# Impact of Hall Current and Nonlinear Thermal Radiation on Jeffrey Nanofluid Flow in Rotating Frame

Authors: Hakeem Ullah; Abdelaziz Alsubie; Mehreen Fiza; Nawaf N. Hamadneh; Saeed Islam; Ilyas Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This research article deals with the nonlinear thermally radiated influences on non-Newtonian nanofluid considering Jeffrey fluid in a rotating system. The governing equations of the nanofluid have been transformed to a set of differential nonlinear equations, using suitable similarity variables. The Homotopy Analysis Method (HAM) and Runge–Kutta Method of order 4 (RK Method of order 4) are used for the solution of the modeled problem. The variation of the skin friction, Nusselt number, Sherwood number, and their impacts on the velocity distribution, temperature distribution, and concentration distribution have been examined. The influence of the Hall effect, rotation, Brownian motion, porosity, and thermophoresis analysis are also investigated. Moreover, for comprehension of the physical presentation of the embedded parameters, Deborah numberβ, viscosity parameter R, rotation parameter Kr, Brownian motion parameter Nb, porosity parameter γ, magnetic parameter M, Prandtl number Pr, thermophoretic parameter Nt, and Schmidt number Sc have been plotted and deliberated graphically. For large values of Brownian parameter, the kinetic energy increases, which in turn increases the temperature distribution, while the thermal boundary layer thickness decreases by increasing the radiation parameter, and the Hall parameter increases the motion of the fluid in horizontal direction. Also, the mass flux has been observed as a decreasing function at the lower stretching plate.


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## Body

## 1. Introduction

The word nanofluid denotes the nanoparticles deferred into the base fluid. Usual examples of nanoparticles contain metals such as copper, aluminum, and silver, nitrides like silicon nitride, carbides such as silicon carbides, oxides like aluminum oxide, and nonmetals such as graphite. The usual liquids are water, oil, and ethylene glycol. The combination of nanoparticles through base liquid enormously develops the thermal qualities of the vile liquid. Choi et al. [1] introduced the term nanofluid and heat transfer features of vile fluids, such as thermal conductivity that is enriched by the addition of nanoparticles into it [2, 3].Magnetohydrodynamics (MHD) are the information of the magnetic assets of electrically conducting fluids. Plasmas, electrolytes, water, and liquid metals are examples of the such magnetofluids. Hannes Alfven [4] was the first to introduce the field of MHD. Magnetohydrodynamics have several applications in the field of industries and engineering such as plasma, crystal growth, magnetohydrodynamic sensors, liquid-metal cooling of reactors, electromagnetic casting, MHD power generation, and magnetic drug targeting. MHD depends on the strength of the magnetic field; the stronger the magnetic field, the greater the magnetohydrodynamic effects, and vice versa. B.J. Gireesha et al. studied the thermal radiation on MHD boundary layer flow of Jeffrey nanofluid over stretching sheet [5]. When the magnetic force becomes stronger, then we cannot neglect the Hall effects produced due to the Hall currents. Hall effect is produced due to the potential difference across an electrical conductor when a magnetic field is acting in a direction vertical to that of the flow of current. Edwin Hall [6] was the pioneer to give the concept of Hall current. It is of substantial status and attractive to investigate hydrodynamical problems improved by the influence of Hall currents with their results. Hall currents change flows to cross flow, making it three-dimensional. Pop and Soundalgekar [7] have studied Hall effect on time independent hydromagnetic viscous fluid flow. Ahmed and Zueco [8] have investigated the heat and mass transmission influence to flow in a rotating porous channel by captivating the influence of Hall current and obtained exact solution of the modeled problem. Abdel Aziz [9] has studied Hall effects on the nanofluid flow of viscid flow with heat transmission through a stretching sheet. Hayat [10] has scrutinized the influence of viscous dissipation on mixed convection Jeffrey fluid flow over a vertical stretching surface under the Hall and ion effects. Sulochana [11] has studied unsteady fluid flow over a permeable medium in a rotating parallel plate with Hall effects considering it in three dimensions. Because eccentric features of the nanoliquids make them proficient in many applications, nanofluids are used in the hybrid powered engines, pharmaceutical procedures, fuel cells, and microelectronics, and currently, they are mostly used in the field of nanotechnologies [12]. Wang et al. [13] have given a brief review on nanofluids on the view of their experiments and applications. It enhances the thermal conduction of the base liquid; hence, in the investigation of the flow of nanofluids in a rotating system, the scientists are intensely interested in it. Especially, the flow of nanofluid between parallel plates is one of the standard problems that have significant applications like in accelerators, MHD power generators and pumps, purification of crude oil, aerodynamic heating, petroleum industry, different automobiles sprays, and designing cooling systems with liquid metal. Goodman [14] was the first to study viscous fluid in parallel plates. Borkakoti and Bharali [15] have investigated hydromagnetic viscous flow between parallel plates, where one of them is being a stretching sheet. Attia et al. [16] have examined viscous flow between parallel plates with magnetohydrodynamics. Sheikholeslami et al. [17] have investigated nanofluid flow of viscous fluids between parallel plates with rotating systems in three dimensions under the magnetohydrodynamics (MHD) effects. For the solution of the modeled problems, they used numerical techniques and described the effects of achieving parameters in detail. Mahmoodi and Kandelousi have [18] investigated the hydromagnetic effect of kerosene−alumina nanofluid flow in the occurrence of heat transfer analysis, and differential transformation method is used in their work. Tauseef et al. [19] and Rokni et al. [20] have observed the magnetohydrodynamics and temperature effects on nanofluids flow in parallel plates with rotating system. M. Fiza et al. Studied three-dimensional MHD rotating flow of viscoelastic nanofluid in porous medium between two parallel plates [21]. In the current situation, different hybrid technology is developing day by day. In order to save the energy, the hybrid technology is developed. Fluid flow in a rotating system is a natural phenomenon. In fact, this rotation exists among the fluid particles internally and increases when fluid starts flowing. So, in the natural phenomenon of fluid, flow rotation exists up to some extent. The experimental idea of the viscous fluid motion in a rotating system was given by Taylor and Geoffrey [22]. Greespan [23] has investigated the detailed study of fluid movement in a rotating system. Vajravelua and Kumar [24] have examined magnetohydrodynamics viscous fluid flow amongst binary horizontal and parallel plates in a rotating system, in which one plate is stretched, and the other is permeable. They obtained numerical solution and studied the effect of physical parameters. Their work was extended by Mehmood and Asif [25]. In everyday life’s industries and technologies, non-Newtonian fluids are used frequently, and extremely less studies of Newtonian nanofluids to rotating system are found. Hayat et al. [26] have studied non-Newtonian fluid flow with rotation using different models and extended their work in two and three dimensions. Nadeem et al. [27] have investigated micropolar nanofluid in two horizontal and parallel plates with rotation. They obtained the analytical solution of the problem and discussed the embedded parameters. Jena et al. [28] have investigated viscoelastic fluid with the effect of MHD and internal heat in porous channel with rotating system. Jeffrey fluid is the significant subclass of non-Newtonian fluids, which were initially studied by Jeffrey [29]. Shehzad et al. [30] have extended the same work using convective conditions. The detailed study of Jeffery fluid other than the rotating system can be seen in [31–37]. In the field of science and engineering, most of the mathematical problems are complex in their nature, and the exact solution is almost extremely difficult or even not possible. So, for the solution of such problems, numerical and analytical methods are used to find the approximate solution. One of the important and popular techniques for the solution of such type problems is the Homotopy Analysis Method. It is a substitute method, and its main advantage is applying to the nonlinear differential equations without discretization and linearization. Liao [38–40] was the first to investigate this technique for the solution of nonlinear high ordered problems and generally proved that this technique is quickly convergent to the approximated series solutions. Z. shah et al. applied successfully this technique to solve the three-dimensional third-grade nanofluid flow in a rotating system between parallel plates with Brownian motion and thermophoresis effects [41]. Also, this technique provides series solutions in the form of a single variable. Solution by this technique is significant, because it involves all of the physical parameters of a problem, and we can easily discuss its behavior. In all these studies, the impact of Hall current with MHD on Jeffrey fluid has not been studied. So, for this aim, the impact of Hall current with MHD Jeffrey nanofluid with nonlinear thermal radiations in the rotating frame is considered.


## 2. Problem Formulation

In this section, we describe the physical description and mathematical description of the proposed model.


### 2.1. Physical Description

The flow of electrically conducting Jeffrey nanofluid between two parallel and horizontal plates is considered. The distance between the upper and lower plates is denoted withh. The coordinate system is selected in such a method that the plate and fluid both are rotated about the y-axis with an angular velocity Ω.. The two forces are assumed to have the same magnitude, but they are opposite in direction, to stretch the lower plate along x-axis, so that the origin 0,0,0 remains constant. The flow of the fluid and temperature transfer is considered in steady state, which is incompressible, laminar, and stable. The surface temperature of the nanofluid between parallel plates is taken after the influence of convective heating process, and its temperature of the hot fluid is T0 under the surface. The free stream is occupied at a uniform ambient nanofluid temperature Th with T0>Th. A magnetic field B0 is acting in y direction, with which the system is rotating. In addition, the effect of Hall current is taken in the nanofluid model. Here, the fluids are electrically conducting, and when the magnetic field becomes stronger, the Hall current is produced, which affects the nanofluids. This effect gives increase to a force in z−direction, which tempts a cross flow in the same direction, and hence, the nanofluid flows in a deflected way into three dimensions, as shown in Figure 1.Figure 1
Physical model of the nanofluid flow problem.


