Nonlinear convective flow of Powell-Erying magneto nanofluid with Newtonian heating

Accepted Manuscript Nonlinear convective flow of Powell-Erying magneto nanofluid with Newtonian heating Sajid Qayyum, Tasawar Hayat, Sabir Ali Shehzad, Ahmed Alsaedi PII: DOI: Reference:

S2211-3797(17)31017-3 http://dx.doi.org/10.1016/j.rinp.2017.08.001 RINP 851

To appear in:

Results in Physics

Please cite this article as: Qayyum, S., Hayat, T., Shehzad, S.A., Alsaedi, A., Nonlinear convective flow of PowellErying magneto nanofluid with Newtonian heating, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp. 2017.08.001

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Nonlinear convective flow of Powell-Erying magneto nanofluid with Newtonian heating Sajid Qayyuma,1 , Tasawar Hayata,b , Sabir Ali Shehzadc and Ahmed Alsaedib a

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University P. O. Box 80207, Jeddah 21589, Saudi Arabia

c

Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan

Abstract: Objective of present article is to describe magnetohydrodynamic (MHD) non-linear convective flow of Powell-Erying nanofluid over a stretching surface. Characteristics of Newtonian heat and mass conditions in this attempt is given attention. Heat and mass transfer analysis is examined in the frame of thermal radiation and chemical reaction. Brownian motion and thermophoresis concept is introduced due to presence of nanoparticles. Nonlinear equations of momentum, energy and concentration are transformed into dimensionless expression by invoking suitable variables. The series solutions are obtained through homotopy analysis method (HAM). Impact of embedded variables on the velocity, temperature and nanoparticles concentration is graphically presented. Numerical values of skin friction coefficient, local Nusselt and Sherwood numbers are computed and analyzed. It is concluded that velocity field enhances for fluid variable while reverse situation is noticed regarding Hartman number. Temperature and heat transfer rate behave quite reverse for Prandtl number. It is also noted that the concentration and local Sherwood number have opposite behavior in the frame of Brownian motion.

Keywords: Powell-Erying nanofluid; magnetohydrodynamic (MHD); nonlinear convec1

Corresponding author. Tel.: +92 51 90642172.

email address: [email protected] (Sajid Qayyum)

1

tion; thermal radiation; chemical reaction; Newtonian heat and mass conditions.

1

Introduction

The present world is a world of faster technology. Cooling of electronic equipments is one of the major requirements of modern industry. The existing ordinary heat transport liquids are unable to provide the proper cooling in the processes of industries. Such circumstances lead to the development of modern technology named as nano-technology. This technology is developed by collides of ultrafine nanoparticles smaller or equal to 100 nm in oil, ethylene glycol or water. Such phenomena is categorized as nano-fluids. The nano-particles involved in nanofluid are generally made of oxides, nitrides, metals, carbides, carbon nanotubes and graphites. Collides of nano-particles have certain chemical and physical features that make them highly reliable in the manufacturing of various engineering and industrial products include coatings, ceramics, paints, drugs delivery systems, foods etc. The nanoliquids have useful implementations transportations, cooling of engines, electronic devices cooling, heat exchanger, biomedicine, manufacturing, and many more. Different mechanisms have been explored in the literature which is responsible for the enhancement of thermal performance of nanoliquid. These mechanism include nano-particles Brownian motion, thermal diffusion, coupling of particle-to-particle via potential of inter-particle, thermophoresis, induced microconvection etc. [1, 2]. Choi [3] made on experimental investigation on nano-particles and reported that the involvement of nano-particles in base liquid improve the thermal conductivity of working liquid remarkably. Buongiorno [4] showed that the thermal performance of base liquid is enhanced due to Brownian motion and thermophoretic factors. Sandeep [5] presented the aligned magnetic field on liquid thin film flow of magnetic-nanofluids in the presence of graphene nanoparticles. Kumaran and Sandeep [6] analyzed the thermophoresis 2

