# Impact of Homogeneous-Heterogeneous Reactions on Flow of Non-Newtonian Ferrofluid over a Stretching Sheet

Authors: H. Zeb; S. Bhatti; U. Khan; H. A. Wahab; A. Mohamed; I. Khan
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

In this study, we considered heterogeneous and homogeneous chemical reactions in Eyring-Powell ferrofluid past a stretchable surface. In addition, thermal radiation and magnetic dipole impacts are also considered. The direct numerical solution of the governing model is complex. For this, we simplified the model into coupled ordinary differential equations (ODEs) with the dimensionless variables. Then, we determined the computational solution of the resulting transformed ODE system via the Runge-Kutta (RK) method. The impacts of interesting engineering parameters on temperature, velocity, and concatenation behaviors are presented through graphs. We concluded that the velocity field is reduced for higher values of ferromagnetic parameterβ and fluid material parameters ß and H and arises the temperature field for higher values of thermal radiation R while the temperature field is reduced for Prandtl number Pr and dissipation parameter λ1. The obtained results of the current work are compared with published work, and we found that between them a good agreement can be seen in the table. The characteristics of skin fraction and Nusselt number are presented.


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## Body

## 1. Introduction

The boundary layer flows are classified into non-Newtonian and Newtonian fluid materials. The fluids in which the stress is linearly proportional to the strain are known as Newtonian fluids while those otherwise are non-Newtonian materials. Mineral oil, gasoline, water, kerosene, organic materials, glycerin, solvents, and alcohol are examples of the Newtonian fluid materials. The modeling of non-Newtonian fluid has high attention due to their high range of uses in manufacturing phenomena, geothermal engineering, and industrial processes. The nuclear reactors, spinning of fibers, metallurgical processes, casting, crystal growth, liquid metal space technology, etc. are uses of these applications. The modeling of non-Newtonian fluid phenomena cannot be captured in a single relation because of their aspects. Different models are characterized for the aspects of non-Newtonian fluids considered. The modeling of non-Newtonian fluids is classified into subclasses such as (i) differential type, (ii) rate type, and (ii) integral type. The Maxwell fluid, Carreau fluid, and Burger fluid do not clarify the properties of the rheological fluid model. So therefore the focus of our present work is an assumed Eyring-Powell fluid model [1].The Powell-Eyring has several advantages of using at low and high shear rates and can be reduced to the Newtonian fluid by the derivation of the kinetic theory of liquids. The heat transfer modeling of an Eyring-Powell fluid has entrusting applications in thermal insulation, geophysical process and environmental pollution. Human blood, ketchup, toothpaste, etc. are the examples of Powell-Eyring fluid. Ghaffar et al. [2] studied the modeling of convection heat transfer in Eyring-Powell fluid past a vertical sheet. Najeeb et al. [3] expressed the homotopy solution for Eyring-Powell fluid flow with nanoparticles over a vertical channel. Hina et al. [4] analyzed the effects of viscose dissipation and thermophysical on Powell-Eyring fluid along the curved channels with nanoparticles. Taseer et al. [5] reported the significance of Arrhenius activation energy on the 3-dimensional flow of an Eyring-Powell fluid with nonlinear thermal radiation. The MHD flow modeling of non-Newtonian fluid with stratified boundary was proposed by Reddy et al. [6]. The activation energy in heat and mass transfer of Eyring-Powell fluid along by taking the stretching surface is explored by Nisar et al. [7]. The modeling of convective heat and momentum slip boundary conditions in nanofluid was determined by Wang et al. [8]. The