Numerical Study and Comparison of Two-dimensional Ferrofluid Flow in Semi-porous Channel under Magnetic Field

In this article, the equations governing the constant ferromagnetic current are investigated. The Lorentz force restrains this ferrofluid flow in a semi-porous valve. Analyzes were performed on three sub-particle fluids: kerosene and blood, water and magnetite. Modeling in the Cartesian coordinate system using the relevant equations was investigated. A slight thinning should be considered in the lower part of this channel. This research has used two Akbari-Ganji methods (AGM) and finite element method (FEM) to solve the equations. Nonlinear differential equations are solved using the above two methods. In the finite element model, the effect of changing the Hartmann number and the Reynolds number on the flow velocity and the derivatives of the velocity and shear stress of the fluid were investigated. As the Hartmann number increases, the velocity decreases in both directions. The Reynolds number changes in different slip parameters, which shows the opposite behavior for the two directions. Also, the insignificant effect of volume fraction of nanoparticles on velocity and its derivatives and shear stress was investigated. The results of solving the equations with the above two methods were compared with HAM. The results obtained using AGM and FEM and their comparison with previous researches have led to complete agreement, which shows the efficiency of the techniques used in this research.


INTRODUCTION 1
Recently, the wide and abundant use of non-Newtonian fluids in industry and engineering sciences has caused *Corresponding Author Email: b.jalili@iau-tnb.ac.ir (B.Jalili) them to pay special attention to these materials.The characteristic and importance of this type of fluid can be seen in polymer compounds and edible substances.Ferrofluid is one of the non-Newtonian fluids that has recently been investigated by scientists.The physical structure of this fluid is very complex and no model can fully show all its features.For these reasons, investigation and analysis of the behavior of ferrofluid is very important.
Abbas et al. [1] investigated second-order fluid in the presence of a chemical reaction in a semi-porous channel.In this research, homotopy method was used to approximate analytical solution of the differential equations.Abdel Rahim and Rahman [2] analyzed heat transfer and fluid mass transfer in a magnetic field, which was assumed to be slow flow under electric conduction.Abbas and Hasnain [3], Abbas et al. [4,5] have conducted studies on a two-phase fluid flow in an entirely porous medium.In this research, magnetite (Fe3O4) fluid was selected, and the sliding flow of nanomaterials in an inclined channel with thermal radiation was investigated and analyzed.
For the investigation of fluid properties, it can be mentioned that the fluid in the semi-porous channel is chemically conductive, and the mass transfer of the fluid is second-order with chemical reaction, Abbas et al. [6] investigated Maxwell fluid flow with radiation in an axisymmetric semi-porous channel.The natural convection micropolar fluid flow problem developed by Ashmawy [7].According to that study, it was observed that the enhancement of the micropolar parameter includes the decrease or an increase in the fluid velocity.There are various medical mechanisms, such as blood circulation in the veins, the presence of oxygen generators in the blood, blood dialysis in the artificial kidney, and various types of engineering activities such as filter design; all of these mechanisms, despite the presence of a fluid flow in a channel semi-porous flows, it can be analyzed and investigated.Ayaz [8] used it for nonlinear ordinary differential equations.Also, in electrical engineering, new problems are presented, which are linear and nonlinear.It should also be noted that solving the governing equations of medical flow using DTM was investigated by Bég et al. [9].One of the most effective analyses of fluid flow inside the channel is presented by Berman [10].This article gives equations governing the smooth flow of a rectangular channel with semi-porous walls.
To fully describe a fluid flow in Manal with porous walls, solving the Navier-Stokes equations in twodimensional laminar flow was necessary.This work was done by Brinkman [11], which is clearly shown the dependence of pressure and velocity components on fluid properties and channel dimensions.Due to the development and evolution of components, researchers are trying to create a variety of advanced fluids.These types of fluids should be closer to the real state in terms of heat exchange.For this purpose, they focused their studies on nanofluids because nanofluids release nanosized particles.Chen and Ho [12] extended DTM to solve partial differential equations.Therefore, using these materials in liquids can achieve lower thermal conductivity.Choi and Eastman [13], Choi et al. [14] proposed the first idea for these advanced fluids and discussed the extraordinary physical and chemical properties of these fluids that ferrofluids are also considered part of these fluids.
Ghasemian et al. [15] investigated forced displacement heat transfer on magnetite and numerically compared them with the presence of fluid under constant and alternating fields.Analysis of the flow of two fluids with the same density and varying viscosity in a horizontal channel and an unsteady state was recorded by Ghosh et al. [16].According to this study, when the viscous fluid is near the channel, it is unstable for many parameters.Koriko et al. [17] conducted studies on the importance of partial slip and buoyancy on the boundary flow of a nanofluid, where the viscosity was considered to be zero.The equations governing nanoparticles were solved using the classic Runge-Kutta method; they found that it can be concluded that the maximum flow velocity occurs in larger values of partial slip and buoyancy parameters.We need analytical approximations often broken nonlinearly to solve nonlinear problems.Liao [18] mentioned the homotopy method in his book and solved difficult nonlinear problems with the help of this method.Mousavi et al. [19] investigated the hydrodynamic behavior of ferrofluid with 3D simulation in a wave channel and used the mathematical model with matching magnetohydrodynamics and ferro hydrodynamics to formulate the problem.Also, this research investigated the efficacy of changes in the Nusselt number and the amount of magnetic gradient.The use of semi-numerical techniques such as the homotopy analysis method (HAM) and differential transform method (DTM) has been presented by Parsa et al. [20].They solved the governing equations of fluid flow in a semi-porous channel and investigated the effect of some parameters, such as Reynolds and Hartmann numbers in the heat transfer rate.