and Brownian motion effects on parabolic flow of MHD Casson and Williamson fluids with cross diffusion. Sheikholeslami and Bhatti [7] reported the forced convection flow of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles. Hayat et al. [8] studied the nonlinear convective flow of Oldroyd-B magneto nanoliquid in presence of heat generation/absorption. Some remarkable investigations about the theory of nanofluids under various aspects are addressed in studies [9-22]. The boundary layer behavior of non-Newtonian fluids is a hot topic of research for the recent investigators due to their commercial importance. Various industrial liquids include shampoos, polymer, solutions, certain oils, drilling muds, paper pulp, paints, slurries and many others have the nature of non-Newtonian liquids. The flow of these liquids significantly appeared in different processes of industries like metal spinning, plastic films, cool oil slurries, glass blowing, continuous casting etc. There is not a single constitutive formula which has an ability to describe all the features of non-Newtonian materials. Various models of nonNewtonian liquids have been formulated according to the nature of liquid. Powell-Eyring liquid model is one sub-category of non-Newtonian models. This fluid model is designed for the systems of chemical engineering. This rheological model has many advantages over other non-Newtonian formulations like physical robustness and simplicity. It is derived by the kinetic theory fluids instead of empirical formula. Khan et al. [23] addressed the phenomena of Powell-Eyring liquid flow generated by the rotation of disk under magnetic field. Jalil et al. [24] constructed the self-similer solutions of Powell-Eyring liquid induced by movement of surface. Ghaffar et al. [25] reported free-convective hydro magnetic Powell-Eyring liquid flow in prous space with hall currents. Hina et al. [26] described the mechanism of peristaltically Powell-Eyring fluid flow induced by curved channel. Malik et al. [27] analyzed the magnetohydrodynamic mixed convection flow of Eyring-Powell nanofluid towards a stretching sheet. 3

Simultaneous effects of thermal radiation and non-uniform heat source/sink in unsteady flow of Powell-Eyring fluid past an inclined stretching sheet is addressed by Krishna et al. [28]. Hayat et al. [29] addressed the behavior of Eyring-Powell nanoliquid flow induced by the non-linearly stretched sheet. Having all the above mentioned aspects in mind, here we are developing a mathematical model of nanoliquids by using the Powell-Eyring fluid as a working liquid. A hydromagnetic flow is taken into consideration under the effects of non-linear convection. The Newtonian heat and mass condition are imposed at the boundary of sheet. All previous researches have been reported only using the condition of Newtonian heating [30-35]. Here we also introduced this condition for mass transport phenomena. To model the physical phenomena, we employed the assumptions of boundary-layer. The resulting expressions are coupled and highly complicated due to nonlinear convection. These expressions are evaluated by the use of homotopy analysis scheme [36-50]. The results are presented and elaborated via graphs and numerical benchmarks.

2

Problems development

We examine the steady two-dimensional flow of an incompressible nonlinear convective flow of Powell-Erying nanofluid over a stretching surface with Newtonian heat and mass conditions. The flow is confined in the domain y > 0. The x-axis is taken along the stretching surface in direction of fluid motion and y-axis is perpendicular to it. It is assumed that velocity of the stretching surface is uw (x) = cx where c is positive constant and its dimension is T −1 . A uniform magnetic field B0 is applied in y−direction (see Fig. 1). Effects of thermal radiation and chemical reaction are introduced. Features of Brownian motion and thermophoresis are taken into account due to presence of nanoparticles. The continuity, momentum, energy 4

and nanoparticles concentration expressions after boundary layer approximations can be expressed as: ∂u ∂v + = 0, ∂x ∂y ∂u ∂u ∂ 2u 1 ∂ 2u 1 u +v = ν 2+ − 2 ∂x ∂y ∂y ρf βa ∂y 2ρf βa2



