A novel analytical approach to micro-polar nanofluid thermal analysis in the presence of thermophoresis, Brownian motion and Hall currents

The present study analyzed micro-polar nanofluid in a rotating system between two parallel plates with electric and magnetic fields. The fluid flow study was performed in a steady state. The governing equations of the present issue are considered coupled and nonlinear equations with proper similar variables. Numerical and new semi-analytical methods have been employed to solve the problem to define the exactness of the results. The influence of physical parameters governing the problem is investigated and illustrated in detail in the diagram. Results show that velocity profile and micro-rotation velocity increased when the magnetic parameter increased. Furthermore, the velocity is increased by increasing the rotation parameter. Also, in the case of the temperature profile, the Reynolds and Schmidt numbers have an inverse effect, and Prandtl number and Brownian motion have a direct effect. Other results indicate that concentration value declines by increasing the thermophoretic parameter and Reynolds number. Results compared to the prior research display good accuracy and efficiency. The study demonstrates that the method provides quantifiable reliable outcomes while requiring less computing work than conventional techniques. This method offers significant advantages in terms of simplicity, applicability, computational efficiency and accuracy.


Introduction
Magnetohydrodynamics (MHD) explores magnetic properties and the action of electrically conductive fluids. Liquid metals, electrolytes, salt water and plasmas are some sample of such magnetofluids. Liquid-metal cooling of reactors (MHDs) have numerous practical uses in the engineering and technologies areas such as (MHD) sensors, MHD power generation, plasma, crystal growth and electromagnetic casting. Because of dependence on the strength of the magnetic inductions, the magnetic force becomes stronger; then, Hall effect produced due to the Hall currents cannot be neglected. Hall effect is an electric crossover field when it holds an electric current and is put in a magnetic area. The governing equations combine Navier-Stokes equations and Maxwell's equations; they must be solved either analytically or numerically at the same time. Krishna and Chamkha (2019) analyzed at skin friction and the Sherwood number as well as the effects of radiation-absorption on the free convective rotational flow of nanofluids in magnetohydrodynamics. They demonstrated that the thermal boundary layer thickness increases with an increase in the radiation-absorption parameter. Micro-polar nanofluid on a stretched surface is studied by Hayat et al. (2017). Their research shows that at increasing Brownian motion parameters, the rates of mass and heat transmission rise. Achara et al. (2019) investigated the impact of Hall current on the motion of a spinning disk on radiative nanofluid flow. The outcome demonstrates an improvement in the Hall parameter's radial skin frictional factor. Raju (2019) illustrates the Hall effect in terms of fluid flow and two-ionized heat transfer in magnetohydrodynamics between two parallel plates. He demonstrated how a peak in the viscosity ratio and Hall parameter causes the temperature diffusion to decrease along with an increase in the Hartmann number. The effects of a slip condition on an unsteady magnetohydrodynamics flow were investigated by Das et al. (2018). They discovered that the Hall parameter caused the boundary layer's momentum thickness to increase. When dealing with an unstable magnetohydrodynamics free convective micropolar fluid flow, Sheri and Shamshudin (2018) used the finite element approach. They calculated how the developing thermophysical factors affected skin friction, surface mass and heat transfer rate. Using HPM and FEM, Jalili et al. (2021a) examined the effects of ferrofluid on a shrinking sheet. The Duan-Rach method was used by Dogonchi and Ganji (2017) to describe the magnetohydrodynamics nanofluid flow between two parallel surfaces. They demonstrated an inverse link between the Nusselt number and the thermal relaxation parameter. Jalili et al. (2019a) studied the inertial and microstructure properties of a magnetite ferrofluid on a stretched plate using two semi-analytical techniques. In the region of the sheet, they showed the greatest degree of speed. DTM and FEM were used by Mosayebidorcheh et al. (2017) to examine magnetohydrodynamics heat transfer and Couette flow of dusty fluid. Convective heat transfer of a magnetohydrodynamics nanofluid flow was examined by Yahyazadeh et al. (2012). They used scaling modifications to convert PDE to ODE. The effectiveness of the system is impacted by nanoparticle mobility, as demonstrated by Malvandi et al. (2015). They described how nanoparticles affect the improvement of heat transmission.
Researchers have used semi-analytical techniques to solve many engineering problems in investigating heat transfer and fluid flow. Pasha et al. (2018) used semi-analytical techniques to Analysis of unsteady heat transfer of specific longitudinal fins with temperature-dependent thermal coefficients by differential transform method DTM. Jalili et al. (2018) Humphries et al. (2021) investigated the analytical approach of Fe3O4-ethylene glycol radiative MHD nanofluid on entropy generation in a shrinking wall with porous medium. Gupta et al. (2018) used an accurate 2D analytical model for transconductance-to-drain current ratio (gm/Id) for a dual halo dual dielectric triple material cylindrical gate all around MOSFETs. Huu et al. (2020Huu et al. ( , 2019Huu et al. ( , 2020 used a novel equal-order mixed polygonal finite element to solve the incompressible fluid problems. Also, these kinds of analytical and numerical methods are used for developing economic and social problems (Meng and Zhang 2022;Li and Alaa 2022), sports (Guan et al. 2022) and magnetic and mechanical properties (Sun et al. 2021;Liu and He 2022). https://www.ije.ir/article_154646.html The majority of mathematical issues in science and engineering are complicated in nature, making a precise solution nearly impossible or even impractical. To solve these problems, numerical and analytical approaches are utilized to determine an approximation of the answer. The computational cost is a covert word used in optimization that might involve CPU and material optimization. The Akbari-Ganji approach is one of the most significant and practical methods for solving these kinds of issues. The fundamental benefit of this alternative approach is that it may be applied to nonlinear differential equations without discretization or linearization. The present study investigates micro-polar nanofluid flow in spinning parallel plates with Hall current and electrical magnetohydrodynamics. The fourth-order Runge-Kutta numerical method and Akbari-Ganji semianalytical method have been used for solving this complicated problem. AGM is used in the present work for the solution of modeled equations which are nonlinear and coupled. The effect of all embading parameters has been studied graphically. The obtained results have been compared with previous research. The results show that these methods are accurate and efficient. This study consists of five sections: In the first section, the topic and literature are discussed. In section two, formulation of the electrical MHD and Hall current on micro-polar nanofluid flow between rotating parallel plates is defined. Also, we provide a review of the AGM and its application in section three and illustrate the applicability and validity of the proposed method in section four. Finally, section five is the conclusion.

