A Computational Model for the Radiated Kinetic Molecular Postulate of Fluid-Originated Nanomaterial Liquid Flow in the Induced Magnetic Flux Regime

Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan LSEC and ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100190, China Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, Almaty, Kazakhstan Department of Mathematical and Computer Modeling, Kazakh British-Technical University, Almaty, Kazakhstan


Introduction
Improving the thermal efficiency of fluid flows under different conditions and applications has always been a famous research area. Besides, the significance of this issue because of the very wide range of applications in industries has made this area attractive to scientists and companies working in this field. To improve effectiveness, any solution proposed for this in any application can have different technical aspects that should be considered and investigated adequately [1]. For example, Ellahi et al. [2] explored the slip effect in the Newtonian fluid two-phase flow. Particles of the nanosized Hafnium are utilized in the base fluid. Two cases are discussed for fluid under consideration, namely, (i) phase of particles and (ii) fluid phase. ree forms of geometries are investigated in both cases. Relevant studies of nanofluids are discussed in recent articles [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].
ere are many engineering applications of the mixed convective boundary layer flow such as food processing, solidification system, and nuclear reactors. Convection also plays an important role in managing the production cycle such as medications and cosmetics. e transverse magnetic field that merged with the boundary layer-mixed convection flow towards an inclined plate with a wave is examined. e retardation inflow far from the magnetic field and leadingedge yield acceleration in the leading-edge close flow of the wavy sheet is observed by Wang and Chi-Chang [20]. Many researchers analyzed the regime of mixed convection flows in their articles [21][22][23][24][25][26].
ere are several uses of free convection in the presence of Lorentz forces, such as fire engineering and geophysics. Newly proposed nanotechnology is a new passive way to enhance heat transfer [27]. e induced magnetic field's influence on temperature curves is represented by Ghosh et al. [28]. Vanita and Kumar [29] have examined transient flow towards a cone with the inclined magnetic field. Akbar et al. [30] investigated nanoparticle interaction for peristaltic flow in an asymmetric channel towards the magnetic field induced. Nadeem and Ijaz [31] explored the impulse of nanoparticles as a drug carrier to investigate theoretically stenosis arteries in the existence of the induced magnetic effect. Hayat et al. [32] observed second-grade nanoliquid flow with the induced magnetic field towards a stretched convectively heated surface. Rashid et al. [33] inspected the induced magnetic field effects of the Williamson peristaltic fluid flow in a curved channel. e radiation may be sunlight, infrared, or visible and the nature of the material emitted by such radiation depends on its exposure. Depending on how solar heat is collected and distributed or converted into solar electricity, a heat source and its systems are also categorized as either passive solar or active solar. ermal radiations are defined as electromagnetic emissions from a sheet with a temperature greater than zero [34]. Viskant and Grosh [35] noted that when considering the power plants, hypersonic flights, cooling systems, and combustion chambers, radiations became an important part. Using Rosseland's approximations, they addressed thermal radiation impacts for the flow of Falkner-Skan. Hayat et al. [36] portrayed the Ag-CuO/H 2 O rotating hybrid nanofluid flow in the existence of partial slip radiation impacts. Hussain et al. [37] noted the non-Newtonian fluid flow with radiation efficacy and timedependent viscosity. Li et al. [38] examined the radiation impacts in the heat storage system by adding nanoparticles. Zeeshan et al. [39] investigated, due to entropy generation, the impacts of electromagnetohydrodynamics radiative diminishing internal energy of the pressure-driven dioxide-water titanium nanofluid flow. e non-Newtonian fluid is more naturalistic to consider because of the rheological characteristics of physiological and industrial fluids. ere is no extensive model that can describe the moving structure of all fluids due to the complex behavior of non-Newtonian fluids.
us, to study non-Newtonian fluid flow characteristics, numerous models have been developed.
e Eyring-Powell model was obtained from a liquid molecular hypothesis. And the inclusion of additional analytical constants was further improved. It accurately reflects the essence of Newtonian for low and high shear values. For example, rubber melts, condensed liquids, toiletries, cosmetics, and vegetable products are included in these fluids [40]. Appropriate studies of Eyring-Powell fluid may be mentioned in these articles [41][42][43][44][45].
is report is to narrate the specifications of radiative mass and heat transfer enhancement and flow analysis of the molecular kinetic theory of liquid-initiated boundary layer stagnation point nanofluid towards a vertical stretched surface. e non-Newtonian nanofluid model is manifested with the induced magnetic field, radiation efficacy, combined convection, Brownian, and thermophoresis diffusion. e flow field describes that, in the form of partial differential equations, the laws of conservation of momentum are considered. By reducing the number of independent variables by using the technique of similarity transformation, these coupled equations are then purified into the system of ordinary differential equations. e results are interpreted by the MATLAB bvp4c technique. Induced magnetic pattern near the wall decreases, and far away, it increases with an increase in the values of the reciprocal of the magnetic Prandtl number.
e concentration curve enhances when the number of magnetic, stretching, and Prandtl characteristics incline. is study of nanofluid is mainly applied in heat transfer devices such as electrical cooling systems, radiators, and heat exchangers.