nonlinear thermoradiator heat flux in the time-dependent flow of an Eyring-Powell fluid with variable thermophysical effects was reported by Khatshwa et al. [9]. The magnetic field non-Newtonian fluid under the assumption of internal energy was proposed by Bilal et al. [10].The ferrofluid is a nanofluid consists of ferrite nanoparticles and base fluid. Initiating the investigation of ferromagnetic fluids is developed by Stephen et al. [11] in 1965. Ferrofluid has enormous uses in industrial engineering, biological sciences, and chemical engineering. Some common uses of ferrofluid generate rotating X-ray tubes, hi-fi speakers, sensors, targeting/tumors cancer cells, accelerometer, magnetized devices, optical applications, and sensors (hard plates and recording procedures), which are used in biological processes. The combination of impacts of thermal radiation and magnetic field on viscous flow of ferrofluid was explored by Neuringer et al. [12]. The computational study of heat transfer modeling in magnetic dipole of ferrofluid past a stretchable surface was considered by Sharma et al. [13]. The heat transfer phenomena and magnetic dipole in a ferrofluid flow over a stretching surface were identified by Majeed et al. [14]. The heat transfer modeling in stratified flow of ferrofluid over a stretching sheet was demonstrated by Noor et al. [15]. Ashraf et al. [16] explored the combination of momentum slip phenomena in hybrid nanofluid by considering curved sheet. The nonlinear thermal radiating flow of ferrofluids over a moving plate with magnetic dipole effects was identified by Ali et al. [17]. The combined effects of thermodiffusion on generalized fluid with nanomaterials was analyzed by Souayeh et al. [18]. The simulation of Cattaneo and Christov heat flux modeling in ferromagnetic fluid with nanoparticles behaviors was demonstrated by Shehzad et al. [19]. The related study of ferrofluid in detail can be seen in Refs. [20–27].Generally, homogeneous and heterogeneous reactions are concerned with natural chemical reaction processes. Some chemical reactions are very slow without a catalyst. The heterogeneous and homogeneous reactions are very complex because of their interaction in the volume of the fluid and catalyst surface, i.e., reactions produced in combustion, catalysis, and biochemical processes. The importance of chemical reactions is found in hydrometallurgical industries, mist development, food processing and diffusing, crop damaging through freezing, hardware configuration, plantation of fruit trees, and many more. Some examples of homogeneous and heterogeneous reactions are represented by Williams [28–31]. The effects of a chemical reaction and stagnation point in the boundary layer flow were presented by Chaudhary et al. [32]. Pop and Khan et al. [33] presented the viscoelastic fluid affected by homogeneous and heterogeneous reactions. Merkin et al. [34] presented that the isothermal boundary layer flow is affected by homogeneous and heterogeneous reactions. MHD flow of a second-grade fluid is affected by homogeneous and heterogeneous reactions with the porous medium reported by Cortell et al. [35]. The Newtonian heating effects on the heterogeneous and homogeneous reactive flow of non-Newtonian fluid over a stretching sheet were reported by Imad et al. [36]. Zero mass flux effects in bioconvective nanofluid over a stretching surface with response surface methodology were explored by [29]. Nanofluid flow is affected by chemical reaction and nonlinear thermal radiation over a curved surface presented by Ramzan et al. [37].In the above literature review, we are interested to explore the heterogeneous and homogeneous reaction in non-Newtonian ferrofluid over a stretching surface. Further, the effects of thermal radiation and magnetic dipole are also determined. Here in this work, we transformed the governing model into a coupled nonlinear ODE system by similarity variables. The governing equations are computationally evaluated using shooting method with RK method.


