Numerical Solutions of Micropolar Nanofluid over an Inclined Surface Using Keller Box Analysis

Department of Mathematics, University of Sialkot, Sialkot 51310, Pakistan Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Department of Mathematics, Faculty of Science, University of Sargodha, Sargodha, Pakistan School of Quantitative Sciences, Universiti Utara Malaysia, Sintok 06010, Kedah, Malaysia Faculty of Mathematics and Statistics, Ton Duc /ang University, Ho Chi Minh City, Vietnam


Introduction
Due to the ideal potential of heat and mass exchange impacts, the nanofluids have pulled in consideration of analysts worldwide. ese fluids are the mixture of nano-sized particles along with base fluids. e main purpose to mix the nanoparticles into base fluids is to enhance the thermal conductivity. Brownian motion and thermophoretic effects are two principal ideas for an abnormal upgrade of thermal conductivity. is model is becoming more widely used since it empowers us to efficiently explore different applications in the marvels of science and technology. Recently, the flow of the Casson nanofluid over an inclined surface was discussed numerically by Rafique et al. [1]. Moreover, the flow of Casson nanofluid on a slanted rotating disk was probed by Saeed et al. [2]. In addition, Rani and Govindarajan [3] investigated the nanofluid flow through an inclined plate numerically. Nowadays, many investigators discussed the flow of nanofluid by considering different geometries [4][5][6][7][8][9][10].
ermal radiations are an active part of the research which is very valuable in the latest technology, cancer cure, missiles, and nanotechnology. Pal and Roy [11] discussed the numerical impact of thermal radiation on the flow of nanofluid along the sheet. More recently, Ghadikolaei et al. [12] scrutinized Casson nanofluid flow along permeable slanted surface via a famous numerical technique. Saidulu et al. [13] considered an exponentially slanted sheet for examining the radiation influences on nanofluid flow. For further details about the literature on the flow of nanofluid by considering different geometries, see [14][15][16][17][18][19][20][21][22][23][24]. e behavior of the flow of non-Newtonian fluid is a study of keen interest of scholars and practical significance.
ere is a little regular and mechanical utilization of such fluids, for occasion volcanic magma, molten polymers, penetrating mud, oils, certain paints, liquid suspensions, food products, and cosmetic and numerous others. In the literature, there exist numerous numerical models with various constitutive conditions including a distinctive set of exact parameters. e micropolar liquid model is suitable for exotic oils, animal blood, fluid crystals with rigid atoms, certain organic liquids, and colloidal or suspensions solutions. e micromotion of liquid constituents, spin inertia, and the influences of the couple stresses are very noteworthy in micropolar liquids. Eringen [25] offered the philosophy of micropolar fluids based on constitution equations. Uddin [26] studied the variable electrical conductivity on the flow of micropolar liquid. Recently, Shamshuddin and umma discussed the flow of micropolar fluid on an inclined sheet numerically [27]. Nandhini and Ramya [28] studied the flow of micropolar fluid by incorporating the radiation effects. For some recent investigations on micropolar fluids with different impacts, see [29][30][31][32][33][34].
is work focuses on establishing the basis for a novel study on thermal investigation in solar magnetohydrodynamic nanotechnology.
e approach empowers us with great flexibility to dissect the Brownian motion and thermophoretic impact on the flow of micropolar nanofluid with thermal radiations and Soret impacts on the inclined geometry. e physical quantities of practical interest such as energy exchange, velocity, skin friction, and concentration species are elaborated in graphical and tabulated form. It is worth stating there is no investigation assumed for the study under concern and all the results are new and contrast to the current outcomes with already available literature [35]. In this article, we employed the Keller box technique for numerical results. Moreover, the current results can be obtained via another numerical technique, but the Keller box method is more easier to program, friendly, and flexible. For complete detail about Keller box scheme, see [36].

Problem Formulation
Our main aim in this analysis is to examine the radiations and Soret impacts on MHD flow of micropolar nanofluid. A uniform external magnetic field of strength B 0 is considered perpendicular to the inclined surface in this study, while induced magnetic field is neglected [37]. e slanted sheet here is stretched with a stretching rate "a" due to which the flow is generated. Moreover, ω is the angle taken with the vertical direction of the stretching sheet. Brownian motion and thermophoretic impacts are considered. In addition, suction or injection impacts on heat and species exchange rates are discussed via graphs.
e Lorentz force in momentum equation is expressed by J × B where J represents the current density, B denotes the total magnetic field: Using Afify [38] × B � σB 2 0 u. e governing equations for the study under concern are zu zx e Rosseland approximation is described by By employing Taylor expansion, approximate value of T 4 is given as By employing (6) and (7), equation (4) becomes e subjected boundary conditions are 2 Journal of Mathematics e stream function ψ � ψ(x, y) for the concerned study are in the following form: e similarity transformations are defined as e converted form of equations (2)-(5) given by employing equation (8) is where where in order to make local Grashof number and local modified Grashof number free from x, the coefficient of thermal expansion β t and coefficient of concentration expansion β c are proportional to x 1 . Hence, we assume that (see references [39][40][41]) where n and n 1 are the constants; thus Gr x and Gc x become e corresponding boundary conditions are Journal of Mathematics e related expressions for the skin friction coefficient

