School of Mathematics, Statistics,
and Computer Science,
University of KwaZulu-Natal,
Private Bag X01, Scottsville 3209,
Pietermaritzburg, South Africae-mail: firstname.lastname@example.org
School of Mathematics, Statistics,
and Computer Science,
University of KwaZulu-Natal,
Private Bag X01, Scottsville 3209,
Pietermaritzburg, South Africae-mail: email@example.com
School of Mathematics, Statistics,
and Computer Science,
University of KwaZulu-Natal,
Private Bag X01, Scottsville 3209,
Pietermaritzburg, South Africae-mail: firstname.lastname@example.org
Manuscript received November 5, 2013; final manuscript received April 9, 2014; published online May 2, 2014. Assoc. Editor: Malisa Sarntinoranont.
The combined effects of viscous and Joule heating on the stagnation point flow of a nanofluid through a stretching/shrinking sheet in the presence of homogeneous–heterogeneous reactions are investigated. The nanoparticle volume fraction model is used to describe the nanofluid. In this study, the density temperature relation is nonlinear which causes a nonlinear convective heat transfer. The surface of the sheet is assumed to be convectively heated with a hot fluid. The governing nonlinear differential equations are solved using the successive linearization method (SLM), and the results are validated by comparison with numerical approximations obtained using the Matlab in-built boundary value problem solver bvp4c and with existing results in literature. The nanofluid problem finds applications in heat transfer devices where the density and temperature relations are complex and the viscosity of the fluid has significant effect on the heat transfer rate.
The study of flow characteristics of viscous, incompressible fluids with suspended nanosized solid particles is highly significant due to the application of such fluids in heat transfer devices. The suspension of nanoparticles affects the thermal conductivity and the convective heat transfer rates in a wide variety of engineering devices. In some cases, such as in advanced nuclear systems, the presence of nanofluids enhances the heat transfer rates , while these can also be used as a coolant in several heat transfer devices. Choi et al.  first showed that the addition of nanoparticles increases the thermal conductivity of the base fluid by up to two times. It was shown by Masuda et al.  that a characteristic feature of the nanoparticles is to increase the thermal conductivity of the fluid. The topic of heat transfer in nanofluids has been surveyed in review articles by Das and Choi  and Wang and Mazumdar  and in a book by Das et al. .
The investigation of stagnation point flow of a viscous and incompressible nanofluid over a stretching/shrinking sheet is of considerable interest due their appearance in many engineering processes, such as extrusion of plastic sheets, continuous stretching of plastic films and artificial fibers, cooling of metallic plate, polymer extrusion, etc. With these facts in mind, the flow characteristics of such flows are studied by several researchers. Bachok et al.  investigated the two-dimensional stagnation point flow of a nanofluid over a stretching/shrinking sheet. In Ref.  they studied the heat transfer characteristics in steady two-dimensional stagnation point flow of a copper-water nanofluid over a permeable stretching/shrinking sheet. In these studies, the nanofluid problem is described using the nanoparticle volume fraction model. The boundary-layer stagnation point flow of a nanofluid toward a stretching/shrinking sheet is investigated by Bachok et al. .
In the above studies, the effect of an external applied magnetic field was not considered. However, it is well known that the use of a magnetic field in fluid flow and heat transfer problems provides a stabilization mechanism. Such type of flows is encountered in polymer industry and metallurgy. The metallurgical applications include the cooling of continuous strips or filaments in, for example, the process of drawing, annealing, and thinning of copper wires. However, there are very few studies which considered the effect of an external magnetic field on the stagnation point flow of a nanofluid over a stretching/shrinking sheet. The effect of magnetic field on stagnation point flow and heat transfer due to nanofluid toward a stretching sheet was studied by Ibrahim et al. . The transport equations employed in the analysis included the effect of Brownian motion and thermophoresis. They observed that the presence of transverse magnetic field decreases the velocity field. Chamkha et al.  investigated the unsteady, laminar, boundary-layer flow with heat and mass transfer of a nanofluid along a horizontal stretching plate in the presence of a transverse magnetic field, melting, and heat generation or absorption effects. Bég et al.  presented an explicit numerical investigation of unsteady magnetohydrodynamic (MHD) mixed convective boundary-layer flow of a nanofluid over an exponentially stretching sheet in a porous media. They also observed that the Lorentzian hydromagnetic drag which appears in the flow field due to the presence of external magnetic tends to retard the flow considerably.