With the help of this method in an analytical function and using unknown and known boundary conditions, the derivative of n can be calculated at a known point.Rashidi et al. [21,22] by using the differential transform method (DTM), were able to provide a solution for studying a non-Newtonian fluid flow inside a channel with semi-porous boundaries.This research was done using the parametric perturbation method.Salehpour and Ashjaee [23] investigated on the ferrofluid flow in a miniature channel under alternating and constant fields in a two-dimensional space.They found that the constant magnetic field enhancement heat transfer and the heat transfer rate decreases with increasing Reynolds number.They also calculated the optimal frequency for maximum heat transfer at high Reynolds numbers.In addition to that, Sanyal and Sanyal [24] studied on the two-dimensional steady flow in an inclined rectangular channel with the presence of a magnetic field.From the outcomes of this research, we can refer to the graphical comparison of velocity, temperature, and magnetic field numerically.Also, the fluid flow has been investigated as a two-dimensional steady Maxwell flow in a symmetric semi-porous channel with a heat transfer rate.By using DTM and a semi-numerical solution with Taylor series expansion, Zhou [25] has formulated such existing problems.
The composition of ferrofluid includes substances such as Ferrum (Fe3O4) and suspension liquid of nanoparticles in several domains.Due to surface tension between solid particles and traditional fluid, a coating such as surfactants can be used to reduce it.If there is a magnetic field next to these particles, their behavior differs from ordinary metals.Also, these particles become magnetized inside the magnetic field.The location of the fluid under the magnetic force can be affected.Insomuch, the response of the ferrofluid to the external magnetic field is evident.One of the characteristics of sub-fluids is that they have very small particles.This makes them maintain their position and not settle down despite the magnetic force for a long time.In fact, papers and researches on ferrofluid flow in nonrepetitive geometries are very limited.
Abbas et al. [26] developed a steady ferrofluid flow study in a semi-porous channel.In this research, fluids such as water, kerosene, blood, and magnetite are subjected to the Lorentz force in a semi-porous channel.The solution methods in this paper are the homotopy analysis method (HAM), differential transformation method (DTM), and Runge-Kutta method.The efficacy of Hartmann and Reynolds number changes on the flow velocity is shown and analyzed.Jalili et al. [27] investigated ferrofluid microstructure and inertial characteristics using homotopy and Akbari-Ganji methods.Also, the efficacy of related parameters on flow performance, temperature, and velocity has been analyzed.It should be noted that the fluid velocity in the vicinity of the sheet and at a distance from the sheet has been compared.In a study with a semi-analytical solution method on the Lorentz force and the efficacy of viscosity on nanofluid, the viscosity is variable, and the temperature parameter changes in a linear function.The semi-analytical AGM is used in this paper presented by Jalili et al. [28].Changes in fluid velocity have been investigated by changing parameters such as Prantel, Hartmann, and Nusselt numbers.In another study conducted by Jalili et al. [29], the performance of magnetohydrodynamic heat transfer in a porous circular chamber was analyzed using Darcy's law.Also, by using the eddy current function formula and solving it numerically, a good comparison has been made with FEM.This material is placed in a magnetic field under a horizontal magnetic force, and its behavior has been investigated with changes in the volume fraction of nanoparticles, inclination angle, Lorentz, and buoyancy forces.The use of three methods of comparing them to investigate the characteristics of a ferrofluid on a shrinking plate was demonstrated in a research conducted by Jalili et al. [30], which are AGM, FEM, and HPM.The base fluid in it is water and Fe3O4.According to this paper, it can be seen that the boundary and magnetic parameters have the same efficacy on the fluid velocity.This is not the case for the micro-rotation parameter.
Recent studies have investigated the nanofluid flow between two parallel plates under magnetic and electric fields in a rotating system.According to the research conducted by Jalili et al. [31], the velocity profile and micro-rotation velocity increased with the magnetic and rotation parameters also increased.Also, the efficacy of changes in Reynolds, Prantel, and Schmidt numbers, thermophoretic parameters, and Brownian motion have been analyzed.
Jalili et al. [32] conducted a study on a twodimensional viscous fluid located between two porous spaces in a calm and incompressible manner.Based on this research, the equations between two discs have been solved using the Akbari-Ganji and finite element methods.With the help of the findings, the velocity, pressure, and temperature parameters have been analyzed.Also, sliding in the boundaries and changing the Reynolds number causes changes in the fluid velocity, which are obvious.
In the article presented by Jalili et al. [33], all heat transfer processes in non-Newtonian fluid flow have been investigated.AGM and FEM methods have been used to solve this fluid flow's governing equations, showing complete agreement.Several values for Hartmann number and electromagnetic force affect the velocity profiles.Also, an extensie analytical research has been done in this field [34][35][36].
As a result, the general approach of this article and research is to provide numerical and semi-numerical solutions for ferrofluid flow in a semi-porous channel under a magnetic field with two basic fluids such as water and kerosene.Finally, the fluid velocity is obtained in its final form by solving the relevant equations with three homotopy methods: the Akbari-Ganji and finite element methods.Also, comparing these three different techniques and the efficacy of fixed parameters on fluid velocity are analyzed.The use of other numerical and experimental solution methods such as Euler, Taylor, and RK-4 methods can also be used to solve the governing equations of fluid flow.The very good agreement of the obtained results from the used methods is one of the advantages of adopting this approach in this article.The error caused by the mentioned methods is very small and negligible and finally it is acceptable.A comprehensive literature above motivated us to investigate and analyze a ferrofluid flow in a transverse magnetic field (see Figure 1).In the continuation of this research, the effect of changes in Reynolds and Hartmann numbers, velocity slip parameter and volume fraction of nanoparticles is shown using the finite element method.The results help the readers to have a better understanding of the effect of the above parameters on the velocity profile, velocity derivatives and ferrofluid shear stress in two directions.The comparison of the values obtained from the present research with other researches is presented in order to show the validity of the proposed solution methods.

