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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 π΄
unsteadiness parameter
πβ
ambient temperature
π
characteristic temperature
π’
velocity component in the π₯-direction
πΆ
nanoparticle concentration
ππ
π₯ -velocity at the edge of the boundary layer
π
positive constant
π
velocity vector
πΆπ€
nanoparticle concentration at the surface
π£
velocity component in the π¦-direction
πΆβ
ambient nanoparticle concentration
π₯
distance along the plate
ππ
volumetric volume expansion of the fluid
π¦
distance perpendicular to the plate
ππ
volumetric volume expansion of the
πβ
ambient temperature
particle π
characteristic nanoparticle concentration
π·π΅
Brownian diffusion coefficient
π
positive constant
π·π
thermophoretic diffusion coefficient
π
independent similarity variable
π
dimensionless stream function
π
stream function
local Grashof number
π
dimensionless temperature
π
acceleration vector due to gravity
π
dimensionless nanoparticle concentration
π»
induced magnetic field vector
πΌ
thermal diffusivity
π»π
π₯-magnetic field at the edge of the
ππ
density of the fluid
πΊππ₯
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Greek symbols
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ππ
boundary layer
density of the particle
π»0
uniform induced magnetic field strength
(π π)π
heat capacity of the fluid
ββ²
dimensionless induced magnetic field
(π π)π
effective heat capacity of the nanoparticle
βπ₯
induced magnetic field in the π₯-direction
βπ¦
induced magnetic field in the π¦-direction
π
dynamic viscosity
mass flux vector
ππ
magnetic diffusivity
πΎπ
thermal conductivity
π0
magnetic permeability
πΏπ
Lewis number
π
kinematic viscosity
π
magnetic parameter
π
ratio between effective heat capacity of the
ππ
Brownian parameter
nanoparticle material and heat capacity of the
ππ
nanofluid buoyancy parameter
fluid
ππ‘
thermophoresis parameter
ππΆ
relaxation time of heat flux
ππ
Prandtl number
ππ»
relaxation time of mass flux
π
heat flux vector
π½
volumetric expansion coefficient of the fluid
π
π
Richardson number
π½1
thermal relaxation parameter
local Reynolds number
π½2
nanoparticle concentration relaxation
π½
π
ππ₯ π
temperature
π‘
time
ππ€
material
parameter πΎ
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 Fourierβs law and the CattaneoβChristov 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 (π₯, π¦) are considered. The π₯ -axis is considered as the coordinate measured along the boundary layer surface. The π¦ -axis is the normal coordinate to that surface. It is assumed that the flow outside the boundary layer moves with a stream velocity ππ that is parallel to the flat plate. As considered by [33], we may choose an unsteady induced π» magnetic field strength as 0 . This field is normal to the flat β1βπ π‘ surface where π»0 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 π»π is applied at the outer edge of the boundary layer which is parallel to the flat plate. Moreover, at the surface, ππ€ and πΆπ€ represent the temperature and nanoparticle concentration, respectively. Meanwhile, far away from the surface, the temperature and nanoparticle concentration are taken as πβ and πΆβ , respectively. It is worthwhile to note that: ππ , π»π , ππ€ and πΆπ€ are functions of π₯ and π‘. The acceleration vector due to the gravity taken as π = (βπ, 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 π». π = 0.
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(1)
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π + ππ» (
ππ ππ‘
+ π. π»π β π. π»π + (π». π)π) = βπΎπ π»π,
(6)
where the relaxation time of heat flux ππ» = ππ»0 (1 β π π‘), ππ»0 is the initial relaxation time of heat. The nanoparticle concentration equation yields [21] π ( + π. π») πΆ ππ‘ = βπ». π½,
(7)
where π½ satisfies the following mass flux equation Gauss's law of magnetism (one of Maxwell's equations) π». π» = 0.