### 2.2. Mathematical Description

Ohm’s law in generalized form containing the Hall current is written as [3–7](1)J+ωeteB0×J×B−σnfE+V×B−σnfPeene=0.Here,J=Jx,Jy,Jz represents the current density, B=0,B0,0 represents the magnetic field, E represents the intensity of electric field, V=u,v,w represents the velocity components, ωe represents the oscillating frequency of the electron, te denotes the time of electron collision, σnf denotes the electrical conductivity, e is the charge of electron, ne is the number density of electron, and Pe is the pressure of the electron. We take E=0, because the applied voltage imposed on the flow of nanofluid is zero. In case of weak ionized molecules, the Law of Ohm in a generalized form in the view of the aforementioned circumstances provides Jy=0 in the flow field. Using these assumptions, we get Jx and Jy as(2)Jx.=σnfB021+m2mu−w,(3)Jz=σnfB021+m2u+mw.Here,m=ωete is the Hall parameter. The rheological model that illustrates the Jeffrey fluids is known as [24–34](4)S=μλ1L+λ2DLDt.S denotes Cauchy stress tensor, μ denotes the dynamic viscosity of the Jeffrey fluid L=∇v+∇vT, and λ1 and λ2 are a ratio of relaxation and retardation time respectively. Observance in light of the above deliberation, the elementary equations of continuity, velocity, energy, and concentration are articulated as(5)∂u∂x+∂v∂y+∂w∂z=0,(6)ρfu∂u∂x+v∂u∂y+2wΩ=−∂P∂x+μ1+λ1∂2v∂x2+∂2v∂y2−σB021+m2u+mw+μγ21+λ1−νκu2∂u∂x∂2u∂x2+2∂v∂x∂2u∂x∂y+u∂3u∂x3+∂3u∂x∂y2+v∂3u∂x2∂y+∂3u∂y3+∂u∂y∂2v∂x2+∂2u∂y∂x+∂2u∂y2+∂2v∂x∂y∂v∂y,(7)ρfu∂v∂x+v∂v∂y=−∂P∂y+μ1+γ1∂2v∂x2+∂2v∂y2+μγ21+γ12∂v∂y∂2v∂y2+2∂u∂y∂2v∂y∂x+∂3v∂x3+∂3v∂x∂y2u+∂3v∂x2∂y+∂3v∂∂y2v+∂2u∂x∂y+∂2v∂x2∂u∂x+∂2u∂y2+∂2v∂x∂y∂v∂x,(8)ρfu∂ω∂y+v∂ω∂y−2Ωu=μ1+λ1∂2ω∂x2+∂2ω∂y2−σnfB021+m2mu−w+μγ21+λ1−νκw∂2ω∂y2+∂2ω∂x∂y∂v∂y+u∂3ω∂y3+v∂3ω∂x2∂yv+∂2ω∂x∂y∂u∂y+∂u∂x∂2ω∂x2+∂3ω∂x3+∂3ω∂x∂y2,(9)u∂T∂x+v∂T∂y+w∂T∂z=α∂2T∂x2+∂2T∂y2+∂2T∂z2−1ρcf∂qr∂y−Q0ρcfT−T0+τDB∂C∂x∂T∂x+∂C∂y∂T∂y+∂C∂z∂T∂z+DTTC∂T∂x2+∂T∂y2+∂T∂z2,(10)u∂C∂x+v∂C∂y+w∂C∂z=DB∂2C∂x2+∂2C∂y2+∂2C∂z2+DTT0∂2T∂x2+∂2T∂y2+∂2T∂z2.Here,u,v and w symbolize the components of the velocity along x,y and z directions. In equations (2)–(10), the symbols υ, μ represent the coefficient of kinematic and dynamic viscosities, respectively, ρ is density, σ denotes electrical conductivity, and Ω is the angular velocity. In equation (9), T represents temperature, α is the thermal diffusivity, cp represents specific heat, thermal conductivity of fluid is represented by k, the coefficient of Brownian diffusion is denoted by DB, and the thermophoretic diffusion coefficient is denoted by DT. The τ=ρcp/ρcf is defined as nanoparticles and effective heat capacity ratio, ρf denotes the base fluid density, ρb represents density of the particles, and C is coefficient concentration of the fluids particles. qr indicates the radioactive heat fluctuation, which is given by Rosseland approximation as(11)qr=−16φ3K∂T4∂y,where φ denoted the Stefan Boltzmann constant and K denoted the mean absorption coefficient. Using Taylor Series, equation, we get(12)T4=T04+4T03T−T02+⋯.By neglecting higher-order term, we get(13)T4=4TT03−3T04.Inserting equations (13) into (11), it is reduced to the form (14)∂qr∂y=−16TC3φ3K∂2T∂y2.For state problem, the boundary conditions are defined as(15)u=ax,v=0,w=0,T=Th,C=Cd,wheny=0,u=v=0,w=0,T=T0,C=C0,wheny=h.The nondimensional variables are presented as(16)u=αxf′η,v=−αhfη,w=αxgη,Θη=T−ThT0−Th,Φη=C−ChC0−Ch,where η=y/h.Substituting the nondimensional variables from the equations (16) in (1)–(10), equation (1) holds identically, and the other governing equations are reduced to the following form:(17)fiv+1+λ1Rff‴−f′f″−2Krg′−M1+m2f″+mg′−γf″+β2f″f‴−ffiv−f′fv=0,(18)g″+1+λ1Rfg′−gf′+2Krf′+M1+m2mf′−g−γg+βg′f″−fg‴=0,(19)ddη1+43RdΘf+Θf−1ΘΘ′+PrRefΘ′+NbΦ′Θ′+NtΘ′2+λθ=0,(20)Φ″+ReScfΦ′+NtNbΘ″=0.Using of equation (16) in (15), it reduced the boundary conditions in the form (21)f=0,f′=1,g=0,θ=1,ϕ=1,η=0,f=0,f′=0,g=0,θ=0,ϕ=0,η=1.The nondimensional physical parameters after generalization are(22)Kr=2Ωh2ν,Re=ah2ν,γ=h2ρκ,M=σB02h2ρυ,Pr=μρfα,β=λ2α,Nb=ρcpDBChαρcf,Nt=ρcpDTT0ρcfTc,Rd=4Tc3φkΚ,Sc=μDρf,where Kr denotes rotation parameter, M is the magnetic parameter, Re is the viscosity parameter, γ is the porosity parameter, β is the Deborah number, Pr is P and tl number, Sc denotes the Schmidt number, Nb is the parameter of Brownian, Rd is the radiation parameter, and Νt is the thermophoretic parameter.The Skin friction is defined as(23)Cf=Sxyy=0ρUw2,where(24)Cf=μ/1+λ1∂2v/∂x2+∂2v/∂y2+λ2u∂2u/∂x∂y+v∂2v/∂x∂y+u∂2v/∂x2+v∂2u/∂y2y=0ρUw2,CfRex=11+λ1f″0+βf″0,where Rex is called local Reynolds number defined as Rex=Uwx/ν. The Nusselt number is defined as Νu=hQw/k^T0−Th. Qw is the heat flux and Qw=−k^∂T/∂yy=0. The Sherwood number is defined as Sh=hJw/DBC0−Ch,Jw is the mass flux, and Jw=−DB∂C/∂yy=0. Here, the dimensionless forms of Νu and Sh are obtained as(25)Νu=−1+43RdΘf+Θf−1ΘΘ′0,Sh=−Φ′0.


## 2.1. Physical Description

The flow of electrically conducting Jeffrey nanofluid between two parallel and horizontal plates is considered. The distance between the upper and lower plates is denoted withh. The coordinate system is selected in such a method that the plate and fluid both are rotated about the y-axis with an angular velocity Ω.. The two forces are assumed to have the same magnitude, but they are opposite in direction, to stretch the lower plate along x-axis, so that the origin 0,0,0 remains constant. The flow of the fluid and temperature transfer is considered in steady state, which is incompressible, laminar, and stable. The surface temperature of the nanofluid between parallel plates is taken after the influence of convective heating process, and its temperature of the hot fluid is T0 under the surface. The free stream is occupied at a uniform ambient nanofluid temperature Th with T0>Th. A magnetic field B0 is acting in y direction, with which the system is rotating. In addition, the effect of Hall current is taken in the nanofluid model. Here, the fluids are electrically conducting, and when the magnetic field becomes stronger, the Hall current is produced, which affects the nanofluids. This effect gives increase to a force in z−direction, which tempts a cross flow in the same direction, and hence, the nanofluid flows in a deflected way into three dimensions, as shown in Figure 1.Figure 1
Physical model of the nanofluid flow problem.


