parameter. However, an analytical solution of viscous flow past a nonlinear stretching sheet was also solved by Vajravelu and Cannon [8]. They reported that the fluid velocity is a decreasing function of the nonlinear stretching parameter. On the other hand, Cortell [9] numerically extended and studied this problem using the Runge-Kutta method by considering the effect of viscous dissipation with non-isothermal boundary conditions. He reported that an increase in the nonlinear stretching parameter increases the rate of heat transfer. Similarly, Cortell [10] studied the effect of thermal radiation on an induced quiescent fluid past a nonlinear stretching sheet. It was observed from his report that an increase in the radiation parameter reduces the thickness of the thermal boundary, and hence, the heat transfer rate grows.

One of the mechanisms that influence the rate at which heat is being transported in the flow system is the presence of a magnetic field (such as liquid metals, plasma, electrolyte, or salt water) in such flow problem. The concept of such electrically-conducting fluids is known as magnetohydrodynamic (MHD) fluids and has many applications in engineering processes such as MHD power generators, thermal insulators, MHD pumps, and cooling of nuclear reactors [11]. Furthermore, the presence of a magnetic field in the flow problems plays a vital role in controlling the rate of cooling. In view of these important applications of MHD flows past a stretching sheet, many studies [12–14] were conducted by incorporating the magnetic field into the flow problem. Prasad et al. [15] examined the effect of heat generation on the MHD power law flow over a nonlinear stretching sheet. Their results showed that an increase in the power-law index parameter increased the momentum boundary layer thickness and reduced the thickness of the thermal boundary layer. On the other hand, Ullah et al. [16] analyzed the effects of a chemical reaction in the presence of heat generation/absorption and thermal radiation with convective boundary conditions on an unsteady mixed convection flow of Casson fluid over a nonlinear stretching sheet. In the same vein, Ullah et al. [17] used the Keller–Box numerical scheme method to study the effect of a chemical reaction on an electrically-conducting Casson fluid flow past a nonlinear stretching sheet. Furthermore, Hayat et al. [18] analyzed the magnetohydrodynamic Walters' B nanofluid past a nonlinear stretching sheet. They discovered that the rate of heat transfer and the thermal field are enhanced with the increase in temperature ratio.

A viscoelastic fluid is one of the classes of a non-Newtonian fluids that possesses double effect properties (i.e., heat transfer reduction and drag reduction properties), in addition to its properties of exhibiting both viscosity and elasticity, thereby leading to its numerous applications in polymer industries, for instance paper production, production of glass fiber, extrusion processes, thinning and annealing of copper wires, and the production of artificial fibers and plastic film [19,20]. One of the pioneering works of the study of the viscoelastic fluid past a stretching surface was investigated by Rajagopal et al. [21]. They observed from their results that there is a decrease in the skin friction coefficient with the increase in the viscoelastic parameter. Later, Dandapat and Gupta [22] extended this work by including heat transfer in the flow problem. Furthermore, Cortell [23] analytically investigated the influence of a magnetic field on viscoelastic fluid induced by a stretching sheet. His results showed that the velocity boundary layer thickness is thicker in second-grade fluid compared to that of Walter's B liquid and observed that the viscoelasticity parameter influences both the viscoelastic fluids.

MHD viscoelastic fluid flow over a stretching sheet plays a vital role in chemical engineering, the metallurgy industry, the polymer extrusion process, the manufacturing of plastic sheets, the drawing of wires and plastic films, the cooling of metallic sheets, and petroleum engineering [23]. However, Andersson [24] studied MHD viscoelastic fluid flow over a stretching sheet. He reported from his findings that both viscoelastic and magnetic parameters have the same effect in the fluid flow problem. The radiative effect on MHD viscoelastic fluid flow past a stretching sheet was examined by Char [25]. Likewise, Prasad et al. [26] examined the behavior of an electrically-conducting viscoelastic fluid and heat transfer over a stretching sheet. They noticed that an increase in the magnetic parameter leads to a significant decrease in the wall temperature profile and the velocity gradient. Moreover, an exact solution of a viscoelastic fluid past a stretching sheet with a heat source and viscous dissipation was studied by Abel et al. [27]. Later, Abel et al. [28] examined MHD viscoelastic fluid flow past a stretching sheet and found that the surface temperature of the flow diminishes with the increase in the viscoelasticity. Furthermore, the flow of a viscoelastic fluid induced by a nonlinear stretching sheet using the optimal homotopy analysis method was studied by Mustapha [29].