FORMULATION OF PROBLEM AND BASIC EQUATION
According to Figure 1, two infinitely rigid parallel plates located at a distance h from each other, where the desired fluid enters this range (see Figure 1).The slip condition is applied on the plate of length Lx along the x*-axis at  * = 0.The rate of transpiration in an infinite porous plate is equal to q.The physical properties of the flow are two-dimensional, steady, and slow flow.
Water and kerosene-carrying magnetite Fe3O4 are two fluids considered nanoparticles in this research.Applying a magnetic field with intensity  is assumed in the desired channel.The magnetic field is transverse to the fluid flow.The effects of the induced magnetic field are not taken into account.Considering the above assumptions, the equations governing the flow are as follows [19,20].
Also, the effective dynamic viscosity is calculated as follows [11]: The nanoparticle volume fraction is represented by the symbol  (<0.05 for most practical cases).The shape of the particles can be considered spherical or non- The physical properties of magnetic nanoparticles and the base fluid are shown in Table 1 [17][18][19].
(1 ) In specific situations, the boundary conditions (BCs) are as follows: Scientists and researchers are trying to investigate the conditions of the sliding velocity of Navier in the walls and their use.Some articles and research that have used these boundary conditions as well [1,3,7,16,24].
Using the following equation, the average velocity () is calculated) With the help of the following non-dimensional variables, solutions to the problem can be found.
By substituting the above variables in Equations ( 1), (3), and ( 6) is obtained (1 ) (1 )  8) and ( 9) are equal to: The term ɛ, is very tiny because it is the ratio of h and Lx.
To remove ɛ from Equations ( 8) and ( 9), Berman [10] similarity transformations are used According to the above relationships, it can be seen that in Equation ( 9), the quantity    ⁄ is independent of the  variable.According to Equation ( 8), it can be said that  2    2  ⁄ is not a function of the longitudinal variable .If the asterisks are ignored for ease of work, the following relationships will appear after changing the variables .