∂u ∂y

(1) 2

∂ 2 u σB02 − u ∂y 2 ρf

+g{Λ1 (T − T∞ ) + Λ2 (T − T∞ )2 } + g{Λ3 (C − C∞ ) + Λ4 (C − C∞ )2 }, (2)  2     2 kf ∂ T ∂T ∂C τ DT ∂T = + τ DB + (ρcp )f ∂y 2 ∂y ∂y T∞ ∂y 1 ∂qr − , (ρcp )f ∂y  2    ∂C ∂C ∂ C DT ∂ 2 T u +v = DB + −k1 (C − C∞ ), ∂x ∂y ∂y 2 T∞ ∂y 2

∂T ∂T u +v ∂x ∂y

(3)

(4)

with

u = uw (x) = cx, v = 0,

∂T ∂C = −ht T, = −hc C at y = 0, ∂y ∂y

u → 0, T → T∞ , C → C∞ as y → ∞.

(5)

where ν = (µ/ρ)f for kinematic viscosity, β and a for fluid parameters, σ for electrical conductivity, g for gravitational acceleration, Λ1 and Λ2 for linear and nonlinear thermal expansions coefficients, Λ3 and Λ4 for linear and nonlinear concentration expansion coefficients, ρf for fluid density, (cp )f for fluid heat capacity, ρp for particle density, (cp )p for particle heat capacity, kf for thermal conductivity, qr for radiative heat flux, τ = (ρcp )p /(ρcp )f for capacity ratio, DB for Brownian diffusion coefficient, DT for themophoretic diffusion coefficient, k1 for reaction rate (k1 > 0 leads to destructive and k1 < 0 for generative reactions), T for fluid temperature, C for fluid concentration, T∞ for ambient fluid temperature, C∞ for nanoparticle concentration far away from the surface, ht for heat transfer coefficient, hc for mass diffusion coefficient and c for stretching rate. 5

Rosseland approximation leads to [50]: qr = −

4σ ∗ ∂T 4 3k ∗ ∂y

(6)

where σ ∗ represents the Stefan-Boltzmann constant and k ∗ shows mean absorption coefficient. Using the Taylor series and neglecting higher-order terms, one obtains 3 4 T 4 u 4T∞ T − 3T∞ .

(7)

Thus using Eq. (7) in Eq. (6), we get qr = −

3 16σ ∗ T∞ ∂T . ∗ 3k ∂y

(8)

Now Eqs. (3) and (8) yield ∂T ∂T u +v ∂x ∂y

 2     2 kf ∂ T ∂T ∂C τ DT ∂T = + τ DB + (ρcp )f ∂y 2 ∂y ∂y T∞ ∂y ∗ 3 2 1 16σ T∞ ∂ T + . (ρcp )f 3k ∗ ∂y 2

(9)

We describe suitable variables are r

c T − T∞ C − C∞ y, θ (η) = , φ (η) = , ν T∞ C∞ √ √ u = cxf 0 (η) , v = − cνf (η) , ψ(η) = cνxf (η). η =

(10)

continuity equation is trivially satisfied while other equations yield 2

2

(1 + α) f 000 + f f 00 − (f 0 ) − αΛ (f 00 ) f 000 − Ha2 f 0 + λ(1 + β t θ)θ + λN ∗ (1 + β c φ)φ = 0, (11) 

 4 2 1 + R θ00 + Pr f θ0 + Pr Nb θ0 φ0 + Pr Nt (θ0 ) = 0, 3 φ00 + Sc(f φ0 − γφ) +

Nt 00 θ = 0, Nb

(12) (13)

f 0 (η) = 1, f (η) = 0, θ0 (η) = −Bt (1 + θ(η)), φ0 (η) = −Bc (1 + φ(η)) at η = 0, f 0 (η) = 0, θ(η) = 0, φ(η) = 0 as η → ∞, 6

