A revised model for stretched flow of third grade fluid subject to magneto nanoparticles and convective condition

    A revised model for stretched flow of third grade fluid subject to magneto nanoparticles and convective condition Tasawer Hayat, Ikram Ullah, Taseer Muhammad, Ahmed Alsaedi PII: DOI: Reference:

S0167-7322(16)33646-7 doi:10.1016/j.molliq.2017.01.074 MOLLIQ 6876

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

16 November 2016 7 January 2017 22 January 2017

Please cite this article as: Tasawer Hayat, Ikram Ullah, Taseer Muhammad, Ahmed Alsaedi, A revised model for stretched flow of third grade fluid subject to magneto nanoparticles and convective condition, Journal of Molecular Liquids (2017), doi:10.1016/j.molliq.2017.01.074

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ACCEPTED MANUSCRIPT

A revised model for stretched flow of third grade fluid subject to magneto nanoparticles and convective condition b

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

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a

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Tasawer Hayata,b , Ikram Ullaha , Taseer Muhammada∗ and Ahmed Alsaedib

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

Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Corresponding author E-mail: taseer [email protected] (Taseer Muhammad)

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∗

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Abstract: Magnetohydrodynamic (MHD) stretched flow of third grade nanoliquid with convective surface condition is examined. Third grade liquid is electrically conducting subject to uniform

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magnetic field. Aspects of Brownian diffusion and thermophoresis have been accounted. Newly suggested condition for zero nanoparticles mass flux is employed. Proper transformations are utilized to convert the partial differential system (PDE) into the non-linear ordinary differential system

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(ODE). The resulting nonlinear systems is solved for the series solutions of velocity, temperature and concentration distributions. Convergence of the developed solutions is verified explicitly through

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tables and plots. Consequences of various influential variables on the non-dimensional velocity, temperature and concentration distributions are interpreted graphically. Skin friction coefficient

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and local Nusselt number are analyzed through plots and numerical data.

Keywords: Third grade fluid; MHD; Nanoparticles; Nonlinear analysis; Convective boundary condition.

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ACCEPTED MANUSCRIPT Nomenclature velocity components (m.s−1 )

x, y

coordinate axes (m)

µ

dynamic viscosity (P a.s)

ρf

density of base fluid (kg.m−3 )

ν

kinematic viscosity (m2 .s−1 )

α1 , α 2 , α 3

material constants (kg.m−1 )

σ

electrical conductivity (S.m−1 )

B0

magnetic field strength (N.m−1 .A−1 )

T

temperature (K)

C

concentration

αm

thermal diffusivity (m2 .s−1 )

k

thermal conductivity (W.m−1 .K −1 )

(ρc)p

effective heat capacity of nanoparticles (J.kg −3 .K −1 )

(ρc)f

heat capacity of fluid (J.kg −3 .K −1 )

DB

Brownian diffusion coefficient (m2 .s−1 )

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thermophoretic diffusion coefficient (m2 .s−1 ) heat transfer coefficient (W.m−2 .K −1 )

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hf

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DT

T

u, v

f′

dimensionless velocity

θ

dimensionless temperature

φ

dimensionless concentration

β 1 , β 2 , ǫ1

fluid parameters

ǫ2

local Reynolds number

Le

Lewis number

Pr

Prandtl number

Nb

Brownian motion parameter

Nt

thermophoresis parameter

M

magnetic parameter

γ

Biot number

2

ACCEPTED MANUSCRIPT 1

Introduction

Analysis of non-Newtonian liquids is of great interest to the recent scientists and engineers. It is because of involvement of non-Newtonian materials in engineering and industrial pro-

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cesses. Paints, cosmetic products, colloidal fluids, suspension fluids, shampoos, blood at low

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shear rate, ice cream, mud and polymers etc. are few examples of non-Newtonian materials. The non-Newtonian materials in view of their diverse rheological characteristics cannot be

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explored through unique constitutive relationship. There exist a non-linear link between the shear stress and shear rate in the case of non-Newtonian liquids. Therefore the scientists and

