# Finite Element Method Solution of Boundary Layer Flow of Powell-Eyring Nanofluid over a Nonlinear Stretching Surface

Authors: Wubshet Ibrahim; Gosa Gadisa
Journal: Journal of Applied Mathematics
 (2019)

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## Abstract

The nonlinear convective flow of Eyring-Powell nanofluid using Catteneo-Christov model with heat generation or absorption term and chemical reaction rate over nonlinear stretching surface is analyzed. The simultaneous nonlinear partial differential equations governing the boundary layer flow are transformed to the corresponding nonlinear ordinary differential equations using similarity solution and then solved using Galerkin finite element method (GFEM). The impacts of pertinent governing parameters like Brownian diffusion, thermophoresis, mixed convection, heat generation or absorption, chemical reaction rate, Deborah numbers, Prandtl number, magnetic field parameter, Lewis number, nonlinear stretching sheet, and Eyring-Powell fluid parameters on velocity field, temperature, and nanoparticle concentration are given in both figures and tabular form. The result shows that the rise in chemical reaction rate will improve mass transfer rate and reduce heat transfer rate and local buoyancy parameter has quit opposite effect. The attributes of local skin friction coefficient, Nusselt number, and Sheer wood number are investigated and validated with existing literatures.


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## Body

## 1. Introduction

Due to wide application of non-Newtonian fluids in the advancement of modern technologies, many researchers engaged to explore the non-Newtonian fluid. The investigators exposed that non-Newtonian fluids are applicable in polymer devolatilization and processing, heat exchangers, extrusion process, wire and fiber coating, chemical processing equipment, etc. Different researchers studied different non-Newtonian models by different solution method. Modeling of the viscoelastic properties of polymers has often been a very hot subject. The viscoelastic constitutive equations have been developed for the understanding of the numerous means of deformation and flow but unluckily have not offered us quantitative prophetic power. Extremely often the predictions depend on the model applied for the computations and are not confirmed with experimental observations. Few viscoelastic boundary layer flow problems can be solved with the suitable constitutive equations, but this is still an area of academic research with partial practical applications at the moment.Most of the products we enjoy today in our daily life are invented from polymers. Some of them are house wares, toys, knobs, appliance parts, electrical fixtures, toothbrushes, cups, handles, lids, and packages. The Eyring-Power law model which is often used in chemical engineering is also capable of modelling these complex fluid flows under general flow conditions. Many fluids do not exhibit the same type of rheological attributes. They exhibit Newtonian character for a range of shear stresses and non-Newtonian character for some other ranges of shear stresses. In some cases, these types of fluids are suitable to be investigated. Among this kind of scenario, the present study paid attention to Eyring-Powell model which has more advantage as a result of its reduction to Newtonian model for smaller and larger shear rates. To give clear insight on non-Newtonian flow especially Eyring-Powell fluid, recently much research has been conducted. We discuss few of them as follows.Satyabanet al. [1] studied mixed convective flow of a Powell-Eyring fluid over a nonlinear stretching surface with thermal diffusion and diffusion thermo. Khan etal. [2] analyzed flow of an Eyring-Powell fluid over a stretching sheet. Gaffaret al.[3] investigated non-Newtonian thermal convection of Eyring-Powell fluid from an isothermal sphere with Biot number effects. Obaidet al. [4] studied entropy generation in MHD Eyring–Powell fluid flow over an unsteady oscillatory porous stretching surface under the impact of thermal radiation and heat source/sink. The effect of Cattaneo-Christov heat flux model for Eyring-Powell fluid over an exponentially stretching sheet is discussed by Ahmad and Iqbal [5]. The boundary layer flow of the non-Newtonian power-law fluid past the nonlinear stretching sheet has been analyzed by Megahed [6, 7]. Later on, Megahed [8] introduced the concepts of flow and heat transfer of non-Newtonian Sisko fluid past a nonlinearly stretching sheet with heat generation and viscous dissipation. Upadhay and Raju [9] explored