(14)

where α and Λ denotes the fluid parameters, Ha for magnetic parameter/Hartman number, λ for mixed convection parameter, β t for nonlinear convection parameter due to temperature, N ∗ for ratio of concentration to thermal buoyancy forces, β c for nonlinear convection parameter due to concentration, R for radiation parameter, Pr for Prandtl number, Nb for Brownian motion parameter, Nt for thermophoresis parameter, Sc for Schmidt number, Gr and Gr∗ for Grashof number in terms of temperature and concentration, γ for chemical reaction, Bt and Bc for thermal and solutal conjugate parameters. Definitions of involved parameters are 1 c3 x 2 σB02 Gr Λ2 T∞ , Λ = 2 , Ha = , λ= , 2 , βt = ρνβc 2c ν ρc Λ1 Rex 3 Gr∗ Λ3 C ∞ Λ4 C ∞ 4σ ∗ T∞ µcp = = , βc = , R= , Pr = , ∗ Gr Λ1 T∞ Λ3 kf k kf r τ DB τ DT ν k1 ν = , Nt = , Sc = , γ = , Bt = ht , ν νT∞ DB c c r ν gΛ1 T∞ x3 gΛ3 C∞ x3 ∗ = hc , Gr = , Gr = . c ν2 ν2

α = N∗ Nb Bc

(15)

Skin friction coefficient Cfx , local Nussult N ux and Sherwood Shx numbers are defined as C fx =

τw , 1 ρu2w 2

N ux =

xqw xjw , Shx = , kf (T − T∞ ) DB (C − C∞ )

(16)

in which the surface shear stress τ w , the surface heat flux qw and the surface mass flux jw satisfy the relations "

τw qw

 3 # ∂u 1 ∂u 1 1 ∂u = µ + − , ∂y βa ∂y 6β a ∂y y=0      ∗ 3 4 4σ T∞ ∂T ∂C = −kf 1 + , jw = −DB . 3 kf k ∗ ∂y y=0 ∂y

(17)

Dimensionless form of skin friction coefficient Cfx , local Nusselt N ux and Sherwood Shx numbers are    1 0.5 1 4 1 3 −0.5 00 00 Re Cfx = (1 + α)f (0) − αΛ (f (0)) , Rex N ux = Bt 1 + R 1+ , 2 x 3 3 θ (0)   1 −0.5 Rex Shx = Bc 1 + , (18) φ(0) where Rex =

uw x ν

denotes local Reynolds number. 7

3

Solutions expressions

Linear operators Lf , Lθ and Lφ and initial guesses f0 (η), θ0 (η) and φ0 (η) are taken in the form Lf (f ) =

d3 f df d2 θ d2 φ − , L (θ) = − θ, L (φ) = − φ, θ φ dη 3 dη dη 2 dη 2 

f0 (η) = 1 − exp (−η) , θ0 (η) =   Bc φ0 (η) = exp (−η) , 1 + Bc

Bt 1 + Bt

(19)

 exp (−η) , (20)

satisfying the following properties Lf [Γ1 + Γ2 exp(−η) + Γ3 exp(η)] = 0, Lθ [Γ4 exp(−η) + Γ5 exp(η)] = 0, Lφ [Γ6 exp(−η) + Γ7 exp(η)] = 0.

(20)

We obtain the general solution through the following procedure of refs. [48-50]. ? fm (η) = fm (η) + Γ1 + Γ2 exp(−η) + Γ3 exp(η),

θm (η) = θ?m (η) + Γ4 exp(−η) + Γ5 exp(η), φm (η) = φ?m (η) + Γ6 exp(−η) + Γ7 exp(η),

(21)

∗ in which fm (η), θ∗m (η) and φ∗m (η) denotes the special functions and Γi (i = 1 − 7) are the

arbitrary constants given by

Γ1 Γ4 Γ6

? ? ∂fm (η) ∂fm (η) ∗ = − − fm (0), Γ2 = , Γ3 = 0, ∂η η=0 ∂η η=0 " # 1 ∂θ?m (η) ? = + Bt θm (0) , Γ5 = 0, 1 − Bt ∂η η=0 " # 1 ∂φ?m (η) ? + Bc φm (0) , Γ7 = 0. = 1 − Bc ∂η η=0 8

(27)