## 2. Mathematical Model

We have considered heterogeneous and homogeneous reaction phenomena of Eyring-Powell ferrofluids over a stretching surface. Further, the characteristics of magnetic dipole and internal energy are assumed. The surface is to be considered along atx,y with velocity components u,v; it is placed normal to the sheet in Figure 1. Here, place a magnetic dipole with the center y-axis by the displacement γ1 from the sheet. The magnetic dipole is assumed along the positive x-axis. The magnetic field strength is arising because leading the magnetic dipole is situated to ferrofluid. Here, consider a uninformed temperature to the sheet Tw and Curie temperature Tc, whereas the fluid element is assumed in the ambient temperatureT∞=Tc. So it is unable to magnetize until they initiate to cooling upon entering the sheet at the boundary layer region.Figure 1
Geometrical configuration of the fluid problem.Here, we assume the interaction between the (homogeneous and heterogeneous reactions) chemical reaction with chemical reactantsB and A seen in the study of boundary layer flows introduced by Merkin et al. [34]. It can be represented as follows. The homogeneous (bulk) reaction is formulated as
(1)A+2B⟶3B,rate=κ1ab2,A⟶Brate=κ2a.Here, in the above formulation,a,b are concentration of the chemical species B and A, where κ1 and κ2 represent rate constants.The mathematical formulation of the two-dimensional law conservation of momentum can be expressed as follows:(2)ρdVdt=∇.T+F⟶,where ρ, V, and T represent the density, velocity, and Cauchy stress tensor. The stress tensor is stated as T=S−PI, where P is taken for pressure, I is the identity, S is the extra stress tensor, and F⟶=μoM·∇H. F⟶ the body force which represents the action of magnetic field and μo magnetic permeability can be seen in [38]. The modeling of an Eyring-Powell fluid is formulated as follows by Siddque et al. [39] in the form
(3)S=μoA1+A11γ̇Γsinh−1γ̇c,where Γ, μ, and c are the fluid material, viscosity, and dynamic constant, γ̇=0.5trA12, V=ux,y,0,vx,y,0,0, and A1 is taken for a Rivlin-Ericksen tensor. It is formulated as
(4)A1=gradV+gradVT∗.The consideration of the rating process is to determine the formulation of an Eyring-Powell model (1994) investigated in [40–42] which also considered the shear stress of an Eyring-Powell ferrofluid model given by
(5)sinh−1γ̇c≅1c∂ui∂uj−16c3∂ui∂uj3.By implementing the above consideration and the boundary layer approximation leads to the governed fluid being also studied in the model [38, 43, 44]. The heat and mass transfer modeling generalized Newtonian fluid with magnetic dipole is stated as follows:
(6)∂v∂y+∂u∂x=0,(7)∂u∂yv+∂u∂xu=Mμoρ∂H∂x+ν+1ρΓc∂2u∂2y−12ρΓc3∂u∂y2∂2u∂y2,(8)∂T∂yv+∂T∂xu+TμoρCp∂M∂Tu∂H∂x+v∂H∂y+1ρcρ∂qr∂y=κρCp∂2T∂y2,(9)v∂a∂y+u∂a∂x=DA∂2b∂y2−Ksab2,(10)v∂b∂y+u∂b∂x=DB∂2b∂y2+Ksab2.Here,u,v represents the velocity components along the x,y-axes, while T is taken for temperature of the fluid. Further, κ,DA,DB,μ, and μo represent the thermal conductivity, diffusion species coefficients, dynamic viscosity, and magnetic permeability, respectively. The boundary conditions are given by
(11)ux,y=uwx=dx,vx,y=0,T=Tc−Tw=Ax12,DA∂a∂y=κda,DB∂b∂y=−κda,aty=0,ux,y⟶0,∂u∂y⟶0,T⟶T∞,a=a0,b=0,asy⟶∞.The radiation termq~r in Equation (8) is expressed in Ref. [45] as
(12)q~r=−4ς3d∗∂T4∂y.Here, ς and d∗ are for the Stefan Boltzmann constant absorption coefficient. We assumed the relation between the fluid temperature and the fluid temperature far away is assumed from the sheet to be a linear function. Utilizing the Taylor series expression of T4 about T∞, we can define
(13)T4≈−34T∞−TT∞3.Utilizing Equation (12) in Equation (13), we obtained
(14)q~r=4ς3d∗T∞3∂T∂y.


## 3. Magnetic Dipole

The magnetic dipole variation under the consideration of magnetic field can be settled by the function known as magnetic scalar potential explored in Refs [12, 14, 46]. It can be expressed as
(15)Φ=γ22πxx2+y+γ12.Here, γ2 represents magnetic field strength; Hx and H take the magnetic field intensity at x,y-axis, and it is given by
(16)Hx=−∂Φ∂x=−γ22π−γ1+y2+x2γ1+y2+x22,(17)Hy=∂Φ∂y=γ22π2xγ1+yγ1+y2+x22.Generally, this is known as the magnetic force which is gradient of a magnetic fieldH. It is expressed by
(18)H=∂Φ∂x2+∂Φ∂y2.Using Equations (15) and (17) in (18), we obtain
(19)∂H∂x=−γ2πxy+γ14,(20)∂H∂y=γ2π2xy+γ15−1y+γ13.We considered that magnetizationM is a linear function at given temperature. It is presented as
(21)M=−KpT−Tc.The geometrical and physical implementation of the ferrofluids can be seen in Figure1.


