A revised model for Darcy-Forchheimer flow of Maxwell nanofluid subject to convective boundary condition

Accepted Manuscript

A revised model for Darcy-Forchheimer flow of Maxwell nanofluid subject to convective boundary condition Taseer Muhammad, Ahmed Alsaedi, Sabir Ali Shehzad, Tasawar Hayat PII: DOI: Reference:

S0577-9073(16)30536-6 10.1016/j.cjph.2017.03.006 CJPH 200

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

14 September 2016 27 February 2017 11 March 2017

Please cite this article as: Taseer Muhammad, Ahmed Alsaedi, Sabir Ali Shehzad, Tasawar Hayat, A revised model for Darcy-Forchheimer flow of Maxwell nanofluid subject to convective boundary condition, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.03.006

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Highlights • Darcy-Forchheimer flow of Maxwell nanofluid is modeled. • Flow is induced by a linear stretching surface.

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• Uniform magnetic field and convective surface are accounted. • Nanofluid model includes the effects of Brownian motion and thermophoresis.

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• Newly proposed condition associated with mass flux is incorporated.

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A revised model for Darcy-Forchheimer flow of Maxwell nanofluid subject to convective boundary condition Taseer Muhammad1∗ , Ahmed Alsaedi2 , Sabir Ali Shehzad3 and Tasawar Hayat1,2 1 2

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

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of

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Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan ∗

Corresponding author E-mail: [email protected] (Taseer Muhammad)

Abstract: This research article provides the magnetohydrodynamic (MHD) boundary-layer flow of

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Maxwell nanomaterial saturating a non-Darcy porous medium. Flow is generated due to a stretching surface. The flow in porous media is characterized by considering the Darcy-Forchheimer based model. Novel features of Brownian motion and thermophoresis are retained. A uniform applied magnetic field is employed. Small magnetic Reynolds number and boundary-layer assumptions are employed in the formulation. Simultaneous effects of convective heat and zero nanoparticles mass flux conditions are imposed. Transformation procedure

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is adopted to convert the partial differential system into the nonlinear ordinary differential system. The governing nonlinear ordinary differential system is solved for the convergent homotopic solutions. Convergence

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analysis is performed through the plot and numerical data. Graphs have been plotted in order to analyze the temperature and concentration profiles by distinct pertinent flow parameters. Local Nusselt number is also computed and examined.

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Keywords: Darcy-Forchheimer flow; Maxwell fluid; Nanoparticles; MHD; Porous medium;

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Convective boundary condition.

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1 Introduction

The flows in porous media are very popular amongst the engineers, mathematicians and modelers due to their occurrence in resource of geothermal energy, modeling of oil reservoir in the insulating processes, production of crude oils, ground water systems, movement of water in reservoirs and many others. Flow in porous media due to heat transfer becomes much more important in the processes of thermal insulation materials, nuclear waste disposal, solar collectors and receivers, energy storage units etc. [1 − 3]. The available literature witnesses that much attention has been given to those

problems of porous media that are modeled and developed by using the classical Darcy’s theory. The classical Darcy’s law is valid under circumstances of lower velocity and smaller porosity. The Darcy’s law is incapable when inertial and boundary effects can take place at higher flow rate. 2

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On the other side, the exceeding of Reynolds number from unity corresponds to nonlinear flow. Under these conditions it is impossible to ignore the effects of inertia and boundary. Forchheimer [4] added a square velocity term in the expression of Darcian velocity to predict the impacts of inertia and boundary. Muskat [5] named this term as “Forchheimer term” which always holds for high Reynolds number. Physically higher filtration velocities produce quadratic drag for porous media in momentum expression. Seddeek [6] explored the impacts of thermophoresis and viscous

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dissipation in Darcy-Forchheimer mixed convective flow saturating porous medium. Pal and Mondal [7] utilized the Darcy-Forchheimer theory to examine the hydromagnetic flow of variable viscosity liquid in a porous medium. Recently Hayat et al. [8] studied Darcy-Forchheimer flow of Maxwell material subject to heat flux via Cattaneo-Christov expression and temperature dependent thermal conductivity.

