# The Velocity Slip Boundary Condition Effects on Non-Newtonian Ferrofluid over a Stretching Sheet

Authors: Hussan Zeb; Hafiz Abdul Wahab; Umar Khan; Amnah S.Al Juhani; Mulugeta Andualem; Ilyas Khan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this work, radioactive heat transfer analysis in non-Newtonian Ferrofluid over a stretchable sheet is considered. Furthermore, the effects of Arrhenius activation energy, magnetic dipole, velocity slip, and mass convective boundary condition are taken into account. The governing model is transformed into coupled ordinary equations (ODEs) via a similarity transformation. The solution of these resulting ODEs systems are computed by Runge–Kutta method (RK-45). The influence of beneficial physical parameters on momentum, pressure, energy, and concentration profiles are presented. The major finding of this study is the variation of the velocity field is reduced for the higher values of velocity slip parametersA1 and β2 and fluid material parameter H. The temperature field increases for higher values of Rd and decreases for Pr. Here we concluded that, the increasing or decreasing of the concentration, temperature, and velocity field for various physical parameters can be seen in the result and discussion section in detail. The physical quantities such as skin friction, Nusselt, and Sherwood numbers are examined.


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## Body

## 1. Introduction

The boundary layer flows are characterized by Newtonian and non-Newtonian fluids. The stress in the Newtonian fluids is linearly proportional to the strain. Examples of Newtonian fluids are mineral oil, water, gasoline, organic matter, kerosene, solvents, glycerin, alcohol, and so on. The boundary layer flow of non-Newtonian fluids is becoming interesting due to an enormous range of uses in industries, manufacturing processes, and geothermal engineering. These applications include nuclear reactors, metallurgical processes, spinning of fibers, casting, liquidating metals space technology and crystal growth, and so on. Such examples of non-Newtonian fluids models are plastics polymer, optical fibers, hot rolling, drilling mud, cooling of metallic plates, paper production, and metal spinning. It is a fact that in contrast to Newtonian fluids, Newtonian (Navier–Stokes equation) fluid flow cannot be expressed by a single reference equation. This is due to the properties of Newtonian fluid physics. It is because of the rheological feature of the non-Newtonian fluids. The non-Newtonian fluids due to a stretching have got a huge achievement due to its large numbers of application manufacturing processes, geothermal engineering, and chemical engineering processes. The properties of non-Newtonian can be examined due to its elasticity, but sometimes, rheological properties of the fluid are identified by their constitutive equations. Generally, non-Newtonian fluids are categorized into three classes such as (i) rate type, (ii) differential type, and (iii) integral type. Carreau fluid, Maxwell fluid, Oldroyd-B, Burger fluids, and Eyring–Powell fluids are the non-Newtonian fluids. Many authors investigated the non-Newtonian fluid different models that can be seen in references [1–3]. Here, the focus of this work is to examine the Eyring–Powell fluid.’The Powell–Eyring fluid materials have many benefits when used at low and high shear rates, and they formed a Newtonian fluid via the formulation of kinetic theory. It is an interesting topic due to the application of the non-Newtonian model in chemical engineering processes. Furthermore, it suitably covers the Newtonian performance at high and low shear rates. Human blood, ketchup, toothpaste, and so on use Powell–Eyring fluid. Thus, we assumed the generalized (Eyring–Powell) fluid models [4] due to the importance of other fluid material models. Instead of the realistic relation, according to the kinetic theory of liquids in generalized fluid materials, Ghaffar et al. [5] reported the computational approaches for the effects of heat convection on Eyring–Powell fluid over a vertical surface. Najeeb et al. [6] explored to perform the analysis of a Powell fluid produced by a vertical channel. Taseer et al. [7] explored a three-dimensional flow analysis of an Eyring–Powell fluid with nanomaterials and radiation effects. Nisar et al. [8] illustrated the heat and mass transfer analysis in a generalized fluid along the non-linear stretching surface. Time-dependent heat flux analysis of an Eyring–Powell fluid with temperature-dependent conductivity was demonstrated by Khatshwa et al. [9]. The numerical approximation for stratified magnetic field flow of a non-Newtonian fluid over the included stretching surface was computed by Bilal et al. [10].The heat and mass transfer analysis got notable attention due to its applications in chemical industries, biological, and manufacturing processes. For example, space heating, air conditioning, control generation, and refrigeration are well-known applications in engineering. These include the transport of certain drugs spread in food processing, distillation, oil distribution, and transport phenomena in the blood (drug delivery) found in many other processes. These applications motivate engineers and scientists to examine heat and mass transfer phenomena theoretically and experimentally. Numerous investigations on mass transfer also establish living matter phenomena, such as sweating, pathological process, nutrition, and so on. The chemical reacting flow in heat and mass transfer phenomena with dissipation effects was demonstrated