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investigators have suggested several models of non-Newtonian liquids. These liquids are generalized mainly into three groups namely integral, differential and rate types. Second grade

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model describing normal stress only is a simple subclass of differential type liquids. Further second grade liquid does not predict shear thinning/thickening aspects. Here we consider third grade fluid (subclass of differential type) which characterizes both shear thickening and

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thinning behavior even in one-dimensional steady flow. Abbasbandy et al. [1] explored the analysis of thin film flow of third grade fluid. He presented both the series and exact solu-

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tions. Hayat et al. [2] elaborated consequences of MHD rotating flow of third grade liquid . Farooq et al. [3] examined the flow of third-grade nanoliquids in a vertical flow configura-

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tion. Li et al. [4] inspected the flow of electro-osmatic rotating third grade fluids bounded by two microparallel plates. Sinha [5] examined the MHD stretched flow of third order liquid in a porous channel. Hussain et al. [6] analyzed the flow of third grade nanoliquid in the existence of Joule heating and solar radiation. Okoya [7] studied the flow of transition and thermal criticality of a reactive third-grade fluid in a pipe. Here Reynolds model of viscosity is considered. Nanofluid is an advanced kind of material comprising suspension of solid particles known as nanoparticles in conventional base fluids (oil, ethylene glycol, H 2 O, bioliquids, polymer solutions and lubricant). Typically the nanoparticles are made of oxides like alumina, titania and copper oxide, carbides and metal including copper and gold. Many investigators also utilized the carbon nanotubes and diamond in base of liquid. Nanoparticles improve the physical properties of conventional liquids. Such fluids have several applications like electronics cooling, transformer cooling, heat exchanger etc [8 − 10] . Moreover magneto nanofluids have tremendous interaction in medicine, engineering and physics. MHD generators, pumps,

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ACCEPTED MANUSCRIPT bearings and boundary layer control are few such applications. The magnetic material has both liquid and magnetic properties of magneto nanofluids. Choi [11] firstly utilized the nanoparticles to advance thermal conductivity of liquids and storage of energy. Owing to

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this several researchers and scientists are engaged in the inspection of flows of nanofluids

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through different aspects. Few contributions in this direction can be consulted through the analyses [12 − 30] and various studies therein.

The prime purpose of present study is to explore the magnetohydrodynamic (MHD)

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stretched flow of third grade nanofluid subject to convective boundary condition. Brownian diffusion and thermophoresis aspects have been accounted. Convective heat and zero

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nanoparticles mass flux conditions are utilized at the surface. Such conditions are very rare and more realistic physically. No doubt the convective heat transfer is an important

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phenomenon in the industrial manufacturing processes like chemical catalytic reactors, electrochemical, heat exchangers, heat storage beds, refrigerators, insulation of nuclear reactors etc. Boundary layer approach is utilized for relevant mathematical formulation. Homotopic

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approach [31 − 40] is implemented for the solution procedure. Corresponding convergence criteria is developed and verified explicitly. Solution expressions are examined via tabular

2

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and analyzed.

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values and graphs. Skin friction coefficient and local Nusselt number have been also computed

Mathematical formulation

Steady two-dimensional (2D) flow of third grade nanoliquid is considered. The sheet is linearly stretched with velocity Uw = cx where c is a positive constant. Further stretching sheet is responsible for flow induction. Third grade fluid is assumed an electrically conducting through a uniform magnetic field B0 applied in the y−direction. Effects of electric field and Hall current are neglected. The induced magnetic field is not considered subject to small magnetic Reynolds number. A system of Cartesian coordinate is considered such that the x−axis is chosen horizontal (along the stretching surface) and y−axis is normal to it. Effects of Brownian motion and thermophoresis are taken into account. Thermal convective and zero nanoparticles mass flux conditions are employed. The temperature at the surface is because of convective heating which is featured by temperature Tf and coefficient of heat transport hf . The boundary layer expressions governing the flow of third grade nanofluid in the absence 4

ACCEPTED MANUSCRIPT of viscous dissipation, thermal radiation and chemical reaction are written as follows [6 , 25] : ∂u ∂v + = 0, ∂x ∂y