## 2.2. Mathematical Description

Ohm’s law in generalized form containing the Hall current is written as [3–7](1)J+ωeteB0×J×B−σnfE+V×B−σnfPeene=0.Here,J=Jx,Jy,Jz represents the current density, B=0,B0,0 represents the magnetic field, E represents the intensity of electric field, V=u,v,w represents the velocity components, ωe represents the oscillating frequency of the electron, te denotes the time of electron collision, σnf denotes the electrical conductivity, e is the charge of electron, ne is the number density of electron, and Pe is the pressure of the electron. We take E=0, because the applied voltage imposed on the flow of nanofluid is zero. In case of weak ionized molecules, the Law of Ohm in a generalized form in the view of the aforementioned circumstances provides Jy=0 in the flow field. Using these assumptions, we get Jx and Jy as(2)Jx.=σnfB021+m2mu−w,(3)Jz=σnfB021+m2u+mw.Here,m=ωete is the Hall parameter. The rheological model that illustrates the Jeffrey fluids is known as [24–34](4)S=μλ1L+λ2DLDt.S denotes Cauchy stress tensor, μ denotes the dynamic viscosity of the Jeffrey fluid L=∇v+∇vT, and λ1 and λ2 are a ratio of relaxation and retardation time respectively. Observance in light of the above deliberation, the elementary equations of continuity, velocity, energy, and concentration are articulated as(5)∂u∂x+∂v∂y+∂w∂z=0,(6)ρfu∂u∂x+v∂u∂y+2wΩ=−∂P∂x+μ1+λ1∂2v∂x2+∂2v∂y2−σB021+m2u+mw+μγ21+λ1−νκu2∂u∂x∂2u∂x2+2∂v∂x∂2u∂x∂y+u∂3u∂x3+∂3u∂x∂y2+v∂3u∂x2∂y+∂3u∂y3+∂u∂y∂2v∂x2+∂2u∂y∂x+∂2u∂y2+∂2v∂x∂y∂v∂y,(7)ρfu∂v∂x+v∂v∂y=−∂P∂y+μ1+γ1∂2v∂x2+∂2v∂y2+μγ21+γ12∂v∂y∂2v∂y2+2∂u∂y∂2v∂y∂x+∂3v∂x3+∂3v∂x∂y2u+∂3v∂x2∂y+∂3v∂∂y2v+∂2u∂x∂y+∂2v∂x2∂u∂x+∂2u∂y2+∂2v∂x∂y∂v∂x,(8)ρfu∂ω∂y+v∂ω∂y−2Ωu=μ1+λ1∂2ω∂x2+∂2ω∂y2−σnfB021+m2mu−w+μγ21+λ1−νκw∂2ω∂y2+∂2ω∂x∂y∂v∂y+u∂3ω∂y3+v∂3ω∂x2∂yv+∂2ω∂x∂y∂u∂y+∂u∂x∂2ω∂x2+∂3ω∂x3+∂3ω∂x∂y2,(9)u∂T∂x+v∂T∂y+w∂T∂z=α∂2T∂x2+∂2T∂y2+∂2T∂z2−1ρcf∂qr∂y−Q0ρcfT−T0+τDB∂C∂x∂T∂x+∂C∂y∂T∂y+∂C∂z∂T∂z+DTTC∂T∂x2+∂T∂y2+∂T∂z2,(10)u∂C∂x+v∂C∂y+w∂C∂z=DB∂2C∂x2+∂2C∂y2+∂2C∂z2+DTT0∂2T∂x2+∂2T∂y2+∂2T∂z2.Here,u,v and w symbolize the components of the velocity along x,y and z directions. In equations (2)–(10), the symbols υ, μ represent the coefficient of kinematic and dynamic viscosities, respectively, ρ is density, σ denotes electrical conductivity, and Ω is the angular velocity. In equation (9), T represents temperature, α is the thermal diffusivity, cp represents specific heat, thermal conductivity of fluid is represented by k, the coefficient of Brownian diffusion is denoted by DB, and the thermophoretic diffusion coefficient is denoted by DT. The τ=ρcp/ρcf is defined as nanoparticles and effective heat capacity ratio, ρf denotes the base fluid density, ρb represents density of the particles, and C is coefficient concentration of the fluids particles. qr indicates the radioactive heat fluctuation, which is given by Rosseland approximation as(11)qr=−16φ3K∂T4∂y,where φ denoted the Stefan Boltzmann constant and K denoted the mean absorption coefficient. Using Taylor Series, equation, we get(12)T4=T04+4T03T−T02+⋯.By neglecting higher-order term, we get(13)T4=4TT03−3T04.Inserting equations (13) into (11), it is reduced to the form (14)∂qr∂y=−16TC3φ3K∂2T∂y2.For state problem, the boundary conditions are defined as(15)u=ax,v=0,w=0,T=Th,C=Cd,wheny=0,u=v=0,w=0,T=T0,C=C0,wheny=h.The nondimensional variables are presented as(16)u=αxf′η,v=−αhfη,w=αxgη,Θη=T−ThT0−Th,Φη=C−ChC0−Ch,where η=y/h.Substituting the nondimensional variables from the equations (16) in (1)–(10), equation (1) holds identically, and the other governing equations are reduced to the following form:(17)fiv+1+λ1Rff‴−f′f″−2Krg′−M1+m2f″+mg′−γf″+β2f″f‴−ffiv−f′fv=0,(18)g″+1+λ1Rfg′−gf′+2Krf′+M1+m2mf′−g−γg+βg′f″−fg‴=0,(19)ddη1+43RdΘf+Θf−1ΘΘ′+PrRefΘ′+NbΦ′Θ′+NtΘ′2+λθ=0,(20)Φ″+ReScfΦ′+NtNbΘ″=0.Using of equation (16) in (15), it reduced the boundary conditions in the form (21)f=0,f′=1,g=0,θ=1,ϕ=1,η=0,f=0,f′=0,g=0,θ=0,ϕ=0,η=1.The nondimensional physical parameters after generalization are(22)Kr=2Ωh2ν,Re=ah2ν,γ=h2ρκ,M=σB02h2ρυ,Pr=μρfα,β=λ2α,Nb=ρcpDBChαρcf,Nt=ρcpDTT0ρcfTc,Rd=4Tc3φkΚ,Sc=μDρf,where Kr denotes rotation parameter, M is the magnetic parameter, Re is the viscosity parameter, γ is the porosity parameter, β is the Deborah number, Pr is P and tl number, Sc denotes the Schmidt number, Nb is the parameter of Brownian, Rd is the radiation parameter, and Νt is the thermophoretic parameter.The Skin friction is defined as(23)Cf=Sxyy=0ρUw2,where(24)Cf=μ/1+λ1∂2v/∂x2+∂2v/∂y2+λ2u∂2u/∂x∂y+v∂2v/∂x∂y+u∂2v/∂x2+v∂2u/∂y2y=0ρUw2,CfRex=11+λ1f″0+βf″0,where Rex is called local Reynolds number defined as Rex=Uwx/ν. The Nusselt number is defined as Νu=hQw/k^T0−Th. Qw is the heat flux and Qw=−k^∂T/∂yy=0. The Sherwood number is defined as Sh=hJw/DBC0−Ch,Jw is the mass flux, and Jw=−DB∂C/∂yy=0. Here, the dimensionless forms of Νu and Sh are obtained as(25)Νu=−1+43RdΘf+Θf−1ΘΘ′0,Sh=−Φ′0.


## 3. Solution by HAM

Liao was the first one who proposed the Homotopy analysis method (HAM). He deduced HAM from one of the fundamental ideas of the topology called Homotopy. He used two Homotopic functions in the derivation of this technique. The functions are called Homotopic functions when one of them can be continuously distorted into another. Consider thatf1, f2 are two functions that are continuous, and X, Yare two topological spaces, where f1 and f2 map from X to Y, and then f1 is said to be homotopic to f2 if they produce continuous function F^(26)F^:X×0,1⟶Y.such that∀,x∈Χ(27)F^x,0=f1x&F^x,1=f2x.Then, this mappingF^ is called Homotopic. Ham is a substitute method, and it is mainly applied to the nonlinear differential equations without discretization and linearization. This technique has several advantages; some of them are as follows: (i) it is free from the values of the parameters, which may be small or large. (ii) It assures the convergence of the solution. (iii) It is self-determining for an assortment of base function and linear operator.The solutions of equations (17)–(20) with the consistent boundary conditions (21) are obtained by the use of analytic technique. Solution results obtained by HAM contained the assisting parameters h, which adjust and control to converge the solutions and bases functions. The initial guesses are(28)f0η=η−2η2+η3,g0η=0,Θ0η=1−η,Φ0η=1−η.The linear operators are selected in the following way:(29)Lff=fηηηη,Lgg=gηη,LΘΘ=Θηη,LΦΦ=Φηη.The above-mentioned differential operators contents are shown as follows:(30)Lfκ1+κ2η+κ3η2+κ4η3=0,Lgκ5+κ6η=0,LΘκ7+κ8η=0,LΦκ9+κ10η=0.Here,∑m=110κm where m=1,2,3,… are arbitrary constants.


### 3.1. Zeroth-Order Deformation Problem

ExpressingΡ∈01 as an embedding parameter with associate parameters ℏf,ℏg,ℏθ and ℏϕ where ℏ≠0, then, the problem in case of zero order deforms to the following form:(31)1−PLff^η,P−f0P=PℏfNff^η,P,g^η,P,1−PLgg^η,P−g0η=PℏgNgf^η,P,g^η,P,1−PLΘΘ^η,P−Θ0η=PℏΘNΘf^η,P,g^η,P,Θ^η,P,1−PLΦΦ^η,P−Φ0η=PℏΦNΦf^η,P,Θ^η,P,Φ^η,P.The boundary conditions in homotopic form are written as(32)f^0,P=f^′0,P=g^0,P=Θ^0,P=Φ^0,P=0,f^1,P=f^′1,P=g^1,P=Θ^1,P=Φ^1,P=0.The resultant nonlinear operators are(33)Nff^η;P,g^η;P=f^ηηηηη;P1+λ1M1+m2f^ηηη;P+mg^ηη;P−γf^ηηη;P+1+λ1Rf^η;Pf^ηηηη;P−f^ηη;Pf^ηηη;P−2Krg^ηη;P+β2f^ηηη;Pf^ηηηη;P−f^η;Pf^ηηηηη;P−f^ηη;Pf^ηηηηηη;P,Ngg^η;P,f^η;P=g^ηηη;P−1+λ1M1+m2mf^ηη;P−g^η;P−γg^η;P+1+λ1Rf^η;Pg^ηη;P−g^η;Pf^ηη;P+2Krf^ηη;P+βg^ηη;Pf^ηηη;P−f^η;Pg^ηηηη;P,NΘΘ^η;P,f^η;P,Φ^η;P=ddη1+43RdΘf+Θf−1ΘΘ^ηη;P+PrRf^η;PΘ^ηη;P+NbΦ^ηη;PΘ^ηη;P+NtΘ^ηη;P2,NΦf^η;P,g^η;P,Θ^η;P,Φ^η;P=Φ^ηηη;P+RScf^ηη;PΦ^ηη;P+NtNbΘ^ηηη;P.Using Taylor’s series expansion to expandf^η;P,g^η;P,Θ^η;P and Φ^η;P in termd of P, we get(34)f^η,P=f0η+∑i=1∞fiη,g^η,P=g0η+∑i=1∞giη,Θ^η,P=Θ0η+∑i=1∞Θiη,Φ^η,P=Φ0η+∑i=1∞Φiη,where(35)fiη=1i!f^ηη,P|P=0,giη=1i!g^ηη,P|P=0,Θiη=1i!Θ^ηη,P|P=0,Φiη=1i!Φ^ηη,P|P=0.


