Brownian Motion and Thermophoretic Di ﬀ usion E ﬀ ects on Micropolar Type Nanoﬂuid Flow with Soret and Dufour Impacts over an Inclined Sheet: Keller-Box Simulations

: The principal objective of the current study is to analyze the Brownian motion and thermophoretic impacts on micropolar nanoﬂuid ﬂow over a nonlinear inclined stretching sheet taking into account the Soret and Dufour e ﬀ ects. The compatible similarity transformations are applied to obtain the nonlinear ordinary di ﬀ erential equations from the partial di ﬀ erential equations. The numerical solution of the present study obtained via the Keller-Box technique. The physical quantities of interest are skin friction, Sherwood number, and heat exchange, along with several inﬂuences of material parameters on the momentum, temperature, and concentration are elucidated and clariﬁed with diagrams. A decent settlement can be established in the current results with previously published work in the deﬁciency of incorporating e ﬀ ects. It is found that the growth of the inclination and nonlinear stretching factor decreases the velocity proﬁle. Moreover, the growth of the Soret e ﬀ ect reduces the heat ﬂux rate and wall shear stress.


Introduction
The investigation of boundary layer flow and heat exchange over a stretching surface has pulled in consideration of numerous specialists because of its immense mechanical and industrial applications in the field of industry metallurgical procedures, tinning of copper wires, assembling of plastic, elastic sheets, and fiber. The viscous fluid flow over uniform surface started by Sakiadis, [1]. Moreover, Crane [2] studied the closed-form solution of the boundary layer flow over a stretching sheet. The boundary layer flow of dusty fluid over an inclined surface with the heat source/sink presented by Ramesh et al. [3]. Singh [4] investigated heat and mass exchange of viscous fluid flow on a porous inclined plate by incorporating the viscous dissipation. Similarity solution of magnetohydrodynamic fluids, creature plasma, and body-liquids as well as numerous different circumstances. A fantastic audit about the micropolar fluid and its uses explored by Ariman et al. [30], Qukaszewicz [31], and Eringen [32]. Rahman et al. [33] discussed the flow of micropolar liquid by considering the variable properties. Micropolar fluid flow by taking different effects over an inclined sheet studied by Das, [34]. Kasim et al. [35] examined the micropolar fluid flow on the inclined plate numerically. Srinivasacharya and Bindu [36] explored micropolar fluid flow through a slanted channel having parallel plates. Hazbavi and Sharhani, [37] examined the flow of micropolar fluid between two parallel plates with a constant pressure gradient. Shamshuddin et al. [38] studied the heat and mass transfer of micropolar fluid flow through the permeable inclined plate. The effect of double dispersion on micropolar fluid flow over an inclined surface discussed by Srinivasacharya et al. [39]. Newly, Rafique et al. [40] examined the micropolar nanofluid flow on an inclined surface via a numerical technique.
Keeping in mind the above literature review, the present problem is focused on micropolar nanofluid flow over an inclined stretching surface with Soret and Dufour effects. Similar work is not reported in the literature, and therefore, the present study will fill this gape. Besides, this field of research has important applications in industry and engineering. The model under concern is novel, and all numerical outcomes found from the present problem are new.