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∂3u ∂ 3u ∂u ∂ 2 u ∂u ∂ 2 u + v + + 3 ∂x∂y 2 ∂x ∂y 2 ∂y ∂x∂y ∂y 3  2 6α3 ∂u ∂ 2 u σB02 2α2 ∂u ∂ 2 u + + − u, ∂y ∂y 2 ρf ∂y ∂x∂y ρf ρf ! "   2 DT ∂T ∂ 2T ∂T ∂T (ρc)p ∂T ∂C + +v = αm 2 + u DB , ∂x ∂y ∂y ∂y ∂y (ρc)f T∞ ∂y  ∂ 2 C DT ∂ 2 T ∂C ∂C +v = DB 2 + . u ∂x ∂y ∂y T∞ ∂y 2 u

u = Uw = cx, v = 0, − k

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The associated boundary conditions are

(2)

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∂u ∂ 2 u α1 ∂u +v = ν 2+ u ∂x ∂y ∂y ρf

(1)

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

u → 0, T → T∞ , C → C∞ as y → ∞.

(3) (4)

(5)

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(6)

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In the above expressions u and v show the flow velocities in the x− and y−directions respectively, µ stands for dynamic viscosity, ν for kinematic viscosity, ρf for density of fluid, α1 , α2 k (ρc)f

for thermal diffusivity,

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and α3 for material constants, σ for electrical conductivity, αm =

k for thermal conductivity, (ρc)f for heat capacity of fluid, (ρc)p for effective heat capacity of nanoparticles, DB for Brownian diffusion coefficient, T for temperature, C for concentration, DT for thermophoretic diffusion and T∞ and C∞ for ambient liquid temperature and concentration respectively. The non-dimensional variables are defined below: u = cxf ′ (η), v = − (cν)1/2 f (η), θ(η) =

# c $1/2 T − T∞ C − C∞ , φ(η) = , η= y. Tf − T ∞ C∞ ν

(7)

Now incompressibility condition is satisfied and Eqs. (2) − (6) become 2

2

2

f ′′′ + f f ′′ − f ′ + β 1 (2f ′′′ f ′ − f f ′′′′ ) + (3β 1 + 2β 2 ) f ′′ + 6ǫ1 ǫ2 f ′′′ f ′′ − M 2 f ′ = 0, 2

θ′′ + Pr(f θ′ + Nb θ′ φ′ + Nt θ′ ) = 0, φ′′ + Le Pr f φ′ +

Nt ′′ θ = 0, Nb

(8) (9) (10)

f = 0, f ′ = 1, θ′ = −γ(1 − θ), Nb φ′ + Nt θ′ = 0 at η = 0,

(11)

f ′ → 0, θ → 0, φ → 0 as η → ∞.

(12)

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ACCEPTED MANUSCRIPT Here β 1, β 2 and ǫ1 stand for material parameters of third grade liquid, ǫ2 for local Reynolds number, M for magnetic parameter, Pr for Prandtl number, γ for Biot number, Nt for thermophoresis parameter, Nb for Brownian motion parameter and Le for Lewis number.

β2 =

cα2 , µ

ǫ1 =

cα3 , µ

2

M =

σB02 , ρf c

Pr =

(ρc)p DT (Tf −T∞ ) , (ρc)f νT∞

ν , αm

Nt = 2 hf % ν p DB C∞ αm , Le = , γ = . ǫ2 = cxν , Nb = (ρc)(ρc) c k DB fν

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β1 =

cα1 , µ

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These quantities are expressed as follows:

τw =



µ ∂u ∂y

+ α1

#

∂2u u ∂x∂y

+

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with

τw xqw , N u = , x 1/2ρf Uw2 k (Tw − T∞ ) ∂u 2 ∂u ∂x ∂y

+ # $

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Cfx =

qw = −k

The dimensionless forms of Eq. (14) yield

∂T ∂y

2 v ∂∂yu2

y=0

.