3.1

Convergence analysis

Here homotopy analysis technique comprises auxiliary variables }f , }θ and }φ . The auxiliary variables are valuable in adjusting and controlling the convergence region for homotopic expressions. Meaningful ranges of these variables are attained by plotting }-curves (see Figs. 2). Suitable values of }f , }θ and }φ are found in the ranges −1.6 ≤ }f ≤ −0.2, −1.6 ≤ }θ ≤ −0.1 and −1.5 ≤ }φ ≤ −0.15. Table 1: Convergence when α = 0.2, Λ = 0.2, Ha = 0.1, λ = 0.1, β t = 0.1, N ∗ = 0.5, β c = 0.1, R = 0.3, Pr = 2.0, Nb = 0.2, Nt = 0.1, Sc = 2.0, γ = 0.9, Bt = 0.2 and Bc = 0.2. Order of approximation

−f 00 (0) −θ0 (0) −φ0 (0)

1

0.9547

0.2648

0.2383

5

0.9151

0.2813

0.2330

6

0.9126

0.2822

0.2329

8

0.9097

0.2829

0.2329

16

0.9065

0.2829

0.2329

20

0.9065

0.2829

0.2329

25

0.9065

0.2829

0.2329

30

0.9065

0.2829

0.2329

Convergence solution through homotopic technique for momentum, energy and nanoparticle concentration expressions are exhibited in Table 1. Here we observed that 16th , 8th and 6th orders of approximation are acceptable for the convergence of momentum, energy and nanoparticle concentration expressions, respectively.

9

4

Results and discussion

For the intention of discussing the results of interesting variables on the velocity, temperature and nanoparticles concentration are studied graphically, skin friction coefficient, local Nusselt and Sherwood numbers are determined numerically in (see Fig. 3-16 and Table 2-4).

4.1

Dimensionless velocity fields

Fig. 3 explores fluid variable α impact on velocity. One can see that velocity and layer thickness are enhanced via larger α. Physically it can be observed that for higher values of α, the viscosity decreases due to which fluid velocity are enhanced. Fig. 4 delineates the characteristics of Λ on velocity. Here velocity and related layer thickness show a decreasing behavior for larger Λ. Hartman number Ha effect on velocity is described in Fig. 5. There is reduction in velocity and associated layer thickness. In fact for higher Ha , the Lorentz force enhances which makes more resisting to fluid motion and thus velocity reduces. Velocity field for λ is captured in Fig. 6. Both velocity and layer thickness are increased for higher λ. In fact that larger λ give rise to more buoyancy force and so velocity and layer thickness are enhanced.

4.2

Dimensionless temperature fields

Fig. 7 is prepared to see the behavior radiation parameter R on temperature field. Temperature and associated layer thickness are increasing functions of R. Physically it is verified because heat is produced due to radiation process in the working fluid so temperature enhances. Variations in temperature for different values of Prandtl number Pr is shown in Fig. 8. Here we noticed that both temperature and layer thickness are decreasing behavior for larger Pr . It is due to the fact that higher Pr corresponds to lower thermal diffusivity 10

which results in decay for temperature. Fig. 9 witnesses that temperature and thickness of associated layer are enhanced in view of Brownian motion parameter Nb . In fact more heat is produce through the random motion of the fluid particles due to larger Nb which results temperature enhances. Fig. 10 represents increasing behavior of temperature with Nt . In thermophoresis phenomenon heated particles are pulled away from hot surface to the cold region. Due to this fact that fluid temperature enhances. Fig. 11 describe the influence of Bt on temperature. Both temperature field and associated layer thickness have increasing behavior for Bt . Physically, higher thermal conjugate parameter increases heat transfer coefficient which enhances the fluid temperature.

