Numerical Analysis of Micro-Rotation Effect on Nanoﬂuid Flow for Vertical Riga Plate

: The investigation of heat and mass transport properties of the ﬂow is a key research area in mathematics, physics, engineering, and computer science. This article focuses on studying the heat and mass transport phenomenon for micropolar nanoﬂuid ﬂow generated by a vertical stretching Riga plate. It is assembled by including a spanwise-aligned array of alternating electrodes and permanent magnets. This technique produces electromagnetic hydrodynamic behavior in ﬂow. Our aim for this article is to examine the inﬂuences of Brownian motion and thermophoresis on a Riga plate. We also explore the micro-rotational effects of the particles. The ﬂow behavior of the modeled problem has also been computed numerically and presented by the graph. It is veriﬁed that the numerical computations show a good approval with the reported earlier studies. The velocity proﬁle is computed and presented by the graph, which shows direct correspondence with the modiﬁed Hartmann number. We also show that energy and mass ﬂux rates increase by increasing modiﬁed Hartmann numbers. The results also revealed that concentration distribution diminishes for larger values of Brownian motion, whereas temperature distribution portrays increases for larger values of both Brownian motion and thermophoresis. Moreover, it is found that concentration distribution shows direct relation with thermophoretic impact.


Introduction
A novel procreation liquid having high thermal performance is helpful in accomplishing engineering and innovative necessities. Earlier, Choi and Eastman [1] investigated nanofluids and found that the scattering of nanoparticles could enlarge the thermal performance of base liquids. Recently, we have seen an explosive growth of activities in developing nanosuspensions for thermal engineering because of their superior and sub-wonders associated with this kind of working liquid. For example, Rafique et al. [2][3][4] reviewed and discussed the analysis of nanofluid flow for a slanted surface, while Alotaibi et al. [5] examined nanofluid flow numerically for a convective heat surface. Furthermore, Rafique et al. [6] investigated Soret and Dufour impacts on nanofluid flow for a slanted surface. The stagnation flow behavior of hybrid nanofluid through a cylinder was explored by Waini et al. [7]. Moreover, recently, Khashi'ie et al. [8] examined nanofluid flow over a shrinking cylinder. Rashad et al. [9] studied the porous medium effect on nanofluid flow on a cylinder. In addition, Reddy et al. [10] studied chemical reaction influence on a cone for nanofluid flow. Rasool et al. [11] examined nanofluid flow via entropy generation by considering thermophoretic impacts. Recently, Hazarika et al. [12] conducted a numerical investigation of nanofluid flow over a porous surface.
The investigation of electro-magneto-hydrodynamics (EMHD) has had a critical effect on innovative and mechanical uses, for example, submarines, micro-coolers, and warm reactors. A Riga plate is an electromagnetic gadget working from magnets and anodes on a plane surface. This gadget can initiate a wall-parallel Lorentz force, i.e., electromagnetohydrodynamic (EMHD) force, and this idea has been proposed by Gailitis and Lielausis [13]. This imaginative actuator is also beneficial in deferring the boundary layer partition and decreasing the turbulence impacts. This article aims to develop and support the investigation of the flow generated by the Riga plate. For example, Ramzan et al. [14] studied numerical treatment of flow overextending the Riga surface. Moreover, Iqbal et al. [15] discussed numerically the Boungiorno model for a stretchable Riga plate. Recently, Khashi'ie et al. [16] investigated the stagnation point flow by incorporating mixed convection effects. Hayat et al. [17] examined the flow overextending a Riga plate for variable thickness. Moreover, Nadeem et al. [18] investigated the energy exchange of micropolar fluid for an exponential Riga plate. Ahmad et al. [19] conducted mixed convection effects on nanofluid flow behavior on the Riga plate numerically. Afridi et al. [20] discussed the behavior of viscous fluid for a horizontal Riga plate via entropy generation. Moreover, Mollah [21] examined Bingham fluid flow by considering parallel Riga plates. Recently, Khatun [22] discussed a vertical Riga plate in a rotating system by incorporating suction effects in the flow of radiating fluid. Wakif et al. [23] examined viscous fluid behavior for the Riga plate numerically.
These days, numerous issues in different fields of mechanics and design should be addressed because of their significant uses in everyday life, for example, adjusting an engine's temperature, machine speed, and, furthermore, building spans and other logical utilizations. The micropolar liquid model might be used to formalize the dust or smoke particle problems in the gas or climate. Moreover, such problems are difficult to elucidate analytically because of nonlinear terms. There are few strategies utilized to examine these problems, including the Keller box technique. Modather et al. [24] conducted an analytical study of micro-rotational flow for the porous vertical surface. Ibrahim [25] studied the micro-rotation and magnetic effects on boundary layer flow over extending surfaces. Yasmin et al. [26] investigated the energy and mass transport phenomenon for extending curves for micropolar fluid flow. Lund et al. [27] investigated multiple solutions for micropolar fluid flow by incorporating viscous dissipation. Abbas et al. [28] studied the stagnation flow by considering thermal slip over a cylinder. Uddin et al. [29] examined energy transport for the micropolar fluid flow towards a shrinking surface.
All the aforementioned studies have investigated the Riga plate and a viscous nanofluid flow via different scenarios. However, no previous work so far on the flow of micropolar nanofluid generated by the Riga plate has been witnessed. We aim, in this article, to fill this gap and study the effect of the involved physical parameters on modeled flow. This novel study is accompanied by Brownian motion, thermophoresis, and mixed convection. By taking advantage of the similarity transformations, the governing problems are transformed into a set of ODEs with suitable boundary conditions, and then resolved numerically by the Keller box technique.

