**1. Introduction**

It is a well-known fact that stretching flows have acquired a lot of attention because of their numerous applications. In industrial and mechanical engineering progressions (rubber and plastic sheets, cooling of electronic chips, glass blowing, metal spinning, production of glass fiber, liquid film crystallization during condensation, etc.), stretching surfaces are used extensively. In addition, various research of boundary layer flow in conjunction with a plane extending surface has already been carried out. However, research journals offer just a few experiments on horizontally stretching sheets with the axisymmetric flow. Crane [1] first studied the flow of viscous materials caused by a stretched sheet. Nadeem and Haq [2] investigated the convective flow of viscous nanoparticles using radiation beyond a stretched sheet. Ahmad et al. [3] explored power-law fluid in the occurrence of axisymmetric flow and heat transfer. Ariel [4] looked at the classic problem of an axisymmetric flow caused by a stretched sheet and gave perturbed, asymptotic, exact, and numerical solutions. Hsiao [5] investigated MHD heat transfer across a stretching surface utilizing Maxwell fluid flow by means of radiative and viscous dissipation properties.

**Citation:** Rooman, M.; Jan, M.A.; Shah, Z.; Vrinceanu, N.; Ferrándiz Bou, S.; Iqbal, S.; Deebani, W. Entropy Optimization on Axisymmetric Darcy–Forchheimer Powell–Eyring Nanofluid over a Horizontally Stretching Cylinder with Viscous Dissipation Effect. *Coatings* **2022**, *12*, 749. https://doi.org/10.3390/ coatings12060749

Academic Editor: Eduardo Guzmán

Received: 12 April 2022 Accepted: 26 May 2022 Published: 30 May 2022

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Researchers are still interested in studying non-Newtonian fluids because they are more suitable for industrial areas such as power engineering, polymer solution industries, food engineering, and petroleum production. A linear association flanked by stress and rate of strain cannot be used to represent non-Newtonian fluids. Because of their various features, non-Newtonian fluids are much more complex than Newtonian fluids. Powell– Eyring fluids [6] are a type of non-Newtonian fluid with several compensations upon the power-law model, for example, they are built on liquid kinetic theory and behave Newtonian at both low and high shear rates. Patel et al. [7] used asymptotic boundary conditions to study a numeric solution of MHD Powell–Eyring fluid flow. Hayat et al. [8] discovered the radiative effects in electrically conducting Eyring–Powell fluid in three dimensions. Ara et al. [9] inspected the effect of radiation on Eyring–Powell fluid boundary layer flow across an exponentially shrinking sheet. Hayat et al. [10] showed boundary layer stagnation-point flow of Powell–Eyring fluid including dissolving heat transfer. In the existence of a double-stratified medium, Rehman et al. [11] numerically measured a flow study with heat generation/absorption influences of Powell–Eyring fluid mixedconvection flow around a stretching cylinder. According to Hayat et al. [12], these fluids had a variety of complicated features that provided them an edge and various applications over Newtonian fluids. Improvements in mud house renovation and the production of clay pots, gels, medical syrups, and fruit juices, such as Delmonte, yoghurt, Afya, and energy drinks are just a few of the benefits. Additionally, they are used in the production of pseudo-plastic fluids, paints, and medications in the pharmaceutical industry.

In thermodynamics, entropy is a key term. The concept of irreversibility is inextricably linked to the concept of entropy. Irreversibility is something that everyone instinctively understands. We may easily comprehend the irreversibility phenomenon by watching a movie in both forward and reverse sequences. Many progressive processes in ordinary life cannot be reversed, such as plastic deformation, pouring water into a glass, unrestrained fluid expansion, gas rising from a chimney, egg unscrambling, and so on. Originally, the term entropy was manipulated to define the loss of energy in numerous mechanical systems and heat engines that could not efficiently transform the energy into work. Many engineers and scientists are working hard in this modern period to find novel ways to control or limit the waste of valuable energy. This energy loss in thermodynamic systems can cause a lot of chaos. Using Bejan number and entropy creation, any system's efficiency can be boosted. Bejan [13] studied if heat transfer and flow mechanism abnormalities might be scrutinized in expressions of entropy formation. Many investigators have inspected entropy production results in heat flow and transmission to back up his claim. In a dissipative Blasius flow, But et al. [14] looked at entropy creation as well as radiative flux. Their findings show that as the heat radiation variable rises, entropy decreases. Entropy formation for mass and heat transmission over an isothermal medium was proposed by San and Laban [15]. Tamayol et al. [16] looked at how entropy affects heat transmission and fluid flow past a leaky material on a stretchy surface. Rashidi et al. [17] used the homotopy approach to entropy production in hydromagnetic flow across a spinning disk. Shit et al. [18] studied the irreversibility of hydromagnetic nanoparticle flow and heat transit on an exponentially speeded sheet. In the existence of radiative heat flux, convective boundary conditions, and MHD, the flow was explored. But and Ali [19] used a radially stretched surface to scrutinize the impact of a magnetic force on entropy formation in heat transfer and flow processes. Munawar et al. [20] deliberate the formation of entropy in viscid flow via an oscillated stretching cylinder. Khan et al. [21] used a radially stretched disk to evaluate the influence of entropy formation on Carreau nanofluid due to nonlinear thermal radiation.

Furthermore, nanotechnology is regarded as one of the most important conduits for the advancement in key manufacturing rebellion in our sector. Nanofluids are mostly employed due to their enhanced thermal properties. They are manipulated as coolants in heat transfer devices such as electronic cooling systems (such as flat plates), radiators, and heat exchangers. Nanofluid is made up of nanoparticles ranging in size from 1 to 100 nanometers. Choi and Eastman [22] were the first to propose the term "nanfluid". Iqbal et al. [23] used the Newtonian Carreau model to do a computational investigation of thin-film flow through a moving surface. Khan et al. [24] investigated the inspiration of Cattaneo–Christov heat flux on Maxwell nanofluid boundary layer hydromagnetic flow using the two-phase Buogiorno model. Ali et al. [25] created the mathematical model of the unsteady and laminar couple stress nanofluid flow using engine oil and molybdenum disulphide nanomaterial as the base fluid and nanoparticles, respectively. They discovered that adding molybdenum disulphide nanoparticles to the base fluid improves the heat transfer rate of engine oil by up to 12.38 percent. Acharya et al. [26], investigated the effect of entropic production of a time-independent radioactive combination nanoliquid flowing through a slip spinning disk. Acharya et al. [27], used an entropy approach to evaluate mixed convection and radiation impacts in non-Newtonian-flowing fluid by a flexible cylinder. Verra Krishna and Chamkha [28] examined the impact of Hall and ion slip on the MHD convective flow of elastico-viscous fluid via a permeable channel between two rigidly rotating parallel plates. Takhar et al. [29], characterized the free stream of a vertically moving cylinder, as well as mass and heat transfer. A significant amount of noteworthy work has recently been accomplished [30–41].

Several scholars have looked into entropy propagation effects in the context of heat and mass transport on stretching surfaces. Although, there are just a few papers on the subject of entropy generation's impacts on inflow on a stretching disk. The consequences of entropy formation in Powell–Eyring nanofluid caused by mass and heat transport on a horizontally stretched disk are investigated in this article. The heat equation was modeled using several factors such as viscous dissipation, heat radiation, thermophoresis, and Brownian diffusion. The equations are numerically solved by the bvp4c method. The velocity, Bejan number, concentration, temperature, and entropy are all graphically explained.