Cattaneo-Christov on heat and mass transfer of unsteady Eyring-Powell dusty nanofluid over sheet with heat and mass flux conditions. Kumaret al. [10] discussed the concept of magnetohydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink. Later on, Kumaret al. [11] introduced impact of nonlinear radiation on MHD nonaligned stagnation point flow of micropolar fluid over a convective surface. Beside this, Kumaret al. [12] investigated the impact of Brownian motion and thermophoresis on bioconvective flow of nanoliquids past a variable thickness surface with slip effects.Hayatet al. [13] described MHD nonlinear stretching flow of Powell–Eyring nanomaterial. Madhu and Kishan [14] derived MHD boundary-layer flow of a non-Newtonian nanofluid past a stretching sheet with a heat source/sink. Rahimiet al. [15] presented solution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linear stretching sheet by collocation method. Akinshilo and Olaye [16] explained the analysis of the Eyring-Powell model based fluid flow in a pipe with temperature dependent viscosity and internal heat generation. Madhu and Kishan [17] analyzed finite element analysis of heat and mass transfer by MHD mixed convection stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface with radiation. Babuet al. [18] described free convective MHD Cattaneo-Christov flow over three different geometries with thermophoresis and Brownian motion.Moreover, large temperature difference between the surface and the ambient fluid forces the researchers to investigate the nonlinear density temperature (NDT) difference in the buoyancy force term due to its major influence on the flow and heat transfer characteristics. Vajravelu and Sastri [19] investigated the flow between two parallel plates in view of the quadratic density temperature (QDT) variation and reported that the flow and heat transfer rates are considerably affected by it. The fully developed free convection flow in circular pipe with nonlinear density temperature variations was studied by Bhargava and Agarwal [20]. Khanet al. [21] investigated entropy generation in radiative motion of tangent hyperbolic nanofluid in presence of activation energy and nonlinear mixed convection. The nonlinear convection effects on the flow past a flat porous plate have been investigated by Vajravelu et al. [22]. Khanet al. [23] studied the numerical simulation of nonlinear thermal radiation and homogeneous-heterogeneous reactions in convective flow by a variable thickened surface.Most recently, Hayat and Nadeem [24] solved flow of 3D Eyring-Powell fluid by utilizing Cattaneo-Christov heat flux model and chemical processes over an exponentially stretching surface. In the same year, Saima and Noreen [25] presented magneto-thermo-hydrodynamic peristaltic flow of Eyring-Powell nanofluid in asymmetric channel. Later on, Wubshet Ibrahim and Bullo Hindebu [26] analyzed MHD boundary layer flow of Eyring-Powell nanofluid past stretching cylinder with Cattaneo-Christov heat flux model. Kumaret al. [27] analyzed impact of Lorentz force on unsteady bioconvective flow of Carreau fluid across a variable thickness sheet with non-Fourier heat flux model. Ramadeviet al. [28] scrutinized MHD flow of Carreau fluid over a variable thickness melting surface subject to Cattaneo-Christov heat flux. Kumaret al. [29] have been studying MHD Carreau fluid flow past a melting surface with Cattaneo-Christov heat flux.From the surveyed literatures above, it has been observed that the problem of nonlinear convective flow of magnetohydrodynamic (MHD) boundary layer flow of an Eyring-Powell nanofluid using Catteneo-Christov model with heat generation/absorption and chemical reaction term past nonlinear stretching surface is still limited. Additional effects of local buoyancy parameter, local heat generation, chemical reaction rate, nonlinear stretching sheet parameter, and the technique used to solve the problem are added features to the uniqueness.Thus, the main target of the present study is to analyze nonlinear convective flow of two-dimensional steady laminar incompressible magnetohydrodynamic (MHD) boundary layer flow of an electrically conducting Eyring-Powell nanofluid using Catteneo-Christov model with heat generation/absorption and chemical reaction over nonlinear stretching surface. The impacts of diverse governing variables on fluid velocity, temperature of the fluid, and nanoparticle concentration profile of laminar boundary layer flow are presented in both graphs and tabular form. The numerical method employed to solve the current problem is Galerkin finite element method (GFEM) [21–30].


