*4.2. Case II: Prescribed Heat Flux*

For this case, the thermal boundary conditions are:

$$-k\frac{\partial T}{\partial y} = q\_w = c\mathbf{x}^{2\mathbf{n}-1} \qquad \text{at } y = 0$$

and:

$$T \to \infty \qquad \text{ as} \quad y \to \infty \tag{34}$$

where *n* stands for the surface temperature parameter. The dimensionless temperature *g*(η) is assumed to be of the form:

$$g(\eta) = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}} \Big|\_{} \qquad \text{where} \qquad T\_w - T\_{\infty} = \frac{c}{k} \mathbf{x}^{2n-1} \sqrt{\frac{2v\mathbf{x}}{(n+1)u}} \tag{35}$$

Substitute Equations (19), (20), (34), and (35) into Equation (26) to get dimensionless energy equation for PHF as:

$$\int \text{g}'' + \text{Pr}\left\{ f\text{g}' - \left(\frac{2(2n-1)}{n+1}\right) f\text{'g} + E\_c(f')^2 \right\} = 0 \tag{36}$$

similarly, the corresponding boundary conditions (34) give:

$$g'(\eta) = -1 \qquad \text{at} \qquad \eta = 0$$

$$g(\eta) = 0 \qquad \text{as} \qquad \eta \to \infty \tag{37}$$

and all other physical parameters are in analogy with those mentioned in case 1, except for the constant *c*, which will be replaced in PHF.

#### **5. Method of the Solution**

An implicit finite difference scheme known as the Keller box method is used in solving the ordinary differential Equations (21), (29), and (36) with their respective boundary conditions (22), (30), and (32). This method is very accurate, useful, efficient, and unconditionally stable. More explanations of this method can be found in [41,42], and it involves four basic steps:


The step size Δη = 0.02 was used to achieve the numerical solution, and the procedures were repeated until the convergence to a specified accuracy was achieved.

#### **6. Results and Discussion**

In order to examine the effects of nonlinear stretching parameter *n*, viscoelastic parameter *K*, magnetic parameter *M*, Eckert number *Ec*, and Prandtl number *Pr*, on boundary flow of an electrically-conducting viscoelastic fluid past a nonlinear stretching sheet, results from the graphs for velocity and temperature profiles, as well as the numerical results of the skin friction coefficient and heat transfer rate (reduced Nusselt number) for the PST and PHF cases are illustrated in tables. To validate the accuracy of the present numerical method, we computed and compared the present results for skin friction coefficient *f* --(0) with different values of *n* and *K* with that of Vajravelu [7] and Arnold et al. [43] as displayed in Table 1, heat transfer rate θ- (0) (for the PST case), and the surface temperature *g*(0) (for the PHF case) for different values of *Pr* with that of Arnold et al. [43] when *Ec* = 0.1 and *K* = 0.02, as shown in Table 2. The current results demonstrated a close agreement with the previous results under some certain conditions. It is also observed from Table 2 that the rate of heat transfer increased with an increase in *Pr* for the PST case. This is because fluid with higher *Pr* has a comparatively lower thermal conductivity, which decreases the conduction, which in turn increases the variation. This phenomenon reduces the thickness of the thermal boundary layer and increases the heat transfer at the surface. Similarly, the surface temperature reduced with the increase in the *Pr* for the PHF case. This implies that an increase in *Pr* has a cooling effect on the surface.

Table 3 is introduced to analyze the influence of some physical parameters for *M* = 0 and *M* = 5 on the coefficient of skin friction *f* --(0), and it is observed from this table that an increase in *n* and *K* increased the coefficient of skin friction *f* --(0) significantly. This behavior is also true with the inclusion of a magnetic field *M* into the flow problem. In Table 4, the variation of the rate of heat transfer θ- (0) and the temperature at the wall *g*(0) with different values *M*, *K*, *Ec*, and *Pr* for *n* = 2 in both PST- and PHF cases is recorded. It is clear that the magnitude of the heat transfer rate decreased with an increase in *M*, *K*, and *Ec* and increased significantly with the increase in the value of Prandtl number *Pr* for the PST case. Similarly, the temperature at the wall increased with the increase in *Ec* and *M* and reduced with respect to the increase in *K*. The effect of *K* was to increase the heat transfer rate for the PST case and the temperature at the surface for the PHF case, while the effect of *Ec* and *M* was to amplify the heat transfer rate and diminish the temperature at the surface for both the PST- and PHF cases. This implies that the thickness of the thermal boundary layer reduced as *M* and *Ec* increased, which led to the higher heat transfer rate at the surface, thereby enhancing the temperature *g*(η). Similarly, an increase in *Pr* led to the reduction in the thickness of the thermal boundary layer in both cases.