### 3.2. ith-Order Deformation Problem

Differentiating zeroth-order equationith time, we obtained the ith order deformation equations with respect to P, dividing by i! and then inserting P=0. So, ith order deformation equations are as follows:(36)Lffiη−ξifi−1η=hfℜifη,Lggiη−ξigi−1η=hgℜigη,LΘΘiη−ξiΘi−1η=hΘℜiΘη,LΦΦiη−ξiΦi−1η=hΦℜiΦη.The resultant boundary conditions are(37)f^i=f′^i=g^i=Θ^i=Φ^i=0,atη=0,f^i=f′^=g^i=Θ^i=Φ^i=0,atη=1,ℜifη=fi−1iv+2Κrgi−1+1+λ1R∑j=0i−1fi−1−jfj″′−fi−1−j′fj″−M1+m2fi−1″+mgi−1′−γfi−1″+β∑j=0i−12fi−1−j″fj″′−fi−1−jfjivf−fi−1−j′fjv,ℜigη=gi−1″−1+λ1R∑j=0k−1fi−1−jgj′−gi−1−jfj′+Krfi−1′−M1+m2mfi−1′−gi−1−γgi−1+β∑j=0i−1gi−1−j′fj″−fi−1−jgj″,ℜiΘη=ddη1+43RdΘf+Θf−1ΘΘi−1′+PrRe∑j=0i−1fi−1−jΘj′+Nb∑j=0i−1Φi−1−j′Θj′+Nt∑j=0i−1Θi−1−j″Θj′,ℜiΦη=Φi−1″+RSc∑j=0i−1fi−1−jΦj′+NtNbΘi−1″,where(38)ξi=1,ifP>1,0,ifP≤1.The overall homotopic solutionsfi,gi,Θi,Φi of equation (27) in terms of special solutions f^i,g^i,Θ^i,Φ^i are specified as(39)fiη=f^iη+κ1+κ2η+κ3η2+κ4η3,giη=g^iη+κ5+κ6η,Θiη=Θ^iη+κ7+κ8η,Φiη=Θ^iη+κ9+κ10η.


## 3.1. Zeroth-Order Deformation Problem

ExpressingΡ∈01 as an embedding parameter with associate parameters ℏf,ℏg,ℏθ and ℏϕ where ℏ≠0, then, the problem in case of zero order deforms to the following form:(31)1−PLff^η,P−f0P=PℏfNff^η,P,g^η,P,1−PLgg^η,P−g0η=PℏgNgf^η,P,g^η,P,1−PLΘΘ^η,P−Θ0η=PℏΘNΘf^η,P,g^η,P,Θ^η,P,1−PLΦΦ^η,P−Φ0η=PℏΦNΦf^η,P,Θ^η,P,Φ^η,P.The boundary conditions in homotopic form are written as(32)f^0,P=f^′0,P=g^0,P=Θ^0,P=Φ^0,P=0,f^1,P=f^′1,P=g^1,P=Θ^1,P=Φ^1,P=0.The resultant nonlinear operators are(33)Nff^η;P,g^η;P=f^ηηηηη;P1+λ1M1+m2f^ηηη;P+mg^ηη;P−γf^ηηη;P+1+λ1Rf^η;Pf^ηηηη;P−f^ηη;Pf^ηηη;P−2Krg^ηη;P+β2f^ηηη;Pf^ηηηη;P−f^η;Pf^ηηηηη;P−f^ηη;Pf^ηηηηηη;P,Ngg^η;P,f^η;P=g^ηηη;P−1+λ1M1+m2mf^ηη;P−g^η;P−γg^η;P+1+λ1Rf^η;Pg^ηη;P−g^η;Pf^ηη;P+2Krf^ηη;P+βg^ηη;Pf^ηηη;P−f^η;Pg^ηηηη;P,NΘΘ^η;P,f^η;P,Φ^η;P=ddη1+43RdΘf+Θf−1ΘΘ^ηη;P+PrRf^η;PΘ^ηη;P+NbΦ^ηη;PΘ^ηη;P+NtΘ^ηη;P2,NΦf^η;P,g^η;P,Θ^η;P,Φ^η;P=Φ^ηηη;P+RScf^ηη;PΦ^ηη;P+NtNbΘ^ηηη;P.Using Taylor’s series expansion to expandf^η;P,g^η;P,Θ^η;P and Φ^η;P in termd of P, we get(34)f^η,P=f0η+∑i=1∞fiη,g^η,P=g0η+∑i=1∞giη,Θ^η,P=Θ0η+∑i=1∞Θiη,Φ^η,P=Φ0η+∑i=1∞Φiη,where(35)fiη=1i!f^ηη,P|P=0,giη=1i!g^ηη,P|P=0,Θiη=1i!Θ^ηη,P|P=0,Φiη=1i!Φ^ηη,P|P=0.


## 3.2. ith-Order Deformation Problem

Differentiating zeroth-order equationith time, we obtained the ith order deformation equations with respect to P, dividing by i! and then inserting P=0. So, ith order deformation equations are as follows:(36)Lffiη−ξifi−1η=hfℜifη,Lggiη−ξigi−1η=hgℜigη,LΘΘiη−ξiΘi−1η=hΘℜiΘη,LΦΦiη−ξiΦi−1η=hΦℜiΦη.The resultant boundary conditions are(37)f^i=f′^i=g^i=Θ^i=Φ^i=0,atη=0,f^i=f′^=g^i=Θ^i=Φ^i=0,atη=1,ℜifη=fi−1iv+2Κrgi−1+1+λ1R∑j=0i−1fi−1−jfj″′−fi−1−j′fj″−M1+m2fi−1″+mgi−1′−γfi−1″+β∑j=0i−12fi−1−j″fj″′−fi−1−jfjivf−fi−1−j′fjv,ℜigη=gi−1″−1+λ1R∑j=0k−1fi−1−jgj′−gi−1−jfj′+Krfi−1′−M1+m2mfi−1′−gi−1−γgi−1+β∑j=0i−1gi−1−j′fj″−fi−1−jgj″,ℜiΘη=ddη1+43RdΘf+Θf−1ΘΘi−1′+PrRe∑j=0i−1fi−1−jΘj′+Nb∑j=0i−1Φi−1−j′Θj′+Nt∑j=0i−1Θi−1−j″Θj′,ℜiΦη=Φi−1″+RSc∑j=0i−1fi−1−jΦj′+NtNbΘi−1″,where(38)ξi=1,ifP>1,0,ifP≤1.The overall homotopic solutionsfi,gi,Θi,Φi of equation (27) in terms of special solutions f^i,g^i,Θ^i,Φ^i are specified as(39)fiη=f^iη+κ1+κ2η+κ3η2+κ4η3,giη=g^iη+κ5+κ6η,Θiη=Θ^iη+κ7+κ8η,Φiη=Θ^iη+κ9+κ10η.


## 4. Convergence of HAM Solution

When we compute the series solutions of the velocity, temperature, and concentration functions to use HAM, the assisting parametersℏf,ℏg,ℏΘ and ℏΦ appear. These assisting parameters are responsible for adjusting the convergence of these solutions. In the possible region of ℏ, ℏ-curves of f‴0,g′0,Θ′0 and Φ′0 for 12th order approximation are plotted in Figures 2 and 3 for different values of embedding parameter. The ℏ-curves consecutively display the valid region. Table 1 displays the numerical values of HAM solutions at different approximations using dissimilar values of parameters. It is clear from the table that homotopy analysis technique is a speedily convergent technique. The region of convergence for the velocity distribution fη and gη is given as −3≤ℏ≤1and−3.8≤ℏ≤1.7, respectively, while, for the temperature and concentration profile, the convergence regions are given as −2.6≤ℏ≤0.5and−2.1≤ℏ≤0.4, respectively.Figure 2
Combinedh curves of functions velocity fη and gη profiles at 12th order approximation.Figure 3
Combinedh curves of temperature and concentration Θη and Φη profiles at 12th order approximation.Table 1
The convergence of HAM up to 20th order approximations whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m=M=0.01,γ=0.1.
Order of approximationf″0g′0Θ′0Φ′013.98886−0.02079210.9537501.0002543.97650−0.03870640.9131691.0737583.97584−0.03964530.9104901.08915123.97583−0.03966770.9103831.09009163.97583−0.03966820.9103791.09014203.97583−0.03966820.9103791.09014