Convective heat transfer in flow of nanofluids is important in different engineering devices, where a change in the heat transfer directly causes a change in the flow field. The natural convective boundary-layer flow of a nanofluid past a vertical plate was studied by Kuznetsov and Nield . They obtained an analytic solution for the flow and heat transfer problem. Das  studied the mixed convection stagnation point flow and heat transfer of a copper-water nanofluid toward a shrinking sheet. Pakravan and Yaghoubi  investigated the combined thermophoresis, Brownian motion, and Dufour effects on the natural convection of nanofluids. The double diffusive natural convective boundary-layer flow a nanofluid past a vertical plate is studied by Kuznetsov and Nield .
In most of research papers, the effect of buoyancy forces is studied assuming that the temperature and density vary linearly. However, there are several reasons for the density temperature relationship to become nonlinear. Thermal stratification and heat released by viscous dissipation (e.g., wall jets) induce significant changes in density gradients . In such cases, when the temperature difference between the surface of the plate and the ambient fluid becomes significantly large, nonlinear density and temperature variations in the buoyancy force term may exert a strong influence on the flow field. The effect of variable coefficient thermal expansion was investigated by Barrow and Rao  and Brown . Partha  investigated the natural convection in a non-Darcy porous medium using a temperature–concentration-dependent density relation. He concluded that the effect of a nonlinear temperature parameter is much more significant in a Darcy medium as compared with a non-Darcy porous medium. Prasad et al.  studied the coupled nonlinearity generated by the density variation with temperature on the non-Darcy flow on a vertical flat plate embedded in a fluid-saturated porous medium in the presence of the lateral mass flux with prescribed constant surface temperature.
In addition to the nonlinearity in the density temperature relation, viscous and Joule dissipation effects on such boundary-layer flows have a significant impact. The heat transfer characteristics of the flow are directly influenced by these effects which include changes in heat transfer within the fluid due to a change in the velocity gradient. Kameswaran et al.  investigated the hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. They used the nanoparticle volume fraction model to describe the nanofluid flow problem. The radiation effect on the viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet was studied by Hady et al. . Khan et al.  considered the unsteady MHD free convection boundary-layer flow of a viscous, incompressible nanofluid along a stretching sheet with thermal radiation and viscous dissipation effects.
In most of the above studies, the surface temperature is considered to be isothermal. However, there are several situations where a constant surface temperature may not be applicable. Makinde and Aziz  studied the boundary-layer flow induced in a nanofluid due to a linearly stretching sheet. The transport equations included the effects of Brownian motion and thermophoresis. The boundary-layer flow and heat transfer of nanofluid over a vertical plate with a convective surface boundary condition was studied by Ibrahim and Shanker . The investigation of boundary-layer stagnation point flow of a nanofluid past a permeable flat surface with Newtonian heating was carried by Olanrewaju and Makinde . They used the Brownian motion and thermophoresis effects to describe the nanofluid model.