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There is a remarkable advancement in the nanotechnology due to its abundant applications in the industrial and physiological processes. The researchers at present are engaged to examine the mechanisms behind the nanomaterials. A solid-liquid mixture of tiny size nanoparticles and base liquid is termed as nanofluid. The colloids of base liquid and nanoparticles have important physical characteristics that enhance their potential role in the applications of ceramics, drug delivery,

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paintings, coatings etc. Nanofluids are declared as super coolants because their heat absorption capacity is much higher than traditional liquids. The reduction of the system and enhancement in its

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performance can be achieved with the implications of nanoliquids. Further a wide range of present and future utilizations of nanofluids have been presented in the work done by Das et al. [9] and Wang and Mujumdar [10, 11]. Choi [12] experimentally explored the mechanism of nanoparticles and

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concluded that the inclusion of nanoparticles into ordinary base liquids is highly useful technique to increase the thermal conductivity of traditional liquids. Buongiorno [13] developed the two-

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phase model of nanoparticles by considering the thermophoretic and Brownian motion aspects. Here we use the Buongiorno model [13] to study the convective heat transfer characteristics in

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nanofluids. This model determined that the homogeneous-flow models are in conflict with the experimental results and tend to underpredict the heat transfer coefficient of nanofluid. While the dispersion effect is totally negligible as a result of nanoparticle size. Thus, Buongiorno proposed an alternative model that ignores the shortcomings of homogeneous and dispersion models. He affirmed that the abnormal heat transfer appears due to particle migration in the fluid. Exploring the nanoparticle migration, he considered the seven slip mechanisms that can produce a parallel velocity between the nanoparticles and base fluid. These are inertia, thermophoresis, Brownian diffusion, diffusiophoresis, Magnus effect, fluid drainage and gravity. He concluded that, of these seven, only Brownian diffusion and thermophoresis are important slip mechanisms in nanofluids. Based on such 3

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findings, he established a two-component four-equation nonhomogeneous equilibrium model for mass, momentum and heat transport in nanofluids. Some recent works on the topic can be visualized in the investigations [14 − 25] and several studies therein.

The magneto nanofluid is a topic of hot spot research now a days. It is due to its involvement in

the industrial manufacturing processes which include gastric medications, sterilized devices, wound treatment, X-rays and many others. Turkyilmazoglu [26] reported the magnetohydrodynamic flow

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of nanoliquid by a permeable stretching/shrinking surface. He presented both exact and analytical solutions. Sheikholeslami et al. [27] employed the least square and Galerkin techniques to analyze the impact of magnetic field on flow of viscous fluid over a porous channel. MHD flow of pseudoplastic nanoliquid over an unsteady stretching surface with internal heat generation is reported by Lin et al. [28]. Khan et al. [29] examined the non-aligned hydromagnetic flow of nanoliquid by

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a radiative stretching surface with variable viscosity. Hayat et al. [30] studied the characteristics of chemical reaction and magnetic field in three-dimensional flow due to a bidirectional radiative surface. The recent development on magneto nanofluids can be seen in the investigations [31 − 35] and various studies therein.

Motivated by the above mentioned literature, our main goal here is to examine the characteris-

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tics of magnetohydrodynamic flow of Maxwell fluid [36 − 40] through Darcy-Forchheimer porous

medium. Convective heat and zero nanoparticles mass flux conditions are employed at the boundary

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[41 − 43] . Impacts of Brownian motion and thermophoresis are taken into account in the temperature

and concentration species expressions. The governing mathematical model is presented through the boundary layer assumptions. The dimensionless momentum, energy and mass species equations are

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computed using the homotopy analysis method (HAM) [44−55]. In fact HAM has three advantages. Firstly it does not require small/large physical parameters in the problem. Secondly it provides a

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simple way to ensure the convergence of series solutions. Thirdly it provides a large freedom to choose the base functions and related auxiliary linear operators. Convergence of computed solutions

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is checked by plots and numerical data. Local Nusselt number is computed in tabular form while the local Sherwood number is zero at the wall.