### 3.1. Simplification of the Problem

By assuming the dimensionless variables as follows,(22)φη,ζ=fηζμρ,ϑη,ζ=Tc−TT∞−To=ϑ1η+ζ2ϑ2ηa=aogη,b=aohη,here the dimensional coordinates are given as
(23)ζ=yϱdμ,η=yϱdμ,where φζ,η and ϑζ,η are the dimensionless stream function of the temperature and pressure. The velocity can be defined as
(24)u=∂φ∂y=df′η,(25)v=−∂φ∂x=−μdϱfη.By using the similarity transformation (22)–(24) and (15)–(21) in (6)–(9), it reduced to highly nonlinear ODEs such as
(26)ff″−f′2+H+1f‴−2βϑ1α+η2−Hßf″2f‴=0,(27)1+Rϑ1″+Prfϑ1′−2f′ϑ1+2fβλ1ϑ1−ϵα+η3+2ϑ2−4λ1f′2=0,(28)ϑ2″1+R−Prfϑ2′+4f′ϑ2+2fβλ1ϑ2−ϵα2+η3−λ1βϑ1−ϵ2f′α+η3−4fα+η5−4λ1f′′2=0,(29)1Scg′′+fg′2=0,(30)δ1Sch′′+hg′2=0.The boundary and initial condition becomes
(31)f′η=1,fη=0,ϑ1η=1ϑ2η=0,aty=0,f′⟶0,ϑ1⟶0,ϑ2⟶0,g′0=Ksg0,g∞=1,δh′0=Kah0,g∞=1,aty⟶∞.Consider the diffusion coefficient DB,DA of reactants B and A is equal (i.e., δ1=1); see in Ref. [44]; therefore, this assumption can be expressed as
(32)gη+hη=1.So Equations (29) and (30) can be seen in Ref. [36] and formulated as
(33)1Scg′′−g′f+Ksgg−12=0,with boundary conditions
(34)g1Ka0=g′0,g∞=1.In the above calculation, f′, ϑ1, and g represent the velocity, temperature, and concentration, respectively. Further, β and λ1 are represented the magnetic dipole and viscous dissipation parameters, respectively, Ka=κ1ao2/d is taken for strength of homogeneous (surface) reaction, δ1=DB/DA is the ratio of diffusion coefficient, Ks=κ2/DAaod/ν represents strength of heterogeneous reaction, Sc=ν/DB is the Schmidt number, H=1/μΓc and ß=ϱuw/2xc2μ are fluid parameters, and Pr=μCp/k∞ is the Prandtl number. Further, λ1=dμ2/ϱkTw−T∞ is the viscous dissipation parameter, ϵ=Tc/Tc−Tw is the Curie temperature, β=γ1ϱμokTc−Tw/2πμ2 is the ferromagnetic parameter, and α=γ1ρd/μ is the curvature parameter.The skin fraction coefficientCf and local Nusselt number Nux are given below:
(35)Cf=τwϱuw2,Nux=xq~wκTw−T∞.Here,qw and τw are the heat flux and skin friction, which are given as follows:
(36)τw=μ1+1Γc∂u∂y+1Γc6∂u∂y3y=0,q~w=−κ∂T∂y−∂q~r∂yy=0.Here, q~m and uw are defined for heat transfer and for wall shear stress. The quantities stated as in Equation (20) are simplified as follows:
(37)CfRex=1+Hf′′0−Hβ3f′′03,NuxRex~=−1+Rϑ1′0+ζ2ϑ2′0.


