**5. Results and Discussions**

This section's focus is on examining the effects of velocity, temperature, concentration profile, entropy generation, and Bejan number. The influence of fluid parameter *α*, porosity parameter *β*0, volume fraction *φ*1, curvature parameter *γ*, and inertia coefficient *Fr* on fluid velocity is examined in Figure 2a–e. The effect of fluid parameter *α* on the velocity profile is presented in Figure 2a. With the higher value of *α*, the fluid velocity and boundary layer thickness increase. In reality, as *α* increases, the viscosity of the fluid declines, resulting in an improvement in the velocity profile. The impression of the porosity factor *β*<sup>0</sup> on the velocity profile of Powell–Eyring nanofluid is shown in Figure 2b. The velocity of the nanofluid declines when permeability of the fluid rises, which is in line with authenticity. Moreover, the permeability of the border has no consequence on the fluid velocity as we move away from it. The impact of volume fraction *φ*<sup>1</sup> on velocity profile is plotted in Figure 2c. By raising the volume fraction, the Powell–Eyring nanofluid velocity decreases. The science here seems to be that when the volume fraction increases, the flow becomes more vicious, resulting in friction forces that slow the nanofluid velocity. The feature of the curvature parameter *γ* on the velocity profile is presented in Figure 2d. As *γ* goes up, the fluid velocity shrinks at the surface and escalates further from the cylinder, according to the results. In reality, as the curvature parameter is increased, the cylinder's radius decreases. As a result, the cylinder's contact surface with the fluid lowers, providing less confrontation to fluid motion. Consequently, the velocity profile rises. The impression of inertia coefficient *Fr* on the velocity profile is shown in Figure 2e. The velocity profile drops as the inertia coefficient rises.

The impact of temperature profile *θ*(*η*) against relevant flow parameters, such as fluid parameter *α*, porosity parameter *β*0, curvature parameter *γ*, Prandtl number Pr, temperature ratio *θω*, thermal radiation *Rd*, Eckert number *Ec*, thermophoresis constraint *Nt*, and Brownian motion constraint *Nb*, are depicted in Figure 3a–j. Figure 3a signifies the inspiration of the fluid constraint *α* on the temperature profile. Higher *α* values result in a lower temperature profile, according to the findings. The viscosity of the thermal boundary layer continues to shrink, as the higher value of fluid parameter *α* viscosity of the fluid decreases.

Accordingly, the temperature profile declines. The variation in temperature profile with *η*. over a range of the porous parameter *β*<sup>0</sup> is shown in Figure 3b. This illustration clearly demonstrates that the heat distribution is a weak function of *β*<sup>0</sup> and that it changes little when it passes through the thermal boundary layer. Therefore, increasing *β*<sup>0</sup> causes a modest thickening of the thermal boundary layer. The outcome of volume fraction parameter *φ*<sup>1</sup> on the temperature profile is depicted in Figure 3c. It is observed that the increasing volume fraction raises the temperature profile. The science behind this mounting temperature pattern is because the temperature rises as the smash flanked by the molecules of the Powell–Eyring nanofluid rises. Figure 3d exhibits the impressions of the curvature constraint on the temperature profile, with temperature showing an increasing trend via *γ*. As *γ* increases, the surface contact area exposed to fluid particles decreases, resulting in less resistance for particles and an increase in their average velocity. The temperature rises because the Kelvin temperature is expressed by an average kinetic energy. The characteristics of the Prandtl number Pr on the temperature distribution are exposed in Figure 3e. The temperature profile and thickness of the thermal boundary layer are found to diminish as the Prandtl number rises. It connects thermal diffusivity to momentum diffusivity. Accordingly, a higher Prandtl number correlates to a reduced thermal diffusivity; consequently, temperature distribution rises up before dropping down. Figure 3f depicts the effects of temperature ratio *θω* on the temperature distribution. It has been realized that advanced temperature ratio enhances the temperature distribution. Figure 3g depicts the

behavior of the radiation parameter *Rd* on a temperature profile. For bulky levels of the radiation parameter, temperature and the accompanying boundary layer thickness rise. Higher values of the radiation parameter reduce the mean absorption coefficient, resulting in an upturn in the temperature distribution. Figure 3h depicts the difference in fluid temperature caused by the change in Eckert number *Ec*. The figure depicts how the fluid temperature rises as the value of *Ec* rises. This happens since frictional heating produces heat in the fluid as the value of *Ec* rises. Physically, the Eckert number is explained as the ratio of kinetic energy to the difference in specific enthalpy flanked by the wall and the fluid. As a result of the effort exerted against the viscous fluid pressures, an upsurge in Eckert number converts kinetic energy into internal energy. As a result, as *Ec* rises, the fluid's temperature rises. As shown in Figure 3i, the thermophoresis parameter *Nt* has an outcome on the temperature. The graph shows that as the number of thermophoresis parameters *Nt* increases, so does the temperature. The temperature of the fluid rises as the temperature variance amid the surface and ambient heat grows. The stimulus of the Brownian motion *Nb* on the temperature is revealed in Figure 3j. This graph shows that augmented *Nb* raises the temperature.

**Figure 2.** Velocity profile variation for distinct values of (**a**) is *α*, (**b**) is *β*0, (**c**) is *φ*1, (**d**) is *γ*, (**e**) is *Fr*.

**Figure 3.** *Cont.*

**Figure 3.** Temperature profile variation for distinct values of (**a**) is *α*, (**b**) is *β*0, (**c**) is *φ*1, (**d**) is *γ*, (**e**) is Pr, (**f**) is *θω*, (**g**) is *Rd*, (**h**) is *Ec*, (**i**) is *Nt*, (**j**) is *Nb*.