π½ + ππΆ (
(2)
ππ½ ππ‘
+ π. π»π½ β π½. π»π + (π». π)π½) = βπ·π΅ π»πΆ,
The momentum equation
(8)
where the relaxation time of mass flux ππΆ = ππΆ0 (1 β The coupling between the fluid velocity, magnetic field and π π‘), π is the initial relaxation time of mass. πΆ0 nanofluid on the conservation of momentum yields π π0 ( + π. π») π β (π». π»)π» ππ‘ 4πππ 1 = β π»π + π π» 2π ππ 1 + (π πΆ ππ π + (1 β πΆ)[ ππ (1 β π½(π β πβ ))])π,
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 ππ£ approximations can be written as follows: π£ βͺ π’ , β 0,
ππ£ ππ¦
β 0,
ππ’ ππ₯
βͺ
ππ’ ππ¦
,
ππ ππ₯
βͺ
ππ ππ¦
,
ππΆ ππ₯
βͺ
ππΆ ππ¦
ππ₯
.
By taking the 2-dimensional Cartesian coordinates with π = (π’, π£) and π» = (βπ₯ , βπ¦ ). Under the Oberbeck-Boussinesq and the standard boundary layer approximations, the system of π where π = π0 + 0 |π»|2 is the magnetohydrodynamic equations that governs the motion can be formulated as follows 8π π pressure, π0 is the pressure of the fluid, and 0 |π»|2 is the 8π The continuity equations magnetic pressure. (3)
ππ’ ππ£ + = 0. ππ₯ ππ¦ Gauss's law of magnetism
Magnetic induction equation gives ππ» ππ‘ = π» β§ (π β§ π») + ππ π» 2 π».
(4)
(9)
πβπ₯ πβπ¦ + = 0. ππ₯ ππ¦
(10)
The energy equation yields [21]
The momentum equation
π (π π)π ( + π. π») π ππ‘ = βπ». π,
According to the boundary layer approximations the ππ π¦ β momentum equation reduced to = 0 , and the (5)
ππ¦
π₯ βmomentum equation can be written as follows
where π satisfies the following heat flux equation
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Science and Engineering Applications 2(3) (2017) 164-176 ππ’ ππ’ ππ£ +π’ +π£ ππ‘ ππ₯ ππ¦
The nanoparticle concentration equation
πππ πππ π0 ππ»π π2π’ + ππ β +π ππ‘ ππ₯ 4πππ ππ₯ ππ¦ 2 π0 πβπ₯ πβπ₯ + (β + βπ¦ ) 4πππ π₯ ππ₯ ππ¦ 1 β (πΆ ππ ππ + (1 β πΆ)[ ππ (1 β π½(π β πβ ))])π, (11) =
ππΆ ππΆ ππΆ +π’ +π£ ππ‘ ππ₯ ππ¦ π2πΆ π2πΆ π2πΆ ππ’ ππΆ ππ£ ππΆ + ππΆ ( 2 + 2π’ + 2π£ + + ππ‘ ππ₯ ππ‘ ππ¦ ππ‘ ππ‘ ππ₯ ππ‘ ππ¦ ππ’ ππΆ ππ£ ππΆ ππ’ ππΆ ππ£ ππΆ π2πΆ +π’ +π’ +π£ +π£ + π’2 ππ₯ ππ₯ ππ₯ ππ¦ ππ¦ ππ₯ ππ¦ ππ¦ ππ₯ 2 2 2 π πΆ π πΆ + 2π’π£ + π£2 ) ππ₯ ππ¦ ππ¦ 2 π 2 πΆ π·π π 2 π = π·π΅ 2 + , (14) ππ¦ πβ ππ¦ 2
Magnetic induction equation where the term (π’
πβπ₯ ππ‘
ππ’ ππ’ πβπ₯ πβπ₯ = βπ₯ + βπ¦ βπ’ βπ£ ππ₯ ππ¦ ππ₯ ππ¦ π 2 βπ₯ + ππ . ππ¦ 2
ππΆ ππ₯
+π£
ππΆ ππ¦
) indicates that nanoparticles can
move homogeneously within the fluid, the term (π·π΅ due to Brownian diffusion, the term (
π·π
π2 π
πβ ππ¦ 2
π2 πΆ ππ¦ 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 π’ = π£ = 0 , π = πβ , πΆ = πΆβ πππ π‘ < 0. (15) thermophoresis into consideration, Eqs. (5) & (7) become as follow The appropriate boundary conditions for π‘ β₯ 0 are [34] The energy equation π¦ = 0:
ππ ππ ππ +π’ +π£ ππ‘ ππ₯ ππ¦ π2π π2π π2π + ππ» ( 2 + 2π’ + 2π£ ππ‘ ππ₯ ππ‘ ππ¦ ππ‘ ππ’ ππ ππ£ ππ ππ’ ππ ππ£ ππ + + +π’ +π’ ππ‘ ππ₯ ππ‘ ππ¦ ππ₯ ππ₯ ππ₯ ππ¦ 2 ππ’ ππ ππ£ ππ π π +π£ +π£ + π’2 ππ¦ ππ₯ ππ¦ ππ¦ ππ₯ 2 2 2 π π π π + 2π’π£ + π£2 ) ππ₯ ππ¦ ππ¦ 2 2 π π =πΌ ππ¦ 2 ππ ππΆ + π (π·π΅ ( ) ππ¦ ππ¦ π·π ππ 2 + ( ) ), (13) πβ ππ¦
π’=π£=0 ,
πβπ₯ = βπ¦ = 0 , ππ¦
ππ₯ , πΆ = πΆπ€ (1 β π π‘)2 ππ₯ = πΆβ + . (16) (1 β π π‘)2
π = ππ€ = πβ +
ππ₯ π»0 π₯ , βπ₯ = π»π (π₯, π‘) = , 1βππ‘ 1βππ‘ π = πβ , πΆ = πΆβ ,
π¦ βΆ β: π’ = ππ (π₯, π‘) =
where π and π are constants and both have dimension (π‘ β1 ) such that (π > 0 and π β₯ 0 , π π‘ < 1), π is a constant with dimension (πΏβ1 ), and π is a constant with dimension (π πΏβ1 ), such that "π > 0 and π > 0" represents the assisting flow (the case of heating plate), "π < 0 and π < 0" is corresponding to ππ ππ where the term (π’ + π£ ) is the heat convection, the term the opposing flow (the case of cooling plate), and "π = 0 and ππ₯ ππ¦ π = 0" represents the forced convection limit which means the π2 π ππ ππΆ (πΌ ) is the heat conduction, the term (ππ·π΅ ) is the absence of the free convection. ππ¦ 2 ππ¦ ππ¦ thermal energy transport due to Brownian diffusion, the term (π
π·π
ππ 2
( ) ) is the energy transport due to thermophoretic
πβ ππ¦
effect [35].
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ππ‘ β²β² The system of partial differential Eqs.(11) β (14) is converted β²β² π + π into a system of four coupled ordinary differential equations by ππ π using the following similarity transforms + πΏπ (ππ β² β π β² π β π΄ (2 π + π β² ) 2 ππ π β πβ 2 π΄ 3 π=β π₯ π(π) , π(π) = , β π½1 [ (π 2 π β²β² + 11 π π β² + 24 π) + π΄ π π β² π β² 1βππ‘ ππ€ β πβ 4 2 πΆ β πΆβ 9 π(π) = , 2 + 5 π΄ π β² π β π΄ π π β² β π΄ π π π β²β² + π β² π β π π β²β² π πΆπ€ β πΆβ 2 π π = π¦β , π(1 β π π‘) βπ¦ = βπ»0 β
β π π β² π β² + π 2 π β²β² ])
π»0 π₯ β² βπ₯ = β (π), 1βππ‘
= 0,
π β(π). π(1 β π π‘)
(17)
where the non-dimensional parameters are defined as follows π
π =
To satisfy the continuity Eq.(9), we may consider a stream ππ ππ function π such that π’ = and π£ = β . Also, βπ₯ and βπ¦ ππ¦
(21)
πΊππ₯ π
ππ₯ 2
=
(1 β πΆβ )(ππ€β β πββ )π₯ π½ π
, ππ 2 (1 β πΆβ )(ππ€β β πββ )π₯ 3 π½ π πΊππ₯ = , π2 ππ π₯ π
ππ₯ = , π
ππ₯
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
(ππ β ππ )(πΆπ€ β πΆβ ) , ππ π½ (1 β πΆβ )(ππ€β β πββ ) π π·π (ππ€ β πβ ) ππ‘ = , πβ π ππ =
The momentum equation π π β²β²β² + 1 + π π β²β² β πβ²2 + π΄ (1 β π β² β π β²β² ) 2 2 + π(ββ² β β ββ²β² β 1) + π
π( π β ππ π) = 0. (18)
π=
π0 π»0 2 , 4 π ππ π 2
π΄=
π π
,
π½2 = π ππΆ0 ,
The magnetic induction 1 π ββ²β²β² + (π ββ²β² β π β²β² β β π΄ ( ββ²β² + ββ² )) πΎ 2 = 0.
ππ =
π π·π΅ (πΆπ€ β πΆβ ) , π
(22) ππ , π π ππ = , πΌ πΎ=
π½1 = π ππ»0 , πΏπ =
π . π·π΅
Also, the initial and boundary conditions of equations (15) and (16) take the following forms
(19)
The appropriate initial conditions become The energy equation π β²β² + ππ (ππ β² β π β² π β π΄ (2 π +
π β² π ) + ππ π β² π β² + ππ‘ πβ²2 2
π΄2 2 β²β² (π π + 11 π π β² + 24 π) 4 3 9 + π΄ π π β² πβ² + 5 π΄ π β²π β π΄ π πβ² 2 2 2 β π΄ π π π β²β² + π β² π β π ππ β²β² π β π π β² π β² β π½1 [
π(π) = 0 ,
π β² (π) = 0 , π(π) = 0 , πππ π(π) = 0 πππ π‘ < 0. (23)
The appropriate boundary conditions for π‘ β₯ 0 become π = 0:
π(π) = 0 , π β² (π) = 0 , β(π) = 0 , ββ²β² (π) = 0 , π(π) = 1 , πππ π(π) = 1.
+ π 2 π β²β² ])
(24)
= 0.
(20)
The nanoparticle concentration equation
π βΆ πβ : π β² (π) = 1 , ββ² (π) = 1, π(π) = 0 , πππ π(π) = 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 Ξ· = 0: f0 (Ξ·) = f1 (Ξ·) = β― = 0 , the mathematical manipulation, we will use the homotopy f0 β² (Ξ·) = f1 β² (Ξ·) = β― = 0 , perturbation technique [32]. This technique is a combination of the traditional perturbation methods and homotopy techniques. h0 (Ξ·) = h1 (Ξ·) = β― = 0 , h0 β²β² (Ξ·) = h1 β²β² (Ξ·) = Through this technique, there is no need for a small parameter. β― = 0 , According to the homotopy technique, a homotopy imbedding ΞΈ0 (Ξ·) = 1 , ΞΈ1 (Ξ·) = ΞΈ2 (Ξ·) = β― = 0, (30) parameter π β [0,1] is considered. Therefore, Eqs. (18-21) may be rewritten as π0 (Ξ·) = 1 , and π1 (Ξ·) = π2 (Ξ·) = β― = 0. π π β²β²β² = βπ [1 + π π β²β² β πβ²2 + π΄ (1 β π β² β π β²β² ) 2 2 + π(ββ² β β ββ²β² β 1) + π
π( π Ξ· βΆ Ξ·β : f0 β² (Ξ·) = 1 , f1 β² (Ξ·) = f2 β² (Ξ·) = β― = 0, β ππ π)]. (25) h0 β² (Ξ·) = 1, h1 β² (Ξ·) = h2 β² (Ξ·) = β― = 0, 1 Ξ· hβ²β²β² = βp [ (f hβ²β² β f β²β² h β A ( hβ²β² + hβ² ))] (26) ΞΈ0 (Ξ·) = ΞΈ1 (Ξ·) = β― = 0 , and π0 (Ξ·) = π1 (Ξ·) = Ξ³ 2 β― = 0. 1
π β²β² = βπ [ππ (1 β ππ π½1 {4π΄2π2βπ΄ π π+π2}) 3 2
β1
1
({π β 2 π΄ π
9 2
β π½1 (11 π + π΄ π π β² β π΄ π β π π β² )} π β² + {π β² β 2 π΄ 2 β π½1 (6 π΄2 + 5 π΄ π β² + π β² β π π β²β² )}π + ππ π β² π β² + ππ‘ πβ²2 )].
(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 π0 =
1
π β²β² = βπ [πΏπ (1 β πΏπ π½2 {4π΄2π2βπ΄ π π+π2}) 3
β1 9
β π½2 (11 π + 2 π΄ π π β² β 2 π΄ π β π π β² )} π β² + {π β 2 π΄ 2 β π½2 (6 π΄2 + 5 π΄ π β² + π β² β π π β²β² )}π ππ‘ + π β²β² )]. (28) πΏπ ππ At this stage, any function β³ may be written as
4
, β0 = π , π0 = 1 β
π 2
π
, πππ π0 = 1 β . (31) 2
1
({π β 2 π΄ π
β²
π2
3.2 First order solution π1 = π1 π 2 + π2 π 3 + π3 π 4 + π4 π 5 , β1 = π5 π + π6 π 3 + π7 π 4 , π1 = π8 π + π9 π 2 + π10 π 3 + π11 π 4 + π12 π 5 + β― + π23 π16 , (32) πππ π1 = π24 π + π25 π 2 + π26 π 3 + π27 π 4 + π28 π 5 + β― + π39 π16 .
β
β³ = β ππ β³π ,
(29)
3.3 Second order solution π2 = π40 π 2 + π41 π 3 + π42 π 4 + π43 π 5 + β― + π57 π19 ,
π=0
where β³ stands for π , β , π, and π.
β2 = π58 π + π59 π 3 + π60 π 5 + π61 π 6 + π62 π 7 + β― + π66 π11 , (33)
The above equation represents the approximation solutions for the functions π , β , π, and π in terms of the power series of the π = π + π π , πππ π = π + π π, 2 1 67 2 2 68 homotopy parameter π. where π1 and π2 are two functions of π which are given in the The initial and boundary conditions which satisfy the above Appendix. system can be written as The coefficients (π1 β π68 ) , (π 1 β π 52 ) , and (π1 β π94 ) are for t < 0 βΆ f0 (Ξ·) = f1 (Ξ·) = β― = 0 , given in the Appendix. f0 β² (Ξ·) = f1 β² (Ξ·) = β― = 0, For the complete solution corresponding to π β 1 in Eq. (29), ΞΈ0 (Ξ·) = ΞΈ1 (Ξ·) = ΞΈ2 (Ξ·) = β― = 0 , the analytical perturbed solutions for the velocity, the induced and π0 (Ξ·) = π1 (Ξ·) = π2 (Ξ·) = β― = 0. magnetic field, the temperature, and the nanoparticle concentration are written as for t β₯ 0 βΆ π β² = π0β² + π1β² + π2β² , (34)
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Science and Engineering Applications 2(3) (2017) 164-176 ββ² = β0β² + β1β² + β2β² ,
(35)
π = π0 + π1 + π2 ,
(36)
πππ π = π0 + π1 + π2 .
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Fig. 2a: π½1 = .1, π½2 = .4, ππ = πΏπ = 1, π
π = 1, ππ‘ = ππ = πΎ = .5, ππ = .05
(37)
To get a good convergence, we choose the length of the semiinfinite plate is limited to 2, i.e. Ξ·β = 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 (π΄) , magnetic parameter(π), reciprocal magnetic Prandtl number(πΎ), thermal relaxation parameter (π½1 ) , nanoparticle relaxation parameter(π½2 ) , Prandtl number (ππ) , Lewis number (πΏπ) , Brownian parameter (ππ) , and thermophoresis parameter Fig. 2b: π΄ = πΎ = .8, ππ = πΏπ = 1, π
π = 1, ππ‘ = ππ = (ππ‘). Moreover, Richardson number (π
π), which represents the .5, ππ = .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 < π
π < 10 . Meanwhile, at π
π < 0.1 the natural convection is negligible, and the forced convection is negligible at π
π > 10. Furthermore, π
π > 0 means that ππ€ > πβ (the assisting flow), π
π < 0 means that ππ€ < πβ (the opposing flow), and π
π = 0 is the case of the forced convection. Figs.2, show the relation between the velocity πβ² and the different physical parameters. In general, the velocity starts at its lowest value at the surface, π β² (0) = 0, then it increases till it approach its free stream value that satisfying the far field boundary condition, π β² (2) = 1. Fig.(2.a) sketches the rising in velocity πβ² with the increasing of π΄ . Also, the increasing in magnetic parameter π accelerates the velocity πβ² [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: π΄ = π½1 = .8, π½2 = πΏπ = .1, π
π = 1, π = .5, ππ‘ = illustrates the effect of the relaxation parameters π½1 and π½2 on ππ = .5, ππ = .05 the velocity. The velocity πβ² reduces with the increasing of π½1 . Meanwhile velocity πβ² enhances with the increasing of π½2 . From Fig.(2.c), the velocity πβ² slightly decreases as Prandtl number ππ increases. In fact Prandtl number ππ is defined as the ratio between the momentum (viscous) diffusivity and the thermal diffusivity. This means growing in ππ enhances the rate of viscous diffusion which in turns decreases the velocity. Whilst, a reduction of πβ² is indicated as the reciprocal magnetic Prandtl number πΎ increases. Fig(2.d) indicates that πβ² reduces as Lewis number πΏπ increases till πΏπ β 3.5, and then it starts to increase with the increasing of πΏπ . Fig.(2.e) shows that πβ² decreases as Richardson number π
π increases.
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Science and Engineering Applications 2(3) (2017) 164-176 Fig. 2d: π΄ = π½1 = .8, π½2 = ππ = .1, π
π = 1, π = .5, ππ‘ = ππ = .5, ππ = .05
Fig. 3a: πΎ = 8, π
π = 1, ππ = .05
Fig. 3b: π΄ = .5, π
π = 1, ππ = .05 Fig. 2e: π΄ = π½1 = πΎ = .8, π½2 = .1, ππ = .3, πΏπ = 2, π = .5, ππ‘ = ππ = .5, ππ = .05
Fig. 3c: π΄ = .5, πΎ = .8, ππ = .05
Figs. 2 Variation of the velocity π β² for different values of π΄, π½1 , π½2 , π, πΎ, ππ, πΏπ, π
π Fig.(3.a) indicates that the induced magnetic field ββ² reduces with the increase in unsteadiness parameter π΄ till π΄ β 0.2, and then ββ² starts to increase as π΄ increases. Fig.(3.b) shows that ββ² increases with the growing in the reciprocal magnetic Prandtl number πΎ. This is because of πΎ is defined as the ratio between the magnetic diffusivity and viscous diffusivity. So, the increasing in πΎ enhances the magnetic diffusivity which in turns enhances ββ². Fig.(3.b) indicates that ββ² increases with the increase in Richardson number π
π.
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Figs. 3 Variation of the induced magnetic field ββ² for different values of π΄, πΎ, π
π
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Science and Engineering Applications 2(3) (2017) 164-176 Fig. 4a:π½1 = .1, π½2 = .4, ππ = πΏπ = 1, ππ‘ = ππ = .5, π
π = 1, ππ = .05
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Fig. 4d:π΄ = π½2 = .8, π½1 = .4, ππ = πΏπ = 1, π
π = 1, ππ = .05
Fig. 4e:π΄ = π½2 = .8, π½1 = .4, ππ = 1, πΏπ = 1.2, ππ‘ = ππ = .5, ππ = .05 Fig. 4b:π΄ = .8, ππ = πΏπ = 1, ππ‘ = ππ = .5, π
π = 1, ππ = .05
Fig. 4c:π΄ = π½2 = .8, π½1 = .4, ππ‘ = ππ = .5, π
π = 1, ππ = .05
Fig. 4d:π΄ = π½2 = .8, π½1 = .4, ππ = πΏπ = 1, π
π = 1, ππ = .05
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Figs. 4 Variation of the temperature π for different values of π΄, π½1 , π½2 , ππ, πΏπ, ππ‘, ππ, π
π Fig.(4.a), plots the variation of temperature π due to the unsteadiness parameter π΄. This figure indicates that for small values of π΄, the temperature decreases as long as π΄ increases till the critical value π΄π = 0.55, and then the temperature slightly increases near the surface with the increasing of π΄. Moreover, the curves dispatcher for the higher values of . In Figs.(4.b) and (4.c), there exists a certain point (π β 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:π½1 = .1, π½2 = .4, ππ = πΏπ = 1, ππ‘ = ππ = .5, π
π = 1, ππ = .05
Fig. 5b:π΄ = .8, ππ = πΏπ = 1, ππ‘ = ππ = .5, π
π = 1, ππ = .05
Fig. 5c:π΄ = .8, π½1 = .6, π½2 = .4, ππ‘ = ππ = .5, π
π = 1, ππ = .05
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Fig. 5d:π΄ = .8, π½1 = .6, π½2 = .4, ππ = πΏπ = 1, π
π = 1, ππ = .05
Fig. 5e:π΄ = .8, π½1 = .6, π½2 = .4, ππ = 1, πΏπ = 1.2, ππ‘ = ππ = .5, ππ = .05
Figs. 5 Variation of the nanoparticle volume fraction π for different values of π΄, π½1 , π½2 , ππ, πΏπ, ππ‘, ππ, π
π It should be noted that the increasing in the relaxation parameters π½1 and π½2 , the Prandtl number ππ , and Lewis number πΏπ 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 ππ , the temperature π decreases till that certain point (π β 1.5), and then it starts to increase. Also, it is depicted that with the growing in ππ‘, the temperature π decreases up to a point ( π β 1 ), and then increases. The temperature π is slightly rises as Richardson number π
π increases, this can be shown in Fig.(4.e). From Figs. (5.a) and (5.b), it is illustrated the decreasing in nanoparticle concentration π with the increasing of each of π΄ and π½1 . Physically, as π΄ 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 π decreases with the increase π½2 till a certain point (π β 1.6) and then it starts to increase.
reduces with the increasing in π΄ , π½1 , ππ, πΏπ , ππ‘ and π
π.
Fig.(5.c) shows that π decreases with the increasing of ππ and ο· The nanoparticle concentration profiles indicate that πΏπ. This is because that the growing of πΏπ reduces the Brownian π decreases with the increase π½2 till a certain point diffusion coefficient that leads the flow to decrease the (π β 1.6) and then it starts to increase. nanoparticle concentration. Fig.(5.d) indicates that π decreases with the increase of ππ‘, but π enhances with the increasing References values of ππ. Fig.(5.e) elucidate the reduction in nanoparticle [1]Keblinski, P.; Eastman, J. A.; Cahill, D. G., Materialstoday concentration π that resulted from the growing of π
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The nanoparticle concentration profiles indicate that [25] Hayat, T.; Imtiaz , M.; Alsaedi, A.; Almezal, S., J. of Mag. π rises with the increasing in ππ . However, it &Mag. Mat. 2016, 401, 296-303.
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