There are several chemically reacting systems which involve both homogeneous and heterogeneous reactions, with examples occurring in combustion, catalysis, and biochemical systems. The interaction between the homogeneous reaction in the bulk of the fluid and heterogeneous reactions occurring on some catalytic surfaces is generally very complex, and is involved in the production and consumption of reactant species at different rates both within the fluid and on the catalytic surfaces. A model for isothermal homogeneous–heterogeneous reactions in boundary-layer flow of a viscous fluid past a flat plate was studied by Merkin . He represented the homogeneous reaction by cubic autocatalysis and the heterogeneous reaction by a first-order process and showed that the surface reaction is the dominant mechanism near the leading edge of the plate. Chaudhary and Merkin  investigated homogeneous–heterogeneous reactions in boundary-layer flow. They obtained the numerical solution near the leading edge of a flat plate. The effects of flow near the two-dimensional stagnation point flow on an infinite permeable wall with a homogeneous–heterogeneous reaction were studied by Khan and Pop . They solved the governing nonlinear equations using the implicit finite difference method and observed that the mass transfer parameter considerably affects the flow characteristics. Khan and Pop  investigated the effects of homogeneous–heterogeneous reactions on a viscoelastic fluid toward a stretching sheet. They observed that the concentration at the surface decreased with an increase in the viscoelastic parameter. Kameswaran et al.  investigated the homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. They showed that the velocity profiles decrease with an increase in nanoparticle volume fraction while the fluid concentration is oppositely affected by nanoparticle volume fraction for both Cu-water and Ag-water nanofluids. Recently, Kameswaran et al.  discussed the effects of homogeneous–heterogeneous reactions on stagnation point of a nanofluid over a porous stretching or shrinking sheet taking into account the effect of an externally applied magnetic field. They showed that dual solutions exist for certain suction/inject, stretching/shrinking, and magnetic parameter values. However, this study did not explain the heat transfer characteristics of the problem which is an important aspect in the flow of nanofluids.
The present study aims to provide a detailed discussion of the combined viscous and Joule dissipation effect on the stagnation point flow of a viscous incompressible electrically conducting nanofluid through a permeable stretching/sinking sheet in the presence of homogeneous–heterogeneous reactions and an applied magnetic field. The nanoparticle volume fraction model is used to describe the nanofluid problem. In the present study, the density temperature relation is considered to be nonlinear which may happen due to thermal stratification and heat released by viscous dissipation and causes a nonlinear convective heat transfer. The permeable sheet is assumed to be convectively heated with a hot fluid. The governing nonlinear partial differential equations are transformed to a set of nonlinear ordinary differential equations which are then solved using successive linearization method. The nanofluid problem, which is not yet considered by other researchers, finds applications in heat transfer devices where the density and temperature relations are more complex and the viscosity of the fluid has significant effect on the heat transfer rate.
Consider the steady two-dimensional boundary-layer flow of a viscous, incompressible, and electrically conducting nanofluid in the region y > 0 driven by a stretching/shrinking surface at y = 0 with a fixed stagnation point as shown in Fig. 1. The fluid flow is also affected by nonlinear convection due to nonlinear temperature and density relation. Two equal but opposite forces are applied along the sheet so that the wall is stretched, keeping the position of the origin unaltered. The fluid flow is permeated with a uniform transverse magnetic field B0. It is assumed that the induced magnetic field produced by the fluid motion is negligible in comparison with the applied one. This assumption is valid for low magnetic Reynolds number fluids . Also, there is no external applied electric field so that the effect of polarization of magnetic field is negligible .
The base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. It is assumed that a simple homogeneous–heterogeneous reaction model exists as proposed by Chaudhary and Merkin  in the following form:Display Formula
while on the catalyst surface, we have the single, isothermal, first order reactionDisplay Formula
where a and b are the concentrations of the chemical species A and B, kc, and ks are the rate constants. It is assumed that both the reaction processes are isothermal. Under these assumptions, the boundary-layer equations describing the nanofluid flow, heat and mass transfer can be written asDisplay Formula
where u and v are the velocity components of the nanofluid in x and y directions, T is the temperature of the nanofluid, a and b are species concentration, g is the acceleration due to gravity, σ is the electrical conductivity of the nanofluid. μnf, ρnf, knf, and (cp)nf are, respectively, viscosity of the nanofluid, nanofluid density, nanofluid thermal conductivity, and specific heat at constant pressure. In Eq. (4), the last term accommodates the linear density temperature variation as well as the quadratic density temperature variation. In this term, βnf and βnf* are the volumetric coefficients of thermal expansion of the first and the second orders, respectively [18-20].