Materials and Methods
The problem under discussion is prepared for the numerical analysis of nanofluid flow with microparticle impacts over the Riga plate. The Riga plate is designed by combining electrodes and permanent magnets. In addition, thermophoresis and Brownian motion impacts are under consideration. Moreover, the micro-rotational particles are considered.  In view of [14,30], the flow equations for this problem are given as: .
The compatible boundary conditions are: In view of [14,30], the flow equations for this problem are given as: .
The compatible boundary conditions are: In view of [14,30], the flow equations for this problem are given as: The compatible boundary conditions are: Here, p stands for electrode and magnet width; magnetization of variable permanent magnets is denoted by M * (= M 0 x); the applied current density is represented by j 0 [31,32].
The similarity transformations are demarcated as: Crystals 2021, 11, 1315 4 of 12 On replacing Equation (7), Equations (2)-(5) are converted to: where, M = πj 0 M 0 8a 2 ρ , denotes the modified Hartmann number; m = π p ν a , is the dimensionless parameter; , stands for the local Grashof number; , if the modified Grashof number; The corresponding boundary settings are changed to: The associated terms are demarcated as:

Results
Employing the MATLAB software (version 2019) via the Keller box scheme, the numerical results were recovered by discretizing Equations (8)- (11). The solver is programed with the Keller box technique explained by [4]. This method can accurately work to predict the solutions by considering an arbitrary initial guess; the average CPU time consumed to calculate the results may vary, which depends on the initial guess. The boundary layer thickness is taken, while the suitable initial guesses for MATLAB code and involved factors are selected till the profile (concentration, temperature, and velocity) satisfies the boundary settings (12) to validate the accuracy of the outcome. Figure 3 indicates that the modified Hartmann number boosts the intensity of the outer electric field, which ultimately strengthens the parallel wall Lorentz force and enhances the velocity profile. It means that the parallel Lorentz force supported the flow in the x-direction. Further, the wall velocity gradient enlarged while the boundary layer thickness declined. Figure 4 represents the effect of parameter m declines in the velocity profile, which shows a similar impact as shown by Ramzan et al. [14]. For the growth of m, the internal forces in the thick wall enhance, which causes a decline in the momentum boundary layer and flow velocity. Physically, the increment in wall thickness factor declines the stretching velocity. As this is mainly concerned with the velocity distribution behavior asymptotically, the liquid velocity upsurges monotonically by improving the wall thickness factor. Moreover, Figure 5 validates the earlier reported results found by Wubshet Ibrahim [25]. The increment in material parameter decreases boundary layer thickness, but increases the behavior of h(η). Figure 6 portrays how Brownian motion impact boosts the temperature profile because the Brownian motion improves the particles' motion, which is responsible for yielding more heat. The influence of thermophoresis on temperature distribution is displayed through Figure 7. The thermophoresis is a gauge to scrutinize the temperature distribution, because an increment in Nt pushes the nanoparticles to transfer from a hotter region to a colder region, which enhances the temperature distribution. Moreover, In Figure 8, a