Nonsimilar Modeling and Numerical Simulations of Electromagnetic Radiative Flow of Nanofluid with Entropy Generation

Department of Mathematics, COMSATS University Islamabad, Park Road Chak Shahzad, Islamabad 44000, Pakistan College of Science, Inner Mongolia University of Technology, Hohhot 010051, China Department of Mathematics, Faculty of Science and Humanities, Shaqra University, Afif, Saudi Arabia Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia Department of Mathematics, University of the Poonch Rawalakot, Rawalakot 12350, Pakistan


Introduction
Carbon nanotubes are an allotrope of carbon with tubal nanostructure. Carbon nanotubes have a broad variety of uses, including oil and gas industry, conductive fabrics, atomic force microscope tips, flat-panel displays, radarabsorbing coating, longer-lasting batteries, structural composite materials, antifouling paint, ultracapacitors, and medical instruments and biosensors due to their higher chemical compatibility with biomolecules such as proteins and DNA, as well as purification of tainted drinking water. Compared with other nanostructured compounds, nanotubes have the best thermal conductivity enhancement property. Iijima [1] discovered carbon nanotubes in 1991 by observing the electrical discharge between two carbon electrodes. CNTs are characterized as single wall carbon nanotubes and the multiwall carbon nanotubes, all of which depend on concentric layers of rolled graphene lamina. Carbon nanotubes' thermal conductivity is found to be higher than that of pure fluids, and as a result, there is rising preference in SWCNTs and MWCNTs applications in industry, electronics, and medicine [2][3][4]. Meanwhile, significant work on CNTs boundary layer flow has been documented by [5][6][7]. Shahsavar et al. [8] examined thermal conductivity of a magnetic hybrid (CNTs + Fe 3 O 4 ) nanofluid flow experimentally. Lu et al. [9] discussed the CNTs-suspended nanofluids flow with binary chemical reactions and Cattaneo-Christov heat flux condition. Haq et al. [10] analyzed the velocity slip effects on two-dimensional (2D), steady, incompressible, nanofluid flow containing CNTs over a flat stretching plate. Mosayebidorcheh and Hatami [11] investigated nanofluid (CNTs-water) flow between two parallel rotating disks. Iqbal et al. [12] numerically analyzed the bioconvection, CNTs nanofluid flow across a stretching surface. Lu et al. [13] evaluated numerically the nanofluid flow comprising CNTs with magnetohydrodynamics (MHD) and homogeneous-heterogeneous reactions. Ahmed et al. [14] examined incompressible, 2D, steady boundary layer flow with base fluid (engine oil) in the existence of CNTs with double stratification effects. Upreti et al. [15] assessed the 3D Darcy-Forchheimer water-CNTs (SWCNT and MWCNT) nanofluid flow across 2D stretchable surface with the consequences of Ohmic heating and nonuniform heat source/sink. Entropy generation (EG) is accounted in technology, heat-related industries, and everyday life events. ermal energy is converted to mechanical work through entropy, which is the indicator of thermal system disruption. In the form of a campfire and the combustion of solid wood into ash, smoke, and fumes, human everyday life is inextricably linked with entropy. Regarding fluid efficiency in thermal systems, liquids are treated in such a manner that energy waste is minimized. In support of this idea, the concept of EG has gained prominence in the thermal world. Bejan [16] initially investigated entropy production and developed a thermodynamic optimization process. Soomro et al. [17] numerically examined EG for mixed convection CNTs fluid flow over an inclined surface. Ishaq et al. [18] studied nanofluid flow with EG across an unsteady stretching surface. Farooq et al. [19] studied viscous dissipation and transpiration effects on EG in the nanofluid flow across a nonlinear stretched porous disk. Suleman et al. [20] discussed analytically, EG, and thermal radiation effects for nanofluid flow across an exponentially stretching surface. Sheikholeslami et al. [21] used a non-Darcy model to study EG and magnetic nanofluid flow in a porous enclosure. Khan et al. [22] evaluated the EG in nanofluid flow through porous medium comprising microorganisms and nanoparticles. Some important studies on EG have been conducted in [23][24][25][26][27][28][29].
Boundary layers theory plays a critical role in an extensive range of engineering and scientific fields. By observing high-temperature processes, such as nuclear power plants, energy storage, and gas turbines, the impact of thermal radiations on boundary-layer flows is crucial. Heat transfer through surface convection across different geometries has obtained a lot of consideration because of its possible uses in a variety of engineering technologies. Many researchers have conducted studies in this region. Ishak et al. [30] studied the convective flow across moving-plate with thermal radiation. Akbar et al. [31] analyzed the incompressible nanofluid flow over a cylinder along with convective boundary condition. Nadeem and Haq [32] explored the magnetohydrodynamic nanoliquids transport across stretching-sheet with the impact of convective-surface. Delouei et al. [33] investigated the boundary layer flow and heat transfer of non-Newtonian fluid over a heated cylinder by employing the direct-forcing IB-TLBM. Oyelakin et al. [34] observed Soret and Dufour effects in Casson nanofluid flow across an unsteady stretched surface with thermal radiation effects. Zaib et al. [35] studied the impacts of mixed convection and thermal radiation on copper-water nanofluid flow over a porous shrinking cylinder with heterogeneous-homogeneous reactions. Jyothi et al. [36] numerically assessed the boundary layer nanofluid flow between two disks employing thermal radiation and convective boundary surface. Reddy et al. [37] examined the influence of Arrhenius activation energy, nonlinear thermal radiations, and binary chemical reaction on MHD boundary layer flow of Eyring-Powell fluid over a stretching surface. Saif et al. [38] reviewed viscous fluid flow across a stretchable curved surface through convective boundary conditions. Raees et al. [39] evaluated mixed convective boundary-layer magnetized second grade nanofluid flow over a vertically stretching sheet. Farooq et al. [40] analyzed nonsimilar boundary-layer flow of Casson nanofluid within non-Darcy porous media.
According to the previously mentioned literature review, nonsimilar EG analysis for an electromagnetic, radiating nanofluids (water-SWCNT/MWCNT) flow across convective heated-wedge, vertical plate, and horizontal-plate has not been published. us, the current work aims to address this void in the literature by conducting a systematic review of boundarylayer flow and EG research. e governing equations are numerically solved by utilizing local nonsimilarity method via in-built MATLAB bvp4c. For SWCNT/MWCNT-water nanofluids, the impact of incorporated physical parameters on velocity, temperature, viscous drag, Nusselt number, EG, and Bejan number has been examined.