The dynamic viscosity of nanofluid μnf was given by Brinkman  asDisplay Formula
where ϕ is the solid volume fraction of nanoparticles.
The values of ρnf, βnf, (ρcp)nf, and αnf are, respectively, defined as Display Formula
where knf is the thermal conductivity of the nanofluid which is given by [37,38]Display Formula
Here, the subscripts nf, f, and s refer to the thermophysical properties of the nanofluid, base fluid, and nanosolid particles, respectively.
It is assumed that the surface of the sheet is stretched/shrinked with a velocity proportional to x keeping the point at the origin O as fixed. The nanofluid which is in contact with the surface of stretching/shrinking sheet also has the same velocity because of no-slip which also causes the fluid motion near to the sheet. The free stream velocity U∞ is assumed to be proportional to x so that there is no velocity at the point O at all. This point is known as the stagnation point of the flow. The nanofluid, which is in contact with the surface of the sheet, is convectively heated by a hot fluid which is on the other side of the sheet and whose temperature is Tf. The temperature of the nanofluid outside the boundary-layer regime is assumed to be T∞.
Under the assumptions made above and following the model for homogeneous–heterogeneous reactions proposed by Chaudhary and Merkin , the boundary conditions for the problem areDisplay Formula
Introducing the following transformation:Display Formula
where η is the dimensionless stream function and the above transformation is chosen in such a way that u=∂ψ/∂y and v=-∂ψ/∂x.
Using the above transformation, the equation of continuity (3) is automatically satisfied and we obtain from Eqs. (4)–(7) asDisplay Formula
The nondimensional variables defined above are, the magnetic parameter M, the mixed convection parameter, λ; the nonlinear density–temperature (NDT) parameter, α; the Prandtl number, Pr; the Eckert number, Ec; the Schmidt number, Sc; and the homogeneous reaction rate, K. The functions ϕ1, ϕ2, ϕ3, ϕ4, and ϕ5 are also nondimensional and are depending upon the thermophysical properties of the nanoparticles and the base fluid. δ is the ratio of diffusion constants. The value of λ > 0 corresponds to the buoyancy assisting flow, while the value of λ < 0 corresponds to the buoyancy opposing flows and λ = 0 corresponds to the case of pure forced convection flow.
The boundary conditions (14) and (15), subject to the transformation (16), are given byDisplay Formula
where ϵ is the stretching/shrinking parameter, Bi=(h/kf)(νf/d) is the Biot number, and Ks=(ks/DA)(νf/d) is the strength of the heterogeneous reaction.
It is expected that the diffusion coefficients of chemical species A and B are of a comparable size which leads us to make a further assumption that the diffusion coefficients DA and DB are equal, i.e., δ = 1 . This assumption leads to the following relation:Display Formula
Equations (19) and (20) under this assumption reduce toDisplay Formula
and are subject to the boundary conditionsDisplay Formula
The problem now reduces to the problem of solving Eqs. (17), (18), and (24) subject to the conditions provided in Eqs. (21), (22), and (25).