dominant behavior of the concentration profile against Nb is sketched. The reason behind this behavior is the collision among nanoparticles. A description of Nt on the concentration profile is portrayed in Figure 9. It is analyzed that the concentration profile and boundary layer thickness enhance for increments in Nt. In addition, growth in numeric values of M (modified Hartmann number) and Nb (Brownian motion). In addition, the energy transmission rate can also be controlled via diminishing the electrode and magnet width. Moreover, Figure 12 exhibits that C f x (0) increases by increasing the modified Hartman number and Brownian motion parameter.
against is sketched. The reason behind this behavior is the collision among nanoparticles. A description of on the concentration profile is portrayed in Figure 9. It is analyzed that the concentration profile and boundary layer thickness enhance for increments in . In addition, Figures 10-12 depict the effects of heat and mass flux alteration against variations in and . Figures 10 and 11 present ′ 0 (Nusselt number) and ′ 0 (Sherwood number) decreases for the growth in numeric values of (modified Hartmann number) and (Brownian motion). In addition, the energy transmission rate can also be controlled via diminishing the electrode and magnet width. Moreover, Figure 12 Table 1 has been prepared to check the accordance of the present problem's numerical outcomes with previously available literature, while Table 2 portrays the impacts of the involved flow factors on energy transport and mass transport flux, along with skin friction coefficient. It can be seen that heat exchange slows down on the growth of Brownian motion impact (see Table 2). In the same vein wall, shear stress shows an inverse relation with Brownian motion and thermophoretic effect. On the other hand, mass flux re-  Table 1 has been prepared to check the accordance of the present problem's numerical outcomes with previously available literature, while Table 2 portrays the impacts of the involved flow factors on energy transport and mass transport flux, along with skin friction coefficient. It can be seen that heat exchange slows down on the growth of Brownian motion impact (see Table 2). In the same vein wall, shear stress shows an inverse relation with Brownian motion and thermophoretic effect. On the other hand, mass flux reflects a direct correspondence with Nb and t. Moreover, the growth of prandtl number strengthens the heat and mass exchange rates, as well as the skin friction. In addition, the skin friction and mass exchange rate improve on the cumulative magnitude of the Lewis number. Meanwhile, the opposite impact can be seen in the case of the heat exchange rate. The magnetic field strengthens the heat flux and skin friction for improving its strength, but reduces the mass flux. Table 1. Comparison of the reduced Nusselt number −θ (0) and the reduced Sherwood number, −ϕ (0) when M, K, m, λ, δ, η = 0, Pr, Le = 0.

Conclusions
This study examined the Brownian motion and thermophoretic effects on nanofluid flow with micro-rotational particles numerically. The stretching Riga plate is considered for this analysis. The Riga plate can be utilized to produce an electromagnetic field, which helps in controlling the weak electrically conducting fluid flow. An appropriate boundary layer thickness is supposed by the MATLAB program using the Keller box technique. The important features of the current analysis are:

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The increment in modified Hartmann number boosts the energy and mass flux rates.