by Billal et al. [11]. The Reiner–Philippoff fluid and heat transfer analysis with Darcy–Forchheimer medium was explored by Ganesh Kumar et al. [12]. The hybrid nanoparticle behavior in heat transfer analysis over a stretching sheet was presented by Tahir et al. [13]. Mabood et al. [14] analyzed the characteristics of chemical reaction on stagnation point with heat and mass transfer analysis with porous medium. The magnetic dipole Maxwell fluid over a stretching sheet was demonstrated by Naveen Kumar et al. [15]. The combined effects of thermodiffusion on Casson nanofluid past a thin needle with the magnetic field were demonstrated by Souayeh [16]. The Fourier heat flux analysis in rate type Maxwell fluid and chemical reaction phenomena over the three-dimensional stretchable surface was reported by Chin et al. [17]. The analytical solution for thermal and concentration analysis of generalized fluid over a stretching surface was concluded by Kashif et al. [18]. The Galerkin finite element analysis of generalized fluid and temperature-dependent conductivity in heat transfer was explored by Qureshi et al. [19]. The stagnation point flow of micropolar fluid and heat transfer properties over a stretching sheet with slip boundary condition was explored by Jawad et al. [20]. The stratified non-Newtonian fluid with magnetic field effects was reported by Reddy et al. [21].Activation energy is the minimal amount of energy that is needed to create a chemical process; initially, it is reported by Swante Arrhenius in 1889. It can be considered as the highest potential energy that is reduced to the potential energy materials. It interests the researchers because of its wide range of applications of the activation energy in industrial engineering, geothermal, manufacturing procedure, food manufacturing, oil storage, and oil emulsified oil in water emulsions. Bestman [22] initially expressed the activation energy in mass transfer analysis under the consideration of porous media. The time-dependent rotating rate type (Maxwell fluid) with the combination of the chemical reaction and eternal energy was published by Shafique et al. [23]. The characteristics of activation energy in time-dependent mass transfer overstretching cylinder by Awad et al. [24]. The time-dependent MHD flow of a non-Newtonian fluid over a contract cylinder with a magnetic field was explored by Reddy [25].Ferrofluids are colloid materials that are prepared by the suspension of ferromagnetic particles in non-conducting Carreau fluids with low viscosity. Initially, the ferromagnetic fluids are invented by Stephen [26] in 1965. Without the magnetic force, these fluids have a property like the usual liquids. Ferromagnetic fluids have a remarkable application in chemical industries and modern technology. Some application of these fluids is in electromechanical devices commercially, for instance, electric engines, transformers, generates rotating X-ray tubes, hard disks, electromagnets, generators, cure arthritis, spondylitis, gout, headaches, and hoarding (recording procedures and hard plats), which is used in biological sciences. The numerical aspects of magnetic dipole hybrid nanofluid over spreading surface under the influence of magnetic dipole were presented by Taza et al. [27]. The combined impact of magnetic field and thermal radiation on the viscose flow of ferrofluid were reported by Neuringer et al. [28]. The approximate solution for the effect of the magnetic dipole on ferrofluid over a stretching surface is analyzed by Sunil et al. [29]. The combined effects of magnetic dipole and stratification on ferrofluid with stagnation point over a stretching sheet were reported by Muhammad et al. [30]. The addition of magnetic dipole and Ohmic dissipation in non-Newtonian (Eyring–Powell) fluid and heat transfer analysis over a stretching surface was described by Sajjad et al. [31]. Shehzad et al. [32] identified the analysis of heat flux in ferromagnetic fluid with titanium no-particles effects. The related work of ferrofluid can be seen in references [33–37].The studies of slip flow of a non-Newtonian fluid have motivated various searchers because of a large number of applications in industrial and engineering sectors in the modern age. Ishak et al. [38] utilized the slip condition for the stagnation point flow of the Maxwell fluid ring surface. The combination of slip and heat convective boundary condition effects on non-Newtonian fluid over a stretchable surface was examined by Ajadi et al. [39]. The comprehensive studies of slip boundary condition in boundary layer flow with heat transfer fluid at a solid surface were explored by et al. [40]. The slip boundary effects on Newtonian and non-Newtonian fluids can be seen in references [41, 42].Motivated by the aforementioned literature survey, the combination of magnetic dipole and temperature-dependent conductivity in non-Newtonian ferrofluids is considered over a stretching sheet. Furthermore, the impacts of mass convection boundary conditions and thermal radiation are determined. Due to the complex nature of the governing model, we transformed the main model into nonlinear ordinary differential equations by utilizing similarity transformation. The computational solution of transformed ODEs is a solved shooting technique with the RK method. The physical parameters’ effects on momentum, temperature, and concatenation profiles are analyzed. The present results are compared with past published data and found an excellent agreement. The physical quantities namely skin friction and Nusselt number are examined and presented.