Mathematical Formulations
Consider the incompressible, steady, two-dimensional (2D) stagnant point flow under the assumption of the induced magnetic field in the molecular kinetic theory of liquidinitiated nanofluid and heat transport enhancement in the existence of combined convection and radiation towards a vertical stretched sheet as shown in Figure 1. e surface stretching with velocity u w (x) � dx and ambient velocity is u ∞ (x) � bx while the origin is fixed at O; see Figure 1.
Taken the Cartesian coordinate structure, the velocity of liquid flow will change through xand y-axes in a way that x − axis is assumed vertically and y − axis is supposed horizontally. e fundamental form of the kinetic molecular postulate of liquid-originated nanofluid is [45] where where δ, c 1 , μ, A 1 , and tr are fluid parameters, dynamic viscosity, first Rivlin-Ericksen tensor, and trace, respectively.
Here, τ is the extra stress tensor and A 1 � [(gradv) t + gradv]. We consider the second-order approximation for sinh − 1 function as en, equation (1) becomes Under these premises, the governing equations of this particular investigation are as follows: u  Figure 1: Geometry of the problem.

Mathematical Problems in Engineering
x velocity and y magnetic field at the edge of the boundary layer and H 0 is the uniform value of the vertical magnetic field at infinity. e radiation heat flux is given by using Rosseland approximation: where σ * , k * are the Stefan-Boltzmann and the mean absorption coefficient, respectively, whereas via extending T 4 about T ∞ in Taylor's series and ignoring the larger terms, Substituting equations (11) and (12) into (9), the heat equation takes the form e relevant boundary conditions are Invoking similarity transformations are defined by e magnetized pressure is described as Equations (5) and (6) are satisfied identically. Equations (7), (8), (10), and (13)-(15) reduce to with boundary conditions 4 Mathematical Problems in Engineering Here, prime denotes derivative for c and other dimensionless parameters are described as Physical quantities are very valuable from an engineering point of view. ese quantities reported the flow behavior which is defined by local Nusselt number Nu x , skin friction C f , and local Sherwood number Sh x definitions as follows: where τ w represents the surface shear stress, q w denotes the surface heat flux, and q m presents the surface mass flux for fluid: Using invoking transformation equation (16), the dimensionless local Nusselt number, skin friction, and the local Sherwood number become