### 3.2. Shooting Method for Our Numerical Approximation

It is a numerical scheme, which is implemented for the approximation of the boundary value problem (BVP). In the beginning, we reduce the BVP to a system of first-order initial value problems. Further, detail can be seen in Ref. [47]. The system (26) is the system of coupled ODEs of order three in fξ, order two in ϑ1ξ, order two in ϑ2ξ, and order two in g, respectively. The rearrangement of Equation (26) with boundary conditions in (31) will be expressed in the form
(38)f‴=−ff″−f′2−2βϑ1/α+η2H+1−βHf″2,ϑ1″=−Prϑ1′f−2ϑ1′f′+2fβλ1ϑ1−ϵ/α+η3+4λ1f′21+R,ϑ2″=−Prϑ2′f−4ϑ2′f′+f2βλ1ϑ2−ϵ/α2+η3+λ1βϑ1−ϵ2f′/α+η3−4f/α+η5+4λ1f″21+R,g″=Scfg′−Ksg1−g2.Now, we transform the above couple of ODEs into a first-order ODE system. So for this, we define new variables as follows:(39)f=u1,f′=u2,f′′=u3,f′′′=u3′,ϑ1=u4,ϑ1′=u5,ϑ1′′=u5′,ϑ2=u6,ϑ2′=u7,ϑ2′′=u7′,g=u8,g′=u9,g′′=u′9.Using these substitutions to determine the first-order ODE system as given by(40)u1′=u2,u2′=u3,u3′=−u1u3−u22−2βu4/α+η2H+1−ßHu32,u4′=u5,u5′=−Pru1u5−2u2u5+2u1βλ1u4−ϵ/α+η3+4λ1u221+R,u6′=u7,u7′=−Pru1u7−4u2u7+2u1βλ1u6−ϵ/α2+η3+λ1βu4−ϵ2u2/α+η3−4u1/α+η5+4λ1u321+R,u8′=u9,u9′=Scu1u9−Ksu91−u92,the boundary conditions are stated as
(41)u1=0,u2=1,u3=S1,u4=1u5=S2u6=0,u7=S3,u8=S4,u9=Ksu8,atη⟶0u2=0,u4=0,u6=0,u8=0,atη⟶∞.Identify the solution of these nine first-order ODE system given in Equation (41) with boundary conditions. For this, we use shooting technique with RK method; for this, we need nine initial guesses required whereas five of them are given and the other five initial guesses u2η,u4η, u6η, u8η, and u9η are defined as η⟶∞. Hence, it is considered that u30,u50,u70,u80,u90=q1,q2,q3,q3,q5. These unknown six initial guesses q1,q2,q3,q3,q5 are simulated by iterative scheme called the Newton Rhaphson technique. The main step of this technique is we choose the suitable finite values for boundary conditions. For the convergence criteria, we are taking the step size h=0.02 and tolerance TOL=10−5.


### 3.3. Validation of the Numerical Solution

In this work, we compared the results with previously published data and found good agreement between them. The comparison of skin fraction coefficient (CfRex) and Nusselt number (Nux) is shown in Tables 1 and 2, respectively. For this, we computed the different values of H and Pr.Table 1
Comparison ofCfRex at β=0.0, β==0, and H=0.2 with published work.
HJaved et al. [48]Waqas et al. [49]Current study0.00.2-1.0954-1.0954-1.095400.10.2-1.0940-1.0940-1.09400.20.2-1.0924-1.0925-1.092520.30.2-1.0909-1.0909-1.090940.40.2-1.0894-1.0894-1.08904Table 2
Comparison ofNux at H=0.0 and R=β==0 for Pr with published work.
PrChen et al. [50]Iskak et al. [51]Current study0.72-1.08862-1.0885-1.08861.0-1.33330-1.3333-1.33333.0-2.50972-2.5097-2.509510.0-4.7968-4.79682-4.7943