## 2. Mathematical Formulation of the Problem

We analyze a nonlinear convective flow of two-dimensional steady laminar incompressible magnetohydrodynamic (MHD) boundary layer flow of Eyring-Powell nanofluid using Catteneo-Christov model with heat generation or absorption and chemical reaction over nonlinear stretching surface. The diagram of the targeted incompressible boundary layer flow is illustrated in Figure1. The x-axis is considered along the nonlinear stretching surface and the y-axis is taken normal to it. The surface is extended with a nonlinear velocity uw(x)=axn, where a>0 is constant.The induced magnetic field is ignored owing to small Reynolds number.Figure 1
Geometry of the problem.The extra stress tensor for Powell-Eyring fluid is given by [30](1)τij=μ∂ui∂xj+1β∗sinh-1⁡1c∗∂ui∂xj,where μ represents dynamic viscosity and β∗ and c∗ stand for materials constants.Assuming(2)sinh-1⁡1c∗∂ui∂xj≅1c∗∂ui∂xj-161c∗∂ui∂xj3,1c∗∂ui∂xj<<1,the governing equations for the boundary layer flow are(3)∂u∂x+∂v∂y=0(4)u∂u∂x+v∂u∂y=ν+1ρfβ∗c∗-12ρβ∗c∗3∂u∂y2∂2u∂y2-σB02ρfu+gΛ1T-T∞+gΛ2T-T∞2+gΛ3C-C∞+gΛ4C-C∞2(5)u∂T∂x+v∂T∂y+λEu∂u∂x∂T∂x+v∂v∂y∂T∂y+u∂v∂x∂T∂y+v∂u∂y∂T∂x+2uv∂2T∂x∂y+u2∂2T∂x2+v2∂2T∂y2=αf∂2T∂y2+τDB∂T∂y∂C∂y+DTT∞∂T∂y2+QρfcpT-T∞(6)u∂C∂x+v∂C∂y+λCu∂u∂x∂C∂x+v∂v∂y∂C∂y+u∂v∂x∂C∂y+v∂u∂y∂C∂x+2uv∂2C∂x∂y+u2∂2C∂x2+v2∂2C∂y2=DB∂2C∂y2+DTT∞∂2T∂y2-k1C-C∞with appropriate boundary condition(7)u=uwx=axn,v=0aty=0u→0asy→∞T=Tw,C=Cwaty=0T→T∞,C→C∞asy→∞,where u and v are the velocity components in the x and y directions, ν is the kinematic viscosity, uw is the stretching velocity, g is the gravitational acceleration, Λ1 and Λ2 are linear and nonlinear thermal expansion coefficients due to temperature and Λ3 and Λ4 are linear and nonlinear thermal expansion coefficients due to concentration, ρf is density of base liquid, cp is specific heat capacity of the base fluid, αf is thermal diffusivity of the base fluid, T is temperature, C is concentration, σ is electric conductivity, B02 is magnetic parameter, k1 is a reaction rate, DB is the Brownian diffusion coefficient, DT is the thermophoresis diffusion coefficient, and τ is the ratio of the effective heat capacity of the nanoparticle material and the heat capacity of base fluid.These simultaneous partial differential equations (PDEs) (3)-(6) with the corresponding boundary conditions (7) are reduced to ordinary differential equations (ODEs) using similarity transformation. In this case, continuity equation and momentum equation are reduced to a single ODE and energy equation and concentration equation are reduced to another ODE.We know set up the similarity transformation as follows [31]:(8)η=yan+12νxn-1/2,v=-aνn+12xn-1/2fη+n-1n+1ηf′ηu=axnf′η,ψ=2νan+1xn+1/2fη,θη=T-T∞Tw-T∞andϕη=C-C∞Cw-C∞,where η is a similarity variable ψ stream function defined asu = ∂ ψ / ∂ y and v=∂ψ/∂x which identically satisfies continuity equation (3), f is dimensionless function, θ is dimensionless temperature, and ϕ is dimensionless nanoparticle concentration.Substituting (8) into (4)-(6), we obtain the following corresponding simultaneous nonlinear ordinary differential equations:(9)ff′′-2nn+1f′2+1+εf′′′-εδn+12f′′2f′′′-2n+1Mf′+2λn+11+βtθθ+2λn+1N∗1+βcϕϕ=0(10)θ′′+Pr⁡fθ′+Pr⁡γEn-32ff′θ′-n+12f2θ′′+2n+1Pr⁡βθ+Pr⁡Nbθ′ϕ′+Pr⁡Ntθ′2=0(11)ϕ′′+Pr⁡Lefϕ′+Pr⁡Leγcn-32ff′ϕ′-n+12f2ϕ′′+NtNbθ′′-2n+1Pr⁡Leβ1ϕ=0with the corresponding boundary conditions(12)f=0,f′=1atη=0,f′→0asη→∞θ=1atη=0,θ→0asη→∞ϕ=1atη=0,ϕ→0asη→∞,where(13)Pr=νfαf,γE=λEaxn-1,γC=λCaxn-1,ε=1μβ∗c∗,δ=a3x3n-12νc∗2,M=σB02xn-1aρf,λ=GrRex2,βt=Λ2Λ1Tw-T∞,βc=Λ4Λ3Cw-C∞,N∗=Gr∗Gr=Λ4Λ3Cw-C∞Tw-T∞,Gr=gΛ1ν2Tw-T∞x3,Gr∗=gΛ3ν2Cw-C∞x3,Nb=DBτCw-C∞ν,Nb=DTτTw-T∞T∞ν,β=Qaxn-1ρcpf,Le=αfDB,β1=ka,where Pr is the Prandtl number, γE is the Deborah number with respect to the relaxation time of the heat flux, γC is the Deborah number due to nanoparticle concentration, ε and δ are Eyring-Powell fluid parameters, M is magnetic field parameter, λ is mixed convection parameter/local buoyancy parameter, βt is nonlinear convection parameter due to temperature, βc is nonlinear convection parameter due to concentration, N∗ is ratio of concentration to thermal buoyancy forces, Gr is Grashof number in terms of temperature, Gr∗ is Grashof number in terms of concentration, Nb is Brownian motion parameter, Nt is thermophoresis parameter, β is heat generation/absorption term, Le is Lewis number, and β1 is chemical reaction rate.The Engineering quantities of interest in the study are local skin coefficient frictionCf, the Nusselt number Nux and the nanoparticle of Sherwood number Shx are given by Cf=τw/ρuw2,Nux=xqw/kTw-T∞,Shx=xqnp/DBCw-C∞, where τw is the wall skin friction, qw is the heat flux and qnp is the nanofluid mass flux at the surface of the sheet defined as follow:(14)τw=μ+1βC∂u∂y-161C∂u∂y3,qw=-k∂T∂yy=0,qnp=-DB∂ϕ∂yy=0.After replacing the values from similarity transformation in (8) into (14), we obtain(15)Rex1/2Cf=n+121+εf′′0-εδ3f′′03,Nur=Nuxn+1/2Rex1/2=-θ′0andShr=Shxn+1/2Rex1/2=-ϕ′0.Here, Nur and Shr are reduced form of Nusselt number and nanoparticle Sherwood number, respectively, and the local Reynolds number depending on the nonlinear stretching surface velocity is given by (16)Rex=uxxν.