Problem formulation
This study concentrates on the micro-polar nanofluid flow between two flat plates. The rotation of fluid is around y axis. The fluid field is considered an incompressible, laminar and steady state. Besides, the impact of the Hall current is assumed in the micro-polar nanofluid model. Equation (1) represents the generalized state of Ohm's law, including the Hall current, which is noted as: Also J y and J x are obtained through Eqs. (2) and (3) (Shah et al. 2018) (Fig. 1): The momentum and continuity equations are reduced as follows (Shah et al. 2018): The energy equation is diminished as follows (Shah et al. 2018): Micro-rotation angular velocity and mass transfer equations are obtained according to Eqs. (9) and (10), respectively: A novel analytical approach to micro-polar nanofluid thermal analysis in the presence… 679 Also the boundary conditions are shown as follows: where k denotes the boundary parameter. Strong concentration happens when k ¼ 0. For weak concentration, k ¼ 1 2 , and for turbulent flow, k ¼ 1. The non-dimensional variables are shown in Eq. (12): By replacing Eq. (12), the governing equations and boundary conditions are converted as follows: Re Àgf 0 þ fg 0 ð Þþð 1 þ N 1 Þg 00 þ 2Krf 0 À MEI After some simplification, the dimensional physical parameters become as follows: Fig. 1 Geometry of the micropolar nanofluid between two parallel plates (Shah et al. 2018) 3 Akbari-Ganji's method (AGM) Equation (20) is assumed as nonlinear differential equations: p k : f p; p 0 ; p 00 ; :::; p n And, the boundary conditions are: The answer of basic equations is assumed as a below polynomial form: Therefore, when x ¼ 0 and x ¼ h: pðhÞ ¼ a 0 þ a 1 h þ a 2 h 2 þ ::: þ a n h n ¼ p h0 p 0 ðhÞ ¼ a 1 þ 2a 2 h þ 3a 3 h 2 þ ::: þ na n h nÀ1 ¼ p h 1 p 00 ðhÞ ¼ 2a 2 þ 6a 3 h þ 12a 4 h 2 þ ::: þ nðn À 1Þa n h nÀ2 ¼ p hmÀ1 : : : : : :