Results and Discussion
e objective behind this section is to elaborate on the numerical results along with graphically results via tables and graphs. In order to check the validity of our numerical values with already published results (Table 1) Table 2. Table 2 demonstrates that the energy flux slows down by improving the Brownian motion effect. Moreover, the mass flux increases with the effect of the Brownian factor for large values. Moreover, the exchange rate of heat reduces with improving Brownian motion factor, and mass exchanges enlarge on enhancing the thermophoretic impacts. Physically, the enhancement in the Brownian motion causes the boundary layer to get thicker. Besides, skin friction shows a direct relation with Brownian motion thermophoretic effects. e magnetic effect factor boosts the skin friction against improving magnetic field magnitude. Consequently, the fluid applies a drag force on the solid boundary layer. In addition, the Nusselt number along with Sherwood number and skin friction reduces for increasing the suction parameter. e Soret effect improves the mass exchange and reduces the skin friction on enhancing the Soret impact on the mass flow.
Figures 1-8 are exhibited to demonstrate the performance of incorporated constraints that impact on the velocity sketch. Figure 1 indicates that the impact of the magnetic field creates resistance in the path of fluid flow because of the Lorentz force which reduces the velocity profile. Moreover, the angular velocity presents opposite behavior against the magnetic field factor in Figure 2. Figure 3 shows that the velocity contour upsurges against material factor K. Physically, increment in factor K declines the viscosity and upturns the velocity. On the contrary, against a higher magnitude of the factor K, the angular velocity upturns (Figure 4). is demonstration corresponds with the outcomes of Rafique et al. [42]. Figure 5 represents as we enhance, the buoyancy force the velocity increases.
Besides, the velocity profile shows the direct relation with Gc in Figure 6. In addition, the velocity field and inclination parameter correspond to an inverse relation drawn in Figure 7. Physically, by considering ω � 0, the gravitational force reaches its maximum value. On the contrary, in the case of ω � 90°, the sheet will be in horizontal position, and that is why the power of the bouncy forces declines which is the reason behind the reduction in the velocity profile. Moreover, in Figure 8, the velocity profile enhances against the higher values of S.   (0), and C fx (0).

Nb
Nt e impacts of incorporated factors in our current study against temperature profile are demonstrated in Figures 9-12. e temperature profile boosts against a stronger magnetic field because the boundary layer thickness upsurge corresponds to a higher magnetic effect (Figure 9). e Prandtl number triggers the temperature profile shown in Figure 10. e thermal boundary layer becomes thinner, and the thermal diffusivity becomes weaker on improving the Prandtl number steadily. e large values of the radiation effect accelerate the temperature profile as shown in Figure 11. Physically, the conductive heat exchange is greater than the radiative heat exchange, which causes reduction in boundary layer thickness and buoyancy force. e recovered outcome is the affirmative proof of the relation q r � − (4σ * /3k * )(zT 4 /zy). e temperature profile relates directly proportional to the Brownian Journal of Mathematics motion effect as shown in Figure 12. Physically, the Brownian motion constraint improves the boundary layer heat which leads to rises in the fluid temperature. e variation between reference temperature and wall temperature is enhanced by growing the thermophoresis influence which corresponds the enhancement in temperature profile ( Figure 13). Figures 14 to 18 indicate concentration profiles against different incorporated parameters. e concentration profile improves on strengthening the magnetic impact as shown in Figure 14. An increment in Brownian motion parameter declines the concentration profile and the boundary layer thickness (Figure 15). Figure 16 presents more nanoparticles that pass away from the hot surface on enhancing      Journal of Mathematics the thermophoretic effects which cause the improvement in the concentration contour. e boundary layer viscosity decreases against the Lewis number which relates to a drop in the concentration profile (Figure 17). e concentration profile corresponds to a direct variation with the Soret factor drawn in Figure 18.

Heat and Mass Exchange.
In order to check the behavior of dimensionless heat and mass exchange rates at the wall along with skin friction against the involved parameters, i.e., Nb and ω, Figures 19 to 21 are drawn. Figure 19 reveals the impact of the Brownian motion factor against different inclination parameters. It is noted that − θ ′ (0) inversely relates the Brownian motion factor and inclination factor. As  Journal of Mathematics we improve the inclination and the Brownian motion factor, the heat transfer rate decreases. e same impacts of the Brownian motion factor along with the inclination parameter on the mass exchange rate have been noticed in Figure 20. However, C fx (0) enhances for higher magnitudes of Brownian motion and inclination depicted in Figure 21.       values of inclination and thermophoresis impacts in Figure 23. In addition, skin friction is improved by enhancing the magnitudes of inclination and thermophoretic impacts ( Figure 24).

Conclusions
e article presents numerical simulations of the micropolartype nanofluid flow over a slanted surface. Similarity results for velocity, temperature, and concentration are recovered through the Keller box technique. e statistical outcomes are compared with the already available literature. In research field, some notable points from this study are as follows: (i) e irregular movement of particles boosts with the increment in Brownian motion; and as a result energy and species exchange rate diminish, whereas the skin friction is enhanced. (ii) e temperature field is more influenced by increasing thermal radiation. (iii) e concentration profile enhances with the growing magnitude of Soret impact. (iv) e boundary layer viscosity decreases against Lewis number which relates to a drop in the concentration profile (v) e conductive heat exchange is greater than the radiative heat exchange, which causes reduction in boundary layer thickness and buoyancy force.