## 3. Method of Solution

We employed the Galerkin finite element method which is a prominent technique to solve nonlinear differential equations through the following five steps that are essential to realize what the approach of the FEM is (see [13, 32–34]):(i)
Discretization of the domain into elements(ii)
Element formulation which is called setup of differential equation in its weak form(iii)
Assembly of element equations known as setup of global problem or obtaining equations for the entire system from the equations for one element(iv)
Impose the appropriate boundary condition(v)
Solve the system of equations.An attribute feature of the finite element method is that, instead of seeking the approximation solution over the entire region, the region is partitioned into smaller elements and the approximation is then carried out over each element. After determining the character of all single elements, the elements is patched together, which hopefully enables us to get an approximate solution over the entire body.Assuming(17)f′=gthe simultaneous nonlinear ordinary differential equations in (9)-(11) with the boundary conditions in (12) are reduced to(18)fg′-2nn+1g2+1+εg′′-εδn+12g′2g′′-2n+1Mg+2λn+11+βtθθ+2λn+1N∗1+βcϕϕ=0(19)θ′′+Pr⁡fθ′+Pr⁡γEn-32fgθ′-n+12f2θ′′+2n+1Pr⁡βθ+Pr⁡Nbθ′ϕ′+Pr⁡Ntθ′2=0(20)ϕ′′+Pr⁡Lefϕ′+Pr⁡Leγcn-32fgϕ′-n+12f2ϕ′′+NtNbθ′′-2n+1Pr⁡Leβ1ϕ=0,with the corresponding boundary conditions(21)f=0,g=1atη=0,g→0asη→∞θ=1atη=0,θ→0asη→∞ϕ=1atη=0,ϕ→0asη→∞.At this step, we discretize the whole domain into elements called Mesh.The second step is writing the weighted integral form associated with (17), (18), (19), and (20) over a typical three-node linear element ηe,ηe+1 as follows:(22)∫ηeηe+1w1f′-gdη=0(23)∫ηeηe+1w2fg′-2nn+1g2+1+εg′′-εδn+12g′2g′′-2n+1Mg+2λn+11+βtθθ+2λn+1N∗1+βcϕϕdη=0(24)∫ηeηe+1w3θ′′+Pr⁡fθ′+Pr⁡γEn-32fgθ′-n+12f2θ′′+2n+1Pr⁡βθ+Pr⁡Nbθ′ϕ′+Pr⁡Ntθ′2dη=0(25)∫ηeηe+1w4ϕ′′+Pr⁡Lefϕ′+Pr⁡LeγCn-32fgϕ′-n+12f2ϕ′′+NtNbθ′′-2n+1Pr⁡Leβ1ϕdη=0,where w1,w2,w3, and w4 are arbitrary weight functions and may be considered as the variation in f,g,θ, and ϕ, respectively, and domain ηe,ηe+1 represents the interval of the boundary layer region.At the third step, we seek approximation solution of the form(26)f=∑j=13fjψj,g=∑j=13gjψj,θ=∑j=13θjψj,ϕ=∑j=13ϕjψjwith w1=w2=w3=w4=ψi(i=1,2,3), the quadratic shape functions ψi defined as(27)ψ1e=ηe+1-ηηe+1+ηe-2ηηe+1-ηe2,ψ2e=4η-ηeηe+1-ηηe+1-ηe2,ψ3e=-η-ηeηe+1+ηe-2ηηe+1-ηe2,where ηe≤η≤ηe+1At the fourth step, replacing approximate solution (26) into (22)-(25), we obtain the finite element model for the equation which is given by(28)KeYe=Fe,where [Ke] denotes the elemental stiffness matrix, [Ye] is the vector of elemental nodal variables (unknowns), and [Fe] is the force vector expressed as follows:(29)Ke=K11K12K13K14K21K22K23K24K31K41K32K42K33K43K34K44,Ye=fgθϕ,Fe=h1h2h3h4,where each [Kmn] is of the order 3x3 and [hm],(m,n=1,2.3,4) is of the order 3x1.These matrices are defined as(30)Kij11=∫ηeηe+1ψi∂ψj∂ηdη,Kij12=-∫ηeηe+1ψiψjdη,Kij13=0,Kij14=0,Kij21=0Kij22=∫ηeηe+1ψif¯∂ψj∂ηdη-2nn+1∫ηeηe+1ψig¯ψjdη-1+ε∫ηeηe+1∂ψi∂η∂ψj∂ηdη-εδn+12∫ηeηe+1ψig′¯g′′¯∂ψj∂ηdη-2n+1M∫ηeηe+1ψiψjdη,Kij23=2λn+1∫ηeηe+1ψiψjdη+2λn+1βt∫ηeηe+1ψiθ¯ψjdηKij24=2λn+1N∗∫ηeηe+1ψiψjdη+2λn+1N∗βc∫ηeηe+1ψiϕ¯ψjdη,Kij31=0,Kij32=0,Kij33=-∫ηeηe+1∂ψi∂η∂ψj∂ηdη+Pr⁡∫ηeηe+1ψif¯∂ψj∂ηdη+Pr⁡γEn-32∫ηeηe+1ψif¯g¯∂ψj∂ηdη-Pr⁡γEn+12∫ηeηe+1ψif¯f¯∂2ψj∂η2dη+2n+1Pr⁡β∫ηeηe+1ψiψjdη+Pr⁡Nb∫ηeηe+1ψiϕ′¯∂ψj∂ηdη+Pr⁡Nt∫ηeηe+1ψiθ′¯∂ψj∂ηdη,Kij34=0,Kij41=0,Kij42=0Kij43=-NtNb∫ηeηe+1∂ψi∂η∂ψi∂ηdη,Kij44=-∫ηeηe+1∂ψi∂η∂ψi∂ηdη+Pr⁡Le∫ηeηe+1ψif¯∂ψi∂ηdη+Pr⁡LeγCn-32∫ηeηe+1ψif¯g¯∂ψj∂ηdη-Pr⁡LeγCn+12∫ηeηe+1ψif¯f¯∂2ψj∂η2dη-2n+1Pr⁡Leβ1∫ηeηe+1ψiψjdηhi1=0,hi2=-1+εψi∂g∂ηηeηe+1,hi3=-ψi∂θ∂ηηeηe+1,hi4=-ψi∂ϕ∂η+NtNbψi∂θ∂ηηeηe+1,where(31)f¯′=∑j=13fj¯∂ψj∂η,g′¯=∑j=13gj¯∂ψj∂η,θ′¯=∑j=13θj¯∂ψj∂η,ϕ′¯=∑j=13ϕj¯∂ψj∂η.