## 2. Mathematical Model

Two-dimensional heat and mass transfer analysis in Eyring–Powell ferrofluids is considered in addition to magnetic dipole over a stretching surface. Furthermore, we take the combination of activation energy, temperature-dependent conductivity, and mass convection boundary condition. We have taken the sheet alongx,y-axes with the velocity components u,v, respectively; it is taken into account normal to the surface seen in Figure 1. Here, a magnetic dipole is placed at positive x−axis and taken at the center along y−axis with displacement γ1 from the surface. The magnetic field strength is arising due to the fact that the magnetic dipole is situated on ferrofluids. Here, the non-uniform temperature at the surface Tw and Tc represent the Curie temperature, whereas the fluid element is considered to be the ambient temperature Tc=T∞. So it is unable to magnetize until they start to cool upon entering the surface of the boundary layer region.Figure 1
Geometrical configuration. The circles configure the magnetic dipole.Rate theory of process is utilized to derive an Eyring–Powell model in 1994 for the characterizing of non-Newtonian fluid. Starlin et al. [43] also considered the shear stress of an Eyring–Powell fluid model as follows:(1)τi,j=1Γsinh−11∂uiϵ∂uj+μ∂ui∂uj,where μ is taken for viscosity and ϵ and Γ are the fluid materials parameters. By taking secondorder approximation [6, 43] as follows:(2)sinh−11∂uiϵ∂uj≅1ϵ∂ui∂uj−16ϵ3∂ui∂uj3,by implementing the above assumption and utilized the boundary layer analysis to governing fluid model [44, 45] (Eyring–Powell fluid ferrofluid, heat and mass transfer with the addition of magnetic dipole) are define Equations (3)–(6) as(3)∂v∂y+∂u∂x=0,(4)∂u∂yv+∂u∂xu=−1ρ∂P∂x+ℳμoρ∂ℋ∂x+ν+1ρΓϵ∂2u∂2y−12ρΓϵ3∂u∂y2∂2u∂y2,(5)∂T∂yv+∂T∂xu+TμoρCp∂ℳ∂T∂ℋ∂xu+∂ℋ∂yv=κρCp∂2T∂y2−1ρcρ∂qr∂y,(6)v∂C∂y+u∂C∂x=D∂2C∂y2−Γ2,where u,v represent the velocity components along the x,y axis, respectively, while C,T,ρ, are taken for the concentration, temperature, and density of the base fluid, respectively. Furthermore, μ,μo,kandkr are taking for dynamic viscosity, magnetic permeability, variable thermal conductivity, and chemical reaction parameter, respectively, subjected to the boundary conditions:(7)u=uwx,y+K1ν+1μϵΓ∂u∂y−16Γϵ3μ∂u∂y3,−T=Tw=Tc−Axβ12,vx,y=0−D∂C∂y=hCw−C,aty=0,ux,y⟶0,∂u∂y⟶0,T⟶T∞C⟶C∞asy⟶∞.The thermal radiation term in equation (5) is obtained in [46] as follows:(8)qr˜=−4ς3d∗∂T4∂y,where d∗ denotes the mean absorption coefficient and ς specifies the Stefan–Boltzman constant. Assume that the relation between the fluid temperature and the fluid temperature far away from the surface is considered to be a linear function. By using Taylor’s expansion of T4 about T∞ we obtain(9)T4≈4T∞3T−3T∞.Substituting (9) in (8), we get(10)qr˜=4ς3d∗T∞3∂T∂y.The Arrhenius term in equation (6) was analyzed by Arrhenius [44, 47].(11)Γ2=KrC−C∞TT∞nexp−EςT,where ς=8.61x10−5eV is assigned for Boltzman constant, while n is stand for unit-less exponent fitted rate parameter that generally lies in the range −1<n<1 and E is assigned for activation energy.