The effects of different dimensionless parameters, such as nonlinear stretching parameter *n*, viscoelastic parameter *K*, magnetic parameter *M*, Eckert number *Ec*, and Prandtl number *Pr*, for prescribed surface temperature (PST) and prescribed heat flux (PHF) on the electrically-conducting viscoelastic fluid and heat transfer are depicted in Figures 1–13.

Figures 2–4 illustrate the behaviors of velocity and temperature profiles for various values of nonlinear stretching sheet parameter *n* with and without a magnetic field in the PST and PHF cases. An increase in *n* decreased the velocity profile, which led to the increase in the coefficient of skin friction. The decrease was high in the presence of a magnetic field, which physically shows the influence of a magnetic field in an electrically-conducting viscoelastic fluid to produce an opposing force known as Lorentz force. This force has the ability of slowing down the fluid flow in the layer region. It is also noticed from Figure 2 that an increase in *M* decreased the velocity profile. Figures 3 and 4 illustrate that the temperature profiles for PST- and PHF cases increased with the increase in the nonlinear stretching sheet parameter *n*. This implies that the rate of heat transfer reduced with the

increase in *n*. This phenomenon showed that the thickness of the momentum boundary layer became thinner and thermal boundary thickness became thicker with an increase in *n*. The same behavior of this trend was reported by Vajravelu [7].

**Table 1.** Comparison of local skin friction *f* --(0) at the wall for the present results and that of Vajravelu [7] when *K* = *R* = *Q* = *Ec* = 0 with that of Arnold et al. [43] for various values of *K* when *M* = *Ec* = 0.


**Table 2.** Comparison of the numerical results of the heat transfer rate −θ- (0) and the surface temperature *g*(0) with the published results of Arnold et al. [43] for the PST- and PHF cases, respectively, for different values of *Pr* with *Ec* = 1.0 and *K* = 0.02.


**Table 3.** Variation of the numerical results for skin friction −*f* --(0) with different values of *M*, *n*, and *K*.


The effect of viscoelastic parameter *K* on the velocity and temperature profiles is illustrated in Figures 5–7 in the presence and absence of a magnetic field. The momentum boundary layer thickness decreased with the increase in the value of *K*, as shown in Figure 5. This implies that tensile stress enhances the viscoelasticity through a large value of *K*, which has a tendency to increase the adherence to the surface of the momentum boundary layer, thereby reducing the velocity of the fluid. In the same vein, the temperature profile increased with the increase in the value of *K* in both PST- and PHF cases. Physically, an increase in the viscoelastic normal stress leads to the increase in the thermal boundary layer thickness. This process was the same in both the PST- and PHF cases. However, it is revealed by these figures that the inclusion of a magnetic field increased the thermal boundary thickness a bit higher compared to the case where *M* = 0. This result coincides with the results reported by Hayat et al. [18].


**Table 4.** Variation of the numerical values for the rate of heat transfer −θ- (0) and the wall temperature *g*(0) for PST- and PHF cases for different values of *n*, *M*, *K*, *Ec*, and *Pr*.

**Figure 2.** Variation of the velocity profile for different values of nonlinear stretching sheet parameter *n* and magnetic parameter *M* for *K* = 1, *Pr* = 0.71, and *Ec* = 0.1.