## 5. Results and Discussion

### 5.1. Graphical Discussion

The present investigation has been carried out to study the flow of non-Newtonian nanofluid (considering the Jeffrey fluid) in a rotating system under the influence of MHD between parallel plates. In addition, the effect of Hall current is given, where the medium between the plates is kept porous. The main aim of this subsection is to study the physical effects of different embedding parameters on the velocity distributionsfη,gη, temperature distribution Θη, and concentration distribution Φη, which are illustrated in Figures (4)–(18). The influence of viscosity parameter Re on the velocity distributions fη and gη is shown in Figures 4(a) and 4(b). It is clear that increasing the viscosity parameter Re decreases the velocity distributions fη and gη. The larger values of Re reduce the viscous forces, which generate the stronger inertial forces, and as a consequence, the velocity field retarded. The strong inertial forces resist the flow, and as a result, the velocity distribution recues. Figure 5(a) displays the effect of viscosity parameter Re on the temperature distribution Θη, and the same effect of temperature distribution has been observed, because the larger values of viscosity parameter Re strengthen the inertial forces and tend to reduce the temperature field. Figure 5(b) shows the effect of Re on the concentration distribution Φη. The rise in Re increases the concentration distribution Φη. It means that the thermal conductivity contributed to improve the heat transfer. Also, these observations have been found to be the same as what has been discussed in Sheikholeslami et al. [17], Mahmoodi et al. [18], and Jena et al. [20]. The influence of Κr on the velocity profiles has been shown in Figures 6(a), and 6(b). It is ostensible that when a rotation parameter Κr increases, it raises the fluid motion, and this effect is clearer at the stretching plate, because the rise in rotation parameter Κr increases the Coriolis force, which results in a rise in rotational velocity. An increase in rotation parameter of the fluid increases the kinetic energy of the fluid particles, which in turn increases the flow motion. The effect of Hall parameter m on the velocity profiles fη and gη is shown in Figures 7(a) and 7(b). The Hall effect is the production of potential difference. Here, the Hall parameter m plays an important role in the nanofluid flow. The large value of Hall parameter m reduces the effective conductivity, which drops the magnetic damping force, and so, the velocity profile along the y direction and greater value of Hall parameter m decrease the velocity profile along z direction. Figures 8(a) and 8(b) describe the effect of λ1on velocity profiles fη and gη. It displays that the rise in ratio of relaxation λ1 decreases the relaxation time. The small values dominate the viscous forces. Actually, the rise in λ1 shows viscoelastic effect, and as a result, the fluid motion is reduced. Figures 9(a) and 9(b) demonstrate the influence of the Deborah number on velocity distributions fη and gη. The Deborah number β was first proposed by Markus Reiner; it is a dimensionless number, frequently used in fluid mechanics, to illustrate the flexibility of materials under definite flow situations. With the increase in Deborah number β, the velocity profile increases and has opposite effect to λ1. The large relaxation time decreases the velocity field. The characteristics of Magnetic parameter M on velocity distributions have been shown in Figures 10(a) and 10(b) along y and z-directions, respectively. It is clear that the velocity distribution fη is inversely varied with magnetic parameter M along the y direction and directly varied with gη along the z-direction. Increasing magnetic parameter M decreases the velocity field along y direction, since the magnetic field is applied perpendicularly to y direction, and hence, the conducting fluid particles feel the opposite force of magnetic field and hence reduce the velocity profile along y direction as shown in Figure 10(a) when it is close to the plates. The effect of magnetic field along z direction is shown in Figure 10(b). The magnetic field is applied parallelly to z direction, and hence, it assists the flow. This is because of the fact that the rise in the M progresses the friction force of the movement, named the Lorentz force. Lorentz force has the affinity to reduce the velocity of the flow in the boundary sheet, where another force known as Coriolis force shows the reverse influence on the velocity along the z-direction. The characteristics of porosity parameter γ on velocity fields are shown in Figures 11(a) and 11(b) in y and z-directions, respectively, which have an imperative character in the flow motion. Increasing γ increases the porous space, which creates resistance in the flow path and reduces the flow motion. In fact, growing values of γ show the large number of porous spaces, which create resistance in the flow path and reduce overall fluid motion. Basically with the increase of the number of holes in the porous plate, the velocity decreases during the flow of nanofluid particles over these holes. The impact of thermal radiation parameter Rd on Θη and Φη is presented in Figures 12(a) and 12(b). The thermal radiation has an imperative role in the inclusive surface heat transmission when the coefficient of convection heat transmission is small. When we increase thermal radiation parameterRd, then it is perceived that it augments the temperature in the boundary layer area in the fluid layer. This increase leads to a drop in the rate of cooling for nanofluid flow. The same effect is observed for the concentration distributions (Figure 12(b)). The influence of Pr on the temperature and concentration distributions Θη and Φη is shown in Figures 13(a) and 13(b). Both temperature and concentration distributions vary inversely with Pr. It is clear that temperature distribution decreases with large numbers of Pr and increases for small values of Pr. Physically, the fluids having a small number of Pr have larger thermal diffusivity, and this effect is opposite for higher Pr and tl number Pr. Due to this fact, large Pr causes the thermal boundary layer to decrease. The effect is even more distinct for the small number of Pr, since the thermal boundary layer thickness is relatively large. The same effect of Pr on concentration distribution is shown in Figure 13(b). Figures 14(a) and 14(b) show the features of Brownian motion parameter Nb on temperature distribution and concentration profile. Brownian motion is the inconsistent random motion of nanofluids particles. It has been initiated that the Brownian motion of nanoparticles at the molecular level is a central mechanism leading the thermal conductivity of nanofluids. The augmentation in the active thermal conductivity of nanofluids is due mostly to the contained convection produced by the Brownian movement of the nanoparticles. It is observed from Figure 14(a) that increasing Nb raises the temperature field. In fact, increasing Nb raises the kinetic energy of the nanoparticles inside the fluid, with which rate of heat transfer and boundary layer thickness rises, which leads to an increase in the temperature field. While the opposite impact has been found for concentration distribution that is increasing, the Nb reduces the concentration profile (Figure 14(b)). This is because the rise of Brownian motion diminishes the boundary layer thicknesses, which leads to reducing concentrations. The thermophoresis parameter Nt of temperature distribution and concentration field is shown in Figures 15(a) and 15(b). Thermophoresis is a process perceived in combinations of mobile particles of nanofluids, where the unlike particle kinds display different retorts to the force of a temperature gradient. It is observed from Figures 15(a) and 15(b) that Nt increases the temperature field when it increased, and the same effects of Nt are observed for concentration field. This is due to the thermophoresis parameter, and Nt depends on the temperature gradient in the surrounding nanofluids molecules. Increasing Nt increases the kinetic energy of the nanofluids molecules, which as a result increases the temperature and concentration profile. Figures 16(a) and 16(b) describe the effect of Schmidt number Sc, where Sc is a dimensionless number, and it is the ratio of momentum diffusivity and mass diffusivity. So, for large values of Sc, the temperature rises and falls for small values, while the opposite tendency is perceived in concentration field. Increasing Schmidt number decreases the concentration profile, which results in reducing the boundary layer thickness. Effect of viscosity parameter Ron Skin friction Cf against β and λ1 is shown in Figures 17(a) and 17(b). There is inverse variation between skin friction and viscosity parameter; increasing R decreases the Skin friction Cf. Figure 17(c) demonstrates the effect of Κr on Skin friction Cf against β. It is observed that Cf increases with large values of rotation parameter Κr. Effect of R on Nusselt number Nu against Nt and Nb is shown in Figures 18(a) and 18(b) and it is found that, for large value of Nt and Nb, the mass flux increased Nu. This is because increasing Nt and Nb increases the kinetic energy of the nanofluids molecules, which as a result increases heat flux Nu. Figure 17 describes the effect of R on mass flux Sh against Nt, which shows that mass flux is increasing function when Nt increases.Figure 4
The influence ofRe on fη and gη when γ=0.4,m=0.7,λ1=0.5,M=1,β=0.4,Κr=0.6.
(a)(b)Figure 5
The influence ofReon Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=γ=0.6,Nb=Sc=Nt=Rd=0.5,Pr=5.
(a)(b)Figure 6
The influence ofΚr on fη and gη when m=λ1=0.8,M=β=1,Re=1,γ=0.4.
(a)(b)Figure 7
The influence ofm on fη and gη when Re=1,λ1=0.8,M=β=0.7,Κr=0.6,γ=0.4.
(a)(b)Figure 8
The influence ofλ1on fη and gη when Re=2,β=0.8,M=1.6,γ=0.6,Κr=1.8,m=0.7.
(a)(b)Figure 9
The influence ofβ on fη and gη when Re=1,λ1=0.7,M=1,γ=0.6,Κr=m=0.6.
(a)(b)Figure 10
The influence ofM on fη and gη when Re=1,λ1=0.8,M=β=0.7,Κr=γ=0.6.
(a)(b)Figure 11
The influence ofγ on fη and gη when Re=1.5,λ1=0.8,M=1,β=0.7,Κr=m=0.6.
(a)(b)Figure 12
The influenceRd on Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=γ=0.6,Nb=Sc=Nt=Re=0.5,Pr=5.
(a)(b)Figure 13
The influence ofPr on Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=0.6,Nb=Sc=Nt=Rd=0.5,Pr=5.
(a)(b)Figure 14
The influenceNb on Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=γ=0.6,Re=1,Sc=Nt=0.5,Rd=0.6,Pr=10.
(a)(b)Figure 15
The influenceNb on Θη and Φη when M=γ=β=0.4,λ1=m=0.8,Sc=0.5,Kr=0.6,Re=1,Nb=0.2,Rd=0.5,Pr=1.
(a)(b)Figure 16
The influenceSc on Θη and Φη when M=γ=β=0.4,λ1=m=0.8,Kr=0.6,Re=1,0.5,Nb=0.4,Rd=0.5,Pr=1,Nt=0.4.
(a)(b)Figure 17
The influenceRd,Re,Nt and Κr on Nusselt numbers Nu=−Θ′0 when M=γ=β=0.4,λ1=m=0.8,Sc=1,Nb=0.3,Pr=1, (a) Kr=1,0.5,Nt=0.01. (b) Re=1,Nt=0.01. (c). Kr=0.2,Re=1.0.
(a)(b)(c)Figure 18
The influence ofΚr,M and Κr on Skin friction Cf when β=0.4,λ1=m=0.8. (a) γ=0.1,M=0.5. (b). γ=0.1,Kr=0.6.
(a)(b)