## 3. Magnetic Dipole

The influence of magnetic dipole by the effects of the magnetic field. Some variation can be prescribed by the well-known function; so-called magnetic scalar potential can be seen in reference [28], which can be formulated as follows:(12)Φ=γ22πxx2+y+γ12,where γ2 represent the strength of the magnetic field and ℋx and ℋ are assigned for the magnetic field intensity along the x,y axis, and it can be written as follows:(13)ℋx=−∂Φ∂x=−γ22π−γ1+y2+x2γ1+y2+x22,(14)ℋy=∂Φ∂y=γ22π2xγ1+yγ1+y2+x22.Generally, it is known as the magnetic body force that is the gradient of magnetic field ℳ; it can be formulated as follows:(15)ℋ=∂Φ∂x2+∂Φ∂y2.Invoking equations (11) and (13) in (14), it gives(16)∂ℋ∂x=−γ2πxy+γ14,(17)∂ℋ∂y=γ2π2xy+γ15−1y+γ13.Considering magnetization ℳ as a linear relation to the temperature, it is given by(18)ℳ=KpTc−T.The physical and geometrical representation of the ferrofluids is shown in Figure 1.


## 4. Simplification of the Problems

The dimensionless variables are introduced as follows:(19)φη,ζ=ημρfζ,ϑη,ζ=Tc−TT∞−To=ϑ1η+ζ2ϑ2ηPη,ζ=pμb=P1η+ζ2P2η.The dimensional coordinates are defined as follows:(20)ζ=xxρNμ,η=yρNμ,where φη,ζ, ϑη,ζ, and Pη,ζ represent the dimensionless stream function, pressure, and temperature, respectively. The velocity is expressed as follows:(21)u=∂φ∗∂y=Nf′η,v=−∂φ∗∂x−μNρfη.By using the similarity transformations in equations (18)(20) and (11)(17) in Equations (3)(6), these are transformed into the following ODEs system.(22)ff″−f′2+2P2+H+1f″′−2β2ϑ1γ+η2−Hβ1f′′2f′′′=0,P1′−f″−ff′−2β2ϑ1γ+η3=0,P2′−2β2ϑ1γ+η3+4β2ϑ1γ+η5=0,1+Rϑ1″+Prfϑ1′−2f′ϑ1′+2fβ2λ1ϑ1−ϵγ+η3=0,1+Rϑ′2′−Prfϑ2′+4f′ϑ2′+2fβ2λ1ϑ2−ϵγ2+η3+−λ1β2ϑ1−ϵ2f′γ+η3−4fγ+η5ϕ″+Scfϕ′−ScKr1+β3ϑ1+λ1ϑ2nexp−E1+β3ϑ1+λ1ϑ2ϕ=0.The boundary and initial condition become(23)f′0=1+A11+Hf″0−Hβ13f″03fη=0,ϑ1η=1ϑ2η=0,ϕ′0+Dt∗1−ϕ0=0aty=0.f′⟶0,ϑ1⟶0,ϑ2P1⟶P∞P1⟶0,ϕ⟶0,aty⟶∞,where f′,ϑ1,ϕ represent the dimensionless form of the velocity, temperature, and concentration, respectively. Furthermore, β2, ϵ, and λ1 are represent the magnetic dipole parameter, the Curie temperature parameter, and viscous dissipation parameter, respectively. Kr=ko/a is a parameter of reaction rate; Sc=ν/DB is the Schmidt number; H=1/μΓb and β1=ρuw/2xb2μ are fluid parameters; Pr=μCp/k∞ is Prandtl number; and Dt=Dℬ/DA ratio of diffusion coefficient. Furthermore, λ1=dμ2/ϱkTw−T∞ is the viscous dissipation parameter; ϵ=Tc/Tc−Tw is the Curie temperature; λ1=γ1ϱμokTc−Tw/2πμ2 is the ferromagnetic parameter; and γ=γ1ϱd/μ is the distance from the center of the magnetic dipole to the origin parameter.The skin friction coefficientCf and Nusselt number Nux are expressed as follows:(24)Cf=τwϱuw2,Nux=xq˜wκTw−T∞,where qw is the heat flux, τw represents the skin friction along the sheet, and jm is the mass flux from the surface and is given as follows:(25)τw=μ1+1Γϵ∂u^∂y+1Γϵ6∂u^∂y3y=0q˜w=−κ∂T∂y−∂qr˜∂yy=0,where q˜m and uw are heat transfer and wall shear stress, respectively. The quantities defined in equation (16) are transformed into dimensionless form as follows:(26)CfRex=1+Hf″0−Hβ13f″03,NuxRex˜=−1+43Rdϑ1′0+ζ2ϑ2′0.