4.3

Dimensionless concentration fields

Graphical illustration for concentration against Nb is captured in Fig. 12. One may see from the Fig. that concentration reduced via larger Nb . In fact Brownian motion appeared in the ratio form of nanoparticle mass species equation due to which decreasing trend is noticed. Fig. 13 shows that both concentration and layer thickness are enhanced for larger Nt . The thermal conductivity of the fluid increases in presence of nanoparticles. Higher Nt give rise to thermal conductivity of the fluid. Such higher thermal conductivity indicates rise in concentration. Here concentration and related layer thickness are reduced via Sc (see Fig. 14). Physically Schmidt number Sc is the momentum to mass diffusivities ratio. Thus for larger Schmidt number Sc the mass diffusivity decreases which is responsible in reduction of concentration field. Features of generative/destructive chemical reaction γ on concentration is delineated in Fig. 15. Both concentration and layer thickness are enhanced when γ > 0. For γ < 0 the situation is reverse. Fig. 16 is sketched to see the characteristics of solutal conjugate parameter Bc on concentration. Both concentration and layer thickness 11

are enhanced via larger Bc .

4.4

Skin friction coefficient and local Nusselt and local Sherwood numbers

Behavior of various variables on skin friction coefficient is expressed in Table 2. It is found that the skin friction coefficient enhances through larger α and Ha , while decreasing function in view of Λ, λ, Bt and Bc . Table 3 is drawn to explore the effect of sundry variables on the local Nusselt number. Here we noticed that Nusselt number is decreasing function for Nb , Nt and Bt . Nusselt number increases via α, λ, R and Pr . Behavior of sundry variables on local Sherwood number is expressed in Table 4. It is recognized that Sherwood number enhances α, λ, Nb , Sc, γ and Bc while it diminished in view of Nt . Table 2: Skin friction coefficient

1 2

∗ Re0.5 x Cfx when β t = 0.1, N = 0.5, β c = 0.1, R = 0.3,

Pr = 2.0, Nb = 0.2, Nt = 0.1, Sc = 2.0 and γ = 0.9.

12

Parameters (fixed values)

Parameters − 12 Re0.5 x Cf x

Λ = 0.2, Ha = 0.1, λ = 0.1, Bt = 0.2, Bc = 0.2

α

α = 0.2, Ha = 0.1, λ = 0.1, Bt = 0.2, Bc = 0.2

α = 0.2, Λ = 0.2, λ = 0.1, Bt = 0.2, Bc = 0.2

Λ

Ha

α = 0.2, Λ = 0.2, Ha = 0.1, Bt = 0.2, Bc = 0.2 λ

α = 0.2, Λ = 0.2, Ha = 0.1, λ = 0.1, Bc = 0.2

α = 0.2, Λ = 0.2, Ha = 0.1, λ = 0.1, Bt = 0.2

Bt

Bc

0.2

1.0776

0.3

1.1228

0.5

1.2085

0.2

1.0776

0.3

1.0761

0.5

1.0730

0.1

1.0776

0.2

1.0931

0.5

1.1971

0.1

1.0776

0.2

1.0553

0.5

0.9889

0.2

1.0776

0.3

1.0547

0.5

0.2122

0.2

1.0776

0.3

1.0767

0.5

1.0742

Table 3: Local Nusselt number Re−0.5 N ux when Λ = 0.2, Ha = 0.1, β t = 0.1, N ∗ = 0.1, x β c = 0.1, Sc = 2.0, γ = 0.9 and Bc = 0.2.

13

Parameters (fixed values)