### 5.2. Table Discussion

The effect of viscosity parameterRe and magnetic parameter M on Nusselt number and Sherwood number Sh is numerically shown in Tables 2 and 3. It is observed that both mass flux and heat flux are decreasing functions when the viscosity parameter Re and magnetic parameter M are increased. To validate our results, the obtained results are compared with the result available in the literature as given in Table 4. The influence of radiation parameter Rd and Schmidt number Sc on Nusselt number Νu and Sherwood numbers Sh is numerically shown in Tables 5 and 6. It is concluded that Nusselt number is reduced when the Schmidt number is increased, where, for radiation parameter, it shows opposite result. The obtained results are verified in comparison with the results of Sheikholeslami [17] given in Table 7 in order to validate our results. The effect of Nb and Nt on mass flux Νu and Sherwood numbers (heat flux) Sh is numerically shown in Tables 8 and 9. These tables show that increasing Nb and Nt increases the Nusselt number, where the opposite trend is found for Sherwood numbers Sh that the heat flux is decreased when Nb and Nt are increased. The obtained results are verified for validation in comparison with Sheikholeslami [17] results given in Table 10 which are in complete agreement with our results. The numerical values of Re,λ1,β and kron skin friction Cf are given in Table 11. From this table, it is clear that increasing values of Re,λ1 and β decreases Cf while increasing kr increases the skin friction. The numerical results (Table 11) and graphical result (Figures 14(a), 14(b), and 15) agree with each other. These results are compared with results of [19] given in Table 12. The obtained results are verified by comparing with [19] and good agreement is observed. The numerical values of heat flux Νu and mass flux Share given in Table 13. We have calculated it at the boundaries of the plates for dissimilar values of Nt,Nb and Sc. It is clear from Table 13 that the heat flux Νu is reduced when parameters Nt and Nb rose, because the mass flux is a reducing function of the lower stretching plate, where it is an growing function of Sc at the upper plate. The mass flux Sh is increasing function at the lower stretching plate, while it is a decreasing function of the upper plate at different increasing values of Nt,Nb and Sc. To validate and verify our results, the obtained results are compared with the RK Method of order 4 as given in Tables 14 and 15 , while the absolute errors of these methods are presented in Table 16 at each point.Table 2
Influence ofRd and M on nusselt number Nu (present results), when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdM0.04.08.016.00.01.429941.397651.368021.316763.01.658341.645901.556131.364606.03.192433.124243.094153.0667312.012.188112.000511.910711.8235Table 3
Influence ofRd and M on sherwood number Sh, when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdM0.04.08.016.00.00.9991280.9983510.9977070.9968203.01.021381.020601.019961.019076.01.043631.042851.042211.0413212.01.088131.087351.086711.08582Table 4
Influence ofRd and M on nusselt number Nu (Sheikholeslami [17]), when Nt=Nb=0.01,R=1,Sc=0.5,Pr=10.
RdM0.04.08.016.00.01.5570441.5236671.49541.4499363.06.8302426.8021486.7783036.7398176.012.158112.1305312.1071312.0693112.017.4893817.4620217.4387717.40121Table 5
Influence ofRd and Sc on nusselt number Nu (present results) when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdSc0.010.22.06.00.01.444021.444011.387301.387223.01.475281.388121.386841.382826.03.292213.271563.271503.2711212.012.980412.915012.900212.8157Table 6
Influence ofRd and Scon sherwood number Sh, when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdSc0.010.22.06.00.00.7510840.7588170.8324411.016873.00.8980841.003821.077441.261876.01.241081.042851.322441.5068712.01.731081.733671.812441.99687Table 7
Influence ofRd and Sc on nusselt numbers Nu (Sheikholeslami [17]), Nt=Nb=0.01,R=1,Sc=0.5,Pr=10.
RdSc0.010.22.06.00.01.5570441.5570211.5567981.5562853.06.8302426.8302256.8300656.8297026.012.158112.2580812.1579212.1575712.017.4893817.4893617.4892117.48886Table 8
Influence ofRd and Nb on nusselt number Nu (present results), when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdNb0.010.10.30.60.01.468131.421621.320671.175513.01.389851.357481.283121.165326.03.271573.073342.625561.9351412.012.91511.84519.450445.81478Table 9
Influence ofRd and schmidt number Sc on sherwood number Sh when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdNb0.010.10.30.60.00.7666150.9989211.016111.020383.01.011611.021171.021861.0226.01.256611.043421.027611.0236312.01.746611.087921.039111.02688Table 10
Influence ofRd and Nb on nusselt number Nu (Sheikholeslami [17]), Nt=Nb=0.01,R=1,Sc=0.5,Pr=10.
RdNb0.010.10.30.60.01.5570441.499111.374631.1993513.06.8302426.7830536.6789476.5247526.012.158112.11912.0096612.8573612.017.4893817.4435617.3420117.19043Table 11
The skin friction coefficient for dissimilar values ofRe,Κr,β, and γ whenNt=Nb=0.1,Sc=0.5,Pr=10,λ1=m=M=0.3.
ReMγΚr−CfRex1/20.10.51.01.53.330270.52.948821.02.642081.50.14.339990.54.321571.04.268971.50.15.642270.55.445761.04.899111.50.14.127430.54.357721.05.130481.55.91612Table 12
Variation in skin friction coefficient for dissimilar values ofSq,ω,β, and λ1 when Pr=1.0,γ=1.0 ([19] results).
sqωλ1β−CfRex1/2−0.10.51.00.52.633120.02.651330.12.639951.00.01.312170.11.259170.51.216941.00.02.385080.11.259170.51.033991.00.00.619110.010.798840.10.877320.22.63312Table 13
The effect of physical parameters on nusselt numbersNu and sherwood number Sh when Nt=kr=0.1,R=0.5,Pr=10,λ1=m=M=0.3.
RdNbScη=+1η=−1NuShNuSh0.10.50.5−0.766570−1.700540.766571.700540.5−0.622278−1.233280.6222781.233281.0−0.459501−1.172530.4595011.172531.50.1−0.414644−1.026770.4146441.026770.5−0.316268−1.150540.3162681.150541.0−0.215401−1.356360.2154011.356361.50.1−0.135599−1.607280.1355991.607280.5−0.135444−1.609220.1354441.609221.0−0.1353148−1.619220.1352511.611651.5−0.135057−1.614080.1350571.61408Table 14
Results obtained by HAM at 20th order approximations whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m==0.1,M=1,γ=0.5.
ηfηgηΘηΦη00.000000.000001.000001.000000.11.325581.234560.9875460.9875420.21.234561.235460.8457630.8445870.32.345602.546860.7452630.7421450.42.458762.548960.6754680.6745780.52.625482.756250.5648630.5645780.62.324562.452360.4206350.4222140.72.012452.245630.3214870.3012450.81.562351.235550.2145870.2012450.91.2548631.214520.114230.1011231.00.0000000.0000000.000000.00000Table 15
RK method of order 4, whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m==0.1,M=1,γ=0.5.
ηfηgηΘηΦη00.000000.000001.000001.000000.11.325581.234560.9875460.9875420.21.234561.235460.8457630.8445870.32.345602.546860.7452630.7421450.42.458762.548960.6754680.6745780.52.625482.756250.5648630.5645780.62.324562.452360.4206350.4222140.72.012452.245630.3214870.3012450.81.562351.235550.2145870.2012450.91.2548631.214520.114230.1011231.00.0000000.0000000.000000.00000Table 16
Absolute errors of RK method of order 4 and HAM results whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m==0.1,M=1,γ=0.5.
ηfηgηΘηΦη00.000000.000000.000000.000000.11.22213×10−91.32145×10−82.14785×10−71.65478×10−70.22.13456×10−103.214560×10−92.85632×10−81.25463×10−80.32.14589×10−111.32456×10−102.45782×10−91.87546×10−90.41.54879×10−112.001245×10−101.24586×10−101.85457×10−100.52.14578×10−121.02145×10−113.12458×10−111.24586×10−110.62.145892×10−131.02145×10−125.45786×10−131.45876×10−130.72.45866×10−141.021456×10−123.54689×10−151.45826×10−150.82.001458×10−151.02145×10−132.47896×10−161.85475×10−160.92.14586×10−182.14586×10−171.24589×10−171.98526×10−171.00.0000000.0000000.0000000.000000