The other physical quantities which need to be investigated are the skin friction coefficient Cf, the local Nusselt number Nux which may be obtained by using the following results:Display Formula
The system of nonlinear equations (17), (18), and (24) subject to the boundary conditions prescribed in Eqs. (21), (22), and (25) are solved using the SLM [39-41]. The SLM algorithm starts with the assumption that the functions f(η), θ(η), and g(η) can be expressed asDisplay Formula
where fi, θi, and gi are unknown functions and Fm, Θm, and Gm are successive approximations which are obtained by recursively solving the linear part of the equation system that results from substituting first Eq. (28) in Eqs. (17), (18), and (24). The initial guesses F0, Θ0, and G0 which are chosen to satisfy the boundary conditions for f, θ, and g are taken asDisplay Formula
Starting from the initial guesses, the subsequent solutions Fi, Θi, and Gi (i ≥ 1) are obtained by successively solving the linearized form of the equations which are obtained by substituting Eq. (28) in to Eqs. (17), (18), and (24). After n number of iterations, the solutions f(η), θ(η), and g(η) can be written asDisplay Formula
The linearized equations were solved using the Chebyshev spectral collocation method. The method is based on the Chebyshev polynomials defined on the interval [ − 1, 1]. The domain of our problem [0, ∞) is first transformed to [−1, 1] using the domain truncation technique where the problem is solved in the interval [0, L] instead of [0, ∞) by using the mappingDisplay Formula
where L is the scaling parameter used to invoke the boundary condition at infinity. The domain [−1, 1] is discretized using the Gauss-Lobatto collocation points defined asDisplay Formula
where N is the number of collocation points used. The functions Fi, Θi, and Gi for i ≥ 1 are approximated at the collocation points asDisplay Formula
where Tk is the kth Chebyshev polynomial given byDisplay Formula
The derivatives at the collocation points are defined asDisplay Formula
where r is the order of differentiation and D=2D/L with D being the Chebyshev differentiation matrix whose entries are defined asDisplay Formula
This leads to the following matrix equation:Display Formula
where Ai−1 is a (3N + 3) × (3N + 3) square matrix and Xi and Ri − 1 are (3N + 3) × 1 column vectors defined byDisplay Formula
In the above definitions, T stands for transpose, ak,i-1,bk,i-1, and ck,i-1(k=1,2,3) are diagonal matrices, I is an identity matrix, and O is zero matrix of order (N + 1) × (N + 1). Finally, the solution is obtained asDisplay Formula
In the present study, the base fluid is considered to be water with suspended nanosized particles of copper (Cu). The thermophysical properties of the base fluid and nanoparticles are mentioned in Table 1.
To validate the SLM code and results, the values of f″(0), θ′(0), g′(0) for different values of magnetic parameter M obtained using SLM are compared with the respective values obtained by bvp4c solver which is an in-built boundary value solver of Matlab (Table 2). The comparison shows an excellent agreement between the SLM and bvp4c results up to eight decimal places. The values of coefficient of skin friction CfRex and the local Nusselt number Nux/Rex are compared with the values obtained by Bachok et al.  in the absence of a magnetic field, mixed convection, viscous and Joule dissipations and are presented in Table 3. Once again the results obtained using the SLM are found to be in excellent agreement with the results of Bachok et al. .
The combined viscous and Joule dissipation effects on the hydromagnetic stagnation point flow of a viscous incompressible electrically conducting nanofluid through a stretching sheet in the presence of homogeneous–heterogeneous reactions and nonlinear convection are studied. The fluid is a water-based nanofluid with suspended nanosized Cu particles. To investigate the effects of the magnetic field, nanoparticle volume fraction, nonlinearity in temperature-density relationship, viscous dissipation, convective heat transfer at the surface, and stretching of the sheet on the nanofluid velocity, temperature and species concentration the profiles of f′(η), θ(η), and g(η) are shown in Figs. 2–7 for various values of the magnetic parameter M, nanoparticle volume fraction parameter ϕ, the nonlinear density temperature (NDT) parameter α, the Eckert number Ec, the Biot number Bi, and the stretching parameter ϵ. The effects of these parameters on the skin friction f″(0) and rate of heat transfer at the surface θ′(0) are presented in Table 4, while the effects of homogeneous–heterogeneous reactions on the species concentration are depicted graphically in Fig. 8. For the computation work, the default value of the mixed convection parameter is taken to be λ = 1 which corresponds to buoyancy assisting flow.