The consequence of significant flow parameters such as the fluid parameter *α*, curvature parameter *γ*, the Schmidt number *Sc*, and volume fraction *φ*<sup>1</sup> on the concentration profile *α* are exposed in Figure 4a–d. Figure 4a demonstrates the influence of fluid parameter *α* on concentration distribution *φ*(*η*). It is detected that the concentration profile declines by rising the value of *α*. The inspiration of curvature constraint *γ* on the concentration profile *φ*(*η*) is shown in Figure 4b. Proof is provided through *γ* rising along with the fluid concentration and the thickness of the resulting boundary layer. The impacts of Schmidt number *Sc* and volume fraction *φ*<sup>1</sup> on the distribution of concentrations are depicted in Figure 4c,d, respectively. As can be observed in both of these diagrams, the concentration distribution diminishes for substantial values of *Sc* and *φ*1. The reason for this phenomenon is that viscous forces increase as the concentration slows down. *Sc* is the ratio of mass diffusion to viscous forces at each end of the scale. As *Sc* increases, viscous forces grow and mass diffusion declines, causing the concentration distribution to decrease.

**Figure 4.** Concentration profile variation for distinct values of (**a**) is *α*, (**b**) is *γ*, (**c**) is *Sc*, (**d**) is *φ*1.

The behavior of entropy optimization and Bejan number is shown in Figures 5 and 6, respectively, for various parameters such as porosity parameter *β*0, volume fraction *φ*1, curvature parameter *γ*, and Brinkman number *Br*. Figures 5a and 6a show the nature of entropy formation and Bejan number for the rising porosity parameter *β*0. As the values of porosity parameter *β*<sup>0</sup> upsurge, the values of entropy generation escalations near the wall slightly decrease; the Bejan number, on the other hand, exhibits the opposite pattern. As the porosity parameter tends to oppose the fluid flow, as a result, it increases the rate of total entropy formation. The influence of nanoparticles' volume fraction parameter *φ*<sup>1</sup> on entropy production and Bejan number is exposed in Figures 5b and 6b. These figures show that entropy production increases, while Bejan number drops with an escalation in volume fraction parameter *φ*1. The increase in thermal conductivity and temperature of nanofluid caused by nanoparticles is directly related to this phenomenon. Figures 5c and 6c show the stimuli of the curvature parameter on entropy formation and Bejan number, respectively. As *γ* upturns, the value of Bejan number and entropy generation increases, because less resisting force is offered when the contact surface of a cylinder with particles is reduced. This allows for more nanoparticle movement, increasing the rate of entropy formation. As a result, more curved bodies produce more entropy. Figures 5d and 6d show the nature of entropy formation and Bejan number for the rising Brinkman number *Br*. The outcomes of entropy formation and Bejan number are utterly opposite when Brinkman number *Br* is changed. Entropy production increases as the Brinkman number increases, as shown in Figure 5d. The ratio of heat creation via viscous heating transition for conduction is known as the Brinkman number. More heat is created in the system to reimburse for increasing Brinkman number. As a result, the overall system's level of disturbance grows. For a high Brinkman number *Br*, Figure 6d shows the exact opposite characteristics.

**Figure 5.** Entropy optimization variation for distinct values of (**a**) is *β*0, (**b**) is *φ*1, (**c**) is *γ*, (**d**) is *Br*.

**Figure 6.** Bejan number variation for distinct values of (**a**) is *β*0, (**b**) is *φ*1, (**c**) is *γ*, (**d**) is *Br*.

Table 2 indicates how the numerous constraints affect the skin friction. It is determined that skin friction coefficient decreases for large values of the volume fraction, fluid parameter, porosity parameter, curvature parameter, and inertia coefficient. Table 3 indicates the numeric data of the Nusselt number for a variety of constraints. It is determined that Nusselt number decreases for large values of the volume fraction, fluid parameter, porosity parameter, and temperature ratio, whereas it upsurges for superior values of the curvature parameter. Table 4 compares the numeric values of skin fraction with the pervious result. Both outcomes are noticed to be highly congruent.


**Table 2.** The skin friction coefficient variation for numerous values of *φ*1, *α*, *β*0, *γ*, and *Fr*.


**Table 3.** The variation in Nusselt number for numerous values of *φ*1, *α*, *β*0, *γ*, and *θω*.

**Table 4.** Comparison of *f* (0) skin friction values for several fluid parameter *α* values.


### **6. Conclusions**

In this research, an entropy generation interpretation for axisymmetric flow of Powell– Eyring nanofluid via a horizontal porous stretching cylinder was performed. By using a similarity variable transformation system, the nonlinear equation describing the flow problem is changed to nonlinear ODEs, which are then solved using bvp4c. The impacts of several factors in the model problem on velocity, temperature, concentration, entropy optimization, Bejan number, drag force, and Nusselt number are analyzed. The following conclusions were derived from the study's findings:


• It is concluded that skin friction decreased for a large number of volume fraction, fluid parameter, porosity parameter, curvature parameter, and inertia coefficient, whereas Nusselt number decreased for a cumulative number of volume fraction, fluid parameter, porosity parameter, and temperature ratio, whereas it rose for a cumulative number of curvature parameter.

**Author Contributions:** Conceptualization, M.R. and N.V.; methodology, Z.S. and M.A.J.; software, M.R.; validation, N.V. and W.D.; formal analysis, S.F.B. and S.I.; investigation, Z.S. and M.R.; resources, N.V.; data curation, W.D.; writing—original draft preparation, M.R.; writing—review and editing, Z.S., N.V. and S.I.; visualization, W.D.; supervision, M.A.J. and Z.S.; project administration, N.V. and S.F.B.; funding acquisition, N.V. and S.F.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** The project was financed by Lucian Blaga University of Sibiu and Hasso Plattner Foundation research Grants LBUS-IRG-2021-07.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