## 4. Results and Discussion

The numerical method employed to solve the present problem is influential method in solving nonlinear partial differential equations and called Galerkin finite element method (GFEM). The nonlinear partial differential equations governing the laminar boundary layer flow were converted to the matching nonlinear ordinary differential equations (ODEs) using appropriate similarity transformation and then passed through five necessary steps for solving differential equations using GFEM. We used Gaussian integration rule to evaluate the integrations in (30) and (31) and Matlab software is employed to produce the numerical result. Moreover, we interpreted the impact of physical parameters like magnetic field, Prandtl number, Brownian motion, thermophoresis, mixed convection/local buoyancy parameter, Deborah number with respect to the relaxation time of the heat flux and nanoparticle concentration, heat generation or absorption term, chemical reaction rate, ratio of concentration to thermal buoyancy forces, and stretching sheet parameter. The results are illustrated by tabular and graphical forms. The default values of the related parameters used for this study are(32)Pr=1.0,λ=0.2,ε=0.3,β1=0.3,δ=0.2,γE=0.2,γC=0.2,n=1.2,M=1.0,Nt=0.1,Nb=0.1,N∗=0.1,βt=0.3,Le=1.0,β=0.1,βc=0.2.These values are carefully chosen in accordance with previous literatures (see [31]).Figures2, 3, and 4 show the impacts of magnetic field variable on nanoparticle concentration, temperature profile, and velocity field, respectively. Figures 2 and 3 show the attribute of both nanoparticle concentration and temperature profile of the fluid when magnetic field varies. The graphs reveal that as magnetic field intensified, both the temperature and concentration graphs rise up. This is in agreement with the physical meanings of magnetic field which is directly related to Lorentz force that has the capability of increasing the nanoparticle volume fraction in the motion of the nanofluid and fluid temperature as well. Opposite to this as seen from Figure 4, the enhancement in magnetic field parameter reduces the fluid velocity. This is due to the enhancement of Lorentz force which opposes the motion of the fluid.Figure 2
Impact of magnetic parameter M on nanoparticle concentration.Figure 3
Impact of magnetic parameter M on temperature profile.Figure 4
Impact of magnetic parameter M on velocity field.It is observed from Figure6 that an increase in thermophoresis parameter increases the temperature of the fluid. This is a result of large thermophoretic force which causes large temperature gradient that produces movement of suspended particle from higher temperature to lower temperature.Figures5 and 7 also indicate that the enhancement in thermophoresis parameter will enhance both velocity of the fluid and nanoparticle concentration profile. This is due to large thermophoresis which causes particle migration to produce nonuniform distribution of nanoparticle concentration as seen from Figure 7.Figure 5
Impact of thermophoresisNt on velocity field.Figure 6
Impact of thermophoresisNt on temperature profile.Figure 7
Impact of thermophoresisNt on nanoparticle concentration.Brownian motion is the unsystematic motion of particles suspended in fluid resulting from their collision with high-speed moving molecules in the fluid. It is crystal clear that Brownian motion parameter enlargement leads to the increase in thermal boundary layer region as indicated in Figure9 and opposite to this the enlargement in Brownian motion parameter which causes in fluid motion the particle deposition away from fluid regime and then produces the nanoparticle volume fraction decrease or nanoparticle concentration reduction as shown in Figure 8.Figure 8
Impact of Brownian motionNb on nanoparticle concentration.Figure 9