## 3.1. Simplification of the Problem

By assuming the dimensionless variables as follows,(22)φη,ζ=fηζμρ,ϑη,ζ=Tc−TT∞−To=ϑ1η+ζ2ϑ2ηa=aogη,b=aohη,here the dimensional coordinates are given as
(23)ζ=yϱdμ,η=yϱdμ,where φζ,η and ϑζ,η are the dimensionless stream function of the temperature and pressure. The velocity can be defined as
(24)u=∂φ∂y=df′η,(25)v=−∂φ∂x=−μdϱfη.By using the similarity transformation (22)–(24) and (15)–(21) in (6)–(9), it reduced to highly nonlinear ODEs such as
(26)ff″−f′2+H+1f‴−2βϑ1α+η2−Hßf″2f‴=0,(27)1+Rϑ1″+Prfϑ1′−2f′ϑ1+2fβλ1ϑ1−ϵα+η3+2ϑ2−4λ1f′2=0,(28)ϑ2″1+R−Prfϑ2′+4f′ϑ2+2fβλ1ϑ2−ϵα2+η3−λ1βϑ1−ϵ2f′α+η3−4fα+η5−4λ1f′′2=0,(29)1Scg′′+fg′2=0,(30)δ1Sch′′+hg′2=0.The boundary and initial condition becomes
(31)f′η=1,fη=0,ϑ1η=1ϑ2η=0,aty=0,f′⟶0,ϑ1⟶0,ϑ2⟶0,g′0=Ksg0,g∞=1,δh′0=Kah0,g∞=1,aty⟶∞.Consider the diffusion coefficient DB,DA of reactants B and A is equal (i.e., δ1=1); see in Ref. [44]; therefore, this assumption can be expressed as
(32)gη+hη=1.So Equations (29) and (30) can be seen in Ref. [36] and formulated as
(33)1Scg′′−g′f+Ksgg−12=0,with boundary conditions
(34)g1Ka0=g′0,g∞=1.In the above calculation, f′, ϑ1, and g represent the velocity, temperature, and concentration, respectively. Further, β and λ1 are represented the magnetic dipole and viscous dissipation parameters, respectively, Ka=κ1ao2/d is taken for strength of homogeneous (surface) reaction, δ1=DB/DA is the ratio of diffusion coefficient, Ks=κ2/DAaod/ν represents strength of heterogeneous reaction, Sc=ν/DB is the Schmidt number, H=1/μΓc and ß=ϱuw/2xc2μ are fluid parameters, and Pr=μCp/k∞ is the Prandtl number. Further, λ1=dμ2/ϱkTw−T∞ is the viscous dissipation parameter, ϵ=Tc/Tc−Tw is the Curie temperature, β=γ1ϱμokTc−Tw/2πμ2 is the ferromagnetic parameter, and α=γ1ρd/μ is the curvature parameter.The skin fraction coefficientCf and local Nusselt number Nux are given below:
(35)Cf=τwϱuw2,Nux=xq~wκTw−T∞.Here,qw and τw are the heat flux and skin friction, which are given as follows:
(36)τw=μ1+1Γc∂u∂y+1Γc6∂u∂y3y=0,q~w=−κ∂T∂y−∂q~r∂yy=0.Here, q~m and uw are defined for heat transfer and for wall shear stress. The quantities stated as in Equation (20) are simplified as follows:
(37)CfRex=1+Hf′′0−Hβ3f′′03,NuxRex~=−1+Rϑ1′0+ζ2ϑ2′0.