## 5. Shooting Method for the Proposed Solution

The shooting method is a numerical technique; generally, it is used for the solution of non-linear boundary value problems. Further detail can be seen in reference [48]. Equation (21) is the system of non-linear coupled ODEs of order three in gη, order two in ϑ1η, order two in ϑ2η, and order two in ϕη. The rearranging equation (21) with boundary condition in equation (22) will take the following form:(27)f″′=−ff″−f′2+2P2−2β2ϑ1/γ+η2H+1−β1Hf′′2,P1′=f″+ff′+2β2ϑ1γ+η3,P2′=2β2ϑ1γ+η3−4β2ϑ1γ+η5,ϑ1″=−Prfϑ1′−2f′ϑ1′+2fβ2λ1ϑ1−ϵ/γ+η31+R,ϑ2″=−Prfϑ2′−4f′ϑ2′+2fβ2λ1ϑ2−ϵ/γ+η3+λ1β2ϑ1−ϵ2f′/γ+η3−4f/γ+η51+R,ϕ″=−Scfϕ′−ScKr1+β3ϑ1+λ1nexp−E1+β3ϑ1+λ1ϑ2ϕ.To compute this system, we need to transform these coupled ODEs into a system of non-linear first-order ODEs. For this, we define these new variables as follows:(28)f=u1,f′=u2,f″=u3f‴=u3′,ϑ1=u4,ϑ1′=u5ϑ1′′=u5′,ϑ2=u6,ϑ2′=u7ϑ2′′=u7′,P2′=u8P1′=u9ϕ=u10,ϕ′=u11andϕ″=u11′.By using these, we obtained the following non-linear first-order ODEs system:(29)u1′=u2,u2′=u3,u3′=−u1u3−u22+2u8−2β2u4/γ+η2H+1−β1Hu32,u4′=u5,u5′=−Pru1u5−2u2u5+2u1β2λ1u4−ϵ/γ+η31+R,u6′=u7,u7′=−Pru1u7−4u2u7+2u1β2λ1u6−ϵ/γ2+η3+λ1β2u4−ϵ2u2/γ+η3−4u1/γ+η51+R,u9=u3+u1u2+2β2u4γ+η3,u8=2β2u4γ+η3−4β2u4γ+η5,u10′=u11,u11′=−Scu1u11−ScKr1+β3β3u6+λ1u4nexp−E1+β3u6+λ1u4u10,where the prime indicates the derivative, and the transform boundary conditions are given as follows:(30)u1=0,u20=1+A11+Hu30−Hβ13u303,u3=S1,u4=1u5=S2u6=0,u7=S3,u8=S4u9=S5u10=1u11=S6atη=0u2=0,u4=0,u6=0,u8=0,u9=−P∞andu11=0atη⟶∞.To evaluate the solution of these seven ODEs systems (equation (29)) with boundary conditions by shooting method, we need eleven initial guesses whereas five of them are given and the other six initial guesses u2η,u4η, u6η, u8η, u9η, and u11η are defined as η⟶∞. Hence, it is considered that u30,u50,u70,u80,u90,u110=q1,q2,q3,q3,q5,q6. These unknown six initial guesses q1,q2,q3,q3,q5,q6 are computed by the Newton iterative scheme through RK-45. The main step of this numerical solution is to select the suitable finite boundary conditions. The step size and convergence criteria are taken h=0.02 and TOL=10−5, respectively, for our numerical solution.