Impact of Brownian motionNb on temperature profile.Figure10 reveals that the increase in heat generation or absorption parameter magnifies the temperature of the fluid and thermal boundary layer thickness of the fluid. This is simply as expected that adding heat to the fluid will rise temperature of the fluid. Figure 11 shows nanoparticle concentration diminishes as Lewis number increases. The Lewis number (Le) characterizes fluid flows where there are simultaneous heat and mass transfer. Thus, it is a measure of the relative thermal and mass-transfer (concentration) boundary layer thicknesses and hence increase in Lewis number also results in diminishing concentration boundary layer thickness.Figure 10
Impact of heat generation/absorptionβ on temperature profile.Figure 11
Impact of Lewis numberLe on nanoparticle concentration.Figures12 and 13 describe the notion that the amplification in Prandtl number results in diminishing both temperature and nanoparticle concentration profile, respectively. It also indicates a shrink in both thermal boundary layer and boundary layer width of concentration. Since Prandtl number itself relates momentum diffusivity and thermal diffusivity, it controls the relative thickness of the momentum and thermal boundary layers. Therefore, a large value of Pr implies heat diffuses slowly compared to the velocity (momentum).Figure 12
Impact of Prandtl numberPr on temperature profile.Figure 13
Impact of Prandtl numberPr on nanoparticle concentration profile.Figures14–19 show mixed convection parameter λ and Eyring-Powell fluid parameter ε have similar influences on velocity field, temperature, and nanoparticle concentration, respectively. Velocity field increases with enlargement in mixed convection parameter λ and both nanoparticle concentration and temperature of the fluid decrease with increase in λ. Eyring-Powell fluid parameter ε also exhibits similar property as shown in Figures 16–19.Figure 14
Impact of mixed convection parameterλ on velocity field.Figure 15
Impact of mixed convection parameterλ on nanoparticle concentration.Figure 16
Impact of mixed convection parameterλ on temperature profile.Figure 17
Impact of Eyring-Powell fluid parameterε on nanoparticle concentration.Figure 18
Impact of Eyring-Powell fluid parameterε on temperature profile.Figure 19
Impact of Eyring-Powell fluid parameterε on velocity profile.Figure 20
Impact of chemical reaction rateβ1 on nanoparticle concentration.As revealed in Figures21 and 22, increase in chemical reaction rate β1 reduces nanoparticle concentration; enhancement in Deborah number γE diminishes temperature profile and large values of Deborah number γC result in smaller value in nanoparticle concentration, respectively.Figure 21
Impact ofγE on temperature profile.Figure 22
Impact ofγC on nanoparticle concentration.Figures23 and 24 are drawn to manifest the effect of stretching sheet parameter on temperature and nanoparticle concentration profile, respectively. Stretching sheet parameter has decreasing impact on both temperature and nanoparticle concentration of the boundary layer flow.Figure 23
Impact of stretching sheet parametern on temperature profile.Figure 24
Impact of stretching sheet parametern on nanoparticle concentration.Tables1–3 pointed out that the results obtained in the present study in terms of engineering physical quantities of interest like local skin friction coefficient, local Nusselt number, and local Sheer wood number are in a good conformity with existing literature.Table 1
Comparison of the Nusselt numberθ′(0) when n=1.0,Pr=10.0,Le=1.0,ε=β1=δ=γE=γC=M=β=βt=βc=λ=0.0.