Problem Formulation
A steady, two-dimensional boundary layer flow of micropolar nanofluid over a nonlinear inclined stretching surface by considering an angle γ. The stretching and free stream velocities are taken as, u w (x) = ax m and u ∞ (x) = 0. An external transverse magnetic field is assumed normal to the flow path. It is supposed that the electric and magnetic field properties are very small as the magnetic Reynolds number is negligible in Mishra et al. [41]. The micropolar finite-size particles, along with nanoparticles, are continuously distributed in the base fluids. The Brownian motion and thermophoresis effects are taken into account. The temperature T and nanoparticle fraction C at the wall take the constant values T w and C w , while the ambient forms for nanofluid mass and temperature fractions C ∞ and T ∞ are attained as y tends to infinity shown in Figure 1. procedures that happen in manufacturing. Such applications incorporate the flow of exotic greases, colloidal interruptions, hardening of fluid precious stones, the expulsion of polymer fluids, creature plasma, and body-liquids as well as numerous different circumstances. A fantastic audit about the micropolar fluid and its uses explored by Ariman et al. [30], Qukaszewicz [31], and Eringen [32]. Rahman et al. [33] discussed the flow of micropolar liquid by considering the variable properties. Micropolar fluid flow by taking different effects over an inclined sheet studied by Das, [34]. Kasim et al. [35] examined the micropolar fluid flow on the inclined plate numerically. Srinivasacharya and Bindu [36] explored micropolar fluid flow through a slanted channel having parallel plates. Hazbavi and Sharhani, [37] examined the flow of micropolar fluid between two parallel plates with a constant pressure gradient. Shamshuddin et al. [38] studied the heat and mass transfer of micropolar fluid flow through the permeable inclined plate. The effect of double dispersion on micropolar fluid flow over an inclined surface discussed by Srinivasacharya et al. [39]. Newly, Rafique et al. [40] examined the micropolar nanofluid flow on an inclined surface via a numerical technique. Keeping in mind the above literature review, the present problem is focused on micropolar nanofluid flow over an inclined stretching surface with Soret and Dufour effects. Similar work is not reported in the literature, and therefore, the present study will fill this gape. Besides, this field of research has important applications in industry and engineering. The model under concern is novel, and all numerical outcomes found from the present problem are new.

Problem Formulation
A steady, two-dimensional boundary layer flow of micropolar nanofluid over a nonlinear inclined stretching surface by considering an angle γ . The stretching and free stream velocities are taken as, . An external transverse magnetic field is assumed normal to the flow path. It is supposed that the electric and magnetic field properties are very small as the magnetic Reynolds number is negligible in Mishra et al. [41]. The micropolar finite-size particles, along with nanoparticles, are continuously distributed in the base fluids. The Brownian motion and thermophoresis effects are taken into account. The temperature T and nanoparticle fraction C at the wall take the constant values w T and w C , while the ambient forms for nanofluid mass and temperature fractions ∞ C and ∞ T are attained as y tends to infinity shown in Figure 1. The flow equations for this study were given by Khan and Pop [42] and Anwar et al. [15]: The flow equations for this study were given by Khan and Pop [42] and Anwar et al. [15]: where u and v are the velocity components in the directions of x and y respectively, g is the gravitational acceleration, strength of magnetic field is defined by B 0 , σ is the electrical conductivity, viscosity is given by µ, density of conventional fluid is given by ρ f , density of the nanoparticle is given by ρ p , thermal expansion factor is denoted by β t , concentration expansion constant is given by β c , D B signifies the Brownian dissemination factor and D T represents the thermophoresis dispersion factor, k denotes the thermal conductivity, α = k (ρc) f denotes thermal diffusivity parameter, and the symbolic representation of the relation among current heat capacity of the nanoparticle and the liquid The boundary settings are: Here ψ = ψ(x, y) is, the stream function, which is demarcated as: Equation (1) is fulfilled. The compatible transformations are demarcated as: On substituting Equation (8), the system of Equations (2)-(5) converted to: 1 Pr where, Here, primes mean the differentiation with respect to η, M denotes the magnetic factor, ν denotes the liquid kinematic viscosity, Prandtl number is symbolized by Pr, Lewis number is denoted by Le, the material parameter represented by K, D f means the Dufour factor, and Sr indicates the Soret factor. The corresponding boundary settings are changed to: The concerned physical quantities including C f , Nu, and Sh (skin friction, Nusselt number, and Sherwood number) are demarcated as: The associated expressions of C f x (0) = (1 + K) f (0), −θ (0), and −φ (0) are defined as: where Re x = u w (x)x/v, is the local Reynolds number based on the stretching velocity. The converted nonlinear differential Equations (9)-(12) with the boundary conditions (14) are elucidated by the Keller-box scheme consisting of the steps as finite-differences technique, Newton's scheme and block elimination process clearly explained by Anwar et al. [15]. This method has been extensively applied via Matlab software and it looks to be the most flexible as compared to common techniques. It has been presented to be much quicker, easier to program, more efficient, and more comfortable to practice.