The influence of the magnetic field *M* on the temperature profiles is shown in Figures 8 and 9. It is observed in these figures that the temperature profile increased with the increase in *M*. Physically, the presence of a magnetic field in an electrically-conducting fluid generates a drag-like body force, which always acts against flow, and as a result, the fluid flow decelerates. This force is known as Lorentz force and has the ability to oppose the fluid motion. The temperature of the fluid is also enhanced due to the resistance offered by this force. This shows the thermal boundary layer thickness was reduced in both the PST- and PHF cases. However, the decrease was more pronounced when *Pr* = 0.71 as compared to when *Pr* = 7.

**Figure 3.** Variation of the temperature profile for different values of nonlinear stretching sheet parameter *n* and magnetic parameter *M* for the prescribed surface temperature (PST) case when *K* = 1, *Pr* = 0.71, and *Ec* = 0.1.

**Figure 4.** Variation of the temperature profile for different values of nonlinear stretching sheet parameter *n* and magnetic parameter *M* for the prescribed heat flux (PHF) case when *K* = 1, *Pr* = 0.71, and *Ec* = 0.1.

**Figure 5.** Variation of the velocity profile for different values of viscoelastic parameter *K* and magnetic parameter *M* for *n* = 3, *Pr* = 0.71, and *Ec* = 0.1.

**Figure 6.** Variation of the temperature profile for different values of viscoelastic parameter *K* and magnetic parameter *M* for the PST case when *n* = 3, *Pr* = 0.71, and *Ec* = 0.1.

**Figure 7.** Variation of the temperature profile for different values of viscoelastic parameter *K* and magnetic parameter *M* for the PHF case when *n* = 3, *Pr* = 0.71, and *Ec* = 0.1.

**Figure 8.** Variation of the temperature profile for different values of magnetic parameter *M* and Prandtl number *Pr* for the PST case when *n* = 3, *K* = 1, and *Ec* = 0.1.

**Figure 9.** Variation of the temperature profile for different values of magnetic parameter *M* and Prandtl number *Pr* for the PHF case when *n* = 3, *K* = 1, and *Ec* = 0.1.

Figures 10 and 11 depict the effects of Eckert number *Ec* on the temperature profiles in the PSTand PHF cases in the presence of a magnetic field. It is noticed that an increase in the value of the Eckert number amplified the temperature profiles in both cases. The enhancement was higher for a small value of the Prandtl number. The influence of augmenting *Ec* is to improve the temperature distribution in the boundary layer region, thereby increasing the thickness of the thermal boundary layer. This is because heat energy is stored in the fluid due to frictional heating, which arises due to the presence of viscous dissipation. A large value of *Ec* produces more heat in the fluid. We can therefore infer that an increase in *Ec* improved the temperature of the fluid at any point for the PST- and PHF cases. However, the temperature distribution remained the same at the surface with the variation of the Eckert number for the PST case. A similar behavior was reported by [27].

**Figure 10.** Variation of the temperature profile for different values of Eckert number *Ec* and Prandtl number *Pr* for the PST case when *n* = 3, *K* = 1, and *M* = 2.

**Figure 11.** Variation of the temperature profile for different values of Eckert number *Ec* and Prandtl number *Pr* for the PHF case when *n* = 3, *K* = 1, and *M* = 0.

Figures 12 and 13 are the plots of θ(η) and *g*(η) with respect to η for various values of *Pr* in the presence and absence of the Eckert number. It is clearly evident from these figures that an increase in *Pr* decreases the temperature profiles in the PST and PHF cases, respectively. This implies that the thickness of the thermal boundary layer is augmented with the decrease in the value of the Prandtl number. This is in close agreement with the results of Arnold et al. [43]. Physically, an increase in the Prandtl number will cause heat transfer enhancement, and this is consistent with the fact that the thickness of the thermal boundary reduces with an increase in *Pr*.

**Figure 12.** Variation of the temperature profile for different values of Prandtl number *Pr*, Eckert number *Ec*, and magnetic parameter *M* on the temperature profile for the PST case when *n* = 3 and *K* = 1.

**Figure 13.** Variation of the temperature profile for different values of Prandtl number *Pr*, Eckert number *Ec*, and magnetic parameter *M* on the temperature profile for the PHF case when *n* = 3 and *K* = 1.