N t Nusselt number N b = 0.1 N b = 0.2 N b = 0.3 N b = 0.4 [35] [36] present [35] [36] Present [35] [36] present [35] [36] present 0.1 0.9524 0.95244 0.95239 0.5056 0.50561 0.50557 0.2522 0.25218 0.25217 0.1194 0.11940 0.11940 0.2 0.6932 0.69318 0.69318 0.3654 0.36536 0.36536 0.1816 0.18159 0.18159 0.0859 0.08588 0.08590 0.3 0.5201 0.52025 0.52019 0.2731 0.27313 0.27313 0.1355 0.13564 0.13554 0.0641 0.06424 0.06409 0.4 0.4026 0.40260 0.40258 0.2110 0.21100 0.21101 0.1046 0.10461 0.10461 0.0495 0.04962 0.04946 0.5 0.3211 0.32105 0.32109 0.1681 0.16811 0.16812 0.0833 0.08342 0.08330 0.0394 0.03932 0.03939Table 2
Comparison of the Sheer wood numberϕ′(0) when n=1.0,Pr=10.0,Le=1.0,ε=β1=δ=γE=γC=M=β=βt=βc=λ=0.0.
N t Sheer wood number N b = 0.1 N b = 0.2 N b = 0.3 N b = 0.4 [35] [36] present [35] [36] Present [35] [36] present [35] [36] Present 0.1 2.1294 2.12949 2.12938 2.3819 2.38186 2.38190 2.4100 2.41009 2.41002 2.3997 2.39970 2.39966 0.2 2.2740 2.27401 2.27400 2.5152 2.51537 2.51521 2.5150 2.51501 2.51500 2.4807 2.48066 2.48072 0.3 2.5286 2.52855 2.52855 2.6555 2.65550 2.65550 2.6088 2.60876 2.60884 2.5486 2.54848 2.54860 0.4 2.7952 2.79520 2.79519 2.7818 2.78181 2.78178 2.6876 2.68758 2.68761 2.6038 2.60380 2.60380 0.5 3.0351 3.03511 3.03511 2.8883 2.88830 2.88830 2.7519 2.75190 2.75188 2.6483 2.64831 2.64831Table 3
Comparison of the skin frictionf′′(0) when n=1.0,Pr=1.0,Le=1.0,β1=γE=γC=β=βt=βc=0.0.
ε λ δ M - f ′ ′ ( 0 ) - f ′ ′ ( 0 ) [31] present 0.1 0.1 0.1 0.5 1.2272 1.22720 0.2    1.2826 1.28261 0.3    1.3361 1.33566 0.1  0.5  1.2216 1.22158 0.3   1.1152 1.11520 0.5   1.0133 1.01330 0.1 0.1  1.2271 1.22720 0.3  1.2245 1.22445 0.5  1.2217 1.22168Table4 reveals that the skin friction coefficient is an increasing function of magnetic field parameter M, stretching sheet parameter n, and Eyring-Powell fluid parameter ε and is a decreasing function of mixed convection parameter λ, and Eyring-Powell fluid parameter δ. Table 4 also indicates that local Nusselt number is an increasing function of Prandtl number Pr, Deborah number γE, mixed convection parameter λ, stretching sheet parameter n, and Eyring-Powell fluid parameters and is a decreasing function of Deborah number γC, heat generation or absorption term β, thermophoresis parameter Nt, and Brownian diffusion parameter Nb.Table 4
Numerical values of skin friction coefficient, Nusselt number and Sheer wood number for different parameters.
n P r λ ε β 1 δ γ E γ C M N t N b L e β - f ′ ′ ( 0 ) - θ ′ ( 0 ) - ϕ ′ ( 0 ) 1.2 1.0 0.0 0.1 0.1 0.1 0.2 0.3 0.5 0.1 0.1 1.0 0.0 1.28652 0.53393 0.38052 1.5            1.28652 0.66299 0.48308 2.0            1.28652 0.76485 0.60785 1.0 0.1           1.23845 0.54212 0.38183 0.2           1.19147 0.54981 0.38327 0.0 0.2          1.34109 0.54197 0.38170 0.3          1.39448 0.54922 0.38300 0.1 0.2         1.28652 0.53208 0.47720 0.4         1.28652 0.52920 0.64069 0.1 0.2        1.28175 0.53423 0.38054 0.5        1.26850 0.53510 0.38060 0.1 0.5       1.28652 0.55509 0.35871 1.0       1.28652 0.61103 0.30691 0.2 0.5      1.28652 0.53359 0.38194 1.0      1.28652 0.39878 0.53207 0.3 1.0     1.46343 0.50976 0.37836 1.5     1.62013 0.49047 0.37790 0.5 0.3    1.28652 0.49660 -0.09250 0.5    1.28652 0.46138 -0.43839 0.1 0.3   1.28652 0.47553 0.58955 0.5   1.28652 0.42186 0.63049 0.1 1.5  1.28652 0.52835 0.59337 2.0  1.28652 0.52440 0.78393 1.0 0.2 1.28652 0.34687 0.55032 0.3 1.28652 0.22448 0.66179 0.5 1.28652 -0.15594 1.01042 0.5 1.0 0.0 0.1 0.1 0.1 0.2 0.3 0.5 0.1 0.1 1.0 0.0 1.26147 0.51567 0.42594 1.5             1.29252 0.54171 0.36815 2.0             1.29941 0.55482 0.35257Table4 also shows sheer wood number is an increasing function of Prandtl number Pr, Deborah number γC, mixed convection parameter λ, Eyring-Powell fluid parameters, chemical reaction rate β1, heat generation or absorption term β, Brownian diffusion parameter Nb, and Lewis number Le and is a decreasing function of Deborah number γE, stretching sheet parameter n, and thermophoresis parameter Nt.