## 3.2. Shooting Method for Our Numerical Approximation

It is a numerical scheme, which is implemented for the approximation of the boundary value problem (BVP). In the beginning, we reduce the BVP to a system of first-order initial value problems. Further, detail can be seen in Ref. [47]. The system (26) is the system of coupled ODEs of order three in fξ, order two in ϑ1ξ, order two in ϑ2ξ, and order two in g, respectively. The rearrangement of Equation (26) with boundary conditions in (31) will be expressed in the form
(38)f‴=−ff″−f′2−2βϑ1/α+η2H+1−βHf″2,ϑ1″=−Prϑ1′f−2ϑ1′f′+2fβλ1ϑ1−ϵ/α+η3+4λ1f′21+R,ϑ2″=−Prϑ2′f−4ϑ2′f′+f2βλ1ϑ2−ϵ/α2+η3+λ1βϑ1−ϵ2f′/α+η3−4f/α+η5+4λ1f″21+R,g″=Scfg′−Ksg1−g2.Now, we transform the above couple of ODEs into a first-order ODE system. So for this, we define new variables as follows:(39)f=u1,f′=u2,f′′=u3,f′′′=u3′,ϑ1=u4,ϑ1′=u5,ϑ1′′=u5′,ϑ2=u6,ϑ2′=u7,ϑ2′′=u7′,g=u8,g′=u9,g′′=u′9.Using these substitutions to determine the first-order ODE system as given by(40)u1′=u2,u2′=u3,u3′=−u1u3−u22−2βu4/α+η2H+1−ßHu32,u4′=u5,u5′=−Pru1u5−2u2u5+2u1βλ1u4−ϵ/α+η3+4λ1u221+R,u6′=u7,u7′=−Pru1u7−4u2u7+2u1βλ1u6−ϵ/α2+η3+λ1βu4−ϵ2u2/α+η3−4u1/α+η5+4λ1u321+R,u8′=u9,u9′=Scu1u9−Ksu91−u92,the boundary conditions are stated as
(41)u1=0,u2=1,u3=S1,u4=1u5=S2u6=0,u7=S3,u8=S4,u9=Ksu8,atη⟶0u2=0,u4=0,u6=0,u8=0,atη⟶∞.Identify the solution of these nine first-order ODE system given in Equation (41) with boundary conditions. For this, we use shooting technique with RK method; for this, we need nine initial guesses required whereas five of them are given and the other five initial guesses u2η,u4η, u6η, u8η, and u9η are defined as η⟶∞. Hence, it is considered that u30,u50,u70,u80,u90=q1,q2,q3,q3,q5. These unknown six initial guesses q1,q2,q3,q3,q5 are simulated by iterative scheme called the Newton Rhaphson technique. The main step of this technique is we choose the suitable finite values for boundary conditions. For the convergence criteria, we are taking the step size h=0.02 and tolerance TOL=10−5.


## 3.3. Validation of the Numerical Solution

In this work, we compared the results with previously published data and found good agreement between them. The comparison of skin fraction coefficient (CfRex) and Nusselt number (Nux) is shown in Tables 1 and 2, respectively. For this, we computed the different values of H and Pr.Table 1
Comparison ofCfRex at β=0.0, β==0, and H=0.2 with published work.
HJaved et al. [48]Waqas et al. [49]Current study0.00.2-1.0954-1.0954-1.095400.10.2-1.0940-1.0940-1.09400.20.2-1.0924-1.0925-1.092520.30.2-1.0909-1.0909-1.090940.40.2-1.0894-1.0894-1.08904Table 2
Comparison ofNux at H=0.0 and R=β==0 for Pr with published work.
PrChen et al. [50]Iskak et al. [51]Current study0.72-1.08862-1.0885-1.08861.0-1.33330-1.3333-1.33333.0-2.50972-2.5097-2.509510.0-4.7968-4.79682-4.7943