2 Formulation We examine the steady two-dimensional (2D) stretched flow of Maxwell nanofluid. An incompressible liquid saturates the porous space characterizing Darcy-Forchheimer model. Flow is generated due to a linear stretched sheet. Uniform magnetic field of strength B0 is imposed in the y−direction (see Fig. 1). Small magnetic Reynolds number justifies the omission of induced magnetic field. The 4

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Cartesian coordinate is adopted in such a way that x−axis is taken along the stretching sheet and y−axis orthogonal to it. The sheet at y = 0 is stretched in the x−direction with velocity Uw . The surface temperature is regulated by a convective heating process which is described by heat transfer coefficient denoted by hf and Tf for temperature of the hot fluid under the sheet. The governing boundary layer expressions for flow of Maxwell nanofluid are

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∂u ∂v + = 0, ∂x ∂y     2 2 2 u2 ∂∂xu2 ∂u ∂u ∂ u ∂u ν σB 0 =ν u +v + λ1  u + λ v − u − F u2 , − 1 2 2 2 ∂ u 2∂ u ∂x ∂y ∂y ρ ∂y K f +v ∂y2 + 2uv ∂x∂y    2 ! ∂T ∂T ∂ 2T (ρc)p ∂T ∂C DT ∂T u +v =α 2 + DB + , ∂x ∂y ∂y (ρc)f ∂y ∂y T∞ ∂y  2    ∂C ∂C ∂ C DT ∂ 2 T +v = DB u + . ∂x ∂y ∂y 2 T∞ ∂y 2

The subjected boundary conditions are

∂T ∂C DT ∂T = hf (Tf − T ) , DB + = 0 at y = 0, ∂y ∂y T∞ ∂y

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u = uw (x) = ax, v = 0, − k

u → 0, T → T∞ , C → C∞ as y → ∞.

(1)

(2)

(3) (4)

(5) (6)

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Note that u and v demonstrate the flow velocities in the horizontal and vertical directions respectively, λ1 denotes the relaxation time, ν(= µ/ρf ) stands for kinematic viscosity, µ represents the dynamic

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viscosity, ρf stands for density of base liquid, σ denotes the electrical conductivity, K shows the permeability of porous medium, F = Cb /xK 1/2 depicts the non-uniform inertia coefficient of porous medium, Cb stands for drag coefficient, α = k/(ρc)f for thermal diffusivity, k represents

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the thermal conductivity, (ρc)f stands for heat capacity of fluid, (ρc)p denotes the effective heat capacity of nanoparticles, DB stands for Brownian diffusivity, C stands for concentration, DT

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for thermophoretic diffusion coefficient, T∞ denotes the ambient fluid temperature, C∞ stands for ambient fluid concentration and a for positive constant. Selecting u = axf 0 (η), v = − (aν)1/2 f (η), η =


a 1/2 ν

y,

θ(η) = (T − T∞ ) / (Tf − T∞ ) , φ(η) = (C − C∞ ) /C∞ .

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Now Eq. (1) is trivially satisfied while Eqs. (2) − (6) are reduced to


f 000 + M 2 β + 1 f f 00 + β 2f f 0 f 00 − f 2 f 000 − M 2 + λ f 0 − (1 + Fr ) f 02 = 0,   2 θ00 + Pr f θ0 + Nb θ0 φ0 + Nt θ0 = 0, 5

(8) (9)

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φ00 + Le Pr f φ0 +

Nt 00 θ = 0, Nb

(10)

f (0) = 0, f 0 (0) = 1, θ0 (0) = −γ (1 − θ (0)) , Nb φ0 (0) + Nt θ0 (0) = 0,

(11)

f 0 (∞) → 0, θ (∞) → 0, φ (∞) → 0.

(12)

Here M stands for magnetic parameter, β for Deborah number, λ for porosity parameter, Fr for

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inertia coefficient, Nb for Brownian motion parameter, Pr for Prandtl number, γ for Biot number, Nt for thermophoresis parameter and Le for Lewis number. These variables are expressed as follows:  σB 2 ν b  M 2 = ρ a0 , β = λ1 a, λ = Ka , Fr = KC1/2 , Pr = αν , f (13) (ρc)p DT (Tf −T∞ ) (ρc)p DB C∞ h p Nb = (ρc) , Nt = , γ = kf νa , Le = DαB .  ν (ρc) νT∞ f

f

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The expression of local Nusselt number N ux is given by ∂T x = − (Rex )1/2 θ0 (0). N ux = − (Tw − T∞ ) ∂y y=0

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Local Reynolds number is represented by Rex = uw x/ν. The nondimensional mass flux is zero

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3 Solutions by HAM

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which is described by Sherwood number Shx [16] .