Parameters Re−0.5 N ux x

λ = 0.1, R = 0.3, Pr = 2.0, Nb = 0.2, Nt = 0.1, Bt = 0.2

α

α = 0.2, R = 0.3, Pr = 2.0, Nb = 0.2, Nt = 0.1, Bt = 0.2 λ

α = 0.2, λ = 0.1, Pr = 2.0, Nb = 0.2, Nt = 0.1, Bt = 0.2

α = 0.2, λ = 0.1, R = 0.3, Nb = 0.2, Nt = 0.1, Bt = 0.2

α = 0.2, λ = 0.1, R = 0.3, Pr = 2.0, Nt = 0.1, Bt = 0.2

α = 0.2, λ = 0.1, R = 0.3, Pr = 2.0, Nb = 0.2, Bt = 0.2

α = 0.2, λ = 0.1, R = 0.3, Pr = 2.0, Nt = 0.1, Nt = 0.1

R

Pr

Nb

Nt

Bt

0.2

0.9572

0.3

0.9616

0.5

0.9689

0.1

0.9572

0.2

0.9594

0.5

0.9652

0.1

0.8488

0.3

0.9572

0.5

1.0495

1.0

0.6975

1.5

0.8330

2.0

0.9572

0.2

0.9572

0.4

0.9315

0.7

0.8936

0.1

0.9572

0.2

0.9312

0.5

0.8470

0.2

0.9572

0.3

0.9374

0.5

0.8185

Table 4: Local Sherwood number Re−0.5 Shx when Λ = 0.2, Ha = 0.1, β t = 0.1, N ∗ = 0.1, x β c = 0.1, R = 0.3, Pr = 2.0 and Bt = 0.2. 14

Parameters (fixed values)

Parameters Re−0.5 Shx x

λ = 0.1, Nb = 0.2, Nt = 0.1, Sc = 2.0, γ = 0.9, Bc = 0.2

α

α = 0.2, Nb = 0.2, Nt = 0.1, Sc = 2.0, γ = 0.9, Bc = 0.2

α = 0.2, λ = 0.1, Nt = 0.1, Sc = 2.0, γ = 0.9, Bc = 0.2

α = 0.2, λ = 0.1, Nb = 0.2, Sc = 2.0, γ = 0.9, Bc = 0.2

α = 0.2, λ = 0.1, Nb = 0.2, Nt = 1.0, γ = 0.9, Bc = 0.2

λ

Nb

Nt

Sc

α = 0.2, λ = 0.1, Nb = 0.2, Nt = 1.0, Sc = 2.0, Bc = 0.2 γ

α = 0.2, λ = 0.1, Nb = 0.2, Nt = 1.0, Sc = 2.0, γ = 0.9

15

Bc

0.2

1.4162

0.3

1.4183

0.5

1.4220

0.1

1.4162

0.2

1.4173

0.5

1.4208

0.2

1.4162

0.4

1.5221

0.7

1.5738

0.1

1.4162

0.2

1.2609

0.5

1.0151

1.0

0.9315

1.5

1.2010

2.0

1.4162

0.5

1.1439

0.7

1.2861

0.9

1.4162

0.2

1.4162

0.3

1.4962

0.5

1.5675

5

Concluding remarks

Nonlinear convective flow of Powell-Erying nanofluid under the influence of chemical reaction and magnetic field is addressed analytically. Key points are listed below: • Velocity via α and Λ is quite opposite behavior. • Larger thermal conjugate parameter Bt yields temperature enhancement. • Opposite behavior of concentration field is noticed in view of Nb and Nt . • Qualitative behavior of velocity and skin friction coefficient are reverse when λ is increased. • Reduction in local Nusselt number is observed for thermophoresis parameter Nt . • Concentration diminishes in view of Sc while Sherwood number enhances.

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Figure captions Fig. 1. Flow configuration and coordinate system. Fig. 2. }−curves for f 00 (0), θ0 (0) and φ0 (0). Fig. 3. f 0 (η) variation via α. Fig. 4. f 0 (η) variation via Λ. Fig. 5. f 0 (η) variation via Ha . Fig. 6. f 0 (η) variation via λ. Fig. 7. θ(η) variation via R. Fig. 8. θ(η) variation via Pr . Fig. 9. θ(η) variation via Nb . Fig. 10. θ(η) variation via Nt . Fig. 11. θ(η) variation via Bt . Fig. 12. φ(η) variation via Nb . Fig. 13. φ(η) variation via Nt . Fig. 14. φ(η) variation via Sc. Fig. 15. φ(η) variation via γ. Fig. 16. φ(η) variation via Bc .

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1. Magnetohydrodynamic (MHD) nonlinear convective flow of Powell-Erying nanofluid is modeled. 2. Velocity via fluid parameters (i.e.,  and  ) is quite opposite behavior. 3. Larger thermal conjugate parameter Bt yields temperature enhancement.. 4. Opposite behavior of concentration field is noticed in view of N b and N t . 5. Reduction in local Nusselt number is observed for thermophoresis parameter N t .