THE GENERALITIES OF AKBARI GANJI'S METHOD (AGM)
Considering the boundary conditions, the differential equations are as follows: : (,  ′ ,  " , … ,  () ) = 0;  = () The  function in a nonlinear differential equation is assumed as a function of .Also, the  parameter is considered a function of  and its derivatives.In this way, the boundary conditions of the equations are established as follows: In Equation ( 17), a series of order  is assumed according to the boundary conditions at  = , where the coefficients are considered constant.This series is written as the answer to the first differential equation as follows: In order to enhance the accuracy in solving Equation ( 16), it is necessary to enhance the series expressions in Equation (18).Due to the order of the above series being n, there are ( + 1) unknown coefficients, which we usually need ( + 1) equations to solve.In Equation (17), boundary conditions are applied for the ( + 1) equation.

2. Solving Equations Governing Fluid Flow with AGM
In this method, using appropriate functions, polynomial series with constant coefficients can be considered: Equations ( 12) and ( 13) are written as follows: )) ( 1) The use of BCs in Equations ( 14) and ( 15) is as follows: (1) 0 0 U a a a a a a = → + + + + + = (32) According to the considered series in Equations ( 25) and ( 26), there are 12 unknown coefficients.Therefore, 12 equations are needed.Six equations are obtained by applying boundary conditions, and six more equations are added as follows: Finally, the values of  and  are equal to:

RESULTS AND DISCUSSION
Equations ( 12) and ( 13) were solved with AGM and FEM to obtain velocity fields (),  ′ (), (),  ′ (),   and   according to Equations ( 14) and (15).Drawing and displaying graphs (),  ′ (), (),  ′ (),   and   and the efficacy of changes in relevant parameters have been done in them.Figure 2 shows (),  ′ (), (),  ′ (),   and   profiles.Also, the comparison of three solution methods AGM, FEM, and HAM has been done when  =  = 1.0,  = 0.2 and  = 0.04.By comparing the figures, it can be concluded that the outcome obtained from all methods are very close to each other.This result proves and confirms the validity of the methods.Based on this, it can be concluded that the values of   and   also behave similar to the derivatives of velocity, because the constant coefficient (µ) is multiplied in the derivatives and shear stresses appear.According to Figure 3, it can be seen that there is no difference in the efficacy of Reynolds number changes in the boundary conditions of sliding and non-slipping.
According to Figure 4, the distribution of fluid flow velocity decreases with the enhancement of the Hartmann number.This decrease is much higher in ().The Lorentz force enters in the -direction because the magnetic field is placed in the -direction.Thus, the magnetohydrodynamic force generated in () and ′() can be ignored, although it plays a significant role in () and ′().With an increase in Hartmann's number in the range of 0 ≤  ≤ 0.6, a decrease in ′() can be seen, and in 0.6 ≤  ≤ 1, the value of ′() increases.This behavior is also shown in shear stress (  ).The algebraic value of ′() in the range of 0 ≤  ≤ 0.4 increases with an increase in Hartmann's number, but its value is negative.In the range of 0.4 ≤  ≤ 1, its behavior is reversed and the same process is repeated for   in a smaller order.
The efficacy of  sub-particle volume fraction and  slip on fluid flow velocity distribution () and () Likewise  and  are shown in Figure 5. Also, in the vicinity of the upper and lower plates, for boundary conditions with slip and no-slip, the opposite behavior can be observed for ′() and ′().If the volume fraction of nanoparticles () increases, () decreases and () increases.This behavior is similar for both velocity profiles with an increase in slip parameter ().As the value of  increases, in the range of 0 ≤  ≤ 0.5, the value of ′() and the algebraic value of ′() decrease, and then at 0.5 ≤  ≤ 1, the algebraic value of ′() and ′() increases.A similar behavior is also shown for shear stresses (  and   ) in two directions.
Figures 6 and 7 show the effect of changes in Reynolds and Hartmann numbers and the volume fraction parameter of nanoparticles.As can be seen, with an increase in Reynolds number, () increases, but if the Hartmann number or volume fraction of nanoparticles increases, () decreases.Based on Figures 8 and 9, it can be concluded that () decreases with an increase in Hartman and Reynolds numbers.But if the volume fraction of nanoparticles is increased, () increases.
Convergence for several values of (), (),   and   in terms of  is listed in Table 2 to 5. In Tables 2  and 3, the results obtained for the values of water velocity  the three methods of HAM, DTM and Numerical are presented.The comparison of the present results and their correspondence with the results of other researches [26] shows the efficiency of the methods used in this article.In addition, in Tables 4 and 5, the values obtained from the current research with two AGM and FEM methods are given.
The changes related to the three parameters of ,  and  in a 3D space in the form of optimization are given in Figures 6 and 7 for ().
The changes related to the three parameters of ,  and  in a 3D space in the form of optimization are given in Figures 8 and 9 for ().

CONCLUSION
One of the most important goals of this research was to investigate and analyze two-dimensional fluid flow in relation to sliding efficiency on a nanofluid with a porous wall.The equations governing the flow were solved with the mentioned simplifications and two methods.Among the achievements of this article, the following factors can be mentioned: ➢ The area of convergence in ′′(0) is larger than that of ′(0).➢ The difference in AGM, FEM, and HAM outcomes is negligible, and all three methods are acceptable.➢ As the Reynolds number enhancement, the velocity of the () fluid increases.This is while it reduces with the enhancement of Hartmann's number.
➢ As the Hartmann and Reynolds numbers increase, () decreases.➢ As the nanoparticle volume fraction parameter increases, () increases and () decreases.➢ If there are changes for Hartmann, Reynolds numbers and volume fraction of nanoparticles, the opposite behavior is reported for the derivatives of velocity and shear stress in the range of 0 ≤  ≤ 1.

Figure 1 .
Figure 1.An overview of the physics of the problem

Flowchart 1 .(
Showing the process of solving equations with AGM 0

Figure 9 .
Figure 9.The efficacy of changes in Reynolds number and Phi number on Hartmann's constant number equal to 7

TABLE 2 .
The analogy of the obtained outcome for the velocity of () when  = 0.04,  = 1.0,  = 1.5,  = 0.3 for the Water base.

TABLE 1 .
Physical properties of magnetic nanoparticles

TABLE 3 .
The analogy of the obtained outcome for the velocity of () when  = 0.04,  = 1.0,  = 1.5,  = 0.3 for the Water base.

TABLE 4 .
The analogy of the obtained outcome for   when  = 0.04,  = 1.0,  = 1.5,  = 0.3 for the Water base

TABLE 5 .
The analogy of the obtained outcome for   when  = 0.04,  = 1.0,  = 1.5,  = 0.3 for the Water base