In Fig. 2, the effect of the magnetic parameter M is shown on the nanofluid velocity f′(η), nanofluid temperature θ(η), and species concentration g(η). The magnetic parameter M measures the strength of the external applied magnetic field. It is shown that the increase in the strength of the magnetic field has a decreasing effect on the nanofluid velocity and species concentration while this has an increasing effect on the nanofluid temperature. This is due to the presence of a resistive Lorentz force which appears due to the presence of the magnetic field and provides a stabilizing effect on the flow field. This result can be utilized in boundary-layer flow control. It is also important to note that the presence of Joule dissipation in the temperature field causes an enhancement in the nanofluid temperature. Thus, the effect of the magnetic field is to reduce the thickness of the momentum and concentration boundary layers and to increase the thickness of thermal boundary layer.
The effect of the nanoparticle volume fraction ϕ on the nanofluid velocity, temperature, and species concentration is depicted in Fig. 3. An increase in ϕ indicates an increase in the nanoparticle volume fraction. It is observed that the presence of nanoparticles in the flow field causes deceleration in the flow field while it has an enhancing effect on the fluid temperature. The velocity of the fluid decreases due to the presence of nanoparticles as they produce a resistive force toward the sheet. The enhancement of fluid temperature is the well-known property of nanoparticles. We also noted that the species concentration tended to decrease as the nanoparticle volume fraction increased. The thicknesses of the momentum and concentration boundary layers decreased while the thickness of the thermal boundary layer increased with the increase in nanoparticle volume fraction.
Figure 4 shows the effect of the NDT parameter α on the flow, heat, and mass transfer. The NDT parameter α measures the nonlinearity in density–temperature relationship. It is shown that an increase in the nonlinear density temperature relation causes the nanofluid velocity and species concentration to increase while the nanofluid temperature decreased. However, the observed effect is very nominal. The momentum and concentration boundary layers get thicker, whereas the thermal boundary layer gets thinner with an increase in the nonlinear density–temperature relation.
Figure 5 shows the effect of the Eckert number, Ec, on the flow, heat and mass transfer. An increase in the Eckert number, Ec, signifies an increase in the dissipative effect due to fluid viscosity. It is found that the viscous dissipation has significant effects on the nanofluid velocity and temperature where it tends to increase these quantities while it has marginal effect on the species concentration which also gets increased with an increase in the viscous dissipation. However, we noted that all the three boundary layers get thicker with an increase in the viscous dissipation.
The effect of an increase in the Biot number Bi, which measures the increase in the rate of convective heat transfer at the surface, on the nanofluid velocity, nanofluid temperature, and species concentration is depicted in Fig. 6. It is found that the nanofluid velocity, nanofluid temperature, and species concentration increase with an increase in the convective heat transfer rate at the surface however the observed effect on the nanofluid velocity and species concentration is nominal. The thickness of all the three boundary layers increased with an increase in convective heat transfer rate at the surface.
The effect of an increase in the value of stretching parameter ϵ on the flow, heat, and mass transfer is presented in Fig. 7. It is observed that the stretching of the sheet has significant effect on the nanofluid velocity, temperature, and species concentration. The nanofluid velocity, temperature, and species concentration increased with an increase in the stretching rate of the sheet. The momentum, thermal, and concentration boundary layers get thicker with an increase in the stretching of the sheet.
The effects of the homogeneous reaction parameter K and the heterogeneous reaction parameter Ks on the species concentration and species concentration at the sheet are shown in Fig. 8. An increase in the value of these parameters corresponds to an increase in the strength of homogeneous and heterogeneous reaction rates, respectively. It is found that an increase in the homogeneous and heterogeneous reactions causes a decrease in the species concentration and the rate of mass transfer at the plate. We also observed that the effect of heterogeneous reaction is more on the species concentration as compared with the homogeneous reaction and the rate of mass transfer at the surface approaches to zero as the rate of heterogeneous reaction approaches infinity.