The homotopy analysis technique (HAM) is used to compute the convergent homotopic solutions of Eqs. (8) − (12) . The suitable initial approximations (f0 , θ0 , φ0 ) , linear operators (Lf , Lθ , Lφ )

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and deformation problems at zeroth and mth ˜ orders are      γ γ Nt f0 = 1 − exp (−η) , θ0 = exp (−η) , φ0 = − exp (−η) , 1+γ 1+γ Nb d3 f df d2 θ d2 φ − , L = − θ, L = − φ, θ φ dη 3 dη dη 2 dη 2       Lf B1∗ + B2∗ eη + B3∗ e−η = 0, Lθ B4∗ eη + B5∗ e−η = 0, Lφ B6∗ eη + B7∗ e−η = 0, h i (1 − þ)Lf f˜(η, þ) − f0 (η) = þ~f Nf [f˜(η, þ)], h i ˜ þ)], (1 − þ)Lθ ˜θ(η, þ) − θ0 (η) = þ~θ Nθ [f˜(η, þ), ˜θ(η, þ), φ(η, h i ˜ þ) − φ0 (η) = þ~φ Nφ [f˜(η, þ), ˜θ(η, þ), φ(η, ˜ þ)], (1 − þ)Lφ φ(η,   f˜(0, þ) = 0, f˜0 (0, þ) = 1, f˜0 (∞, þ) = 0,      0 ˜θ (0, þ) = −γ 1 − ˜θ(0, þ) , ˜θ(∞, þ) = 0,   0 0  ˜ ˜ ˜ Nb φ (0, þ) + Nt θ (0, þ) = 0, φ(∞, þ) = 0, 

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Lf =

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(15) (16) (17) (18) (19) (20)

(21)

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h i
∂ 2 f˜ ∂ 3 f˜ 2 Nf f˜(η; þ) = + M β + 1 f˜ 2 + β ∂η 3 ∂η − M2 + λ i



˜ ˜ ˜ ˜ þ) = ∂ θ + Pr f˜∂ θ + Nb ∂ θ ∂ φ + Nt f˜(η; þ), ˜θ(η, þ), φ(η, ∂η 2 ∂η ∂η ∂η 2˜

∂ ˜θ ∂η

!2 

h i 2˜ 2˜ ˜ ˜ þ) = ∂ φ + Le Pr f˜∂ φ + Nt ∂ φ , Nφ f˜(η; þ), ˜θ(η, þ), φ(η, ∂η 2 ∂η Nb ∂η 2 ˜ ˇm Lf [fm˜ (η) − χm˜ fm−1 (η)] = ~f R ˜ f ,

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ˇ m˜ , Lθ [θm˜ (η) − χm˜ θm−1 (η)] = ~θ R ˜ θ   ˜ ˇm Lφ φm˜ (η) − χm˜ φm−1 (η) = ~φ R ˜ φ, 0 0 fm˜ (0) = fm ˜ (∞) = 0, ˜ (0) = fm

θ0m˜ (0) − γθm˜ (0) = 0, θm˜ (∞) = 0,

Nb φ0m˜ (0) + Nt θ0m˜ (0) = 0, φm˜ (∞) = 0,

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ˇ m˜ (η) = θ00 + Pr R θ m−1 ˜

k X

fm−1−k ˜

fk−l fl000

l=0

m−1 ˜ X

fm−1−k fk00 ˜

fm−1−k θ0k ˜

m−1 ˜ X

      

fm−1−k ˜

(22)

(23)

(24) (25) (26) (27)

(28)

k X

0 fk−l fl00

l=0

k=0

m−1 ˜ X
0 0 − M + λ fm−1 − (1 + Fr ) fm−1−k fk0 , (29) ˜ ˜ 2

k=0

+ Pr Nb

k=0

m−1 ˜ X

θ0m−1−k φ0k ˜

+ Pr Nt

k=0

00 ˜ ˇm R + Le Pr φ (η) = φm−1 ˜

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+ 2β

k=0

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k=0

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−β

m−1 ˜ X

˜ X
m−1

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˜ 000 ˇm R + M 2β + 1 f (η) = fm−1 ˜