Besides the importance of a magnetic field in the boundary layer flow and heat transfer is the presence of viscous dissipation in the energy equation. Viscous dissipation plays a significant role similar to that of the energy source, which changes the distribution of the temperature and thereby the rate of heat transfer. This process finds its applications in the flow of oil products through ducts and in polymer processing. The viscous dissipation effect and variable surface temperature on viscous flow past a stretching sheet were examined by Cortell [9]. He explained that the temperature of the fluid rises with an increase in the Eckert number. In the same vein, Abel et al. [30] analyzed the Ohmic and viscous dissipation effect on the MHD boundary layer flow of a viscoelastic fluid past a linear stretching sheet. They observed from their study that the fluid temperature in both the PST and PHF cases amplified with the increase in viscous dissipation parameter. The effects of viscous dissipation and thermal radiation on two-dimensional viscous flow over a nonlinear stretching sheet was addressed numerically through a similarity solution by Cortell [31]. He showed that an increase in thermal radiation and Eckert number leads to the increase in the temperature distribution. A Casson fluid flow over a nonlinear stretching sheet was studied by Medikare et al. [32]. The effects of the heat source/sink and viscous dissipation on magnetohydrodynamic non-Newtonian fluid flow in the presence of Cattaneo-Christov heat flux was examined by Ramandevi et al. [33]. Recently, the effect of joule heating and viscous dissipation on an electrically-conducting tangent hyperbolic nanofluid was examined by Atif et al. [34]. It was reported in their work that the Eckert number and slip parameter enhanced the thermal and concentration fields.

From the above reviews, it is perceived that no consideration has been given to an electrically-conducting viscoelastic fluid over a nonlinear stretching sheet with the viscous dissipation effect. This provides the enthusiasm for the present work, in which the effects of power law surface temperature and power law surface heat flux on the characteristics of the heat transfer of an MHD viscoelastic fluid past a nonlinear stretching sheet in the presence of viscous dissipation are investigated. The unconditionally stable Keller box method was employed in solving the transformed ordinary differential equation by considering non-isothermal boundary conditions. This boundary condition is applicable in an engineering process where the temperature in not constant.

#### **2. Constitutive Equation**

The rheological equation of state, which is also known as the constitutive equation, describes the relationship between strain, stress, and their time dependence. According to Newtons's law of viscosity "the stress is often proportional to the strain rate",

$$
\pi = \mu \frac{\partial \mu}{\partial y} \tag{1}
$$

where, τ, μ, and *<sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>y</sup>* respectively represent the shear stress, the dynamic viscosity, and the velocity gradient or rate of the strain. Thus, any fluid that does not obey Newtons's law of viscosity is termed as a non-Newtonian fluid. Examples of such fluids are a Casson fluid, a viscoelastic fluid, a power law fluid, and many more. There is no single constitutive equation that describes the behaviors of

non-Newtonian fluids due their diverse nature. However, amongst the numerous non-Newtonian fluids' models is Walter's B viscoelastic fluid model. This model has a constitutive equation of the form [35,36]:

$$\mathbf{T} = -p\mathbf{I} + 2\mu\_0 \mathbf{e} - 2k\_0 \frac{\delta \mathbf{e}'}{\delta t} \tag{2}$$

The Cauchy stress tensor **T** is expressed in terms of scalar pressure *p*, identity tensor **I**, dynamic viscosity μ0, short memory coefficient *k*0, and the convected differentiation of the strain rate <sup>δ</sup>**e**- <sup>δ</sup>*<sup>t</sup>* , while the strain rate tensor **e** is defined in terms of the velocity vector **u** as:

$$\mathbf{e} = \nabla \mathbf{u} + (\nabla \mathbf{u})^T \tag{3}$$

It is convenient to represent (3) in the form of matrix suffix notation as:

$$\varepsilon\_{ij} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial x\_j} + \frac{\partial u\_j}{\partial x\_i} \right) \tag{4}$$

where *i* and *j* can take values of one and two for two-dimensional flows, and <sup>δ</sup>**e**- <sup>δ</sup>*<sup>t</sup>* is expressed as:

$$\frac{\delta \mathbf{e}^{\prime}}{\delta t} = \frac{\delta \mathbf{e}}{\delta t} + \mathbf{u} \cdot \nabla \mathbf{e} - \mathbf{e} \cdot \nabla \mathbf{u} - (\nabla \cdot \mathbf{u})^{T} \cdot \mathbf{e} \tag{5}$$

Here, **u** denotes the velocity vector. Furthermore, the generalized constitutive equation for Walter's B viscoelastic fluid can be expressed as:

$$\mathbf{T} = \begin{pmatrix} \tau\_{xx} & \tau\_{xy} & 0\\ \tau\_{yx} & \tau\_{yy} & 0 \end{pmatrix} = -p\mathbf{I} + 2\mu\_0 \mathbf{e} - 2k\_0 \left[ \frac{\delta \mathbf{e}}{\delta t} + \mathbf{u} \cdot \nabla \mathbf{e} - \mathbf{e} \cdot \nabla \mathbf{u} - (\nabla \cdot \mathbf{u})^T \cdot \mathbf{e} \right] \tag{6}$$

Lastly, the Cauchy equation of motion is given by (see Jaluria [37]):

$$
\rho \frac{D\mathbf{u}}{Dt} = \nabla \cdot \mathbf{T} + \mathbf{F} \tag{7}
$$

where ρ is the fluid density, *<sup>D</sup>***<sup>u</sup>** *Dt* is material derivative, **T** is the Cauchy stress tensor and **F**=(*Fx*, *Fy*, 0) is the body force. Following Ahmad et al. [38], the body force **F** can be expressed as **F** = ρ**g** + **J** × **B**, where **g** is the gravitational field, **J** is the current density, **B** = (0, *B*, 0) is the magnetic force and **J** × **B** is the Lorentz force and is simplified as

$$\mathbf{F} = \sigma B^2 \mathbf{u} \tag{8}$$

where σ is the electrical conductivity.