## 6. Validation of the Numerical Solution

In this section, we validate our numerical results with the previously published work. The comparison ofϑ1′0 is shown in Table 1, and we found a good agreement between these two methods. We have to evaluate the accuracy of the current method. For this, we computed the different values of Pr. We used a shooting method programming in MATLAB with step size h=0.02 to get the velocity concentration and temperatures profiles.Table 1
Validation forϑ1′0 at H=0.0 and R=β1=β2=0 for Pr with the published work.
PrIshak et al. [49]Chen [50]Present work0.721.08851.088621.08861.01.33331.33331.33333.02.50972.509722.509510.04.79684.796824.7943


## 7. Results and Discussions

In this section, we solved the governing equations (13)–(15) with the shooting method. Here, we took h=0.002. The tolerance for convergence has been taken TOL=10−5. Moreover, we compared the current result with the previously published work (see Table 1). We present the characteristics of parameters involved in equations (13)–(15). The characteristics of velocity, temperature, and concentration are presented in Figures 2–13. Also, the computational results of the skin friction, heat, and mass transfer rate are presented in Table 2.Figure 2
The result ofH on f′η.Figure 3
The result ofβ1 on f′η.Figure 4
The distribution ofϑη for β2.Figure 5
The gradient ofϑ1η for Pr.Figure 6
The impact ofRd for ϑ1η.Figure 7
The distribution ofϑ1η for λ1.Figure 8
Concentration distributionϕη for Krϕη.Figure 9
The impact ofn for ϕη.Figure 10
The characteristics ofSc for the distribution of ϕη.Figure 11
The properties ofE for the gradient of ϕη.Figure 12
The impact ofDt on concentration field ϕη.Figure 13
The properties ofβ3 for the gradient of ϕη.Table 2
The properties of−f″0,−ϑ10, and ϕη for λ1,Pr,β2, R,β,E,β3,Kr,H,γ, and Dt.
Prβ2λ1RdβEβ3KrHγDt−f″0ϑ10ϕ012110.10.210.2110.51.04411.34850.432691.31.04461.47160.436631.61.45801.58200.439621.51.01411.36290.434852.01.04411.34850.432692.51.07251.33420.4305911.04411.34850.432691.31.03281.46320.403281.61.02181.58590.0429111.04411.34850.432691.31.39701.22860.430411.61.03551.13100.425080.21.01411.36290.434850.31.02901.36000.452340.41.06431.35380.432340.21.04411.34850.432690.41.04411.34850.450990.81.06431.35380.480601.01.04411.34850.482982.01.04411.34850.432693.01.06431.35380.384920.01.04411.34850.577300.21.04411.34850.482980.41.06431.35380.297631.01.04411.34850.432691.50.92041.41410.449052.08.31141.46630.460711.01.04411.34850.432691.50.96241.38650.437721.80.94451.39440.390100.51.04411.34850.432691.01.04411.34850.355731.51.04411.34850.30201The characteristics of ferromagnetic interaction parameterβ2i for axial velocity f′η in Figure 14. The viscosity of the fluid arises due to the existence of ferrite particles; therefore, the velocity reduces for higher values of β2. Figure 2 represents the velocity field for (material fluid parameter H). It increases the variation of velocity profile by enlarging the values of H. The parameter H is inversely proportional to the non-Newtonian fluid dynamic viscosity. The enhancement in H will reduce the resistance of the flow. Therefore, the velocity of the fluid is increased due to the decreasing flow resistance.Figure 14
Characteristics ofβ2 for the velocity distribution f′.Figure3 demonstrates the velocity profile for β1. It is seen that the impacts for β1 on the velocity field increase both the film thickness. Velocity of the sheet is directly related to the fluid parameter β1, and because of the larger stretching speed, the velocity of the fluid also increases. Figure 4 represents the characteristics of β2 on the temperature distribution. Here, the temperature field increases for the higher number of β2. This is due to the interaction in the action of the ferromagnetic field and the motion of the fluids. Reduction occurs in the interaction for velocity field, thereby arising the frictional heating in the boundary layers; therefore, thermal boundary layer thickness increases.Figure15 shows the effects of the velocity slip parameter on the velocity of the fluid. Here, the velocity of the fluid is reduced due to the higher values of slip parameter γ1. This reduction occurs because the velocity of fluid near the sheet is not the same as the velocity stretching sheet. Figure 5 represents the characteristics of the temperature for the higher values of (Prandtl number, Pr). This is due to the fluid containing a small value of Pr having a high thermal diffusivity. Figure 6 is made for the temperature field with the unlike numbers radiation parameter Rd. If we increase the values of Rd, then the temperature profile increases. This increase occurs due to the increase in temperature T¯∞3 in radiation.Figure 15
