*3.2. Convergence Inspection*

As pointed out by Liao [59], the convergence of homotopic results can be controlled by the auxiliary parameters *u* and θ. In Figures 2 and 3, it is observed that the minimum error for the velocity and temperature profiles can be achieved at *u* = −0.5 and θ = −0.65.

In addition, the residual error norms have been utilized to further ensure the accuracy of the obtained series solutions. The residual errors of the velocity and temperature profiles for two succeeding *Eu* = 12120

(*u*(*i*/20))<sup>2</sup> and *E*θ =

approximations of temperature *E*θ and velocity *Eu* up to the 20th order iteration as given in Table 3 can be obtained by the following expressions:

> 12120

 (48)

residual error for θ.

**Figure 3.** Graph of

**Table 3.** Residual error of series solutions when *Gr* = 3.1556, *Br* = 1, *Re* = 442.956, β = 1, *M* = 0.1 and ρ*e* = 1.


## *3.3. Graphical Illustration*

The current research is about flow and entropy on a magnetized power law nanofluid along the horizontal walls. The effects of the nanofluid can be determined at the same pressure with the electric body force and the motion of the upper plate. Flow and entropy generation on the magnetized power law of a nanofluid in the horizontal channel are studied systematically. The lower wall is heated, and the upper wall maintains the temperature. The influence of different important parameters such as Grashof number, electro-osmosis, Reynolds number, magnetic field, Brinkman number, entropy generation, nanoparticle volume fraction and Bejan number on temperature and velocity distributions were illustrated graphically in Figures 4–23. By using different parameters, we ge<sup>t</sup> different results which can be explained as *n* = 0.764 (PVC 3%), φ = 0.03, *M* = 2, β*u* = 0.3, γ = 10, κ = 8, *Gr* = 2.366, *Re* = 442.956 and *Br* = 1.

The impact of *M* on temperature and velocity is demonstrated in Figures 4 and 5. They show that when the magnetic parameter increases, the velocity profile decreases while the temperature profile increases. When one applies a magnetic field on electrically treated nanofluid, then it produces a force opposite to the flow direction. Consequently, the Lorentz force increases the magnetic parameter which opposes the fluid flow, and due to that, the velocity distribution decreases and the temperature distribution upsurges. The effects of volume fraction φ on temperature are depicted in Figure 6. The temperature profile increases by increasing the φ volume fraction. Various concentrations of polyvinylchloride (PVC) on temperature and velocity are illustrated in Figures 7 and 8. From Figure 7, it is observed that the velocity distribution rises with the increase of PVC. The temperature for various values of PVC are portrayed in Figure 8. It can be seen that temperature is a decreasing function between the channel. Figure 9 shows the effect of β*u* on velocity profile. β*u* can be expressed as the ratio of *UHs* and *um* of a nanofluid. Figure 9 shows that flow of a channel exceeds and gains its high value. Figure 10 reveals a minute change for increasing values of the electro-osmotic parameter for velocity and temperature. The effects of the electro-osmotic value of κ on temperature and velocity profile are shown in Figures 11 and 12. It can be seen from Figure 11 that by increasing κ, the velocity profile increases. This is due to a larger value of κ for the velocity profiles that display EDL layers. The effect of electro-osmotic κ on temperature is illustrated in Figure 12, which gives the deficiency of the Joule effect. It is also noted that if the pseudo-plastic increases in electro-osmotic parameter κ, it results in a notable increase in temperature. The effect of *Br* on temperature is shown in Figure 13. It is perceived that by increasing *Br*, the temperature decreases, from which we conclude that *Br* increases as compared to bulk mean temperature. The effect of Brinkman number with respect to volume friction is expressed in Figure 14. It is perceived that for a developing Brinkman number with respect to volume fraction, the Nusselt number increases. Results indicate that temperature increases for higher values of nanoparticle volume fraction. The influence of *UHs* and *um* of nanofluid β*u* and κ on the Nusselt number are shown in Figure 15. It is seen that near the heated wall, the Nusselt number reduces; this is because first heat is moved to the fluid and then transferred into a separated plate.

The entropy generation profiles for various important parameters such as volumetric volume expansion β, magnetic field *M*, dimensionless temperature parameter *Br*Ω−1, and Brinkman number *Br* are illustrated in Figures 16–21. The impact of *M* on entropy generation is displayed in Figure 16. It can be observed that when *M* increases, the entropy generation decreases on the left side with minimum energy loss at *y* = −0.25; after this, increasing behavior is detected. The effect of β on entropy generation is depicted in Figure 17, where it can be observed that when β is increasing, the entropy generation is also increased. The energy loss on the upper wall is comparatively greater when compared to the lower wall. The significance of *Br* on entropy generation in Figure 18 shows that when *Br* increases, entropy generation decreases on the left side and has minimum energy loss at *y* = 0; after this, an increasing trend on the left side is seen. The influence of *Br*Ω−<sup>1</sup> on entropy generation is given in Figure 19 and it is observed that entropy generation decreases on the lower wall and has minimum energy loss at *y* = 0.15. Figure 20 shows that when the electro-osmotic parameter increases, entropy generation also increases. From Figure 21, it is observed that when *R*ζ increases, the entropy generation at the lower wall decreases, but after *y* = 0, it increases at the upper wall. Figure 22 shows that when the magnetic parameter increases, Bejan number also increases. It is observed that there is a dominant effect on the lower as well as the upper wall. Figure 23 shows that when *Br*Ω−<sup>1</sup> increases, the Bejan number also increases. It is noted that the Bejan number shows a dominant role on the upper wall.

The impact of the different parameters on skin friction such as the electro-osmotic parameter, the ratio between *UHs* and *um*, volume concentration, Brinkman number, and magnetic field are given in Table 4. It is found that by increasing the Brinkman number and the magnetic parameter, the skin friction decreases.

**Figure 4.** Performance of *M* on velocity.

**Figure 5.** Performance of *M* on temperature.

**Figure 6.** Performance of φ on temperature.

**Figure 7.** Performance of PVC concentration on velocity.

**Figure 8.** Performance of PVC concentration on temperature.

**Figure 9.** Performance of β*u* on velocity.

**Figure 11.** Effect of the electro-osmotic parameter on velocity.

**Figure 12.** Performance of the electro-osmotic parameter on temperature.

**Figure 13.** Performance of Brinkman number *Br* on temperature.

**Figure 14.** Performance of *Br* w.r.t φ on Nusselt number.

**Figure 15.** Performance of κ w.r.t β*u* on Nusselt number.

**Figure 16.** Performance of the magnetic parameter on entropy generation.

**Figure 17.** Performance of β for various values on entropy generation.

**Figure 18.** Performance of Brinkman number *Br* on entropy generation.

**Figure 19.** Performance of *Br*Ω−<sup>1</sup> on entropy generation.

**Figure 20.** Performance of κ on entropy generation.

**Figure 21.** Performance of *<sup>R</sup>*ζ on entropy generation.

**Figure 22.** Performance of the magnetic parameter on Bejan number.

**Figure 23.** Performance of *Br*Ω−<sup>1</sup> on Bejan number.


**Table 4.** Coefficient of skin friction *Cf* with *M*, *Br* and β*u* along *n* = 0.764 (PVC 3%).