Results and Discussion
This portion of the study managed the calculated results of converted nonlinear ordinary differential Equations (9)-(12) with boundary settings (14) elucidated via the Keller-box method. For the numerical result of physical parameters of our concern including Brownian motion parameter Nb, thermophoresis parameter Nt, magnetic factor M, buoyancy factor λ, solutal buoyancy factor δ, inclination factor γ, Prandtl number Pr, Lewis number Le, Dufour effect D f , Soret effect Sr, non-linear stretching parameter m, and material factor K several figures and tables were prepared. In Table 1, in the deficiency of the Dufour effect, buoyancy parameter, solutal buoyancy constraint, magnetic factor, and material factor with m = 1, Sr = 0.1, and γ = 90 0 outcomes of reduced Nusselt number −θ (0) reduced Sherwood number −φ (0) were equated by existing findings of Khan and Pop [42]. The consequences established a brilliant settlement. The effects of −θ (0), −φ (0) and C f x (0) beside changed values of involved physical factors Nb, Nt, Le, Sr, D f , M, m, K, δ, λ, γ, and Pr are shown in Table 2. From Table 2 it is seen that −θ (0) declines for growing the values of Nb, Nt, M Le, Pr, Sr, D f , m, γ, and increased by enhancing the numerical values of λ, δ, and K. Moreover, it is perceived that −φ (0) enhanced with the larger values of Nb, λ, δ, Nt, Le, K, Pr, D f , Sr, and drops for bigger values of M, m, and γ. On the other hand, C f x (0) rises with the growing values of Nb, Nt, Le,M, K, Pr, m, D f , γ, and drops with the higher values of λ, δ, and Sr. Table 1. Contrast of the reduced Nusselt number −θ (0) and the reduced Sherwood number −φ (0) when M, K, D f , λ, δ = 0, and Sr = 0.1, m = 1, Le = Pr = 10, and γ = 90 0 .          The velocity outline shows a direct correspondence with the material factor K (see Figure 6). Besides, Figure 7 points out that the angular velocity profile upsurged by the growth of K . Physically, viscidness of the boundary layer decreased by improving the values of K . The velocity outline shows a direct correspondence with the material factor K (see Figure 6). Besides, Figure 7 points out that the angular velocity profile upsurged by the growth of K. Physically, viscidness of the boundary layer decreased by improving the values of K.           The impact of λ on velocity contour is exhibited in Figure 11. Figure 11 demonstrates that the velocity profile increased by increasing the buoyancy force factor λ. Physically, the rise in the buoyancy force reasoned the lessening viscous force due to which the fluid particles moved quicker. In summary, the enhancement in the buoyancy force tended to enhance the velocity profile. On the other hand, in Figure 12   The impact of λ on velocity contour is exhibited in Figure 11. Figure 11 demonstrates that the velocity profile increased by increasing the buoyancy force factor λ. Physically, the rise in the buoyancy force reasoned the lessening viscous force due to which the fluid particles moved quicker. In summary, the enhancement in the buoyancy force tended to enhance the velocity profile. On the other hand, in Figure 12 the temperature profile decreased by improving λ and similar effect prominent in the case of the concentration profile shown in Figure 13. The impact of λ on velocity contour is exhibited in Figure 11. Figure 11 demonstrates that the velocity profile increased by increasing the buoyancy force factor λ. Physically, the rise in the buoyancy force reasoned the lessening viscous force due to which the fluid particles moved quicker. In summary, the enhancement in the buoyancy force tended to enhance the velocity profile. On the other hand, in Figure 12 the temperature profile decreased by improving λ and similar effect prominent in the case of the concentration profile shown in Figure 13.      Figure 14, the velocity profile upturned by enhancing the constraint δ. Figure 14 reveals the effect the solutal buoyancy impact on the velocity profile. The concentration difference, length, and viscosity of the fluid affected the solutal buoyancy parameter. Therefore, as we enhanced the solutal buoyancy parameter, the viscosity declined and the concentration increased due to which the velocity of the fluid increased. Whereas the opposite impact (falls) was shown for temperature and concentration contours against large values of δ shown in Figures 15 and 16.   Figure 14, the velocity profile upturned by enhancing the constraint δ. Figure 14 reveals the effect the solutal buoyancy impact on the velocity profile. The concentration difference, length, and viscosity of the fluid affected the solutal buoyancy parameter. Therefore, as we enhanced the solutal buoyancy parameter, the viscosity declined and the concentration increased due to which the velocity of the fluid increased. Whereas the opposite impact (falls) was shown for temperature and concentration contours against large values of δ shown in Figures 15 and 16.    The velocity profile decreased for higher values m of depicts in Figure 17. It is observed that the velocity field was not much different in the case of a linear or nonlinear stretching sheet as an associate to the uniformly moving surface. Physically, the momentum boundary layer thickness reduced with the growth of the nonlinear stretching parameter. Moreover, temperature and concentration outlined an increase for large values of m present in Figures 18 and 19, respectively. Figures 20 and 21 show the effect of Brownian motion on the temperature and concentration profiles, respectively. The temperature sketch enlarged on improving Nb, whereas concentration distribution shows the opposite behavior. Physically, the boundary layer heated up due to the development in Brownian motion, which inclined to travel nanoparticles from the extending sheet to the motionless liquid. Therefore, the absorption nanoparticle lessened. The velocity profile decreased for higher values m of depicts in Figure 17. It is observed that the velocity field was not much different in the case of a linear or nonlinear stretching sheet as an associate to the uniformly moving surface. Physically, the momentum boundary layer thickness reduced with the growth of the nonlinear stretching parameter. Moreover, temperature and concentration outlined an increase for large values of m present in Figures 18 and 19, respectively. Figures 20 and 21 show the effect of Brownian motion on the temperature and concentration profiles, respectively. The temperature sketch enlarged on improving Nb, whereas concentration distribution shows the opposite behavior. Physically, the boundary layer heated up due to the development in Brownian motion, which inclined to travel nanoparticles from the extending sheet to the motionless liquid. Therefore, the absorption nanoparticle lessened. the velocity field was not much different in the case of a linear or nonlinear stretching sheet as an associate to the uniformly moving surface. Physically, the momentum boundary layer thickness reduced with the growth of the nonlinear stretching parameter. Moreover, temperature and concentration outlined an increase for large values of m present in Figures 18 and 19, respectively. Figures 20 and 21 show the effect of Brownian motion on the temperature and concentration profiles, respectively. The temperature sketch enlarged on improving Nb, whereas concentration distribution shows the opposite behavior. Physically, the boundary layer heated up due to the development in Brownian motion, which inclined to travel nanoparticles from the extending sheet to the motionless liquid. Therefore, the absorption nanoparticle lessened.                    Df . It can be justified as an increase in the Dufour parameter caused an increase in the concentration gradient resulting in mass diffusion taking place more rapidly. In this way, the rate of energy transfer related to the particles became higher. That is why the temperature profile was enhanced. The impacts of Soret number on the concentration profile were observed opposite to the effect of Dufour number on the temperature profile. As the parameter Sr increased, concentration profiles reduced as displayed in Figure 27.  Figure 26 shows that the temperature profile became large for larger values in the parameter D f . It can be justified as an increase in the Dufour parameter caused an increase in the concentration gradient resulting in mass diffusion taking place more rapidly. In this way, the rate of energy transfer related to the particles became higher. That is why the temperature profile was enhanced. The impacts of Soret number on the concentration profile were observed opposite to the effect of Dufour number on the temperature profile. As the parameter Sr increased, concentration profiles reduced as displayed in Figure 27.