,

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Nθ

h


∂ f˜ − (1 + Fr ) ∂η

! 3˜ ˜ ∂ 2 f˜ f ∂ f ∂ 2 2f˜ − f˜ ∂η ∂η 2 ∂η 3 !2 ∂ f˜ , ∂η

m−1 ˜ X k=0

fm−1−k φ0k + ˜

m−1 ˜ X

θ0m−1−k θ0k , ˜

(30)

k=0

Nt 00 θ˜ , Nb m−1

  0, m ˜ ≤ 1, χm˜ =  1, m ˜ > 1.

(31)

(32)

Here þ ∈ [0, 1] stands for embedding parameter, ~f , ~θ and ~φ for non-zero auxiliary parameters and Nf , Nθ and Nφ for nonlinear operators. General expressions (fm, ˜ θm ˜ , φm ˜ ) for Eqs. (25) − (27)

∗ ∗ ∗ in terms of special solutions (fm ˜ , θm ˜ , φm ˜ ) are presented by the following expressions: ∗ η ∗ −η ∗ ∗ fm˜ (η) = fm ˜ (η) + B1 + B2 e + B3 e ,

(33)

θm˜ (η) = θ∗m˜ (η) + B4∗ eη + B5∗ e−η ,

(34)

φm˜ (η) = φ∗m˜ (η) + B6∗ eη + B7∗ e−η ,

(35)

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in which the constants Bj∗ (j = 1 − 7) through the boundary conditions (28) are ∗ ∂fm ˜ (η) ∗ ∗ ∗ ∗ ∗ , B1∗ = −B3∗ − fm B2 = B4 = B6 = 0, B3 = ˜ (0), ∂η η=0

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! ∗ 1 (η) ∂θ m ˜ − γθ∗m˜ (0) , B5∗ = 1+γ ∂η η=0 ! ∗ ∗ (η) ∂φ (η) ∂θ N t m ˜ m ˜ ∗ B7∗ = −B + + . 5 ∂η η=0 Nb ∂η η=0

4 Convergence analysis

(36)

(37)

(38)

No doubt the nonzero auxiliary parameters ~f , ~θ and ~φ in homotopic solutions stimulate the

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convergence. The ~−curves for velocity, temperature and concentration distributions are depicted in Fig. 2. Here the intervals of convergence for f, θ and φ are [−1.55, −0.20] , [−1.85, −0.30] and [−1.90, −0.10] respectively. Table 1 indicates that the 16th order of deformations are enough for convergent homotopic solutions of velocity, temperature and concentration fields. Nb = γ = 0.3, Le = 1.0 and Pr = 1.2.

−θ0 (0)

φ0 (0)

1

1.18833

0.21479

0.07160

1.17424

0.20285

0.06762

10

1.17422

0.20180

0.06727

16

1.17422

0.20169

0.06723

25

1.17422

0.20169

0.06723

35

1.17422

0.20169

0.06723

50

1.17422

0.20169

0.06723

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Order of approximations −f 00 (0) 5

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Table 1: Homotopic solutions convergence when M = Fr = Nt = 0.1, β = λ = 0.2,

5 Discussion This portion examines the impacts of various interesting parameters including Deborah number β,

magnetic parameter M, porosity parameter λ, inertia coefficient Fr , Biot number γ, Lewis number Le, Prandtl number Pr, thermophoresis parameter Nt and Brownian motion parameter Nb on the dimensionless temperature distribution θ (η) and concentration distribution φ (η) . Fig. 3 shows the effect of Deborah number β on the temperature profile θ (η) . Both the temperature profile θ (η) and thickness of thermal layer are higher for the larger values of Deborah number β. Deborah number 8