$

+ 2α3

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$ # 1/2 3 Re Cfx = f ′′ + β 1 (3f ′ f ′′ − f f ′′′ ) + 2ǫ1 ǫ2 f ′′ x

η=0

#

∂u ∂y

(13)



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Skin friction coefficient (Cfx ) and local Nusselt number (N ux ) are defined by

 

(14)

$3

y=0

   , 

(15)

  

−1/2

, Re N ux = −θ′ (0). x

(16)

3

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It is noticed that the third grade fluid model reduces to second order fluid case when ǫ1 = 0.

Homotopic solutions

The initial guesses, linear operators and deformation problems are given by f0 (ζ) = 1 − e−η , θ0 (η) =

γ −η γ Nt −η e , e , φ0 (η) = − 1+γ 1 + γ Nb

Lf = f ′′′ − f ′ , Lθ = θ′′ − θ, Lφ = φ′′ − φ, * + * + * + ∗∗ η ∗∗ −η η ∗∗ −η η ∗∗ −η Lf A∗∗ = 0, Lθ A∗∗ = 0, Lφ A∗∗ = 0, 1 + A2 e + A3 e 4 e + A5 e 6 e + A7 e , (1 − p˜)Lf fˆ(η, p˜) − f0 (η) = p˜f Nf [fˆ(η, p˜)], , ˆ p˜)], (1 − p˜)Lθ ˆθ(η, p˜) − θ0 (η) = p˜θ Nθ [fˆ(η, p˜), ˆθ(η, p˜), φ(η, , ˆ p˜) − φ (η) = p˜φ Nφ [fˆ(η, p˜), ˆθ(η, p˜), φ(η, ˆ p˜)], (1 − p˜)Lφ φ(η, 0  ′ ′ ˆ ˆ ˆ f (0, p˜) = 0, f (0, p˜) = 1, f (∞, p˜) = 0,     # $ ′ ˆθ (0, p˜) = −γ 1 − ˆθ(0, p˜) , ˆθ(∞, p˜) = 0,    ′ ′ ˆ ˆ ˆ N φ (0, p˜) + N θ (0, p˜) = 0, φ(∞, p˜) = 0,  b

t

6

(17) (18) (19) (20) (21) (22)

(23)

ACCEPTED MANUSCRIPT ,

! " ! "2 ∂ fˆ ∂ 3 fˆ ˆ∂ 4 fˆ ∂ 3 fˆ ˆ∂ 2 fˆ ∂ fˆ = +f 2 − + β1 2 −f 4 ∂η 3 ∂η ∂η ∂η ∂η 3 ∂η ! "2 ! "2 ˆ ∂ 2 fˆ ∂ 2 fˆ ∂ 3 fˆ 2 ∂f + 6ǫ ǫ − M + (3β 1 + 2β 2 ) , 1 2 ∂η 2 ∂η 2 ∂η 3 ∂η

(24)

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Nf fˆ(η; p˜)

-

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, 2ˆ ˆ ˆ ˆ ˆ p˜), fˆ(η; p˜) = ∂ θ + Pr fˆ∂ θ + Pr Nb ∂ θ ∂ φ + Nθ ˆθ(η, p˜), φ(η, ∂η 2 ∂η ∂η ∂η ! "2 ∂ ˆθ Pr Nt , ∂η 2ˆ 2ˆ ˆ ˆ p˜), ˆθ(η, p˜), fˆ(η; p˜)] = ∂ φ + Le Pr fˆ∂ φ + Nt ∂ θ , Nφ [φ(η, ∂η 2 ∂η Nb ∂η 2 ˇm Lf [fm (η) − χm fm−1 (η)] = f R f (η),

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(26)

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(27)

ˇ m (η), Lθ [θm (η) − χm θm−1 (η)] = θ R θ

(28)

* + ˇ m (η), Lφ φm (η) − χm φm−1 (η) = φ R φ

(29)

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 ′ ′ ′ fm (0) = fm (0) = fm (∞) = 0, θm (0) − γθm (0) = 0,  θ (∞) = 0, N φ′ (0) + N θ′ (0) = 0, φ (∞) = 0,  b m

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m

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ˇ m (η) = f ′′′ (η) + R f m−1 −β 1

m−1 .

t m

fm−1−k fk′′

−

k=0

m−1 .

fm−1−k fk′′′′ + 3β 1

k=0 m−1 .

+6ǫ1 ǫ2

ˇ m (η) = θ′′m−1 (η) + Pr R θ

′′ fm−1−k

k=0

m−1 .

=

m−1 .

′ fm−1−k fk′

+ 2β 1

k=0 m−1 .

′′ fm−1−k fk′′ + 2β 2

m−1 .

′ fm−1−k fk′′′

k=0 m−1 .

′′ fm−1−k fk′′

k=0

′′ ′ fk−l fl′′′ − M 2 fm−1 (η),

(31)

l=0

fm−1−k θ′k

φ′′m−1 (η)

(30)

m

k=0

k .

+ Pr Nb

k=0

ˇ m (η) R φ

(25)

m−1 .

φ′m−1−k θ′k

+ Pr Nt

k=0

+ Le Pr

m−1 .

fm−1−k φ′k

k=0

  0, m ≤ 1, χm =  1, m > 1,

m−1 .

θ′m−1−k θ′k ,

(32)

k=0

+



Nt Nb

θ′′m−1 (η),

(33)

(34)

where p˜ stands for the embedding parameter, f , θ and θ for auxiliary parameters and Nf , Nθ and Nθ for nonlinear operators. The expression of general solutions (fm (η), θm (η),

7

ACCEPTED MANUSCRIPT ∗ ∗ gm (η)) of the governing equations in the form of special solutions (fm (η), θ∗m (η), gm (η)) are

presented as follows: (35)

η ∗∗ −η θm (η) = θ∗m (η) + A∗∗ 4 e + A5 e ,

(36) (37)

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η ∗∗ −η φm (η) = φ∗m (η) + A∗∗ 6 e + A7 e ,

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∗ ∗∗ η ∗∗ −η fm (η) = fm (η) + A∗∗ 1 + A2 e + A3 e ,

values

4

1 1+γ

2 ∂θ ∗m (η) 2 ∂η 2

= −

η=0

A∗∗ 6

= 0,

A∗∗ 3

γ θ∗ (0), 1+γ m

=

A∗∗ 7

2

∗ (η) 2 ∂fm ∂η 2

η=0

2 ∗ m (η) 2 = ∂φ∂η 2

,

η=0

A∗∗ 1 +

#

= Nt Nb

−A∗∗ 3 $

−

∗ fm (0),

−A∗∗ 5 +

2

∂θ ∗m (η) 2 ∂η 2

η=0

  

 . 

(38)

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A∗∗ 5 =

=

A∗∗ 4

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A∗∗ 2

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in which the constants A∗∗ i (i = 1 − 7) through the boundary conditions (30) contain the

Convergence analysis

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In homotopic solutions, the convergence of homotopic solutions strongly depends upon the auxiliary parameters f , θ and φ . The proper values of these parameters play a key role

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to develop the convergent series solutions. A case has been considered when M = 0.5, Nt = 0.2, Le = Pr = 1.2, Nb = 0.7, β 1 = β 2 = 0.1, ǫ1 = ǫ2 = 0.2 and γ = 0.3. For suitable values

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of f , θ and φ , the h−curves are plotted at 16th order of homotopic approximations. It is clearly seen from Figs. 1 and 2 that the acceptable ranges of f , θ and φ are [−0.8, −0.4], [−1.4, −0.1] and [−1.6, −0.6] respectively. Furthermore the series solutions converge in the whole region of η when f = −0.5 and θ = φ = −1.0. Table 1 depicts that 20th order of approximation up to 4 decimal places are adequate regarding the convergence analysis.

Fig. 1 : The −curve for f (η) .

Fig. 2 : The −curves for θ (η) and φ (η) . 8

ACCEPTED MANUSCRIPT Table 1. Homotopic solutions convergence when M = 0.5, Nt = 0.2, Le = Pr = 1.2,

Order of approximations

−f ′′ (0)

−θ′ (0)

φ′ (0)

1

0.4038

0.2196

0.0627

5

0.2115

0.2129

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Nb = 0.7, β 1 = β 2 = 0.1, ǫ1 = ǫ2 = 0.2 and γ = 0.3.

10

0.2080

0.2158

0.0617

15

0.2094

0.2151

0.0615

20

0.2090

0.2153

0.0615

25

0.2090

0.2153

0.0615

35

0.2090

0.2153

0.0615

0.2090

0.2153

0.0615

5

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50

0.0608

Discussion

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Here impacts of several physical flow variables on non-dimensional temperature θ (η) and concentration φ (η) profiles are examined. These outcomes are sketched in the Figs. (3) −(16).

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The features of magnetic parameter M on temperature θ (η) is sketched in Fig. 3. Here rising values of M result in an increment of temperature θ (η) field. It is quite clear that

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magnetic filed intensity tends to create drag resistive Lorentz force which limited the fluid motion and consequently temperature field enhances. Fig. 4 illustrates the variation in θ (η) for distinct values of material parameter β 1 . Here it is revealed that an increment in β 1 reduces the temperature field θ (η) . The consequences of material parameter β 2 on temperature θ (η) is sketched in Fig. 5. It is found that higher values of β 2 yield reduction in temperature θ (η) . Fig. 6 exhibits impacts of thermophoresis parameter Nt on temperature field. Here it is analyzed that an increment in Nt enhances the temperature field. In fact thermophoresis is a mechanism in which heated particles are pulled away from hot surface to the cold region. Due to this fact the temperature of the fluid increases. The variation of Prandtl number Pr on temperature distribution is elucidated in Fig. 7 . Here clearly the penetration depth of temperature is much greater at Pr = 0.7 in comparison to Pr = 1.3. Physically for increment in Pr, the thermal diffusivity decreases which substantially decays the temperature. Fig. 8 portrays the characteristics of Biot number γ on temperature θ(η). Temperature and thermal layer thickness are enhanced for larger Biot number. Effects of material parameters β 1 and β 2 on concentration field φ (η) are plotted in Figs. 9 and 9

ACCEPTED MANUSCRIPT 10. Here both concentration field and corresponding layer thickness are decreased for larger β 1 and β 2 . The variation of magnetic field on concentration field is illustrated in Fig. 11 . Concentration profile dominants for higher values of M. Fig. 12 is interpreted to see the

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variations in concentration field for varying Nt . It is revealed that an increment in Nt causes an

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enhancement in φ(η). Fig. 13 demonstrates the consequences of Brownian motion parameter Nb on concentration distribution φ(η). Concentration field is a decreasing function of Nb . Fig. 14 is drawn to characterize the consequences of Le on concentration. Lewis number

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has inverse relation for coefficient of Brownian diffusion. Higher values of Le cause a smaller Brownian diffusion coefficient which is responsible to reduce the concentration field. The

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characteristics of Pr on concentration field are elucidated in Fig. 15 . We noticed that higher values of Pr (Prandtl number) causes a decay in concentration field. Fig. 16 is interpreted

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to analyze the behavior of Biot number γ on φ(η). Dominant behavior of concentration field is seen for larger values of γ. Fig. 17 shows the impacts of Biot γ and Prandtl P r numbers on local Nusselt number. Local Nusselt number is increasing function of γ and Pr . Fig. 18 is

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interpreted to understand the variations of local Nusselt number for varying Prandtl number Pr and magnetic parameter M . Here heat transfer rate enhances for larger values of M

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while it reduces for larger P r. The numerical values of surface drag coefficient − Re1/2 x Cf x for pertinent flow parameters like β 1 , β 2 , ǫ1 , ǫ2 and M are tabulated in Table 2. Here the

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surface drag coefficient enhances for increasing values of β 1 , M, ǫ1 and ǫ2 while reverse feature is seen for higher values of β 2 .

Fig. 3. Effect of M on θ(η).

Fig. 4. Effect of β 1 on θ(η).

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ACCEPTED MANUSCRIPT Table 2. Numerical data of skin friction coefficient − Re1/2 x Cf x for various values of β 1 , β 2 , ǫ1 , ǫ2 and M. β1

β2

ǫ1

ǫ2

M

− Re1/2 x Cf x 0.2724

0.2

0.2920

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0.1

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0.0 0.1 0.2 0.2 0.5 0.2390

0.1 0.0 0.2 0.2 0.5 0.3313 0.2164

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0.2 0.3

0.1630

0.2

0.2724 0.2759

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0.4

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0.1 0.1 0.0 0.2 0.5 0.2705

0.1 0.1 0.2 0.0 0.5 0.2705

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0.3 0.3 0.2714 0.5

0.2718

0.1 0.2 0.1 0.1 0.0 0.0730

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0.5 0.2731

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0.8 0.6361

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Conclusions

Magnetohydrodynamic (MHD) flow of third grade nanofluid bounded by a convectively heated stretching surface is investigated. The major outcomes of presented analysis are

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listed below.

• Larger values of magnetic parameter M show increasing trend for temperature θ (η)

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and concentration φ (η) .

• Effects of fluid parameters β 1 and β 2 on the temperature θ (η) and concentration φ (η)

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fields are qualitatively similar.

mophoresis parameter Nt .

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• Both temperature θ (η) and concentration φ (η) show similar behavior for larger ther-

• Concentration φ (η) and associated layer thickness are reduced for larger Brownian

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motion parameter Nb .

• By increasing the Prandtl number Pr, a reduction is noticed in both temperature θ (η)

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and concentration φ (η) fields.

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• Skin friction coefficient is enhanced for larger values of magnetic parameter M. • Heat transfer rate at the surface (local Nusselt number) is higher for larger values of Prandtl number Pr while it is lower for magnetic parameter M.

References

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ACCEPTED MANUSCRIPT [4] S.X. Li, Y.J. Jian, Z.Y. Xie, Q.S. Liu and F.Q. Li, Rotating electro-osmatic flow of third grade fluids between two microparallel plates, Collids and Surfaces A, 470 (2015) 240-247.

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[5] A. Sinha, MHD flow and heat transfer of third order fluid in a porous channel with

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and Joule heating in flow of third grade nanofluid, Z. Naturforsch. A, 70a (2015) 177-184.

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[7] S.S. Okoya, Flow, thermal criticality and transition of a reactive third-grade fluid in a pipe with Reynolds’ model viscosity, J. Hydrodynamics, Ser. B., 28 (2016) 84-94.

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[8] Z. Hu, W. Lu and M.D. Thouless, Slip and wear at a corner with Coulomb friction and an interfacial strength, Wear, 338 (2015) 242-251.

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[9] Z. Hu, M.D. Thouless and W. Lu, Effects of gap size and excitation frequency on the vibrational behavior and wear rate of fuel rods, Nuclear Eng. Design, 308 (2016) 261-268.

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[10] Z. Hu, W. Lu, M.D. Thouless and J.R. Barber, Effect of plastic deformation on the

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[11] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, USA, ASME, FED 231/MD 66 (1995) 99-105. [12] M. Turkyilmazoglu, Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids, Chem. Eng. Sci., 84 (2012) 182-187. [13] A. Malvandi and D.D. Ganji, Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, Int. J. Thermal Sci., 84 (2014) 196-206. [14] M. Sheikholeslami, D.D. Ganji, M.Y. Javed and R. Ellahi, Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model, J. Magn. Magn. Mater., 374 (2015) 36-43.

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ACCEPTED MANUSCRIPT [15] T. Hayat, A. Aziz, T. Muhammad and B. Ahmad, Influence of magnetic field in threedimensional flow of couple stress nanofluid over a nonlinearly stretching surface with convective condition, Plos One, 10 (2015) e0145332.

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[16] A. Malvandi, M.R. Safaei, M.H. Kaffash and D.D. Ganji, MHD mixed convection in a

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vertical annulus filled with Al 2 O3 –water nanofluid considering nanoparticle migration, J. Magn. Magn. Mater., 382 (2015) 296-306.

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[17] C. Zhang, L. Zheng, X. Zhang and G. Chen, MHD flow and radiation heat transfer of

Math. Modell., 39 (2015) 165-181.

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nanofluids in porous media with variable surface heat flux and chemical reaction, Appl.

[18] T. Hayat, T. Muhammad, A. Alsaedi and M.S. Alhuthali, Magnetohydrodynamic three-

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dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation, J. Magn. Magn. Mater., 385 (2015) 222-229.

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[19] M. Sheikholeslami and D.D. Ganji, Entropy generation of nanofluid in presence of mag-

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netic field using Lattice Boltzmann Method, Physica A: Stat. Mech. Appl., 417 (2015)

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[21] M. Turkyilmazoglu, A note on the correspondence between certain nanofluid flows and standard fluid flows, ASME J. Heat Transfer, 137 (2015) 024501. [22] R. Ellahi, M. Hassan and A. Zeeshan, Shape effects of nanosize particles in Cu-H 2 O nanofluid on entropy generation, Int. J. Heat Mass Transfer, 81 (2015) 449-456. [23] T. Hayat, I. Ullah, T. Muhammad, A. Alsaedi and S.A. Shehzad, Three-dimensional flow of Powell-Eyring nanofluid with heat and mass flux boundary conditions, Chin. Phys. B, 25 (2016) 074701. [24] K.L. Hsiao, Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet, Appl. Thermal Eng., 98 (2016) 850-861.

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ACCEPTED MANUSCRIPT [25] T. Hayat, A. Aziz, T. Muhammad and B. Ahmad, On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet, J. Magn. Magn. Mater., 408 (2016) 99-106.

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[26] A. Malvandi, D.D. Ganji and I. Pop, Laminar filmwise condensation of nanofluids over

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a vertical plate considering nanoparticles migration, Appl. Thermal Eng., 100 (2016) 979-986.

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[27] J. Sui, L. Zheng and X. Zhang, Boundary layer heat and mass transfer with CattaneoChristov double-diffusion in upper-convected Maxwell nanofluid past a stretching sheet

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with slip velocity, Int. J. Thermal Sci., 104 (2016) 461-468. [28] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, On magnetohydrodynamic three-

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dimensional flow of nanofluid over a convectively heated nonlinear stretching surface, Int. J. Heat Mass Transfer, 100 (2016) 566-572.

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[29] C.S.K. Raju, N. Sandeep and A. Malvandi, Free convective heat and mass transfer of MHD non-Newtonian nanofluids over a cone in the presence of non-uniform heat

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source/sink, J. Mol. Liq., 221 (2016) 108-115.

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[30] T. Hayat, T. Muhammad, S.A. Shehzad and A. Alsaedi, On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: A numerical study, Comp. Methods Appl. Mech. Eng., 315 (2017) 467-477. [31] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004) 499-513. [32] M. Turkyilmazoglu, Solution of the Thomas-Fermi equation with a convergent approach, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012) 4097-4103. [33] A. Malvandi, F. Hedayati and G. Domairry, Stagnation point flow of a nanofluid toward an exponentially stretching sheet with nonuniform heat generation/absorption, J. Thermodynamics, 2013 (2013) 764827. [34] 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 (2014) 390-401. 18

ACCEPTED MANUSCRIPT [35] J. Sui, L. Zheng, X. Zhang and G. Chen, Mixed convection heat transfer in power law fluids over a moving conveyor along an inclined plate, Int. J. Heat Mass Transfer, 85 (2015) 1023-1033.

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[36] A. Zeeshan, A. Majeed and R. Ellahi, Effect of magnetic dipole on viscous ferro-fluid

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stretching surface, J. Mol. Liq., 220 (2016) 1004-1012. [39] 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

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surface thickness, J. Mol. Liq., 221 (2016) 1121-1127.

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heat generation/absorption, Int. J. Thermal Sci., 111 (2017) 274-288.

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ACCEPTED MANUSCRIPT Highlights

Boundary layer flow of third grade nanofluid is constructed.

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Flow is bounded by a linear stretching surface.

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Brownian motion and thermophoresis effects are considered.

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Thermal convective and zero nanoparticles mass flux conditions are utilized.

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Computations and analysis are made through homotopy analysis method (HAM).

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·