Conclusions
This study explored the heat and mass exchange of micropolar nanofluid flow over a nonlinear inclined extending sheet. The Soret and Dufour effects were taken into account. For numerical simulation, the Keller-box scheme was utilized. For the validity of the statistical outcomes of the current problem, Table 1 was prepared. The Brownian motion and thermophoretic impact on the temperature and concentration distributions were drawn by graphs. More exactly, this part of the

Conclusions
This study explored the heat and mass exchange of micropolar nanofluid flow over a nonlinear inclined extending sheet. The Soret and Dufour effects were taken into account. For numerical simulation, the Keller-box scheme was utilized. For the validity of the statistical outcomes of the current problem, Table 1 was prepared. The Brownian motion and thermophoretic impact on the temperature and concentration distributions were drawn by graphs. More exactly, this part of the research had significant uses in biomedicine, hot rolling, nuclear reactors, electronics, and glass fiber. The primary outcomes of our study were: The skin friction was enhanced by enhancing the Dufour effects and decreased for large values of Soret effects. The Sherwood number increased as we enhanced the Soret and Dufour effects. The Nusselt number declined for increasing the Soret and Dufour effects. The temperature profile was enhanced by increasing the Dufour effects. The concentration profile decreased by enhancing the Soret effect. The velocity profile decreased with the growth of the nonlinear stretching factor.