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depends upon the relaxation time. Larger values of Deborah number leads to higher relaxation time which produces an enhancement in the temperature profile and related thickness of thermal layer. Fig. 4 presents that the larger values of magnetic parameter M causes an enhancement in the temperature profile θ (η) and thickness of thermal layer. Here M 6= 0 corresponds to hydromagnetic flow case and M = 0 is for hydrodynamic flow situation. It is clearly observed that the temperature

profile is higher for hydromagnetic flow case (M 6= 0) when compared with the hydrodynamic flow

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situation (M = 0) . Impact of porosity parameter λ on the temperature profile θ (η) is shown in Fig. 5. Here the temperature profile θ (η) and thickness of thermal layer are higher with an increase in

porosity parameter λ. Physically the presence of porous media is to increase the resistance to fluid flow which causes a stronger temperature profile θ (η) and more thickness of thermal layer. Fig. 6 depicts the variations in temperature profile θ (η) for different values of inertia coefficient Fr . An

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increase in inertia coefficient Fr leads to a stronger temperature profile θ (η) and more thickness of thermal layer. Fig. 7 presents that an increase in the Biot number γ creates an enhancement in the temperature profile θ (η) and related thickness of thermal layer. Larger values of Biot number γ leads to a higher convection which causes a stronger temperature profile and more thickness of thermal layer. Effect of Prandtl number Pr on the temperature profile θ (η) is sketched in Fig.

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8. Both the temperature profile and thickness of thermal layer are decreasing functions of Prandtl number Pr . Prandtl number has an inverse relationship with the thermal diffusivity. Larger values

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of Prandtl number leads to a lower thermal diffusivity. Such lower thermal diffusivity creates a reduction in the temperature profile and related thickness of thermal layer. Fig. 9 shows that the larger values of thermophoresis parameter Nt produces an enhancement in the temperature profile

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θ (η) and associated thickness of thermal layer. Impact of Deborah number β on the concentration profile φ (η) is displayed in Fig. 10. Here we observed that the concentration profile φ (η) is higher

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when we enhance the values of Deborah number β. Fig. 11 presents that the larger values of magnetic parameter M shows an enhancement in the concentration profile φ (η) . Fig. 12 shows the

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impact of porosity parameter λ on the concentration profile φ (η) . Here the concentration profile φ (η) and thickness of concentration layer are increasing functions of porosity parameter λ. Impact

of inertia coefficient Fr on the concentration profile φ (η) is plotted in Fig. 13. An increase in inertia coefficient Fr creates an enhancement in the concentration profile and related thickness of concentration layer. Fig. 14 depicts that the larger values of Lewis number Le shows a reduction in the concentration profile φ (η) . Lewis number has an inverse relationship with the Brownian diffusion coefficient. An increase in Lewis number Le corresponds to lower Brownian diffusion coefficient. Such lower Brownian diffusion coefficient produces a reduction in the concentration profile and related thickness of concentration layer. Fig. 15 presents that an increase in Prandtl 9

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number Pr leads to weaker concentration profile φ (η) and less thickness of concentration layer. Fig. 16 shows the variations in the concentration profile φ (η) corresponding to various values of Biot number γ. Here the concentration φ (η) and related thickness of concentration layer are enhanced when the larger values of Biot number γ are taken into account. Fig. 17 presents that the larger values of Brownian motion parameter Nb corresponds to weaker concentration profile φ (η) and less thickness of concentration layer. Effect of thermophoresis parameter Nt on the concentration

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profile φ (η) is sketched in Fig. 18. Both the concentration profile φ (η) and associated thickness of concentration layer are higher for the larger values of thermophoresis parameter Nt . Table 2 shows the numerical values of local Nusselt number (−θ0 (0)) for various values of β, M, λ, Fr , Le, Pr, γ, Nb and Nt . It is clearly observed that the local Nusselt number is reduced for larger Deborah number β, magnetic parameter M and porosity parameter λ. However opposite behavior

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is observed for Prandtl number Pr and Biot number γ. It is also noticed that the effect of Brownian motion parameter Nb on the local Nusselt number is constant.