## 5. Conclusion

We investigated the nonlinear convective flow of Eyring-Powell nanofluid using Catteneo-Christov heat and mass flux model over a nonlinear stretching surface in the presence of heat generation/absorption and chemical reaction. The Galerkin finite element method (GFEM) is used to solve the coupled highly nonlinear partial differential equations resulting from the problem formulation. The attributes of the laminar follow pertinent to different governing parameters are indicated as follows:(i)
Both nanoparticle concentration and temperature of the fluid are increasing functions of the magnetic field parameter and the velocity field is decreasing function of the magnetic field parameter(ii)
Velocity field, temperature, and nanoparticle concentration profile are increasing functions of thermophoresis parameter(iii)
The enlargement in Brownian motion yields large values in temperature and decrease in nanoparticle concentration(iv)
Both nanoparticle concentration and temperature profile of the fluid are decreasing functions of the Lewis and Prandtl numbers(v)
The temperature and nanoparticle concentration are reduced and opposite to this the velocity field increases when mixed convection parameter increases(vi)
Velocity field is increasing function of Eyring-Powell fluid parameterε and both temperature and concentration profiles are decreasing functions of ε(vii)
The rise in reaction rateβ1 and the Deborah number due to nanoparticle concentration γC diminishes nanoparticle concentration of the boundary layer flow(viii)
The larger stretching sheet parameter and the Deborah number with respect to relaxation time of heat fluxγE results in reduction in temperature profile of the flowing fluid(ix)
The rise in chemical reaction rate will improve mass transfer rate and reduce heat transfer rate and local buoyancy parameter has quit opposite effect.



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