Application of Akbari-Ganji's method (AGM)
Concerning the basic idea of AGM, the next step of AGM solving is to substitute constant coefficients in the polynomials with such as: þ a 51 g 5 þ a 6 g 6 þ a 7 g 7 ; U g ð Þ ¼ X 4 i¼0 e i g i ¼ e 0 þ e 1 g 1 þ e 2 g 2 þ e 3 g 3 þ e 4 g 4 ; ð29Þ For obtaining the constant coefficients of Eqs. (25-29), the boundary conditions are utilized as follows: So by applying the boundary conditions: The total number of our constants is 30 (according to Eqs. 25-29), and we have 12 equations according to Eq. 31, so we require 18 more equations. We provide 18 equations by substituting Eqs. (25-29) in main Eqs. (13-17). Now we can suppose five basic equations (Eqs. 13-17) as D1 to D5, respectively, and use boundary conditions and their derivation to get equations as follows: 4 Results and discussion Figure 2A, b shows the effect of Kr on velocity profile. The graphs illustrate the velocity profile has a downward trend and then an upward trend. Actually, increasing the rotational parameter Kr causes the carioles force to increase, which in turn causes the rotational velocity to increase. This fluid rotation also causes an increase in kinetic energy, which in turn causes an increase in flow motion. Figure 2c, d displays a comparison between AGM and numeric solutions with good accuracy. The impact of m on the velocity profile and micro-rotation velocity is shown in Fig. 3a, b and e. It is evident that the velocity distributions A novel analytical approach to micro-polar nanofluid thermal analysis in the presence… 681 f 0 g ð Þ are directly and inversely varied with g g ð Þ along the z-direction and M along the y-direction, respectively. When M is near to the plates, it reduces the velocity field; however, when M is working along the z-direction, it has the reverse effect. This effect of M on the velocity field results from the fact that as M increases, the Lorentz force-the force that prevents movement-progresses. It has a tendency to lower fluid velocities in boundary sheets, where the Carioles force, another force, exhibits the opposite effect on velocities in addition to the z-direction. The generation of potential difference is the Hall current. Here, the micro-nanofluid flow is significantly influenced by the Hall parameter m. The magnetic damping force decreases due to the large value of m, which also causes the effective conductivity to decrease, causing the velocity profile along the y-and z-axes to rise. When the magnetic parameter m is raised, the micro-rotation profile between parallel plates diminishes. Also results of AGM and numeric solution (present study) with Zahirshah (Shah et al. 2018) are plotted in Fig. 3c, d and f. The results indicate that AGM is efficient and precise. The effect of the Hall parameter on velocity in y-and z-directions is depicted in Fig. 4a, b. In addition, Fig. 4c, d presents the contrast between AGM and fourth-order Runge-Kutta method. Figure 5a, b shows that the concentration and temperature increase by the distance between two parallel plates. Also, increasing the Brownian motion (Nb) increases the amount of concentration and temperature. This occurs as a result of increasing Brownian motion decreasing boundary layer thicknesses, which causes concentrations to fall. The random, erratic motion of particles in nanofluids is known as Brownian motion. It has been proposed that a key factor contributing to the thermal conductivity of nanofluids is the Brownian motion of micro-polar nanofluid at the molecular level. Figure 5c and d compares the findings of the current investigation to those of Zahirshah (Shah et al. 2018) for validation. It is observed that the coupling parameter N1 increases from the center toward the top plate and decreases velocity profile when it is close to the lower plate. The outcomes display that the utilized methods are efficient and reliable. The thermophoretic parameter (Nt) effect on concentration and temperature profiles is shown in Fig. 6a, b. The graphs show that the concentration and temperature decline with the increasing length between parallel disks. This is because the thermophoretic parameter, Nt, is influenced by the temperature differential in the nearby molecules of nanofluids. The temperature and concentration profile rise as a result of increasing Nt because it raises the kinetic energy of the nanofluids molecules. Inverse relationships exist between temperature and concentration distributions and Pr. It is evident that the temperature distribution rises for low values of Pr and decreases for big numbers of Pr. Physically, fluids with a low Prandtl number have more thermal diffusivity, and the reverse is true for fluids with a high Prandtl number.
Because of this, a large value of Pr causes a drop in the thermal boundary layer. Also, the comparison between the semi-analytical method and numerical technique is accomplished in Fig. 6c and d. In Table 1, numerically, the convergence of HAM and AGM solutions for the model equations at various approximations using different parameter values are shown. The table makes it abundantly evident that the Akbari-Ganji method is rapidly convergent by comparison between the AGM, HAM Shah et al. (2018) and Runge-Kutta of f 0 g ð Þ; g for M = 1 and Nb = 0.5. According to the obtained results, the percentage error of the present study compared to numerical results is equal to 1.57% and that obtained in Shah et al. (2018) compared to numerical results is equal to 6.38%. This paper's approximate solution to the governing equation is obtained by applying the AGM. The approximate solutions accepted that provided the AGM are reliable and effective methods. A good agreement has been achieved by comparing the numerical solution obtained using the fourth-order Runge-Kutta method and the proposed method. Figure 7 and Table 1 show that the maximal error remainder is negligible by this suggested method.

Conclusion
This paper analyzes micro-polar nanofluid flow between two rotating parallel plates under magnetohydrodynamic field and Hall current. The governing equations are solved through an analytical solution known as AGM. The study shows that the technique requires less computational work than existing approaches while supplying quantifying reliable results. This method offers significant advantages in straightforward, applicability, computational effectiveness and accuracy. Variations of thermophoretic parameter (Nt), Hall parameter (m), Brownian motion (Nb) and magnitude parameter (M) have been studied numerically and analytically. Results have been obtained as follows: • An increase in the y-direction and a decrease in the zdirection occurred in the velocity profile with a higher value of Kr. • The velocity profile is reduced in the y-direction by decreasing the M parameter. Also, the velocity profile on z-direction and micro-rotation velocity are increased by raising the M parameter. • Temperature profile HðgÞ increased by increasing thermophoretic parameter (Nt) and the Brownian motion parameter (Nb). • The concentration profile UðgÞ is decreased by increasing the thermophoretic parameter (Nt). • The concentration profile UðgÞ is increased by raising (Nb).
Author contributions It is also declared that all the authors have equal contribution in the manuscript. Furthermore, the authors have checked and approved the final version of the manuscript.
Funding The authors state that they have not received any financial support for this manuscript.
Data availability The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Declarations
Conflict of interest It is declared that all the authors have no conflict of interest regarding this manuscript.
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