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Table 2: Numerical data for local Nusselt number (−θ0 (0)) for various values of β, M, λ, Fr ,

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Le, Pr, γ, Nb and Nt . β

M

λ

Fr

Le

Pr

γ

Nb

Nt

−θ0 (0)

0.0

0.1

0.2

0.1

1.0

1.2

0.3

0.3

0.1

0.20321

0.5

0.19945

1.0

0.19587 0.0

0.2

0.1

1.0

1.2

0.3

0.3

0.5 0.1

0.19633

0.0

0.1

1.0

1.2

0.3

0.3

0.4 0.2

0.19656

0.0

1.0

0.5 0.1

0.2

0.1

1.2

0.3

0.3

0.1

0.19895

0.5

1.2

0.3

0.3

0.1

1.5

AC

0.1

PT 0.2

0.1

0.2

1.0

ED

0.2

0.1

CE

0.2

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.1

1.0

1.0

1.0

0.20185 0.20169

M

1.0 0.2

0.20203 0.20042

1.0 0.2

0.20358

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0.1

0.1

0.19990

0.8 0.2

0.20178

0.19954

0.8 0.2

0.1

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0.2

0.5

0.20160 0.3

0.3

0.1

0.15592

1.0

0.19306

1.5

0.21147

1.2

1.2

1.2

0.2

0.3

0.1

0.15100

0.7

0.32691

1.2

0.40532

0.3

0.3

0.5

0.1

0.20169

1.0

0.20169

1.5

0.20169

0.3

0.0

0.20204

0.5

0.20022

1.0

0.19828

6 Conclusions Magnetohydrodynamic (MHD) stretched flow of Maxwell nanofluid with nonlinear porous medium and convective condition effects is investigated. The key findings of this analysis are summarized 11

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below: • Larger values of Deborah number β causes an enhancement in the temperature and concentration. • Temperature and concentration are higher for the larger values of magnetic parameter M.

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• An increase in porosity parameter λ shows an enhancement in the temperature and concentration.

• Larger values of Biot number γ and inertia coefficient Fr show an increasing behavior for both temperature and concentration.

• Both the temperature and concentration are reduced when the larger values of Prandtl number

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Pr are accounted.

• Concentration distribution is weaker for larger values of Brownian motion parameter Nb while opposite behavior is observed for thermophoresis parameter Nt .

• Heat transfer rate at the sheet (local Nusselt number) is lower for the larger values of ther-

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mophoresis parameter Nt while it is constant for Brownian motion parameter Nb .

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References

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[43] T. Hayat, I. Ullah, T. Muhammad and A. Alsaedi, A revised model for stretched flow of third grade fluid subject to magneto nanoparticles and convective condition, J. Mol. Liq., 230 (2017) 608-615. [44] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004) 499-513.

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Theoret. Nanosci., 11 (2014) 486-496. [50] S. Abbasbandy, T. Hayat, A. Alsaedi and M.M. Rashidi, Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid, Int. J. Numer. Methods Heat Fluid Flow, 24

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[53] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness, J. Mol. Liq., 221 (2016) 1121-1127. [54] T. Hayat, M.I. Khan, M. Waqas, T. Yasmeen and A. Alsaedi, Viscous dissipation effect in flow of magnetonanofluid with variable properties, J. Mol. Liq., 222 (2016) 47-54. 16

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[55] T. Hayat, T. Abbas, M. Ayub, T. Muhammad and A. Alsaedi, On squeezed flow of Jeffrey

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nanofluid between two parallel disks, Appl. Sci., 6 (2016) 346.

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Figure 1: Geometry of the problem.

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Figure 2: The ~−curves for f, θ and φ.

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Figure 3: Plots of θ (η) for β.

Figure 4: Plots of θ (η) for M.

Figure 5: Plots of θ (η) for λ.

Figure 6: Plots of θ (η) for Fr . 18

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Figure 8: Plots of θ (η) for Pr .

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Figure 7: Plots of θ (η) for γ.

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Figure 9: Plots of θ (η) for Nt .

Figure 11: Plots of φ (η) for M.

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Figure 10: Plots of φ (η) for β.

Figure 12: Plots of φ (η) for λ.

Figure 13: Plots of φ (η) for Fr . 19

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Figure 14: Plots of φ (η) for Le.

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Figure 15: Plots of φ (η) for Pr .

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Figure 16: Plots of φ (η) for γ.

Figure 17: Plots of φ (η) for Nb .

Figure 18: Plots of φ (η) for Nt .

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