## 5.1. Graphical Discussion

The present investigation has been carried out to study the flow of non-Newtonian nanofluid (considering the Jeffrey fluid) in a rotating system under the influence of MHD between parallel plates. In addition, the effect of Hall current is given, where the medium between the plates is kept porous. The main aim of this subsection is to study the physical effects of different embedding parameters on the velocity distributionsfη,gη, temperature distribution Θη, and concentration distribution Φη, which are illustrated in Figures (4)–(18). The influence of viscosity parameter Re on the velocity distributions fη and gη is shown in Figures 4(a) and 4(b). It is clear that increasing the viscosity parameter Re decreases the velocity distributions fη and gη. The larger values of Re reduce the viscous forces, which generate the stronger inertial forces, and as a consequence, the velocity field retarded. The strong inertial forces resist the flow, and as a result, the velocity distribution recues. Figure 5(a) displays the effect of viscosity parameter Re on the temperature distribution Θη, and the same effect of temperature distribution has been observed, because the larger values of viscosity parameter Re strengthen the inertial forces and tend to reduce the temperature field. Figure 5(b) shows the effect of Re on the concentration distribution Φη. The rise in Re increases the concentration distribution Φη. It means that the thermal conductivity contributed to improve the heat transfer. Also, these observations have been found to be the same as what has been discussed in Sheikholeslami et al. [17], Mahmoodi et al. [18], and Jena et al. [20]. The influence of Κr on the velocity profiles has been shown in Figures 6(a), and 6(b). It is ostensible that when a rotation parameter Κr increases, it raises the fluid motion, and this effect is clearer at the stretching plate, because the rise in rotation parameter Κr increases the Coriolis force, which results in a rise in rotational velocity. An increase in rotation parameter of the fluid increases the kinetic energy of the fluid particles, which in turn increases the flow motion. The effect of Hall parameter m on the velocity profiles fη and gη is shown in Figures 7(a) and 7(b). The Hall effect is the production of potential difference. Here, the Hall parameter m plays an important role in the nanofluid flow. The large value of Hall parameter m reduces the effective conductivity, which drops the magnetic damping force, and so, the velocity profile along the y direction and greater value of Hall parameter m decrease the velocity profile along z direction. Figures 8(a) and 8(b) describe the effect of λ1on velocity profiles fη and gη. It displays that the rise in ratio of relaxation λ1 decreases the relaxation time. The small values dominate the viscous forces. Actually, the rise in λ1 shows viscoelastic effect, and as a result, the fluid motion is reduced. Figures 9(a) and 9(b) demonstrate the influence of the Deborah number on velocity distributions fη and gη. The Deborah number β was first proposed by Markus Reiner; it is a dimensionless number, frequently used in fluid mechanics, to illustrate the flexibility of materials under definite flow situations. With the increase in Deborah number β, the velocity profile increases and has opposite effect to λ1. The large relaxation time decreases the velocity field. The characteristics of Magnetic parameter M on velocity distributions have been shown in Figures 10(a) and 10(b) along y and z-directions, respectively. It is clear that the velocity distribution fη is inversely varied with magnetic parameter M along the y direction and directly varied with gη along the z-direction. Increasing magnetic parameter M decreases the velocity field along y direction, since the magnetic field is applied perpendicularly to y direction, and hence, the conducting fluid particles feel the opposite force of magnetic field and hence reduce the velocity profile along y direction as shown in Figure 10(a) when it is close to the plates. The effect of magnetic field along z direction is shown in Figure 10(b). The magnetic field is applied parallelly to z direction, and hence, it assists the flow. This is because of the fact that the rise in the M progresses the friction force of the movement, named the Lorentz force. Lorentz force has the affinity to reduce the velocity of the flow in the boundary sheet, where another force known as Coriolis force shows the reverse influence on the velocity along the z-direction. The characteristics of porosity parameter γ on velocity fields are shown in Figures 11(a) and 11(b) in y and z-directions, respectively, which have an imperative character in the flow motion. Increasing γ increases the porous space, which creates resistance in the flow path and reduces the flow motion. In fact, growing values of γ show the large number of porous spaces, which create resistance in the flow path and reduce overall fluid motion. Basically with the increase of the number of holes in the porous plate, the velocity decreases during the flow of nanofluid particles over these holes. The impact of thermal radiation parameter Rd on Θη and Φη is presented in Figures 12(a) and 12(b). The thermal radiation has an imperative role in the inclusive surface heat transmission when the coefficient of convection heat transmission is small. When we increase thermal radiation parameterRd, then it is perceived that it augments the temperature in the boundary layer area in the fluid layer. This increase leads to a drop in the rate of cooling for nanofluid flow. The same effect is observed for the concentration distributions (Figure 12(b)). The influence of Pr on the temperature and concentration distributions Θη and Φη is shown in Figures 13(a) and 13(b). Both temperature and concentration distributions vary inversely with Pr. It is clear that temperature distribution decreases with large numbers of Pr and increases for small values of Pr. Physically, the fluids having a small number of Pr have larger thermal diffusivity, and this effect is opposite for higher Pr and tl number Pr. Due to this fact, large Pr causes the thermal boundary layer to decrease. The effect is even more distinct for the small number of Pr, since the thermal boundary layer thickness is relatively large. The same effect of Pr on concentration distribution is shown in Figure 13(b). Figures 14(a) and 14(b) show the features of Brownian motion parameter Nb on temperature distribution and concentration profile. Brownian motion is the inconsistent random motion of nanofluids particles. It has been initiated that the Brownian motion of nanoparticles at the molecular level is a central mechanism leading the thermal conductivity of nanofluids. The augmentation in the active thermal conductivity of nanofluids is due mostly to the contained convection produced by the Brownian movement of the nanoparticles. It is observed from Figure 14(a) that increasing Nb raises the temperature field. In fact, increasing Nb raises the kinetic energy of the nanoparticles inside the fluid, with which rate of heat transfer and boundary layer thickness rises, which leads to an increase in the temperature field. While the opposite impact has been found for concentration distribution that is increasing, the Nb reduces the concentration profile (Figure 14(b)). This is because the rise of Brownian motion diminishes the boundary layer thicknesses, which leads to reducing concentrations. The thermophoresis parameter Nt of temperature distribution and concentration field is shown in Figures 15(a) and 15(b). Thermophoresis is a process perceived in combinations of mobile particles of nanofluids, where the unlike particle kinds display different retorts to the force of a temperature gradient. It is observed from Figures 15(a) and 15(b) that Nt increases the temperature field when it increased, and the same effects of Nt are observed for concentration field. This is due to the thermophoresis parameter, and Nt depends on the temperature gradient in the surrounding nanofluids molecules. Increasing Nt increases the kinetic energy of the nanofluids molecules, which as a result increases the temperature and concentration profile. Figures 16(a) and 16(b) describe the effect of Schmidt number Sc, where Sc is a dimensionless number, and it is the ratio of momentum diffusivity and mass diffusivity. So, for large values of Sc, the temperature rises and falls for small values, while the opposite tendency is perceived in concentration field. Increasing Schmidt number decreases the concentration profile, which results in reducing the boundary layer thickness. Effect of viscosity parameter Ron Skin friction Cf against β and λ1 is shown in Figures 17(a) and 17(b). There is inverse variation between skin friction and viscosity parameter; increasing R decreases the Skin friction Cf. Figure 17(c) demonstrates the effect of Κr on Skin friction Cf against β. It is observed that Cf increases with large values of rotation parameter Κr. Effect of R on Nusselt number Nu against Nt and Nb is shown in Figures 18(a) and 18(b) and it is found that, for large value of Nt and Nb, the mass flux increased Nu. This is because increasing Nt and Nb increases the kinetic energy of the nanofluids molecules, which as a result increases heat flux Nu. Figure 17 describes the effect of R on mass flux Sh against Nt, which shows that mass flux is increasing function when Nt increases.Figure 4
The influence ofRe on fη and gη when γ=0.4,m=0.7,λ1=0.5,M=1,β=0.4,Κr=0.6.
(a)(b)Figure 5
The influence ofReon Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=γ=0.6,Nb=Sc=Nt=Rd=0.5,Pr=5.
(a)(b)Figure 6
The influence ofΚr on fη and gη when m=λ1=0.8,M=β=1,Re=1,γ=0.4.
(a)(b)Figure 7
The influence ofm on fη and gη when Re=1,λ1=0.8,M=β=0.7,Κr=0.6,γ=0.4.
(a)(b)Figure 8
The influence ofλ1on fη and gη when Re=2,β=0.8,M=1.6,γ=0.6,Κr=1.8,m=0.7.
(a)(b)Figure 9
The influence ofβ on fη and gη when Re=1,λ1=0.7,M=1,γ=0.6,Κr=m=0.6.
(a)(b)Figure 10
The influence ofM on fη and gη when Re=1,λ1=0.8,M=β=0.7,Κr=γ=0.6.
(a)(b)Figure 11
The influence ofγ on fη and gη when Re=1.5,λ1=0.8,M=1,β=0.7,Κr=m=0.6.
(a)(b)Figure 12
The influenceRd on Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=γ=0.6,Nb=Sc=Nt=Re=0.5,Pr=5.
(a)(b)Figure 13
The influence ofPr on Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=0.6,Nb=Sc=Nt=Rd=0.5,Pr=5.
(a)(b)Figure 14
The influenceNb on Θη and Φη when M=1,λ1=m=0.8,β=0.4,Kr=γ=0.6,Re=1,Sc=Nt=0.5,Rd=0.6,Pr=10.
(a)(b)Figure 15
The influenceNb on Θη and Φη when M=γ=β=0.4,λ1=m=0.8,Sc=0.5,Kr=0.6,Re=1,Nb=0.2,Rd=0.5,Pr=1.
(a)(b)Figure 16
The influenceSc on Θη and Φη when M=γ=β=0.4,λ1=m=0.8,Kr=0.6,Re=1,0.5,Nb=0.4,Rd=0.5,Pr=1,Nt=0.4.
(a)(b)Figure 17
The influenceRd,Re,Nt and Κr on Nusselt numbers Nu=−Θ′0 when M=γ=β=0.4,λ1=m=0.8,Sc=1,Nb=0.3,Pr=1, (a) Kr=1,0.5,Nt=0.01. (b) Re=1,Nt=0.01. (c). Kr=0.2,Re=1.0.
(a)(b)(c)Figure 18
The influence ofΚr,M and Κr on Skin friction Cf when β=0.4,λ1=m=0.8. (a) γ=0.1,M=0.5. (b). γ=0.1,Kr=0.6.
(a)(b)