Results and Discussion
Coupled nonlinear differential equations (18)- (21) and their boundary conditions (22) and (23) are numerically worked out by employing the MATLAB scheme. is portion illustrates the impact of nondimensional sundry characteristics on induced magnetic, temperature, velocity, and concentration flow characteristics numerically and graphically. Figure 2 portrays the impact of η versus h ′ (c). It is noticeable that the induced magnetic spectrum falls when increasing in η. Figure 3 designates the effects of α 1 on the induced magnetic pattern. Dual behavior has been seen for α 1 , near the wall, it is getting down, and far away, it moves upward. e field of h ′ (c) rises by enhancing the amount of β in Figure 4. With larger values of mixed convection characteristic, the flow of induced magnetic expands as shown in Figure 5. Figures 6 and 7 demonstrate the effect of Brownian and thermophoresis diffusion on the induced magnetic field, respectively. When there is increase in the amount of Brownian diffusion, the induced magnetic field decreases, and the induced magnetic field increases by growing quantity of thermophoresis. Figure 8 shows the consequence of the Prandtl number on the induced magnetic curve, and profile falls by a bigger amount of Prandtl. e impacts of thermal radiation flux on the magnetic field are described in Figure 9. e variation of h ′ (c) near the wall moves upward and very far away it moves downward by increasing the amount of thermal radiation. Figure 10 exhibits the outcome of the fluid parameter on h ′ (c)expands in the values of ε which cause rise inh ′ (c). Figure 11 exhibits the deviation of β on velocity curve. When there is expansion in the amount of β, then there is increase in Mathematical Problems in Engineering 5 velocity flow. Figure 12 portrays the effect of λ 1 on the flow of velocity. e diagram shows that the fluctuation shoots up by increment in the mixed convection parameter. Nanofluid behaviour is shown by Brownian and thermophoretic characteristics in in Figures 13 and 14. It is depicted that the contrary attitude showed variation in velocity decline for Nb and incline for Nt by enhancing the quantity of these parameters. Figure 15 investigates that the velocity graph expands if the fluid parameter ε rises. Figure 16 demonstrates that the field of velocity shrinks by rising the amount of the reciprocal of the magnetic Prandtl number. Figure 17 shows that when increasing stretching parameter, the field of F ′ (c) enhances. Figures 18  and 19 show opposite behavior against the parameters of α 1 and β, respectively. e temperature profile increases and decreases against the characteristics ofα 1 and β. Figure 20 explores the impact of Lewis number on temperature distribution, θ(c) rises near the wall, and far away, it declines compared to Le. e temperature reduction against the Brownian motion diffusion is shown in Figure 21. Figure 22 describes that temperature sketch grows by inclining the quantity of thermophoresis diffusion. It is easily noticed that enlarging the amount of Prandtl number causes the figure of temperature decline in Figure 23. Figure 24 depicts the influence of η on the temperature field. e amplitude of temperature diminishes when the size of the stretching parameter boosts up. Figure 25 explores the influence of radiation characteristics on temperature portraits; field enhances by rising the amount of radiation parameter. Figure 26shows the influence of buoyancy ratio parameter on temperature distribution field increases by increasing the amount of buoyancy ratio parameter. e nanoparticle concentration field decreases if the reciprocal of the magnetic Prandtl number rises in Figure 27. Figure 28 exhibits the outcome of β on the concentration field. e figure of concentration inclines when the number of β grows. Figure 29 portrays the result of buoyancy ratio characteristics on nanoparticle concentration. It is easily observed that the field of concentration reduces if the number of N r increases. Figures 30  and 31 define the matching outcomes on the concentration profile. When the size of Pr and η expands, the nanoparticle concentration grows. Streamlines graphs against the distinct amount ofα 1 andεare shown in Figures 32-35. Table 1 shows the impact of different characteristics on drag friction C f Re (1/2) x . e Nusselt number for noticeable amounts of R d , Nt, Nb, ε, λ 1 , m 1 , β, and α 1 is analyzed and characterized in Table 2. Table 3 portrays the deviation of different amounts of parameters on the local Sherwood number.

Concluding Remarks
We studied the molecular theory of liquid-originated non-Newtonian nanofluids which are commonly used in heat transfer devices such as heat exchangers, engine oils, electrical cooling systems (such as flat plates), nuclear

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Mathematical Problems in Engineering     reactors, biomedicine lubricants, and radiators. e key points of observation in the recent analysis are as follows: (i) Induced magnetic pattern near the wall declines, and far away, it inclines when (α 1 ) intensifies. e variation of h ′ (c) field near the wall goes up and very far away it decays when the size (R d ) ascends. (ii) h ′ (c) (nondimensionless induced magnetic function) falls, whereas (η), Brownian diffusion (Nb), and Prandtl number (Pr) rise. e field h ′ (c) expands by enhancing the amount of magnetic parameter (β), mixed convection (λ 1 ), thermophoresis parameter (Nt), and fluid parameter (ε). e variation of h ′ (c) profile near the wall moves upward and very far away it moves down when the size (R d ) ascends. (iii) e velocity amplitude expands by enlargement in the amount of magnetic parameter (β), mixed convection (λ 1 ), thermophoresis parameter (Nt), fluid characteristic (ε), and stretching parameter (η).F ′ (c) collapses by Brownian motion (Nb) and (α 1 ). (iv) e temperature spectrum increases when the values of (α 1 ), radiation parameter(R d ), and buoyancy ratio(N r ) increases and decreases by Prandtl number, magnetic parameter(β), Brownian motion diffusion(Nb), and stretching parameter(η).