The properties ofA1 for the gradient of f′η.Figure7 demonstrates the characteristics of λ1 on the temperature field. Clearly it is noticed that the temperature declines for higher values of λ1. The physical facts behind the extraordinary characteristics of ferrofluid. Impacts for larger (viscous dissipation parameter Impacts for larger λ1) thermal conductivity of fluid reduces and as a result temperature field is declined.Figure8 represents the characteristics of the concentration gradient for the chemical reaction (Kr). Here reduction occurs in the concentration gradient for the improving numbers of Kr. Here, it is observed that the concentration field decreases due to the higher number of Kr. Figure 9 reports the distribution of the concentration field for the fitted rate constant n. The result has shown that reduction occur in the concentration field due to the improving values of fitted rate constant n. Figure 10 designates the concentration gradient for higher values of Sc. The results show that the concentration profile is decreased by enlarging the values of Sc. Figure 11 demonstrates the influence of (activation energy E) on concentration field ϕ. Here, it is seen that the concentration field is improving for higher valves of E. Figure 12 shows the characteristics of the concentration field for unlike values of Dt. It is noted that the concentration field arises from higher values of Dt. The properties of temperature difference parameter (β3) on the concentration field are shown in Figure 13. Here, the result shows that the concentration field reduces due to larger values of β3.Figure16 is plotted for β2 for pressure gradient. When the concentration increases near the surface, the velocity of the ferrofluids declines. The velocity of field and pressure are inversely proportional to each other. Therefore, when the velocity in a fluid declines, then the pressure arises. Thus, the same will be the behavior of larger parameter of β2.Figure 16
Characteristics ofPr for the velocity distribution P2η.The characteristics ofPr on pressure field have plotted Figure 17. When Prandtl number Pr increases, it leads to in larger of the viscosity or viscous forces. Therefore, the pressure enlarges for higher values of Prandtl number Pr.Figure 17
The result ofβ2 on P2η.Figure18 demonstrates the characteristics of λ1 on pressure field. It is seen that from this figure, the pressure reduces due to higher values of λ1.Figure 18
The result ofλ1 on P2η.Table2 presents the properties of Sherwood numbers, Nusselt numbers, and skin frictions for different physical parameters. Skin friction increases for the higher values H and λ1. Skin friction reduces for larger values β1 and β2. Nux increases for the higher values of H, Rdγ, and β1. Nux Nusselt number reduces for the higher values of Rd, Pr, and λ1. The Sherwood number steps up due to the higher values of Kr, β3, Dt, and Rd. Sherwood numbers Shx reduce for the higher values of γ,HPr, λ1, and E.


## 8. Conclusions

In this work, we analyzed the effects of activation energy in Eyring–Powell ferrofluid fluid in the existence of velocity slip boundary conditions. Furthermore, the impacts of thermal radiation, velocity slip, and mass convective boundary conditions are considered. Here, we transformed the governing model into a non-linear coupled ODEs by using similarity transformation. We determined the computational solution of the transform ordinary differential system by applying the shooting method with the Runge–Kutta method (RK-45). The characteristics of concentration, temperature, pressure, and velocity profiles for various physical parameters are presented through graphs. The behavior of the Sherwood numbers, heat transfer rate, and skin friction coefficients are displayed in tables. In this article, we arrive at the following conclusion [50]:(1)
The reduction occurs in velocity fieldf′0 due to enlarging values of β2, H, and M(2)
There is an increase in temperature gradientsϑ1η due to the higher values of Rd and H(3)
There is a decrease in temperature gradientsθη due to the higher values of Pr(4)
There is an increment in the concentration fieldϕη for higher values of E(5)
There is an increase in the concentration fieldϕη for larger values of Sc, Kr, n, and β3(6)
There is an increase in the pressure fieldP2η for larger values of λ1 while having reduction for higher values of β2 and Pr



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