## 5.2. Table Discussion

The effect of viscosity parameterRe and magnetic parameter M on Nusselt number and Sherwood number Sh is numerically shown in Tables 2 and 3. It is observed that both mass flux and heat flux are decreasing functions when the viscosity parameter Re and magnetic parameter M are increased. To validate our results, the obtained results are compared with the result available in the literature as given in Table 4. The influence of radiation parameter Rd and Schmidt number Sc on Nusselt number Νu and Sherwood numbers Sh is numerically shown in Tables 5 and 6. It is concluded that Nusselt number is reduced when the Schmidt number is increased, where, for radiation parameter, it shows opposite result. The obtained results are verified in comparison with the results of Sheikholeslami [17] given in Table 7 in order to validate our results. The effect of Nb and Nt on mass flux Νu and Sherwood numbers (heat flux) Sh is numerically shown in Tables 8 and 9. These tables show that increasing Nb and Nt increases the Nusselt number, where the opposite trend is found for Sherwood numbers Sh that the heat flux is decreased when Nb and Nt are increased. The obtained results are verified for validation in comparison with Sheikholeslami [17] results given in Table 10 which are in complete agreement with our results. The numerical values of Re,λ1,β and kron skin friction Cf are given in Table 11. From this table, it is clear that increasing values of Re,λ1 and β decreases Cf while increasing kr increases the skin friction. The numerical results (Table 11) and graphical result (Figures 14(a), 14(b), and 15) agree with each other. These results are compared with results of [19] given in Table 12. The obtained results are verified by comparing with [19] and good agreement is observed. The numerical values of heat flux Νu and mass flux Share given in Table 13. We have calculated it at the boundaries of the plates for dissimilar values of Nt,Nb and Sc. It is clear from Table 13 that the heat flux Νu is reduced when parameters Nt and Nb rose, because the mass flux is a reducing function of the lower stretching plate, where it is an growing function of Sc at the upper plate. The mass flux Sh is increasing function at the lower stretching plate, while it is a decreasing function of the upper plate at different increasing values of Nt,Nb and Sc. To validate and verify our results, the obtained results are compared with the RK Method of order 4 as given in Tables 14 and 15 , while the absolute errors of these methods are presented in Table 16 at each point.Table 2
Influence ofRd and M on nusselt number Nu (present results), when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdM0.04.08.016.00.01.429941.397651.368021.316763.01.658341.645901.556131.364606.03.192433.124243.094153.0667312.012.188112.000511.910711.8235Table 3
Influence ofRd and M on sherwood number Sh, when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdM0.04.08.016.00.00.9991280.9983510.9977070.9968203.01.021381.020601.019961.019076.01.043631.042851.042211.0413212.01.088131.087351.086711.08582Table 4
Influence ofRd and M on nusselt number Nu (Sheikholeslami [17]), when Nt=Nb=0.01,R=1,Sc=0.5,Pr=10.
RdM0.04.08.016.00.01.5570441.5236671.49541.4499363.06.8302426.8021486.7783036.7398176.012.158112.1305312.1071312.0693112.017.4893817.4620217.4387717.40121Table 5
Influence ofRd and Sc on nusselt number Nu (present results) when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdSc0.010.22.06.00.01.444021.444011.387301.387223.01.475281.388121.386841.382826.03.292213.271563.271503.2711212.012.980412.915012.900212.8157Table 6
Influence ofRd and Scon sherwood number Sh, when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdSc0.010.22.06.00.00.7510840.7588170.8324411.016873.00.8980841.003821.077441.261876.01.241081.042851.322441.5068712.01.731081.733671.812441.99687Table 7
Influence ofRd and Sc on nusselt numbers Nu (Sheikholeslami [17]), Nt=Nb=0.01,R=1,Sc=0.5,Pr=10.
RdSc0.010.22.06.00.01.5570441.5570211.5567981.5562853.06.8302426.8302256.8300656.8297026.012.158112.2580812.1579212.1575712.017.4893817.4893617.4892117.48886Table 8
Influence ofRd and Nb on nusselt number Nu (present results), when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdNb0.010.10.30.60.01.468131.421621.320671.175513.01.389851.357481.283121.165326.03.271573.073342.625561.9351412.012.91511.84519.450445.81478Table 9
Influence ofRd and schmidt number Sc on sherwood number Sh when Nt=Nb=0.01,R=1,β=γ=0.1,Sc=0.5,Pr=10,λ1=m=M=0.01.
RdNb0.010.10.30.60.00.7666150.9989211.016111.020383.01.011611.021171.021861.0226.01.256611.043421.027611.0236312.01.746611.087921.039111.02688Table 10
Influence ofRd and Nb on nusselt number Nu (Sheikholeslami [17]), Nt=Nb=0.01,R=1,Sc=0.5,Pr=10.
RdNb0.010.10.30.60.01.5570441.499111.374631.1993513.06.8302426.7830536.6789476.5247526.012.158112.11912.0096612.8573612.017.4893817.4435617.3420117.19043Table 11
The skin friction coefficient for dissimilar values ofRe,Κr,β, and γ whenNt=Nb=0.1,Sc=0.5,Pr=10,λ1=m=M=0.3.
ReMγΚr−CfRex1/20.10.51.01.53.330270.52.948821.02.642081.50.14.339990.54.321571.04.268971.50.15.642270.55.445761.04.899111.50.14.127430.54.357721.05.130481.55.91612Table 12
Variation in skin friction coefficient for dissimilar values ofSq,ω,β, and λ1 when Pr=1.0,γ=1.0 ([19] results).
sqωλ1β−CfRex1/2−0.10.51.00.52.633120.02.651330.12.639951.00.01.312170.11.259170.51.216941.00.02.385080.11.259170.51.033991.00.00.619110.010.798840.10.877320.22.63312Table 13
The effect of physical parameters on nusselt numbersNu and sherwood number Sh when Nt=kr=0.1,R=0.5,Pr=10,λ1=m=M=0.3.
RdNbScη=+1η=−1NuShNuSh0.10.50.5−0.766570−1.700540.766571.700540.5−0.622278−1.233280.6222781.233281.0−0.459501−1.172530.4595011.172531.50.1−0.414644−1.026770.4146441.026770.5−0.316268−1.150540.3162681.150541.0−0.215401−1.356360.2154011.356361.50.1−0.135599−1.607280.1355991.607280.5−0.135444−1.609220.1354441.609221.0−0.1353148−1.619220.1352511.611651.5−0.135057−1.614080.1350571.61408Table 14
Results obtained by HAM at 20th order approximations whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m==0.1,M=1,γ=0.5.
ηfηgηΘηΦη00.000000.000001.000001.000000.11.325581.234560.9875460.9875420.21.234561.235460.8457630.8445870.32.345602.546860.7452630.7421450.42.458762.548960.6754680.6745780.52.625482.756250.5648630.5645780.62.324562.452360.4206350.4222140.72.012452.245630.3214870.3012450.81.562351.235550.2145870.2012450.91.2548631.214520.114230.1011231.00.0000000.0000000.000000.00000Table 15
RK method of order 4, whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m==0.1,M=1,γ=0.5.
ηfηgηΘηΦη00.000000.000001.000001.000000.11.325581.234560.9875460.9875420.21.234561.235460.8457630.8445870.32.345602.546860.7452630.7421450.42.458762.548960.6754680.6745780.52.625482.756250.5648630.5645780.62.324562.452360.4206350.4222140.72.012452.245630.3214870.3012450.81.562351.235550.2145870.2012450.91.2548631.214520.114230.1011231.00.0000000.0000000.000000.00000Table 16
Absolute errors of RK method of order 4 and HAM results whenRd=0.5,Re=Nt=Nb=β=Sc=Pr=λ1=M=m==0.1,M=1,γ=0.5.
ηfηgηΘηΦη00.000000.000000.000000.000000.11.22213×10−91.32145×10−82.14785×10−71.65478×10−70.22.13456×10−103.214560×10−92.85632×10−81.25463×10−80.32.14589×10−111.32456×10−102.45782×10−91.87546×10−90.41.54879×10−112.001245×10−101.24586×10−101.85457×10−100.52.14578×10−121.02145×10−113.12458×10−111.24586×10−110.62.145892×10−131.02145×10−125.45786×10−131.45876×10−130.72.45866×10−141.021456×10−123.54689×10−151.45826×10−150.82.001458×10−151.02145×10−132.47896×10−161.85475×10−160.92.14586×10−182.14586×10−171.24589×10−171.98526×10−171.00.0000000.0000000.0000000.000000


## 6. Conclusion

The present study explores the magnetohydromagnetic flow of Jeffrey nanofluid between two parallel plates with Hall current effect. The governing equations are transformed to obtain a set of nonlinear ODEs. The optimal solution is obtained by HAM, while the numerical solution is obtained by RK Method of order 4. The observation of this work depends on the influence of Hall current, porosity, and rotation of non-Newtonian nanofluid flow between two parallel plates. The mathematical formulation of the model has been carried out in such a manner that one of the plates is stretched, and the other is fixed. The constant magnetic field is considered, and it is acting perpendicularly to the direction of the flow field. The modeled equations have been solved through analytical homotopy analysis method HAM. The convergence of the HAM method has been shown numerically. Also, the system of equation is solved numerically by RK Method of order 4 for validating the obtained results. The effect of the embedded parameter is observed and studied graphically. The influence of skin frictionCf is shown graphically as well as numerically, and also its effects on different parameter like β, λ1, M and kr are observed graphically and numerically. The influence of the Nuslet number Nu and the Sherwood number Sh on the temperature and concentration fields has been observed. The central concluded key points are as follows:(i)
The larger values of Brownian motion parameterNb raise the kinetic energy of the nanoparticles inside the fluid, and as a result, the profile of the temperature rises.(ii)
Thermal boundary layer thickness reduces with the rise of radiation parameterRd. So, Nusselt number Νu rises with rise of radiation parameter Rd.(iii)
The higher values of the Hall parameterm increase the velocity of the nanofluid because Hall force transfers forward the nanoliquids in horizontal direction.(iv)
For skin frictionCf it is found that it is increased when the viscosity parameter Re is decreased.(v)
The mass flux has been observed as a decreasing function at the lower stretching plate and an increasing function ofSc at the upper static plate.(vi)
The heat fluxSh has been concluded as an increasing function at the lower stretching plate and decreasing function at upper static plate.(vii)
HAM is fastly convergent when compared to other methods.(viii)
The absolute errors decrease by increasingη.



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*Source: 9930017-2021-07-31.xml*