Science and Engineering Applications - PAYAM SCIENTIFIC

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Science and Engineering Applications 2(3) (2017) 164-176

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Unsteady Cattaneo- Christov double diffusion of conducting nanofluid Galal M. Moatimid, Mona A. A. Mohamed, Mohamed A. Hassan, and Engy M. M. El-Dakdoky* Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt *Email : [email protected]

Abstract This paper investigates the unsteady mixed convection flow of an incompressible electrically conducting nanofluid. The Cattaneo- Christov double diffusion with heat and mass transfer are taken into account. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations by using suitable similarity transformations. These equations are solved analytically by a homotopy perturbation technique. The profiles of the velocity, the temperature, the induced magnetic field and the nanoparticles concentration are sketched with various parameters. The influences of these parameters are discussed in details. Keywords: Nanofluid; Mixed convection; Relaxation time; Magnetohydrodynamics DOI: http://dx.doi.org/10.26705.xxx.xxx.xxxx

Nomenclature 𝐴

unsteadiness parameter

𝑇∞

ambient temperature

𝑏

characteristic temperature

𝑢

velocity component in the 𝑥-direction

𝐶

nanoparticle concentration

𝑈𝑒

𝑥 -velocity at the edge of the boundary layer

𝑐

positive constant

𝑉

velocity vector

𝐶𝑤

nanoparticle concentration at the surface

𝑣

velocity component in the 𝑦-direction

𝐶∞

ambient nanoparticle concentration

𝑥

distance along the plate

𝑐𝑓

volumetric volume expansion of the fluid

𝑦

distance perpendicular to the plate

𝑐𝑃

volumetric volume expansion of the

𝑇∞

ambient temperature

particle 𝑑

characteristic nanoparticle concentration

𝐷𝐵

Brownian diffusion coefficient

𝜆

positive constant

𝐷𝑇

thermophoretic diffusion coefficient

𝜂

independent similarity variable

𝑓

dimensionless stream function

𝜓

stream function

local Grashof number

𝜃

dimensionless temperature

𝑔

acceleration vector due to gravity

𝜙

dimensionless nanoparticle concentration

𝐻

induced magnetic field vector

𝛼

thermal diffusivity

𝐻𝑒

𝑥-magnetic field at the edge of the

𝜌𝑓

density of the fluid

𝐺𝑟𝑥

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Greek symbols

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164



𝜌𝑃

boundary layer

density of the particle

𝐻0

uniform induced magnetic field strength

(𝜌 𝑐)𝑓

heat capacity of the fluid

ℎ′

dimensionless induced magnetic field

(𝜌 𝑐)𝑃

effective heat capacity of the nanoparticle

ℎ𝑥

induced magnetic field in the 𝑥-direction

ℎ𝑦

induced magnetic field in the 𝑦-direction

𝜇

dynamic viscosity

mass flux vector

𝜇𝑒

magnetic diffusivity

𝐾𝑇

thermal conductivity

𝜇0

magnetic permeability

𝐿𝑒

Lewis number

𝜈

kinematic viscosity

𝑀

magnetic parameter

𝜏

ratio between effective heat capacity of the

𝑁𝑏

Brownian parameter

nanoparticle material and heat capacity of the

𝑁𝑟

nanofluid buoyancy parameter

fluid

𝑁𝑡

thermophoresis parameter

𝜏𝐶

relaxation time of heat flux

𝑃𝑟

Prandtl number

𝜏𝐻

relaxation time of mass flux

𝑞

heat flux vector

𝛽

volumetric expansion coefficient of the fluid

𝑅𝑖

Richardson number

𝛽1

thermal relaxation parameter

local Reynolds number

𝛽2

nanoparticle concentration relaxation

𝐽

𝑅𝑒𝑥 𝑇

temperature

𝑡

time

𝑇𝑤

material

parameter 𝛾

reciprocal of the magnetic Prandtl number

temperature at the surface

Received : 04/10/2017

Published online : 25/11/2017

1. Introduction Nanofluids are defined as composition of solid and liquid materials. This solid (nanoparticles or nanofibers) with diameter 1-100 nm dispersed in the base fluid [1]. As the nanoparticles are so small, they can flow smoothly without clogging through micro channels. Choi and Eastman [2] was the first to suggest the term of nanofluid to define designed colloids that consist of nanoparticles suspended in the base fluid. Nanofluids have many advantages like they have more stability. Also, they have suitable viscosity, properties of spreading and dispersion on solid surface. Masuda et al. [3] observed that to enhance the heat transfer, nanofluids should consist of nanoparticles with 5% vol. Choi et al. [4] indicates that adding a small amount (less than 1% by volume) of nanoparticles to a liquid rises the thermal conductivity of it up to approximately two times. For instance, a small amount of adding copper to ethylene glycol enhances the thermal conductivity of the liquid by 40% [5]. This phenomenon enables us to use nanofluids in advanced nuclear systems [6]. Researchers have defined some mechanisms based on the


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thermal enhancements properties of nanofluids. Buongiorno [7] discussed seven possible slip mechanisms and he indicated that Brownian diffusion, and thermophoresis are the two effective mechanisms. One of the significant factors which affect the nanofluids' flow is nanoparticles' charges. Nanofluids content on negatively charged particles generates an electric field that affects the velocity profile [8]. Also, the size and type of nanoparticles play an important role in natural convection heat transfer enhancement [9]. The nanometer sized materials have unrivaled properties. Therefore, nanofluids are used in different industry and engineering applications. Some of these applications [10-12] are food productions, microelectronics, microfluidics, transportation and manufacturing. Also, they are used in power generation in nuclear reactor coolant, fuel cells, hybrid-powered engines, space technology, defence and ships. In addition, medical applications: gold nanoparticle probes that detect DNA, cancer treatment. Mnyusiwella [13] mentioned some nanotechnology dangers for environmental health.


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There are three main types of convection; free, force, and mixed. A free (Natural or Buoyant) convection flow field is a continuous self flow resulted from temperature gradients. This type can be found in many physical phenomena. For example, the free convection occurs when food is placed inside freezer with no circulation assisted by fans [14]. On the other hand, the forced convection is generated by an external source. When free and forced convections take place together, this status is known as the mixed convection. Cheng and Minkowycz [15] considered the problem of free convection through a vertical plate with a porous medium. Bejan and Khair [16] investigated the same problem with the addition of heat and mass transfer. Bejan [17] wrote a book on mixed convection heat and mass transfer. Das et al. [18] studied the MHD mixed convection flow in a vertical channel. They indicated that the magnetic field enhances the velocity and the temperature. Subhashini et al. [19] studied different effects of thermal and concentration diffusions on a mixed convection boundary layer flow across a permeable surface. Some of applications on mixed convection are electronic devices, heat exchanger, and nuclear reactors [20].

The purpose of the current work is to explore the unsteady boundary layer flow over a surface embedded in a nanofluid. Cattaneo- Christov double diffusion with heat and mass transfer is taken into account. The influence of the induced magnetic field is also investigated. Our problem can be clarified as follows: the physical description of the paper at hand is investigated in section 2. This section involves many items like; the mathematical formulation of the problem, the governing equations of motion, the related initial and appropriated boundary conditions. Finally, the suitable similarity transformations are also added in this section. Section 3 is devoted to introduce the method of solution. This technique depends mainly on the homotopy perturbation [32]. The required distribution functions such as velocity, induced magnetic field, temperature, and nanoparticle concentration are given in this section. The influence of the various parameters on the above distribution functions are presented in section 4. Finally, the concluding remarks are introduced in section 5.

The heat transfer process has various applications like energy production, and power generation [21]. Cattaneo [22] modified Fourier law of heat conduction by introducing the term of relaxation time. This term represents the finite velocity of heat propagation. Han et al. [23] made a comparison of Fourier’s law and the Cattaneo–Christov heat flux model. Nadeem et al. [24] used Cattaneo-Christov heat flux model instead of Fourier's law of heat conduction. Also, Hayat et al. [25] used the same model to study the steady two-dimensional MHD flow of an Oldroyd-B fluid over a stretching surface. Rubab and Mustafa [26] studied the MHD three-dimensional viscoelastic flow over stretching surface with the CattaneoChristov heat flux model. This model tends to clarify the characteristics of thermal relaxation time. Hayat et al. [21] used the Cattaneo-Christov double diffusion, which are the generalized Fourier's and Fick's laws, in studying the boundary-layer flow of viscoelastic nanofluids.

The two-dimensional, unsteady, laminar, incompressible mixed convective boundary layer flow over a semi-infinite vertical plate is taken into account. The fluid is considered as a Newtonian and electrically conducting nanofluid flow. The Cartesian coordinates (𝑥, 𝑦) are considered. The 𝑥 -axis is considered as the coordinate measured along the boundary layer surface. The 𝑦 -axis is the normal coordinate to that surface. It is assumed that the flow outside the boundary layer moves with a stream velocity 𝑈𝑒 that is parallel to the flat plate. As considered by [33], we may choose an unsteady induced 𝐻 magnetic field strength as 0 . This field is normal to the flat √1−𝜆 𝑡 surface where 𝐻0 the initial strength of the induced magnetic field. The flat plate is considered to be electrically nonconducting so that no surface current sheet occurs. In addition, a magnetic field 𝐻𝑒 is applied at the outer edge of the boundary layer which is parallel to the flat plate. Moreover, at the surface, 𝑇𝑤 and 𝐶𝑤 represent the temperature and nanoparticle concentration, respectively. Meanwhile, far away from the surface, the temperature and nanoparticle concentration are taken as 𝑇∞ and 𝐶∞ , respectively. It is worthwhile to note that: 𝑈𝑒 , 𝐻𝑒 , 𝑇𝑤 and 𝐶𝑤 are functions of 𝑥 and 𝑡. The acceleration vector due to the gravity taken as 𝑔 = (−𝑔, 0,0), is taken into account. A schematic diagram of the physical model is shown in Fig.1.

Magnetohydrodynamics (MHD) is the study of the magnetic properties of electrically conducting fluids. This branch has many applications in geophysics, astrophysics, sensors, and engineering. Chen [27] combined the boundary layer heat and mass transfer of an electrically conducting fluid in MHD natural convection from a vertical surface. For the importance of MHD, many works were made to study the flow characteristics over a stretching sheet under different conditions of MHD. Hayat et al. [28] studied MHD flow and heat transfer characteristics of the boundary layer flow over a permeable stretching sheet. Further, the induced magnetic field has many applications such as liquid metals and ionized gases [29]. Kumari et al. [30] add the effect of the induced magnetic field to the problem of MHD boundary layer flow and heat transfer over a stretching sheet. Ali et al. investigated some works in this field which can be found in [31].


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2. Mathematical formation

The basic equations that governing the motion [29, 34] may by listed as follows: The continuity equation 𝛻. 𝑉 = 0.


(1)

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𝑞 + 𝜏𝐻 (

𝜕𝑞 𝜕𝑡

+ 𝑉. 𝛻𝑞 − 𝑞. 𝛻𝑉 + (𝛻. 𝑉)𝑞) = −𝐾𝑇 𝛻𝑇,

(6)

where the relaxation time of heat flux 𝜏𝐻 = 𝜏𝐻0 (1 − 𝜆 𝑡), 𝜏𝐻0 is the initial relaxation time of heat. The nanoparticle concentration equation yields [21] 𝜕 ( + 𝑉. 𝛻) 𝐶 𝜕𝑡 = −𝛻. 𝐽,

(7)

where 𝐽 satisfies the following mass flux equation Gauss's law of magnetism (one of Maxwell's equations) 𝛻. 𝐻 = 0.

𝐽 + 𝜏𝐶 (

(2)

𝜕𝐽 𝜕𝑡

+ 𝑉. 𝛻𝐽 − 𝐽. 𝛻𝑉 + (𝛻. 𝑉)𝐽) = −𝐷𝐵 𝛻𝐶,

The momentum equation

(8)

where the relaxation time of mass flux 𝜏𝐶 = 𝜏𝐶0 (1 − The coupling between the fluid velocity, magnetic field and 𝜆 𝑡), 𝜏 is the initial relaxation time of mass. 𝐶0 nanofluid on the conservation of momentum yields 𝜕 𝜇0 ( + 𝑉. 𝛻) 𝑉 − (𝐻. 𝛻)𝐻 𝜕𝑡 4𝜋𝜌𝑓 1 = − 𝛻𝑃 + 𝜈 𝛻 2𝑉 𝜌𝑓 1 + (𝜌 𝐶 𝜌𝑓 𝑃 + (1 − 𝐶)[ 𝜌𝑓 (1 − 𝛽(𝑇 − 𝑇∞ ))])𝑔,

In 1904, Ludwig Prandtl introduced an approximate form of the Navier-Stokes equations. He considered the no-slip condition at the surface and the frictional effects occur only in a thin region near the surface called boundary layer. Outside the boundary layer, the flow is inviscid flow. The boundary layer 𝜕𝑣 approximations can be written as follows: 𝑣 ≪ 𝑢 , ≈ 0,

𝜕𝑣 𝜕𝑦

≈ 0,

𝜕𝑢 𝜕𝑥

≪

𝜕𝑢 𝜕𝑦

,

𝜕𝑇 𝜕𝑥

≪

𝜕𝑇 𝜕𝑦

,

𝜕𝐶 𝜕𝑥

≪

𝜕𝐶 𝜕𝑦

𝜕𝑥

.

By taking the 2-dimensional Cartesian coordinates with 𝑉 = (𝑢, 𝑣) and 𝐻 = (ℎ𝑥 , ℎ𝑦 ). Under the Oberbeck-Boussinesq and the standard boundary layer approximations, the system of 𝜇 where 𝑃 = 𝑃0 + 0 |𝐻|2 is the magnetohydrodynamic equations that governs the motion can be formulated as follows 8𝜋 𝜇 pressure, 𝑃0 is the pressure of the fluid, and 0 |𝐻|2 is the 8𝜋 The continuity equations magnetic pressure. (3)

𝜕𝑢 𝜕𝑣 + = 0. 𝜕𝑥 𝜕𝑦 Gauss's law of magnetism

Magnetic induction equation gives 𝜕𝐻 𝜕𝑡 = 𝛻 ∧ (𝑉 ∧ 𝐻) + 𝜇𝑒 𝛻 2 𝐻.

(4)

(9)

𝜕ℎ𝑥 𝜕ℎ𝑦 + = 0. 𝜕𝑥 𝜕𝑦

(10)

The energy equation yields [21]

The momentum equation

𝜕 (𝜌 𝑐)𝑓 ( + 𝑉. 𝛻) 𝑇 𝜕𝑡 = −𝛻. 𝑞,

According to the boundary layer approximations the 𝜕𝑃 𝑦 − momentum equation reduced to = 0 , and the (5)

𝜕𝑦

𝑥 −momentum equation can be written as follows

where 𝑞 satisfies the following heat flux equation


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Science and Engineering Applications 2(3) (2017) 164-176 𝜕𝑢 𝜕𝑢 𝜕𝑣 +𝑢 +𝑣 𝜕𝑡 𝜕𝑥 𝜕𝑦

The nanoparticle concentration equation

𝜕𝑈𝑒 𝜕𝑈𝑒 𝜇0 𝑑𝐻𝑒 𝜕2𝑢 + 𝑈𝑒 − +𝜈 𝜕𝑡 𝜕𝑥 4𝜋𝜌𝑓 𝑑𝑥 𝜕𝑦 2 𝜇0 𝜕ℎ𝑥 𝜕ℎ𝑥 + (ℎ + ℎ𝑦 ) 4𝜋𝜌𝑓 𝑥 𝜕𝑥 𝜕𝑦 1 − (𝐶 𝜌𝑃 𝜌𝑓 + (1 − 𝐶)[ 𝜌𝑓 (1 − 𝛽(𝑇 − 𝑇∞ ))])𝑔, (11) =

𝜕𝐶 𝜕𝐶 𝜕𝐶 +𝑢 +𝑣 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕2𝐶 𝜕2𝐶 𝜕2𝐶 𝜕𝑢 𝜕𝐶 𝜕𝑣 𝜕𝐶 + 𝜏𝐶 ( 2 + 2𝑢 + 2𝑣 + + 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑦 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑦 𝜕𝑢 𝜕𝐶 𝜕𝑣 𝜕𝐶 𝜕𝑢 𝜕𝐶 𝜕𝑣 𝜕𝐶 𝜕2𝐶 +𝑢 +𝑢 +𝑣 +𝑣 + 𝑢2 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 2 2 2 𝜕 𝐶 𝜕 𝐶 + 2𝑢𝑣 + 𝑣2 ) 𝜕𝑥 𝜕𝑦 𝜕𝑦 2 𝜕 2 𝐶 𝐷𝑇 𝜕 2 𝑇 = 𝐷𝐵 2 + , (14) 𝜕𝑦 𝑇∞ 𝜕𝑦 2

Magnetic induction equation where the term (𝑢

𝜕ℎ𝑥 𝜕𝑡

𝜕𝑢 𝜕𝑢 𝜕ℎ𝑥 𝜕ℎ𝑥 = ℎ𝑥 + ℎ𝑦 −𝑢 −𝑣 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕 2 ℎ𝑥 + 𝜇𝑒 . 𝜕𝑦 2

𝜕𝐶 𝜕𝑥

+𝑣

𝜕𝐶 𝜕𝑦

) indicates that nanoparticles can

move homogeneously within the fluid, the term (𝐷𝐵 due to Brownian diffusion, the term (

𝐷𝑇

𝜕2 𝑇

𝑇∞ 𝜕𝑦 2

𝜕2 𝐶 𝜕𝑦 2

) is

) is due to

thermophoresis effect [35].

(12)

The appropriate initial conditions may be taken as [34]

By taking the influence of the Brownian motion and 𝑢 = 𝑣 = 0 , 𝑇 = 𝑇∞ , 𝐶 = 𝐶∞ 𝑓𝑜𝑟 𝑡 < 0. (15) thermophoresis into consideration, Eqs. (5) & (7) become as follow The appropriate boundary conditions for 𝑡 ≥ 0 are [34] The energy equation 𝑦 = 0:

𝜕𝑇 𝜕𝑇 𝜕𝑇 +𝑢 +𝑣 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕2𝑇 𝜕2𝑇 𝜕2𝑇 + 𝜏𝐻 ( 2 + 2𝑢 + 2𝑣 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑦 𝜕𝑡 𝜕𝑢 𝜕𝑇 𝜕𝑣 𝜕𝑇 𝜕𝑢 𝜕𝑇 𝜕𝑣 𝜕𝑇 + + +𝑢 +𝑢 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑦 2 𝜕𝑢 𝜕𝑇 𝜕𝑣 𝜕𝑇 𝜕 𝑇 +𝑣 +𝑣 + 𝑢2 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 2 2 2 𝜕 𝑇 𝜕 𝑇 + 2𝑢𝑣 + 𝑣2 ) 𝜕𝑥 𝜕𝑦 𝜕𝑦 2 2 𝜕 𝑇 =𝛼 𝜕𝑦 2 𝜕𝑇 𝜕𝐶 + 𝜏 (𝐷𝐵 ( ) 𝜕𝑦 𝜕𝑦 𝐷𝑇 𝜕𝑇 2 + ( ) ), (13) 𝑇∞ 𝜕𝑦

𝑢=𝑣=0 ,

𝜕ℎ𝑥 = ℎ𝑦 = 0 , 𝜕𝑦

𝑏𝑥 , 𝐶 = 𝐶𝑤 (1 − 𝜆 𝑡)2 𝑑𝑥 = 𝐶∞ + . (16) (1 − 𝜆 𝑡)2

𝑇 = 𝑇𝑤 = 𝑇∞ +

𝑐𝑥 𝐻0 𝑥 , ℎ𝑥 = 𝐻𝑒 (𝑥, 𝑡) = , 1−𝜆𝑡 1−𝜆𝑡 𝑇 = 𝑇∞ , 𝐶 = 𝐶∞ ,

𝑦 ⟶ ∞: 𝑢 = 𝑈𝑒 (𝑥, 𝑡) =

where 𝑐 and 𝜆 are constants and both have dimension (𝑡 −1 ) such that (𝑐 > 0 and 𝜆 ≥ 0 , 𝜆 𝑡 < 1), 𝑑 is a constant with dimension (𝐿−1 ), and 𝑏 is a constant with dimension (𝑇 𝐿−1 ), such that "𝑏 > 0 and 𝑑 > 0" represents the assisting flow (the case of heating plate), "𝑏 < 0 and 𝑑 < 0" is corresponding to 𝜕𝑇 𝜕𝑇 where the term (𝑢 + 𝑣 ) is the heat convection, the term the opposing flow (the case of cooling plate), and "𝑏 = 0 and 𝜕𝑥 𝜕𝑦 𝑑 = 0" represents the forced convection limit which means the 𝜕2 𝑇 𝜕𝑇 𝜕𝐶 (𝛼 ) is the heat conduction, the term (𝜏𝐷𝐵 ) is the absence of the free convection. 𝜕𝑦 2 𝜕𝑦 𝜕𝑦 thermal energy transport due to Brownian diffusion, the term (𝜏

𝐷𝑇

𝜕𝑇 2

( ) ) is the energy transport due to thermophoretic

𝑇∞ 𝜕𝑦

effect [35].


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𝑁𝑡 ′′ The system of partial differential Eqs.(11) – (14) is converted ′′ 𝜙 + 𝜃 into a system of four coupled ordinary differential equations by 𝑁𝑏 𝜂 using the following similarity transforms + 𝐿𝑒 (𝑓𝜙 ′ − 𝑓 ′ 𝜙 − 𝐴 (2 𝜙 + 𝜙 ′ ) 2 𝑐𝜈 𝑇 − 𝑇∞ 2 𝐴 3 𝜓=√ 𝑥 𝑓(𝜂) , 𝜃(𝜂) = , − 𝛽1 [ (𝜂 2 𝜙 ′′ + 11 𝜂 𝜙 ′ + 24 𝜙) + 𝐴 𝜂 𝑓 ′ 𝜙 ′ 1−𝜆𝑡 𝑇𝑤 − 𝑇∞ 4 2 𝐶 − 𝐶∞ 9 𝜙(𝜂) = , 2 + 5 𝐴 𝑓 ′ 𝜙 − 𝐴 𝑓 𝜙 ′ − 𝐴 𝜂 𝑓 𝜙 ′′ + 𝑓 ′ 𝜙 − 𝑓 𝑓 ′′ 𝜙 𝐶𝑤 − 𝐶∞ 2 𝑐 𝜂 = 𝑦√ , 𝜈(1 − 𝜆 𝑡) ℎ𝑦 = −𝐻0 √

− 𝑓 𝑓 ′ 𝜙 ′ + 𝑓 2 𝜙 ′′ ])

𝐻0 𝑥 ′ ℎ𝑥 = ℎ (𝜂), 1−𝜆𝑡

= 0,

𝜈 ℎ(𝜂). 𝑐(1 − 𝜆 𝑡)

(17)

where the non-dimensional parameters are defined as follows 𝑅𝑖 =

To satisfy the continuity Eq.(9), we may consider a stream 𝜕𝜓 𝜕𝜓 function 𝜓 such that 𝑢 = and 𝑣 = − . Also, ℎ𝑥 and ℎ𝑦 𝜕𝑦

(21)

𝐺𝑟𝑥 𝑅𝑒𝑥 2

=

(1 − 𝐶∞ )(𝑇𝑤∗ − 𝑇∞∗ )𝑥 𝛽 𝑔

, 𝑈𝑒 2 (1 − 𝐶∞ )(𝑇𝑤∗ − 𝑇∞∗ )𝑥 3 𝛽 𝑔 𝐺𝑟𝑥 = , 𝜈2 𝑈𝑒 𝑥 𝑅𝑒𝑥 = , 𝜈

𝜕𝑥

are defined as the previous forms to satisfy Eq.(10). Substitute from Eq.(17) into Eqs.(11-14), we get the following system of four coupled ordinary differential equations

(𝜌𝑃 − 𝜌𝑓 )(𝐶𝑤 − 𝐶∞ ) , 𝜌𝑓 𝛽 (1 − 𝐶∞ )(𝑇𝑤∗ − 𝑇∞∗ ) 𝜏 𝐷𝑇 (𝑇𝑤 − 𝑇∞ ) 𝑁𝑡 = , 𝑇∞ 𝜈 𝑁𝑟 =

The momentum equation 𝜂 𝑓 ′′′ + 1 + 𝑓 𝑓 ′′ − 𝑓′2 + 𝐴 (1 − 𝑓 ′ − 𝑓 ′′ ) 2 2 + 𝑀(ℎ′ − ℎ ℎ′′ − 1) + 𝑅𝑖( 𝜃 − 𝑁𝑟 𝜙) = 0. (18)

𝑀=

𝜇0 𝐻0 2 , 4 𝜋 𝜌𝑓 𝑐 2

𝐴=

𝜆 𝑐

,

𝛽2 = 𝑐 𝜏𝐶0 ,

The magnetic induction 1 𝜂 ℎ′′′ + (𝑓 ℎ′′ − 𝑓 ′′ ℎ − 𝐴 ( ℎ′′ + ℎ′ )) 𝛾 2 = 0.

𝑁𝑏 =

𝜏 𝐷𝐵 (𝐶𝑤 − 𝐶∞ ) , 𝜈

(22) 𝜇𝑒 , 𝜈 𝜈 𝑃𝑟 = , 𝛼 𝛾=

𝛽1 = 𝑐 𝜏𝐻0 , 𝐿𝑒 =

𝜈 . 𝐷𝐵

Also, the initial and boundary conditions of equations (15) and (16) take the following forms

(19)

The appropriate initial conditions become The energy equation 𝜃 ′′ + 𝑃𝑟 (𝑓𝜃 ′ − 𝑓 ′ 𝜃 − 𝐴 (2 𝜃 +

𝜂 ′ 𝜃 ) + 𝑁𝑏 𝜃 ′ 𝜙 ′ + 𝑁𝑡 𝜃′2 2

𝐴2 2 ′′ (𝜂 𝜃 + 11 𝜂 𝜃 ′ + 24 𝜃) 4 3 9 + 𝐴 𝜂 𝑓 ′ 𝜃′ + 5 𝐴 𝑓 ′𝜃 − 𝐴 𝑓 𝜃′ 2 2 2 − 𝐴 𝜂 𝑓 𝜃 ′′ + 𝑓 ′ 𝜃 − 𝑓 𝜃𝑓 ′′ 𝜃 − 𝑓 𝑓 ′ 𝜃 ′ − 𝛽1 [

𝑓(𝜂) = 0 ,

𝑓 ′ (𝜂) = 0 , 𝜃(𝜂) = 0 , 𝑎𝑛𝑑 𝜙(𝜂) = 0 𝑓𝑜𝑟 𝑡 < 0. (23)

The appropriate boundary conditions for 𝑡 ≥ 0 become 𝜂 = 0:

𝑓(𝜂) = 0 , 𝑓 ′ (𝜂) = 0 , ℎ(𝜂) = 0 , ℎ′′ (𝜂) = 0 , 𝜃(𝜂) = 1 , 𝑎𝑛𝑑 𝜙(𝜂) = 1.

+ 𝑓 2 𝜃 ′′ ])

(24)

= 0.

(20)

The nanoparticle concentration equation

𝜂 ⟶ 𝜂∞ : 𝑓 ′ (𝜂) = 1 , ℎ′ (𝜂) = 1, 𝜃(𝜂) = 0 , 𝑎𝑛𝑑 𝜙(𝜂) = 0. 3.

Method of solution

Now, the governing equations of motion are converted to the nonlinear ordinary differential equations (18-21). Their


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solutions should satisfy the conditions (23) and (24). To relax η = 0: f0 (η) = f1 (η) = ⋯ = 0 , the mathematical manipulation, we will use the homotopy f0 ′ (η) = f1 ′ (η) = ⋯ = 0 , perturbation technique [32]. This technique is a combination of the traditional perturbation methods and homotopy techniques. h0 (η) = h1 (η) = ⋯ = 0 , h0 ′′ (η) = h1 ′′ (η) = Through this technique, there is no need for a small parameter. ⋯ = 0 , According to the homotopy technique, a homotopy imbedding θ0 (η) = 1 , θ1 (η) = θ2 (η) = ⋯ = 0, (30) parameter 𝑝 ∈ [0,1] is considered. Therefore, Eqs. (18-21) may be rewritten as 𝜙0 (η) = 1 , and 𝜙1 (η) = 𝜙2 (η) = ⋯ = 0. 𝜂 𝑓 ′′′ = −𝑝 [1 + 𝑓 𝑓 ′′ − 𝑓′2 + 𝐴 (1 − 𝑓 ′ − 𝑓 ′′ ) 2 2 + 𝑀(ℎ′ − ℎ ℎ′′ − 1) + 𝑅𝑖( 𝜃 η ⟶ η∞ : f0 ′ (η) = 1 , f1 ′ (η) = f2 ′ (η) = ⋯ = 0, − 𝑁𝑟 𝜙)]. (25) h0 ′ (η) = 1, h1 ′ (η) = h2 ′ (η) = ⋯ = 0, 1 η h′′′ = −p [ (f h′′ − f ′′ h − A ( h′′ + h′ ))] (26) θ0 (η) = θ1 (η) = ⋯ = 0 , and 𝜙0 (η) = 𝜙1 (η) = γ 2 ⋯ = 0. 1

𝜃 ′′ = −𝑝 [𝑃𝑟 (1 − 𝑃𝑟 𝛽1 {4𝐴2𝜂2−𝐴 𝜂 𝑓+𝑓2}) 3 2

−1

1

({𝑓 − 2 𝐴 𝜂

9 2

− 𝛽1 (11 𝜂 + 𝐴 𝜂 𝑓 ′ − 𝐴 𝑓 − 𝑓 𝑓 ′ )} 𝜃 ′ + {𝑓 ′ − 2 𝐴 2 − 𝛽1 (6 𝐴2 + 5 𝐴 𝑓 ′ + 𝑓 ′ − 𝑓 𝑓 ′′ )}𝜃 + 𝑁𝑏 𝜃 ′ 𝜙 ′ + 𝑁𝑡 𝜃′2 )].

(27)

To obtain a good solution series for Eq.(29). We solved it till second order. The solutions for the various orders are lengthy but straight forward, away from the detail, these solutions may be written as follows: 3.1 Zero order solution 𝑓0 =

1

𝜙 ′′ = −𝑝 [𝐿𝑒 (1 − 𝐿𝑒 𝛽2 {4𝐴2𝜂2−𝐴 𝜂 𝑓+𝑓2}) 3

−1 9

− 𝛽2 (11 𝜂 + 2 𝐴 𝜂 𝑓 ′ − 2 𝐴 𝑓 − 𝑓 𝑓 ′ )} 𝜙 ′ + {𝑓 − 2 𝐴 2 − 𝛽2 (6 𝐴2 + 5 𝐴 𝑓 ′ + 𝑓 ′ − 𝑓 𝑓 ′′ )}𝜙 𝑁𝑡 + 𝜃 ′′ )]. (28) 𝐿𝑒 𝑁𝑏 At this stage, any function ℳ may be written as

4

, ℎ0 = 𝜂 , 𝜃0 = 1 −

𝜂 2

𝜂

, 𝑎𝑛𝑑 𝜙0 = 1 − . (31) 2

1

({𝑓 − 2 𝐴 𝜂

′

𝜂2

3.2 First order solution 𝑓1 = 𝑙1 𝜂 2 + 𝑙2 𝜂 3 + 𝑙3 𝜂 4 + 𝑙4 𝜂 5 , ℎ1 = 𝑙5 𝜂 + 𝑙6 𝜂 3 + 𝑙7 𝜂 4 , 𝜃1 = 𝑙8 𝜂 + 𝑙9 𝜂 2 + 𝑙10 𝜂 3 + 𝑙11 𝜂 4 + 𝑙12 𝜂 5 + ⋯ + 𝑙23 𝜂16 , (32) 𝑎𝑛𝑑 𝜙1 = 𝑙24 𝜂 + 𝑙25 𝜂 2 + 𝑙26 𝜂 3 + 𝑙27 𝜂 4 + 𝑙28 𝜂 5 + ⋯ + 𝑙39 𝜂16 .

∞

ℳ = ∑ 𝑝𝑖 ℳ𝑖 ,

(29)

3.3 Second order solution 𝑓2 = 𝑙40 𝜂 2 + 𝑙41 𝜂 3 + 𝑙42 𝜂 4 + 𝑙43 𝜂 5 + ⋯ + 𝑙57 𝜂19 ,

𝑖=0

where ℳ stands for 𝑓 , ℎ , 𝜃, and 𝜙.

ℎ2 = 𝑙58 𝜂 + 𝑙59 𝜂 3 + 𝑙60 𝜂 5 + 𝑙61 𝜂 6 + 𝑙62 𝜂 7 + ⋯ + 𝑙66 𝜂11 , (33)

The above equation represents the approximation solutions for the functions 𝑓 , ℎ , 𝜃, and 𝜙 in terms of the power series of the 𝜃 = 𝜉 + 𝑙 𝜂 , 𝑎𝑛𝑑 𝜙 = 𝜉 + 𝑙 𝜂, 2 1 67 2 2 68 homotopy parameter 𝑝. where 𝜉1 and 𝜉2 are two functions of 𝜂 which are given in the The initial and boundary conditions which satisfy the above Appendix. system can be written as The coefficients (𝑙1 → 𝑙68 ) , (𝑠1 → 𝑠52 ) , and (𝑟1 → 𝑟94 ) are for t < 0 ∶ f0 (η) = f1 (η) = ⋯ = 0 , given in the Appendix. f0 ′ (η) = f1 ′ (η) = ⋯ = 0, For the complete solution corresponding to 𝑝 → 1 in Eq. (29), θ0 (η) = θ1 (η) = θ2 (η) = ⋯ = 0 , the analytical perturbed solutions for the velocity, the induced and 𝜙0 (η) = 𝜙1 (η) = 𝜙2 (η) = ⋯ = 0. magnetic field, the temperature, and the nanoparticle concentration are written as for t ≥ 0 ∶ 𝑓 ′ = 𝑓0′ + 𝑓1′ + 𝑓2′ , (34)


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Science and Engineering Applications 2(3) (2017) 164-176 ℎ′ = ℎ0′ + ℎ1′ + ℎ2′ ,

(35)

𝜃 = 𝜃0 + 𝜃1 + 𝜃2 ,

(36)

𝑎𝑛𝑑 𝜙 = 𝜙0 + 𝜙1 + 𝜙2 .


Fig. 2a: 𝛽1 = .1, 𝛽2 = .4, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑅𝑖 = 1, 𝑁𝑡 = 𝑁𝑏 = 𝛾 = .5, 𝑁𝑟 = .05

(37)

To get a good convergence, we choose the length of the semiinfinite plate is limited to 2, i.e. η∞ = 2.

4. Results and discussion This section is devoted to discuss the influence of the various physical parameters on the velocity, the induced magnetic field, the temperature, and the nanoparticle concentration. These parameters are the unsteadiness parameter (𝐴) , magnetic parameter(𝑀), reciprocal magnetic Prandtl number(𝛾), thermal relaxation parameter (𝛽1 ) , nanoparticle relaxation parameter(𝛽2 ) , Prandtl number (𝑃𝑟) , Lewis number (𝐿𝑒) , Brownian parameter (𝑁𝑏) , and thermophoresis parameter Fig. 2b: 𝐴 = 𝛾 = .8, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑅𝑖 = 1, 𝑁𝑡 = 𝑁𝑏 = (𝑁𝑡). Moreover, Richardson number (𝑅𝑖), which represents the .5, 𝑁𝑟 = .05 ratio of the buoyancy term to the shear stress term, its values are taken according to the type of convection. For mixed convection case, we took 0.1 < 𝑅𝑖 < 10 . Meanwhile, at 𝑅𝑖 < 0.1 the natural convection is negligible, and the forced convection is negligible at 𝑅𝑖 > 10. Furthermore, 𝑅𝑖 > 0 means that 𝑇𝑤 > 𝑇∞ (the assisting flow), 𝑅𝑖 < 0 means that 𝑇𝑤 < 𝑇∞ (the opposing flow), and 𝑅𝑖 = 0 is the case of the forced convection. Figs.2, show the relation between the velocity 𝑓′ and the different physical parameters. In general, the velocity starts at its lowest value at the surface, 𝑓 ′ (0) = 0, then it increases till it approach its free stream value that satisfying the far field boundary condition, 𝑓 ′ (2) = 1. Fig.(2.a) sketches the rising in velocity 𝑓′ with the increasing of 𝐴 . Also, the increasing in magnetic parameter 𝑀 accelerates the velocity 𝑓′ [36]. Physically, when the induced magnetic field is normal to the surface, the Lorentz force acts in the upwards direction to enhance the flow and increase the fluid velocity. Fig.(2.b) Fig. 2c: 𝐴 = 𝛽1 = .8, 𝛽2 = 𝐿𝑒 = .1, 𝑅𝑖 = 1, 𝑀 = .5, 𝑁𝑡 = illustrates the effect of the relaxation parameters 𝛽1 and 𝛽2 on 𝑁𝑏 = .5, 𝑁𝑟 = .05 the velocity. The velocity 𝑓′ reduces with the increasing of 𝛽1 . Meanwhile velocity 𝑓′ enhances with the increasing of 𝛽2 . From Fig.(2.c), the velocity 𝑓′ slightly decreases as Prandtl number 𝑃𝑟 increases. In fact Prandtl number 𝑃𝑟 is defined as the ratio between the momentum (viscous) diffusivity and the thermal diffusivity. This means growing in 𝑃𝑟 enhances the rate of viscous diffusion which in turns decreases the velocity. Whilst, a reduction of 𝑓′ is indicated as the reciprocal magnetic Prandtl number 𝛾 increases. Fig(2.d) indicates that 𝑓′ reduces as Lewis number 𝐿𝑒 increases till 𝐿𝑒 ≈ 3.5, and then it starts to increase with the increasing of 𝐿𝑒 . Fig.(2.e) shows that 𝑓′ decreases as Richardson number 𝑅𝑖 increases.


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Science and Engineering Applications 2(3) (2017) 164-176 Fig. 2d: 𝐴 = 𝛽1 = .8, 𝛽2 = 𝑃𝑟 = .1, 𝑅𝑖 = 1, 𝑀 = .5, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑁𝑟 = .05

Fig. 3a: 𝛾 = 8, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 3b: 𝐴 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05 Fig. 2e: 𝐴 = 𝛽1 = 𝛾 = .8, 𝛽2 = .1, 𝑃𝑟 = .3, 𝐿𝑒 = 2, 𝑀 = .5, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑁𝑟 = .05

Fig. 3c: 𝐴 = .5, 𝛾 = .8, 𝑁𝑟 = .05

Figs. 2 Variation of the velocity 𝑓 ′ for different values of 𝐴, 𝛽1 , 𝛽2 , 𝑀, 𝛾, 𝑃𝑟, 𝐿𝑒, 𝑅𝑖 Fig.(3.a) indicates that the induced magnetic field ℎ′ reduces with the increase in unsteadiness parameter 𝐴 till 𝐴 ≈ 0.2, and then ℎ′ starts to increase as 𝐴 increases. Fig.(3.b) shows that ℎ′ increases with the growing in the reciprocal magnetic Prandtl number 𝛾. This is because of 𝛾 is defined as the ratio between the magnetic diffusivity and viscous diffusivity. So, the increasing in 𝛾 enhances the magnetic diffusivity which in turns enhances ℎ′. Fig.(3.b) indicates that ℎ′ increases with the increase in Richardson number 𝑅𝑖.


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Figs. 3 Variation of the induced magnetic field ℎ′ for different values of 𝐴, 𝛾, 𝑅𝑖


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Science and Engineering Applications 2(3) (2017) 164-176 Fig. 4a:𝛽1 = .1, 𝛽2 = .4, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05


Fig. 4d:𝐴 = 𝛽2 = .8, 𝛽1 = .4, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 4e:𝐴 = 𝛽2 = .8, 𝛽1 = .4, 𝑃𝑟 = 1, 𝐿𝑒 = 1.2, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑁𝑟 = .05 Fig. 4b:𝐴 = .8, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 4c:𝐴 = 𝛽2 = .8, 𝛽1 = .4, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 4d:𝐴 = 𝛽2 = .8, 𝛽1 = .4, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑅𝑖 = 1, 𝑁𝑟 = .05


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Figs. 4 Variation of the temperature 𝜃 for different values of 𝐴, 𝛽1 , 𝛽2 , 𝑃𝑟, 𝐿𝑒, 𝑁𝑡, 𝑁𝑏, 𝑅𝑖 Fig.(4.a), plots the variation of temperature 𝜃 due to the unsteadiness parameter 𝐴. This figure indicates that for small values of 𝐴, the temperature decreases as long as 𝐴 increases till the critical value 𝐴𝑐 = 0.55, and then the temperature slightly increases near the surface with the increasing of 𝐴. Moreover, the curves dispatcher for the higher values of . In Figs.(4.b) and (4.c), there exists a certain point (𝜂 ≈ 1.6) called crossing over point in which temperature profile has a conflicting behavior before and after that point.


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Science and Engineering Applications 2(3) (2017) 164-176 Fig. 5a:𝛽1 = .1, 𝛽2 = .4, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 5b:𝐴 = .8, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 5c:𝐴 = .8, 𝛽1 = .6, 𝛽2 = .4, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑅𝑖 = 1, 𝑁𝑟 = .05


Fig. 5d:𝐴 = .8, 𝛽1 = .6, 𝛽2 = .4, 𝑃𝑟 = 𝐿𝑒 = 1, 𝑅𝑖 = 1, 𝑁𝑟 = .05

Fig. 5e:𝐴 = .8, 𝛽1 = .6, 𝛽2 = .4, 𝑃𝑟 = 1, 𝐿𝑒 = 1.2, 𝑁𝑡 = 𝑁𝑏 = .5, 𝑁𝑟 = .05

Figs. 5 Variation of the nanoparticle volume fraction 𝜙 for different values of 𝐴, 𝛽1 , 𝛽2 , 𝑃𝑟, 𝐿𝑒, 𝑁𝑡, 𝑁𝑏, 𝑅𝑖 It should be noted that the increasing in the relaxation parameters 𝛽1 and 𝛽2 , the Prandtl number 𝑃𝑟 , and Lewis number 𝐿𝑒 leads to a decrease in the temperature before that point and an increase in the temperature after that point. Fig.(4.d) show that the increasing in 𝑁𝑏 , the temperature 𝜃 decreases till that certain point (𝜂 ≈ 1.5), and then it starts to increase. Also, it is depicted that with the growing in 𝑁𝑡, the temperature 𝜃 decreases up to a point ( 𝜂 ≈ 1 ), and then increases. The temperature 𝜃 is slightly rises as Richardson number 𝑅𝑖 increases, this can be shown in Fig.(4.e). From Figs. (5.a) and (5.b), it is illustrated the decreasing in nanoparticle concentration 𝜙 with the increasing of each of 𝐴 and 𝛽1 . Physically, as 𝐴 increases, the mass transfer rate reduces from the fluid to the plate. This action caused a decrease in nanoparticle concentration. Also, the nanoparticle concentration


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Science and Engineering Applications 2(3) (2017) 164-176 𝜙 decreases with the increase 𝛽2 till a certain point (𝜂 ≈ 1.6) and then it starts to increase.

reduces with the increasing in 𝐴 , 𝛽1 , 𝑃𝑟, 𝐿𝑒 , 𝑁𝑡 and 𝑅𝑖.

Fig.(5.c) shows that 𝜙 decreases with the increasing of 𝑃𝑟 and  The nanoparticle concentration profiles indicate that 𝐿𝑒. This is because that the growing of 𝐿𝑒 reduces the Brownian 𝜙 decreases with the increase 𝛽2 till a certain point diffusion coefficient that leads the flow to decrease the (𝜂 ≈ 1.6) and then it starts to increase. nanoparticle concentration. Fig.(5.d) indicates that 𝜙 decreases with the increase of 𝑁𝑡, but 𝜙 enhances with the increasing References values of 𝑁𝑏. Fig.(5.e) elucidate the reduction in nanoparticle [1]Keblinski, P.; Eastman, J. A.; Cahill, D. G., Materialstoday concentration 𝜙 that resulted from the growing of 𝑅𝑖. 2005, 8(6), 36-44. [2] Choi, S. U. S.; Eastman, J., A. ASME Int. Mech. Eng. Con. 4. Conclusions & Exp. 1995, 231(66), 99-105. [3] Masuda, H.; Ebata, A. ; Teramae, K. ; Hishinuma, N., An analytical study is investigated for an unsteady boundary layer flow of nanofluid over a vertical plate. The boundary NetsuBussei (Japan) 1993, 7(4), 227–233. layer is affected by an induced magnetic field. The model of [4] Choi, S.; Zhang, Z. G.; Lockwood, F. E.; Yu, W.; Grulke, E. A., Appl. Phys. Lett. 2001, 79(14), 2252-2254. Cattaneo- Christov double diffusion with heat and mass transfer is taken into account. The governing partial [5] Eastman, J. A.; Choi, S.; Yu, S. ; Li, W.; Thompson, L., J. differential equations are transformed to ordinary differential Appl. Phys. Lett. 2001, 78(6), 718-720. equations by using suitable similarity transformations. The [6] Buongiorno, J.; Hu, W. J., Appl. Math. Phys. 2005, 15–19. resulted system is solved analytically by the homotopy [7] Buongiorno, J., ASME J. Heat Trans. 2006, 128(3), 240– 250. perturbation method. The obtained functions such as the [8] Buongiorno, J., ASME J. Heat Trans. 2006, 128(3), 240–25 velocity, the induced magnetic field, the temperature, and the nanoparticle concentration are plotted graphically. The [9] Reddy, P. S.; Chamkha, A. J., Alex. Eng. J. 2016, 55(1), influences of the various parameters, in these functions, are 331–341. [10] Baharanchi, A. A., Florida Int. Uni. 2013, IEEE-5525. sketched. The important results are summarized as follow: [11] Wong, K. V.; Leon, O. D., Adv. In Mech. Eng. 2010, ID:  The velocity profiles indicate that an increase in each 519659 of 𝐴 , 𝑀 and 𝛽2 increases𝑓′. However, a reduction in [12] Park, S. J.; Taton, T. A.; Mirkin, C. A., Sci. (AAAS) 2002, the velocity 𝑓′appears with the increase in each of 295(5559):1503-6. [13] Mnyusiwalla, A.; Abdallah, S. D.; Peter, A. S., Nanotech. 𝛽1 , 𝑃𝑟 and 𝑅𝑖. 2003, 14, 9-13.  The increasing in 𝐿𝑒 reduces the velocity 𝑓′ till 𝐿𝑒 ≈ [14] Khalid, A.; Khan, I.; Shafie, S., Eur. Phys. J. Plus 2015. [15] Cheng, P.; Minkowycz, W. J., J. Geophys. Res. 1977, 3.5, and then it starts to enhance as 𝐿𝑒 increases. 82(14), 2040–2044.  The induced magnetic field profiles indicate that ℎ′ [16] Bejan, A.; Khair, K. R., Int. J. Heat Mass Trans. 1985, 28(5), 909–918. increases as 𝛾 and 𝑅𝑖 increase. [17] Bejan, A., Convection Heat Transfer; John Wiley: New  The induced magnetic field ℎ′ decreases as 𝐴 York, 2004. increases till 𝐴 ≈ 0.2, and then ℎ′ starts to increase as [18] Das, S.; Jana, R. N.; Makinde, O. D., Int. J. Eng. Sci. and Tech. 2015, 18, 244-255. 𝐴 increases. [19] Subhashini, S. V.; Nancy, S.; Pop, I., Int. Commun. Heat  The temperature profiles indicate that 𝜃 decreases Mass Trans. 2011, 38, 1183- 1188. with the increase of 𝑃𝑟, 𝐿𝑒, 𝑁𝑏, 𝛽1 and 𝛽2 till a [20] Fakour, M.; Vahabzadeh, A.; Ganji, D. D., Case Studies in certain point (𝜂 ≈ 1.6) and then it starts to increase. Thermal Eng. 2014, 4, 15–23. [21] Hayat, T.; Aziz, A.; Muhammad, T.; Alsaedi, A., Plos one  The temperature 𝜃 decreases with the increasing of 2017, 12(1). [22] Cattaneo, C., Atti Semin Mat Fis Univ Modena Reggio 𝑁𝑡 up to a point (𝜂 ≈ 1), and then increases. Emilia 1948, 3, 83-101.  The temperature is a decreasing function for small [23] Han, S.; Zheng, L.; Li, C.; Zhang, X., Appl Math Lett. values of 𝐴, while it increases near the surface for 2014, 38, 87-93. 𝐴 > 0.55. Further, the range of increasing extents as [24] Nadeem, S.; Ahmed, S.; Muhammad, N., J. of Molecular long as 𝐴 increases. liquid 2017, 237, 180-184. 

The nanoparticle concentration profiles indicate that [25] Hayat, T.; Imtiaz , M.; Alsaedi, A.; Almezal, S., J. of Mag. 𝜙 rises with the increasing in 𝑁𝑏 . However, it &Mag. Mat. 2016, 401, 296-303.


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