The effects of increase in the values of M, ϕ, α, Bi, and ϵ on the skin friction, which is proportional to f″(0), and the Nusselt number which measures the rate of heat transfer at the plate and can be measured as a variation in θ′(0), are presented in Table 4. It is observed that an increase in the magnetic field and stretching rate causes an increase in the skin friction, whereas an opposite effect on the skin friction is encountered with an increase in nonlinear density–temperature relation, viscous dissipation, and convective heat transfer rate at the surface. An increase in the nanoparticle volume fraction tends to increase the skin friction first and then it tends to decrease it. Thus by increasing the nanoparticle volume fraction, one can obtain a higher heat transfer with lesser skin friction. The rate of heat transfer at the surface decreases with an increase in the Joule dissipation, nanoparticle volume fraction, viscous dissipation, and stretching rate, whereas it is oppositely affected by nonlinear density–temperature relation and convective heat transfer rate.
The present study investigated the effects of viscous and Joule dissipation on the hydromagnetic stagnation point flow of an incompressible and electrically conducting Cu-water nanofluid through a stretching sheet in the presence of nonlinear convection and homogeneous–heterogeneous reactions. The significant findings of this parametric study are summarized as below:
The effect of the external magnetic field is to reduce the nanofluid velocity and species concentration and to increase the nanofluid temperature. The magnetic field tends to enhance the shear stress at the surface while it reduces the heat transfer rate at the surface.
An increase in nonlinear convection reduces the skin friction while this has the opposite effect on the rate of heat transfer at the surface.
Viscous dissipation causes an enhancement of the nanofluid velocity, temperature, and species concentration, whereas it has reverse effect on the skin friction and the rate of heat transfer at the surface.
An increase in the rate of homogeneous and heterogeneous reactions has effect of reducing the species concentration and the rate of mass transfer at the plate. The effect of heterogeneous reaction is more significant on the species concentration as compared with the homogeneous reaction.
Authors are highly thankful to University of KwaZulu-Natal, South Africa for financial assistance.
Physical model of the problem
Effect of magnetic parameter M on (a) f′, (b) θ, and (c) g when ϕ = 0.2, λ = 1, α = 0.2, Bi = 5, ɛ=2, Sc = 1, K = 0.5, Ks = 0.5, and Ec = 0.1
Effect of nanoparticle volume fraction ϕ on (a) f′, (b) θ, and (c) g when M = 2, λ = 1, α = 0.2, Bi = 5, ɛ=2, Sc = 1, K = 0.5, Ks = 0.5, and Ec = 0.1
Effect of NDT parameter α on (a) f′, (b) θ, and (c) g when M = 2, λ = 1, ϕ = 0.2, Bi = 5, ɛ=2, Sc = 1, K = 0.5, Ks = 0.5, and Ec = 0.1
Effect of Eckert number Ec on (a) f′, (b) θ, and (c) g when M = 2, λ = 1, ϕ = 0.2, Bi = 5, ɛ=2, Sc = 1, K = 0.5, Ks = 0.5, and α = 0.2
Effect of Biot number Bi on (a) f′, (b) θ, and (c) g when M = 2, λ = 1, ϕ = 0.2, Ec = 0.1, ɛ=2, Sc = 1, K = 0.5, Ks = 0.5, and α = 0.2
Effect of stretching parameter ɛ on (a) f′, (b) θ, and (c) g when M = 2, λ = 1, ϕ = 0.2, Ec = 0.1, Bi = 5, Sc = 1, K = 0.5, Ks = 0.5, and α = 0.2
Effect of homogeneous reaction parameter K and heterogeneous reaction parameter Ks on (a) and (b) g and (c) g(0) when M = 2, λ = 1, ϕ = 0.2, Ec = 0.1, Bi = 5, Sc = 1, ɛ = 2, and α = 0.2
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