## 4. Results and Discussions

In this work, Equations (17)–(19) are solved through shooting technique. The velocity, temperature, and concentration are analyzed and simulated in Figures 2–10. Further, the computational results of skin fraction and heat and mass transfer rate are shown in Table 3.Figure 2
The properties ofβ on velocity field f′.Figure 3
The velocityf′η for H.Figure 4
The characteristics ofϑ1η for β.Figure 5
The characteristics of velocity fieldß.Figure 6
The variation ofϑ1η for Pr.Figure 7
The impact ofR for ϑ1η.Figure 8
The impacts ofϑ1η for λ1.Figure 9
The impacts ofSc on g.Figure 10
The impact ofKa for the distribution of temperature gradient g.Table 3
The characteristics of skin fraction and Nusselt number for different values ofPr, β, R, λ1, ß, H, and α.
PrβRλ1ßHα−f′′0ϑ1040.20.30.10.40.61.60.593672.703510.523991.092920.520601.562630.518431.92500.00.490072.23540.20.498972.23410.40.507892.23280.00.497703.13050.30.498222.72890.60.498972.23410.10.498222.72890.30.498222.72910.50.498222.72940.00.495612.72961.00.527542.72322.00.590382.71320.01.333222.51442.00.792422.65464.00.603802.70221.00.603802.70221.30.596232.70321.60.593672.7035The importance of ferromagnetic interaction parameterβ on velocity f′η is shown in Figure 2. It is found that there is reduction in the velocity profile for the improved values of β. It is a fact that generally the ferromagnetic fluid is the motion of the fluid with tiny-sized firrite particles arising the viscosity of the fluid. So, the velocity of the fluid reduces for larger values of β.Figure3 is plotted for the characteristics of material fluid parameter H on velocity field. It is noticed that there are increases in the velocity field for H, which means the velocity and momentum boundary layer thickness. Figure 5 illustrates the velocity gradient for higher values ß. It is highlighted boosting the velocity field for ß. It is due to the direct relation between velocity of the sheet and the fluid parameter ß; thus, the velocity of the fluid as also increased.Figure4 presents the characteristics of β on temperature distribution. Here, the temperature distribution is increased for higher values of β. It is due to the interaction between the action of the ferromagnetic field and the motion of the fluids. This interaction reduces the velocity field thereby arising the frictional heating in the boundary layers. Therefore, thickness of the thermal boundary is increased.Figure6 represents the characteristics of the temperature field for higher values (Prandtl number (Pr)). It is concluded that it declined the temperature profile for Pr. This reason is due to the fact that fluid having a small number of Pr has a high thermal diffusivity. Figure 7 shows the variation of temperature field is decreased for radiation parameter R. It is increased in temperature profile for larger values of R. This increasing occurs because of the higher temperature T¯∞3.The outcome of frictional heating due to magnetic dipole and viscosity on the distribution of temperature imported by dissipation parameterλ1 is shown in Figure 8. It noticed the properties of λ1 on the temperature field. The result represents that the temperature distribution is reduced for λ1. The reason behind that is the extraordinary behavior of ferrofluid. However, this is contrary to the case of hydrodynamic (β=0) although shown here that there is a rise in temperature distribution near the boundary layer region by enlarging the values ofλ1. Figure 9 is designated for variation concentration gradient for the higher values of Sc. The result shows that the concentration field decreases as the Schmidt number Sc increases. Here, this impact is decreased in nanoparticle volume fraction boundary layer thickness due to the higher values of Schmidt number.Figures10 and 11 are made to highlight the characteristics of homogeneous and heterogeneous reactions parameters Ka and Ks on concentration behavior. It is highlighted that reduces occur in the concatenation field. It is seen that the thickness of the concentration boundary layer arises with arising the values Ks. Figure 11 is illustrated for concentration field reduced by the improving value of heterogeneous reaction Ka.Figure 11
The properties ofKs on g.Table3 is made for the numerical properties of Nusselt numbers Nux and skin fractions Cf for different physical parameters. The skin friction increases for higher values H and λ1 while skin friction is reduced for ß and β. The Nusselt number Nux is increased for H, R, α, and ß while it is declined for Pr and λ1.


## 5. Conclusions

In this work, we analyzed heterogeneous and homogeneous reactions in Eyring-Powell ferrofluid in the existence of magnetic dipole. Further, the impacts of thermal radiation and activation energy are analyzed. The governing fluid model is transformed into higher-order ODEs by using similarity variables. The characteristics of concentration, temperature, and velocity field for interesting engineering parameters are analyzed. The heat transfer rate and skin friction coefficients are presented. The main results of the current work are listed below.Then velocity field f′0 reduces by higher values of β and H.There are increases in temperature fieldϑ1η for R and H while the temperature field ϑ1η declines for the improved values of Pr.There are increases in the concentration fieldgη for Sc.



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*Source: 2501263-2022-09-02.xml*