**1. Introduction**

Several industrial processes, such as the growing of crystals, the manufacture of rubber and plastic sheets, paper and glass fiber production, and processes of polymer and metal extrusion are affected by the flow problem with heat transport provoked through stretched sheets; thus, this issues is extremely important. The cooling rate plays a significant role concerning the quality of the finished product through these procedures; where a moving sheet materializes via an incision, as a result, a boundary layer flow (BLF) emerges in the track of the surface progress. Crane [1] scrutinized the 2D steady flow of viscous fluid from a stretched sheet. After this study, the pioneering effort on the flow field through a stretched sheet achieved substantial interest; as a result, an excellent quantity of literature has been engendered on this work [2–9].

In recent times, nanotechnology has magnetized researchers' attention owing to its several distinct applications in the modern era, such as cancer therapy and diagnosis, interfaces in neuroelectronics, chemical production, and molecular and in vivo therapy applications such as kinesis and surgery, etc. In addition, there have been enhancements in the heat transfer in mechanical as well as thermal systems. Several regular fluids (ethylene glycol, oil, polymer solutions, water, etc.) have low thermal conductivity. Thus, augmenting the performance of such heat transport fluids appears imperative to achieve the expectations

**Citation:** Khan, U.; Zaib, A.; Khan, I.; Nisar, K.S. Entropy Generation Incorporating γ-Nanofluids under the Influence of Nonlinear Radiation with Mixed Convection. *Crystals* **2021**, *11*, 400. https://doi.org/10.3390/ cryst11040400

Academic Editors: Rajratan Basu and Arcady Zhukov

Received: 26 February 2021 Accepted: 31 March 2021 Published: 10 April 2021

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of scientists and researchers. Choi [10] primarily developed the concept of nanofluids for the purpose of augmenting the performance of regular fluids. Sheikholeslami et al. [11] scrutinized forced convective flow with nanofluids from a stretchable sheet with magnetic function. Mutuku and Makinde [12] examined the influences of dual stratification on time-dependent flow from a smooth sheet with nanofluid and magnetic function. The effect of entropy generation (EG) on the thin fluid flow with nanofluids via a stretched cylinder was scrutinized by Khan et al. [13]. Gireesha et al. [14] implemented a KVL (Khanafer-Vafai-Lightstone) model to explore the influence of nanofluids via dusty fluids with Hall effects. Recently, the influences of nanofluids rendering to assorted surfaces have been studied by numerous researchers [15–20].

The alumina nanofluids are another aspect that has recently attracted the attention of researchers due to their application in numerous procedures of cooling [21–26]. The alumina nanofluids are identified in accordance with their dimension, e.g., alpha and gamma aluminize, etc. The surface properties in well-described forms of gamma and eta alumina were examined in [27]. The entropy influence on the flow of ethylene- and water-based γ-alumina through stretched sheets, as determined using the effective Prandtl model, was explored by Rashidi et al. [28]. The authors claimed that the fluid temperature decelerates owing to effective Pr and accelerates without effective Pr. A comparative investigation considering γ − Al2O<sup>3</sup> with distinct base fluids was scrutinized by Ganesh et al. [29]. They showed that similar nanoparticles have opposite effects on temperature. Moghaieb et al. [30] employed the γ − Al2O<sup>3</sup> particles in their research as an engine coolant. Ahmed et al. [31] examined the unsteady radiative flow comprising ethylene- and water-based γ − Al2O<sup>3</sup> nanomaterials through a thin slit with magnetic function. Recently, Zaib et al. [32] developed the model of effective Prandtl to examine the mixed convective flow through a wedge by nanofluids. They achieved multiple results for the opposite flow.

The second law of thermodynamics is more consistent than the first law of thermodynamics because of the restriction of the effectiveness of the first law in engineering systems of heat transport. To find the best method for thermal structures, the second law is employed through the curtailing of irreversibility [33,34]. A larger entropy generation (EG) signifies a larger scope of irreversibility. Hence, EG can be utilized to ascertain criteria for the manufacturing of devices in engineering. An assessment of EG can be used to augment the performance of a system [35–41]. In addition, entropy generation can be utilized in analysis of the brain and its diseases from both a psychiatric as well as a neurological perspective. Rashidi et al. [42] examined the stimulus of magnetic function on the fluid flow in a rotated permeable disk with nanofluid. Dalir et al. [43] surveyed the effect of entropy on the force convective flow from a stretching surface containing viscoelastic nanofluid. The Keller-box algorithm was utilized to find the numerical result. Shit et al. [44] discussed the effect of EG on convective magneto flow using nanofluid in porous medium with radiation impact. They employed FDM (finite difference method) along with Newton's technique of linearization. The influences of radiation and viscous dissipation on the flow of copper and silver nanomaterials through a rotated disk with entropy were studied by Hayat et al. [45]. Recently, Shafee et al. [46] scrutinized the stimulus of nanofluid via a tube with entropy generation by involving swirl tools of the flow.

The above-mentioned investigations were dependent on steady- state behavior. However, in certain situations, the flow depends on time, owing to unexpected changes in temperature or the heat-flux of the surface, and as a result, it becomes vital to take timedependent (unsteady) flow conditions into consideration. In addition, the phenomena of time-dependent flow is significant in numerous areas of engineering, such as rotating parts in piston engines, the turbo machinery and aerodynamics of helicopters, etc. Thus, the intention of the current research is to explore the impact of time-dependent mixed convective flow incorporating H2O\C2H6O<sup>2</sup> based γ-nanofluids. The influences of nonlinear radiation and viscous dissipation with entropy are also analyzed. The Lobatto IIIA formula is used to find the numerical solutions of the transmuted ODEs (ordinary differential equations).