Mathematical Modeling
Consider the 2D, incompressible, viscous carbon nanotubesbased flow over a wedge. Water is considered as base fluid, while SWCNTs and MWCNTs are picked as nanoparticles. Free stream and wedge velocities are U e � U ∞ x m and U w � U W x m , respectively, where 0 < m � β/(2 − β) < 1, U ∞ and U w are constant . β � ω/π is Hartree pressure gradient parameter; ω represented a total wedge angle. e case β � 0 relates to a horizontal plate and β � 1 indicates a vertical plate. We presume that x− axis is parallel to the wedge and y− axis in the normal direction of flow. Furthermore, it is anticipated that the wedge's lower surface is heated by heat convection from hot fluid with transfer coefficient h f at a temperature T f . e base fluid (water) and the nanoparticles (SWCNTs and MWCNTs) are likewise considered to be in equilibrium. Figure 1 reveals the schematic sketch of the problem.
Governing equations for the problems under consideration [41] are as follows: where u and v are the x− and y− directions velocity components, respectively. T is the nanofluid temperature and μ nf , ρ nf , α nf , k nf , and (ρc p ) nf are the nanofluid's dynamic viscosity, density, thermal diffusivity, effective thermal conductivity, and heat capacitance.
In an inviscid flow, velocity at edge of the boundary layer and mainstream velocity (U e ) are identical, and pressure is treated as constant. From (2), we have Employing equation (4) into equation (2), we get Rosseland approximation is considered [36]: where (k * (m − 1 )) and (σ * � 5.67 × 10 − 9 (W/m 2 k 2 )) represents Rosseland absorption coefficient and Stephan-Boltzmann constant. Consider that the temperature variations in the flow are minimal, so T 4 can be written as Employing equations (6)-(7) into equation (3), we get e linked boundary conditions (BCs) are Dimensionless transformations [41] are as follows:

Mathematical Problems in Engineering
Dynamical & thermal boundary layers Hot convective fluid

Mathematical Problems in Engineering
Equation (1) is verified automatically, while (2)-(9) become With boundary conditions, the following is obtained: Here, m denotes the dimensionless wedge angle; λ represents the velocity ratio parameter; λ � 0 depicts static wedge; λ > 0 and λ < 0 show moving-wedge situation in the same and opposite direction to free stream direction, respectively. Nr expresses the radiation parameter, Re indicates the Reynolds-number, Bi is the Biot number (surface convection parameter), Ec and Pr represent Eckert and Prandtl numbers, respectively, E 1 is the electric field parameter, and M is the Hartmann number defined as whereφ is the nanofluid volume fraction, and ρ bf , k bf , and σ bf are the density, thermal conductivity, and electrical conductivity of the base fluid (see Table 1).

Mathematical Problems in Engineering
Boundary conditions are as follows:

Second Truncation System.
For the second truncation system, we assumed the followings: erefore, equations (11)-(13) take the following form: Boundary conditions are as follows: Frictional drag on the surface (local skin friction coefficient C f ) and rate of heat transfer (local Nusselt number Nu x ) are defined as where τ w � μ nf (zu/zy) y�0 and q w � − [k nf + 16σ * T 3 ∞ /3k * ](zT/zy) y�0 are the shearing stress and heat flux at the surface of the wall, respectively. Putting in (15), we get

Entropy Generation Analysis
Entropy generation (EG) per unit volume for the problem under consideration is [15]
Mathematical Problems in Engineering e dimensionless entropy generation number (N G ) is defined as EcPr Ω where Ω − 1 � T ∞ /ΔT represents the dimensionless temperature difference and S GO � k bf ΔT 2 /T 2 ∞ l 2 denotes the characteristic entropy generation rate.
Dimensionless form of Bejan number (Be) for the considered problem can be stated as Be � entropy generation due to heat transfer total entropy generation ,

Numerical Solution Procedure
In this section, we will discuss the numerical solution procedure for the dimensionless equations (15) to (21) using Matlab algorithm bvp4c. e ordinary differential equations (ODEs) solver technique employs the Lobatto IIIA formula, which necessitated the initial guesses to justify the boundary conditions. Hereafter, we employ FDM, which updates the initial predictions for the subsequent iterations. We accomplish the process with the following assumptions:

Results and Discussion
Electromagnetic, radiating water-CNTs nanofluid flow over convective heated-wedge and vertical/horizontal surface under impact of entropy optimization has been investigated by employing the method of local nonsimilarity [42] Table 2 implies the thermophysical properties of water and both CNTs. To assess the authenticity of the present solutions, Table 3 reports a comparison examination of f ″ (0) for several values of with earlier published results, which validates our method. Table 4 depicts the skin friction coefficient's C f R 1/2 e behavior in response to various estimations of ϕ, m, λ, M, and E 1 for both MWCNT + H 2 O and SWCNT + H 2 O. Viscous drag is reduced as λ and M are increased. Further, rise in ϕ, m, and E 1 upgrades the viscous drag for both MWCNT/SWCNT-water nanofluid. e numerical value of the Nusselt number is shown in Table 5 for both SWCNTs and MWCNTs. Nusselt number (Nu) decreases for greater values of magnetic parameter and Eckert number, while it rises for increasing the values of dimension less electric field, surface convection, velocity ratio parameter, solid volume fraction, and radiation parameters.
Velocity (f ′ (η)) for several values of the (ϕ) are shown in Figure 2. It shows that when ϕ rises, f ′ (η) falls for both conducting SWCNT/MWCNT-water nanofluids flow by wedge (m � 1/3) and a horizontal (m � 0)/vertical plate (m � 1). By raising ϕ, the fluid's viscosity enhances and so resists the flow under the impact of shear stress. As a result, fluid velocity diminishes. e significance of magnetic parameter (M) on the velocity field for all cases are outlined in Figure 3. For greater estimations of M, velocity decreases. Increase in M indicates a rise in resistive power (Lorentz force), and so, the velocity of the liquid decreases. Figure 4 indicates that the rise in E 1 upgrades velocity field irrespective of m � 0, 1/3, 1. is is because an increase in E 1 adds to a Mathematical Problems in Engineering      stronger electric field, which accelerates the flow of CNTs nanofluids. Figure 5 demonstrates the variance in velocity for growing velocity ratio parameter (λ). It shows that velocity of nanofluids within the boundary layer rises with λ.
Influences of nanoparticles fraction (ϕ) on (θ(η)) are seen in Figure 6. Clearly, graphs show that increasing the nanoparticles concentration raises the temperature inside the boundary layer for all considered cases. is is because as ϕ increases, the thermal conductivity of the nanofluid rises, and as a result, the temperature of the fluid grows. Figure 7 outlines the significance of the temperature by considering an estimation of the radiation parameter (Nr). Clearly, increasing the radiation estimator value improves the temperature distribution and corresponding boundary layer thickness. It is evident that thermal radiation causes an upsurge in surface heat flux. Consequently, the temperature rises within the boundary layer region. Figure 8 portrays the effects of Hartmann (M) on temperature profile θ(η). It is clear that as the value of M increases, so does the temperature. e Lorentz force obstructs fluid flow, whereas rise in the Hartmann number increases the Lorentz force, resulting in a portion of energy being transferred as heat. Temperature profile rises as a result. An analysis of Eckert number (Ec) on θ(η) is portrayed in Figure 9. It is worth noting that when Ec rises, θ(η) increases for both SWCNT and MWCNT. Increase in the Ec results in higher drag forces between the fluid materials. Consequently, more heat is generated, causing the temperature distribution to increase. e behaviors of θ(η) in response to different E 1 for electromagnetic flow of CNTs nanofluids transport by convective heated-horizontal/vertical plates and wedge surface are drawn in Figure 10. In this case, the temperature distribution for both nanofluids declines as E 1 increases. e variations of θ(η) with the strength of local Biot number (Bi) is seen in Figure 11. ermal boundary layer thickness was shown to rise slightly with an escalation in the Bi for all the considered cases. Biot number is defined physically as the relationship between conduction and convection at the surface. As predicted, stronger convection results in higher surface temperatures. Physically, the inclination in Bi generates significant heat transfer through convection, which rises the thermal field. Figure 12 displays the variation in entropy generation (N G ) versus nanofluid volume fraction parameter (ϕ), which are stated in terms of m. It is obvious that as the ϕ increases, so does the N G . is is due to the high thermal conductivity (k nf ) of SWCNT/MWCNT nanoparticles. Also, the entropy generation number N G increases with m. Figure 13 represents the thermal radiation (Nr) impact on the entropy generation number (N G ). An increasing trend in the N r causes an upsurge in the N G . e effects of entropy generation number (N G ) versus Hartmann number (M) for electromagnetic SWCNT/MWCNT-H 2 O nanofluids flow are depicted in Figure 14.
e increase in M causes an escalation in N G . Physically, this means that the increasing estimations of M accelerates the drag force that enhances the dissipation energy, which is the primary cause of irreversibility. Figure 15 shows the Eckert number's (Ec) impact on the entropy generation number (N G ). For all considered cases, the N G rises with the rise in Ec. Figure 16 describes the impacts of (Bi), on (N G ). We discover that N G have a minimum value when Bi is small, indicating poor convection in the surfaces, and then begin to improve until stability is obtained when Bi ⟶ ∞ (isotherm-surface). All three flow instances exhibit the same behavior. As a result, minimizing convection via boundaries can reduce entropy generation. Figures 17(a)  thermal radiation's effect on the Bejan number (B e ). B e increases as the radiation parameter (Nr) is increased. Greater radiation-absorption inside boundary zone raises system's energy, which causes an increase in B e . Figure 19 reveals the influence of the convective surface parameter (Bi) on the Bejan number (B e ). When Bi is small, we discover that B e has a minimal value. e behavior is consistent across all three flow cases.

Conclusions
In this study, we explored numerically the nonsimilar electromagnetic, radiating water/CNTs nanofluid flow across convective heated-moving horizontal/vertical plates and wedge with entropy generation influence. e following are the study's key findings: