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Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion
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Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion A.S. Dogonchi∗, D.D. Ganji Mechanical Engineering Department, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran
a r t i c l e
i n f o
Article history: Received 16 June 2016 Revised 3 September 2016 Accepted 23 September 2016 Available online xxx Keywords: Nanofluid Brownian motion Stretchable/shrinkable walls Duan–Rach approach (DRA) KKL
a b s t r a c t In this paper, the nanofluid flow and heat transfer between non-parallel stretching walls with Brownian motion effect is investigated. The governing radial momentum and energy equations are solved by Duan–Rach Approach (DRA). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The effective viscosity and thermal conductivity of nanofluid are calculated via KKL (Koo–Kleinstreuer–Li) correlation in which influence of Brownian motion on the thermal conductivity is considered. The effects of various parameters such as the stretching/shrinking parameter, the radiation parameter, Reynolds number, the opening angle and the heat source parameter are investigated on the velocity and temperature. Also, the value of the Nusselt number is calculated and presented through figures. The results show that the fluid velocity, temperature profile and Nusselt number increase with the increasing of stretching parameter. The results also reveal that the temperature profile increases with the increasing of the heat source parameter and it decreases with the rising of radiation parameter for both divergent and convergent channel. In addition, the results were compared with the previous works and found proposed method has high accuracy to solve this nonlinear problem. © 2016 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1. Introduction The fluid flow and heat transfer between parallel/non-parallel plates have received notable consideration in recent years as they are most applicable in the mechanical engineering and industrial applications. One of the first studies of this type of flow can be traced back to Jeffery [1] and Hamel [2]. There are newly developed methods such as Collocation Method [3], Differential Transformation Method [4–10], Homotopy Analysis Method [11–14], Adomian Decomposition Method [15–17], Homotopy Perturbation Method [18–20] and Optimal Homotopy Analysis Method [21] to solve the equation of Jeffery–Hamel fluid flow and many nonlinear problems in the engineering field. The unsteady motion of a rigid spherical particle in a quiescent shear-thinning power-law fluid was studied by Rahimi-Gorji et al. [3]. Their results show that the time of reaching the terminal velocity in a falling procedure significantly increases with growing of the particle size. The falling of a spherical particle in a plane Couette Newtonian fluid flow was
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (A.S. Dogonchi), [email protected] (D.D. Ganji).
studied by Dogonchi et al. [6]. They applied the DTM-Padé to solve governing equations. They concluded that the horizontal and vertical velocities of spherical particles in water fluid are higher than the ethylene-glycol and glycerin fluids. Dogonchi et al. [7] studied the motion of a vertically falling non-spherical solid particle in an incompressible Newtonian fluid. They applied the DTM-Padé to solve governing equations. They observed that the velocity of the gold particle is higher than the aluminum and copper particles. The entropy generation for the MHD nanofluid flow past a stretching permeable surface was analyzed by Abolbashari et al. [11]. Their results show that the rising of the unsteadiness parameter, nanoparticle volume fraction parameter, magnetic parameter, Reynolds number, suction parameter, Hartmann number and Brinkman number lead to an increase of the entropy generation number. Rashidi and co-workers [12] studied the heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation effect. They found that the heat transfer to the fluid increases with the increasing value of the wedge angle. The effect of thermal radiation on the classical Jeffery–Hamel flow from a source or sink in convergent/divergent channels was studied by Barzegar Gerdroodbary et al. [22]. Their results show that the temperature profile increases with the rising of thermal radiation parameter. The classical Jeffery–Hamel flow from a source or sink in
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Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Nomenclature A1 , A2 , A3 , A4 C Ec Cp f
P Re Pr s qrad . T Tw T∞ r,θ u uc uw Hs N Nu Cf DRA ADM k k∗ Rf dp
constant parameter stretching/shrinking parameter Eckert number specific heat (J/kg K) dimensionless velocity dimensionless temperature pressure term Reynolds number Prandtl number stretching/shrinking rate (m2 /s) radiative heat flux temperature (K) surface temperature of channels (K) free stream temperature cylindrical coordinates velocity component in the radial direction (m/s) rate of movement in the radial direction (m2 /s) wall velocity component in the radial direction (m/s) heat source parameter radiation parameter Nusselt number skin fraction coefficient Duan–Rach approach Adomian decomposition method thermal conductivity mean absorption coefficient thermal interfacial resistance particle size
Greek symbols α angle of the channel η dimensionless angle σ electrical conductivity θ any angle ρ density μ dynamic viscosity ν kinematic viscosity φ solid volume fraction σ∗ Stefan–Boltzmann constant Subscripts nf nanofluid f base fluid p nano-solid-particles
convergent/divergent channels with stretchable walls was investigated by Turkyilmazoglu [23]. He observed that the temperature profile increases with the rising of stretching parameter for the convergent channel. The primary obstacle to enhance the heat transfer in engineering systems is the low thermal conductivity of customary fluids such as water, air, oil, and ethylene glycol mixture. Solid usually has a higher thermal conductivity than liquids. Addition of Nanoparticles to the conventional fluid, the so called “Nano-fluid”, can improve the thermal conductivity of the mixture. Recently, several authors studied nanofluid flow and heat transfer [24–45]. Rashidi et al. [24] studied heat transfer of nanofluid flow over a stretching sheet in the presence of transverse magnetic field considering thermal radiation and buoyancy effects. They concluded that the increasing of buoyancy parameter increases the velocity profile and decreases the temperature profile of the nanofluid. The
effect of magnetic field on the nanofluid flow inside a sinusoidal two-tube heat exchanger was investigated numerically by Valiallah Mousavi et al. [25]. They applied finite volume method to solve governing equations. Their results show that the sinusoidal formation of the internal tube significantly increases the Nusselt number inside a two-tube heat exchanger. Ferrofluid flow and heat transfer in the presence of an external variable magnetic field were studied by Kandelousi [26]. His results indicate that the Nusselt number is an increasing function of Magnetic number, Rayleigh number and nanoparticle volume fraction while it is a decreasing function of the Hartmann number. The influence of magnetic field dependent (MFD) viscosity on free convection heat transfer of nanofluid in an enclosure was studied by Sheikholeslami et al. [27]. They applied Control Volume based Finite Element Method to solve governing equations. They observed that the Nusselt number is an increasing function of Rayleigh number and volume fraction of nanoparticle while it is a decreasing function of viscosity parameter and Hartmann number. The effect of an electric field on Fe3 O4 –water nanofluid flow and heat transfer in a channel was studied by Safarnia et al. [28]. They found that the Nusselt number has a direct relationship with the Reynolds number and voltage supply. Nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation was investigated by Sheikholeslami et al. [29]. They proved that skin friction coefficient increases with increase of Reynolds number and suction parameter, but it decreases with increase of the nanoparticle volume fraction. Effect of electric field on the hydrothermal behavior of nanofluid in a complex geometry was studied by Sheikholeslami et al. [30]. Their results show that supplied voltage can change the flow shape. Impact of non-uniform magnetic field on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects was investigated by Sheikholeslami and Rashidi [31]. They reported that the Nusselt number has a direct relationship with Rayleigh number, buoyancy ratio number and Lewis number while it has a reverse relationship with Hartmann number. The effect of non-uniform magnetic field on nanofluid forced convection heat transfer in a lid driven semi-annulus considering Brownian motion was studied by Sheikholeslami et al. [32]. Force convection heat transfer in a lid driven semi annulus enclosure in presence of non-uniform magnetic field was studied by Sheikholeslami et al. [33]. They applied control volume based finite element method (CVFEM) to solve the governing equations. The effect of spatially variable magnetic field on ferrofluid flow and heat transfer was investigated by Sheikholeslami and Rashidi [34]. They applied control volume based finite element method (CVFEM) to solve the governing equations. Their results indicate that the Nusselt number is an increasing function of Magnetic number, Rayleigh number and nanoparticle volume fraction, while it is a decreasing function of Hartmann number. Effect of Lorentz forces on forced-convection nanofluid flow over a stretched surface was studied by Sheikholeslami et al. [35]. Their results illustrate that the coefficient of skin friction enhances with enhancing magnetic parameter while reduces with enhancing velocity ratio parameter. Ferrofluid flow in a semi annulus enclosure considering radiative heat transfer was studied by Sheikholeslami et al. [36]. Their results show that the Nusselt number is a rising function of Rayleigh number and Magnetic number while it is a decreasing function of radiation parameter. MHD convective nanofluid flow over a vertical stretching sheet considering the thermal radiation and buoyancy effects was studied by Pourmehran et al. [42]. Their results indicate that the reduced Nusselt number has an inverse relation with increasing the nanoparticle concentrations. An analytical investigation of the heat transfer for the microchannel heat sink (MCHS) cooled by different nanofluids (Cu, Al2 O3 , Ag, TiO2 in water and ethylene glycol as base fluids) was studied by Rahimi-Gorji et al. [43]. They reported that by increasing the nanoparticles volume fraction, the Brownian
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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movement of the particles, which carries the heat and distributes it to the surroundings, increases and, consequently, the difference between the coolant and wall temperature becomes less. The unsteady flow of a nanofluid squeezing between two parallel plates was investigated by Pourmehran et al. [44]. Their results demonstrated that when the two plates move together, the Nusselt number has a direct relationship with nanoparticle volume fraction and Eckert number while it has a reverse relationship with the squeeze number. Many of engineering problems, especially some of the heat transfer equations are nonlinear. As mentioned before, some of these non-linear equations can be solved using analytical methods such as Adomian Decomposition Method, Differential Transformation Method and Homotopy Perturbation Method. One of the analytic methods which does not need discretization or linearization is the Adomian Decomposition Method (ADM). The benefits of this method are that it can provide an approximate solution for many non-linear equations without perturbation, linearization, closure approximation, or discretization methods. Also, ADM results are more realistic because it gives an approximate solution of the problem without any simplification. Adomian Decomposition Method (ADM) was proposed by Adomian in 1986 [15]. Hashim [46] applied the Adomian Decomposition Method for solving boundary value problems for fourth-order integro-differential equations and the Blasius equation. Esmaili et al. [47] used the ADM to solve the governing equation of classical Jeffery–Hamel fluid flow. Comparison of the results illustrates that the analytical and numerical methods are in an excellent agreement with each other. The effect of nanoparticle and magnetic field on the Jeffery–Hamel fluid flow was investigated by Sheikholeslami et al. [48]. They used the Adomian Decomposition Method (ADM) to solve governing equations. They reported that when the Hartmann number increases, backflow will reduce. Many authors have tried to modify the ADM. Jin and Liu [49] modified ADM for solving a kind of evolution equation. Jafari and Daftardar-Gejji [50] modified the standard ADM for solving a system of nonlinear equations. All these methods need to find the unknown initial values of the problem that the final solution depends on the accuracy of the initial values determined by numerical method. Duan and Rach [51] have introduced a new modification of the Adomian Decomposition Method to solve many nonlinear boundary value problems (BVP). This modified method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients and in fact the final solution does not contain undetermined coefficients. The non-Newtonian fluid flow in a channel with a porous wall was studied by Dogonchi and Ganji [52]. They applied this modified method (DRA) to solve governing equations. They proved that the Nusselt number has a direct relationship with Prandtl number, Reynolds number and power law index. The heat transfer of a steady, viscous incompressible water based MHD nanofluid flow between two stretchable walls with thermal radiation effect was investigated by Dogonchi and Ganji [53]. Their results indicate that the fluid velocity and temperature profile increase with the increasing of stretching parameter. The unsteady squeezing flow and heat transfer of MHD nanofluid between the infinite parallel plates with thermal radiation effect was studied by Dogonchi et al. [54]. They used the DRA to solve governing equations. Their results show that the temperature profile and Nusselt number increase with the increasing of radiation parameter. This analytical method was successfully applied to different engineering problems [55,56]. In this paper, we have applied this modified method to solve the equation of nanofluid flow and heat transfer between nonparallel stretchable walls in the presence of thermal radiation and heat source using Koo-Kleinstreuer-Li (KKL) model. In this model the impact of Brownian motion on the effective thermal conductivity and the effective viscosity is considered. The effects of the heat
3
Fig. 1. Schematic view of problem.
source parameter, the stretching/shrinking parameter, the radiation parameter and Reynolds number on the flow and heat transfer profiles are investigated. 2. Problem description The steady flow of an incompressible conductive viscous fluid due to a source or sink between two stretchable non-parallel walls that meet at an angle 2α is considered (Fig. 1). The walls are assumed to radially stretch or shrink in accordance with [23,53]:
u = uw =
s r
(1)
with s being the stretching/shrinking rate. The walls are considered to be convergent if α < 0 and divergent if α > 0. We suppose that the velocity is purely radial and depends on r and θ and moreover there is no magnetic field in the z-direction. The reduced forms of continuity, Navier–Stokes and energy equations in polar coordinates are [23,48,53]:
ρn f ∂ (ru ) = 0, r ∂r
(2)
μn f 1 ∂P ∂u u =− + ∂r ρn f ∂ r ρn f
∂ 2u 1 ∂ u 1 ∂ 2u u + + − , ∂ r2 r ∂ r r2 ∂ θ 2 r2
2μnf ∂ u 1 ∂P − = 0. ρnf r ∂θ ρnf r2 ∂θ 2 ∂T knf ∂ T 1 ∂ T 1 ∂ 2T u = + + ∂r r ∂r r2 ∂ θ 2 (ρCp )nf ∂ r2 1 1 ∂ ∂ − r qr,rad. − qθ ,rad. 2 (ρCp )nf r ∂ r (ρCp )nf r ∂θ 2 2 ∂ u u 2 ∂r + r μnf Q0 + + T 2 ρ C ρ ( p )nf + 1 ∂ u ( Cp )nf
(3) (4)
(5)
r ∂θ
where u = u(r, θ ) is the velocity, T is the temperature of the fluid, P is the fluid pressure and qrad is the radiative heat flux. Using Rosseland approximation for radiation (see Rashidi et al. [24]) we have
∂T4 ∂r
∂T4 ∂θ
qr,rad. = − 4σ ∗ /3k∗nf
qθ ,rad. = − 4σ ∗ /3k∗nf
(6) (7)
Here σ ∗ is the Stefan–Boltzmann constant and is nanofluid mean absorption coefficient. Further we assume that temperature difference within the flow is such that T4 may be panded in a Taylor series. Hence, expanding T4 about T∞ and glecting higher order terms we get 3 4 T4 ∼ T − 3T∞ = 4T∞
k∗n f
the the exne-
(8)
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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u
Therefore Eq. (5) is simplified to
∂T ∂r
=
Table 1 The coefficient values of CuO–water and Al2 O3 –water nanofluids [70].
∂ 2T 1 ∂ T 1 ∂ 2T + + ∂ r2 r ∂ r r2 ∂ θ 2 2 2 3 3 16σ ∗ T∞ 16σ ∗ T∞ ∂ T 1 ∂T ∂ T + ∗ + + 3kn f (ρC p )n f ∂ r 2 r ∂ r 3k∗n f (ρC p )n f r 2 ∂ 2 θ 2 2 2 ∂∂ur + ur μn f Q0 + + T (9) ( ρ C p ) n f + 1 ∂ u 2 (ρCp )n f kn f (ρCp )n f
r ∂θ
where ρ nf is the effective density of the nanofluid and (ρ Cp )nf is the heat capacitance of the nanofluid given as:
ρn f = (1 − φ )ρ f + φρ p , (ρCp )n f = (1 − φ )(ρCp ) f + φ (ρCp ) p .
(10)
The used effective thermal conductivity by some authors in their studies [16–19,53,54] to simulate nanofluids depends only on the thermal conductivity of the base fluid and particles and on the volume fraction of particles whereas numerous studies [57– 60] show that the temperature, the particle size, random motion and the kind of nanoparticle can also affect the effective thermal conductivity of a nanofluid. Hence, in this paper the effective viscosity and thermal conductivity of nanofluid are calculated via KKL (Koo-Kleinstreuer-Li) [61–64] correlation in which impacts of particle size, particle volume fraction and temperature dependence as well as types of particle and base fluid combinations on the thermal conductivity are considered.
ke f f = kstatic + kBrownian
(11)
μe f f = μstatic + μBrownian = μstatic + in which
μstatic =
(12)
kp kf
kp kf
+2 −
−1
μf (1 − φ )2.5
kBrownian = 5 × 10
4
φρ f c p, f
52.81348876 6.115637295 0.695574508 0.041745555 0.1769193 –298.1981908 –34.53271691 –3.922528928 –0.235432963 –0.999063481
∂u ∂T uc =0= ,u = ∂θ ∂θ r
(18)
and due to the stretching/shrinking convergent/divergent wall condition at the plates (θ = ±α )
s Tw u = uw = , T = 2 r r
(19)
with uc being the centerline rate of movement and Tw is the constant wall temperature. We should remark that in general conditions where asymmetry is also allowed, infinitely many solutions exist [23]. To retain the symmetry with respect to the centerline, we impose the same stretching/shrinking rates on both walls. Considering only radial flow, the Eq. (1) implies that
f (θ ) = ru(r, θ )
(20)
f (θ ) T , (η ) = r 2 where fmax Tw
η=
θ α
(21)
(13)
Substituting these into the governing equations and eliminating the pressure term yields the nonlinear third order ordinary differential equation for the flow from the radial momentum equation:
(14)
f (η ) + 2α Re
A1 f ( η ) f ( η ) + 4α 2 f ( η ) = 0 A2
(22)
And second order differential equation for the heat
kb T g (T , φ , d p ) ρpd p
(15)
It should be noted that by introducing a thermal interfacial resistance [65,66] (Rf = 4 × 10−8 km2 /W) the kp was replaced by a new kp,eff in the form:
Rf +
Al2 O3 –water
–26.59331085 –0.403818333 –33.3516805 –1.915825591 6.42185846658E–02 48.40336955 –9.787756683 190.24561 10.92853866 –0.720099837
The governing equations are accompanied with the boundary conditions, because of the symmetry assumption at the channel centerline (θ = 0)
f (η ) =
φ
CuO–water
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
using dimensionless parameters
kstatic =1+ kf
k 3 kp − 1 φ f
μf kBrownian × kf Pr f
Coefficient values
(1 + N ) (η ) + 2α 2 +
A3 Hs P r f ( η ) + 2 + 2N + (η ) A4 2A4
A2 EcP r 2 4α f (η )2 + f (η )2 = 0 A4 Re
(23)
Here A1 , A2 , A3 and A4 are constants given by:
dp dp = kp k p,e f f
(16)
The empirical gʹ-function for various nanoparticles and base fluids is different. In this paper for water based nanofluid includes Al2 O3 and CuO (as nanoparticles) in its structure, this function follows the format [61]:
g (T , φ , d p ) = (a1 + a2 ln (d p ) + a3 ln (φ ) + a4 ln (φ ) ln (d p ) +a5 ln (d p ) ) ln (T ) + (a6 + a7 ln (d p ) + a8 ln (φ ) 2
+a9 ln (φ ) ln (d p ) + a10 ln (d p ) ) 2
(17)
in which, the coefficients ai (i = 0…10) are based on the type of nanoparticles and furthermore with these coefficients, Al2 O3 – water nanofluids and CuO–water nanofluids have an R2 of 96% and 98%, respectively [63,67–70] (See Table 1).
A1 =
( ρ C p )n f ρn f μn f ke f f , A2 = , A3 = , A4 = ρf μf kf (ρ C p ) f
(24)
Subject to the boundary conditions
f (0 ) = 1, f (0 ) = 0, f (1 ) = C, (0 ) = 0, (1 ) = 1
(25)
where C = s/uc is the stretching (C > 0) or shrinking (C < 0) parameter, Re = α uc /ν f is the Reynolds number, Pr = uc ρC p /k f is the Prandtl number, Ec = α u2c /C p f Tw is the Eckert number, N = 3 /3k∗ k 16σ ∗ T∞ is the radiation parameter and Hs = r2 Q0 /kf is the nf nf heat source parameter. It is noted that setting C to zero at this stage leads to stationary wall condition for the traditional Jeffery–Hamel flow. The elliptic function solutions as given in the literature may be fulfilled here for nonzero C, but, it is thought that this is not very convenient.
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Fig. 2. Velocity (f) and temperature () profiles for different values of stretching/shrinking parameter (C) in divergent/convergent channel (CuO–water).
Physical quantities of interest are the skin fraction coefficient and Nusselt number which are defined as:
C ∗f
μn f = ρ f u2c
N u∗ =
r k f Tw
1 ∂u r ∂θ
kn f +
θ =α
σ
3 16 ∗ T∞ ∗ 3kn f
3.1. Description of the Duan–Rach approach (DRA)
1 ∂ T r ∂θ θ =α
Consider a third-order non-linear differential equation:
(26)
Lu = Nu + g(x ),
(28)
subject to the mixed set of Dirichlet and Neumann boundary conditions
After simplification, we obtain:
μn f f 1 ( ) μf kn f ( 1 + N ) Nu = (1 ) kf α
3. Applied method
u(x1 ) = α0 , u (x1 ) = α1 , u (x2 ) = α2 , x2 = x1
Cf =
(29)
3
(27)
where L = ddx3 is the linear differential operator to be inverted, Nu is an analytic nonlinear operator, and g(x) is the system input.
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Fig. 3. Velocity (f) and temperature () profiles for different values of Reynolds number (Re) in stretching/shrinking divergent/convergent channel (CuO–water).
We differentiate Eq. (31), then let x = x2 and solve for u (ξ ), hence
We take the inverse linear operator as
L−1 (· ) =
x
x0
x
x1
ξ
x
(· )dxdxdx,
(30)
where ξ is a prescribed value in the specified interval. Thus we have:
L−1 Lu = u(x ) − u(x0 ) − u (x1 )(x − x0 ) 1 − u (ξ )[(x − x1 )2 − (x0 − x1 )2 ] 2
1 u ( x2 ) − u ( x1 ) − x2 − x1 x2 − x1
(31)
(32)
x2
x1
ξ
x
[Nu + g]dxdx.
(33)
Substituting Eq. (33) into Eq. (32) yields,
u(x ) = u(x0 ) + u (x1 )(x − x0 ) +
Applying the inverse operator L − 1 to both sides of Eq. (28) yields:
L−1 [Nu + g] = u(x ) − u(x0 ) − u (x1 )(x − x0 ) 1 − u (ξ )[(x − x1 )2 − (x0 − x1 )2 ] 2
u (ξ ) =
1 (x − x1 )2 − (x0 − x1 )2 2
u ( x2 ) − u ( x1 ) + L−1 g + L−1 Nu x2 − x1 2 2 x2 x 1 ( x − x1 ) − ( x0 − x1 ) − gdxdx 2 x2 − x1 x1 ξ 2 2 x2 x 1 ( x − x1 ) − ( x0 − x1 ) − Nudxdx. 2 x2 − x1 x1 ξ
(34)
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Fig. 4. Velocity (f) and temperature () profiles for different values of opening angle parameter (α ) in stretching/shrinking divergent/convergent channel (CuO–water).
Thus in Eq. (34) the three known boundary values u(x0 ), u (x1 ) and u (x2 ) are included and the undetermined coefficient was replaced. Next, the solution is decomposed and the nonlinearity ∞ u (x ) = ∞ m=0 um (x ), Nu (x ) = m=0 Am (x ) where Am (x) = Am (u0 (x), u1 (x), … , um (x)) are the Adomian polynomials. From Eq. (33), the solution components are determined by the modified recursion scheme:
u0 = u(x0 ) + u (x1 )(x − x0 ) + u ( x2 ) − u ( x1 ) + L−1 g x2 − x1
1 ( x − x1 ) − ( x0 − x1 ) 2 x2 − x1 2
−
1 (x − x1 )2 − (x0 − x1 )2 2
2
x2 x1
ξ
um+1 = L−1 Am −
gdxdx,
(35)
2
2
x2 x1
x
ξ
Am dxdx.
(36)
3.2. Implementation of the method In our study, the Duan–Rach Approach must be modified. We do not use the prescribed value ξ . According to Eq. (28), Eqs. (22) and (23) can be written as follows:
A1 f ( η ) f ( η ) − 4α 2 f ( η ) A2
(37)
1 2α 2 A3 Pr f (η ) + 2 + 2N+ Hs (η ) A4 2A4 2 L2 (η ) = − N+1 + AA24 EcPr 4α f (η )2 + f (η )2 Re
(38)
L3 f (η ) = −2α Re x
1 ( x − x1 ) − ( x0 − x1 ) 2 x2 − x1
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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where N1 u and N2 u are introduced as:
N1 u = −2α Re
A1 f ( η ) f ( η ) − 4α 2 f ( η ) A2
(42)
1 2α 2 A3 Pr f (η ) + 2 + 2N + Hs (η ) A4 2A4 N2 u = − N+1 + AA24 EcRePr 4α 2 f (η )2 + f (η )2
(43)
Evidently, we do not have the values of f (0) and (0). In the standard Adomian Decomposition Method (ADM), we need to evaluate those unknown conditions with numerical methods. Hence, the boundary value problem (BVP) is turned into an initial value problem (IVP). The accuracy of the solution depends on the accuracy of the two unknown parameters. In our study, we use the Duan–Rach Approach [51] to find a totally analytical solution. By putting η = 1 in Eq. (40), we have:
f (0 ) = 2C − 2 where
L−1 3 N1 u
η=1
=
1 0
L−1 3 N1 u
η=1
η η 0
+1
(44)
(Nu )dηdηdη
0
(45)
Substituting Eq. (44) into Eq. (40) yields,
2 −1 f (η ) = 1 − η2 + C η2 + L−1 3 N1 u − η L3 N1 u
(46)
η=1
Thus the right hand side of Eq. (46) does not contain the undetermined parameter f (0). Finally, we have the modified recursive scheme:
f 0 (η ) = 1 − η 2 + C η 2
2 −1 fm+1 (η ) = L−1 3 Am ( η ) − η L3 Am ( η )
(47)
η=1
where Am (η) are the Adomian polynomials, which can be determined by the formula:
m 1 dm Am ( η ) = N λi Fi (η ) m! dλm
(48)
i=0
λ=0
that was first published by Adomian and Rach [71]. Applying Eq. (48), we obtain the terms of the Adomian polynomials and put them in Eq. (47), and we determine fm (η) as follows:
f 0 (η ) = 1 − η 2 + C η 2 1 (C − 1 )αη2 f 1 (η ) = − 30 A2 Fig. 5. Velocity profile (f) for different values of nanoparticles volume fraction (CuO–water).
3
where the differential operator L3 and L2 are given by L3 = ddη3 and 2 L2 = ddη2 respectively. Assume that the inverse operator L−1 and 3 L−1 exist, then we have: 2
L−1 3 (· ) =
η η η 0
0
0
(· )dηdηdη, L−1 2 (· ) =
η η 0
0
( · )d η d η
(39)
Operating with L−1 in Eq. (37) and after exerting boundary con3 ditions on it:
η f (η ) = f (0 ) + f (0 )η + f (0 ) + L−1 3 (N1 u ). 2 2
(40)
−10A2 α − 4A1 Re − A1C Re +10A2 αη2 + 5A1 Reη2 −A1 Reη4 + A1C Reη4
The functions f2 (η), f3 (η), … can be determined in a similar way from Eq. (47). For convenience, we do not represent all terms of fm (η). Using f (η ) = ∞ m=0 f m (η ) = f 0 (η ) + f 1 (η ) + f 2 (η ) + . . . , thus
f (η ) = 1 − η 2 + C η 2 −
1 (C − 1 )αη2 30 A2
−10A2 α − 4A1 Re − A1C Re +10A2 αη2 + 5A1 Reη2 −A1 Reη4 + A1C Reη4
+ −···
(50)
According to Eq. (50), the accuracy increases by increasing the number of solution terms (m). For (η), we proceed in the same manner. We get the following recursive scheme:
Operating with L−1 in Eq. (38) and after exerting boundary con2 ditions on it:
0 ( η ) = 1
−1 m+1 (η ) = − L−1 2 N2 u η=1 + L2 N2 u
(η ) = (0 ) + (0 )η + L−1 2 (N2 u ).
where [L−1 N2 u]η=1 = 2
(41)
(49)
1η 0
0
(51)
(N2 u )dηdη
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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9
Fig. 6. Temperature profile () for different values of heat source parameter (Hs) and radiation parameter (N) (CuO–water).
(η ) =
Using thus
∞
m=0
m ( η ) = 0 ( η ) + 1 ( η ) + 2 ( η ) + . . . ,
1 1 (η ) = 1 − 30(N + 1 ) A4 Re
−4A2 Ec Pr C 2 α 2 − 10A2 Ec Pr C 2 + − · · · + − · · ·
(52)
Evidently, the temperature field (η) depends on the number of terms of the velocity field f(η) found in Eq. (50).
4. Results and discussion The nanofluid flow and heat transfer between non-parallel stretching walls with Brownian motion effect is studied analytically by using the modified standard ADM called Duan–Rach Approach (DRA). The Prandtl number is fixed at 6.2. The effects of different parameters such as Reynolds number (Re), the heat source parameter (Hs), the radiation parameter (N), the stretching/shrinking parameter (C), etc. on the velocity and temperature profiles are investigated. The nanofluid thermo-physical properties have been
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Fig. 7. Velocity (f) and Temperature () profiles for different nanofluids structure.
Table 2 Thermo-physical properties of water and nanoparticles [69].
CuO Al2 O3 Pure water
ρ (kg/m3 )
Cp (J/kgK)
k(W/mK)
dp(nm)
6500 3970 997.1
540 765 4179
18 25 0.613
29 38.4 –
abridged in Table 2. To verify the analytical results, we compared our results with the previous works [23]. As seen in Tables 3–6, they are in an excellent agreement. In these tables, an error is in-
troduced as follows:
DRA Results − Numerical Results × 100. %Error = Numerical Results The effect of the stretching/shrinking parameter on the velocity and temperature profiles in convergent/divergent channel is shown in Fig. 2. It is important to note that the stretching/shrinking parameter (C) describes the movement of the walls (C > 0 corresponds to the walls are stretching, while C < 0 corresponds to the walls are shrinking). It is clear that with the increasing of stretching parameter, the fluid flow moves faster near the walls. Physically speaking, when C = 1 the fluid flow in the channel moves at
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Table 3 Comparison between DRA and numerical results for F (1) when α = −5◦ , Re = 50 for different values of stretching/shrinking parameter.
C
DRA results ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–2 –1 0 1 2
–5.130921689 –4.652183982 –2.833915413 0 3.654305033
–5.1309222926 –4.6521591354 –2.8339514330 0 3.6697111853
0.0 0 0 01177 0.0 0 053412 0.00127102 – 0.41981919
Table 4 Comparison between DRA and numerical results for F (1) when α = 5◦ , Re = 50 for different values of stretching/shrinking parameter.
C
DRA results ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–1.5 –1 0 1 1.5
–5.082942399 –3.508090102 –1.109360533 0 –0.1423152590
–5.0829256568 –3.5081031667 –1.1093265266 0 –0.1464298835
0.0 0 032941 0.0 0 037242 0.0030655 – 2.80996228
Table 5 Comparison between DRA and numerical results for − (1)when α = −5◦ , Re = 50, Pr = 1, Ec = 0 for different values of stretching/shrinking parameter.
C
DRA results the absence of N, Hs and ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–2 –1 0 1 2
0.03157845854 0.03734696604 0.04214811723 0.04640127099 0.05052578617
0.0315761821 0.0373226368 0.0421517243 0.0464015106 0.0502423154
0.0072094 0.06518632 0.00855734 0.0 0 051638 0.5642072
Table 6 Comparison between DRA and numerical results for − (1)when α = 5◦ , Re = 50, Pr = 1, Ec = 0 for different values of stretching/shrinking parameter.
C
DRA results the absence of N, Hs and ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–1.5 –1 0 1 1.5
0.03244718254 0.03475109986 0.03998210836 0.04640127099 0.05019038094
0.0324575970 0.0347758169 0.0399820121 0.0464015106 0.0504570638
0.03208635 0.07107537 0.0 0 02408 0.0 0 051638 0.52853424
the same speed with the centerline velocity (uc ) while when C > 1 the fluid flow in the channel moves faster than the centerline velocity. This means that the fluid velocity increases near the wall with the increasing of stretching parameter. Also, it is observed that stretching warm up the system by increasing the thickness of thermal layer while shrinking cools down the system by thinning of the thermal boundary layer. Hence, where the cooling is necessary, wall shrinking is advisable. The variation of the velocity and temperature profiles for different values of the Reynolds number (Re) is shown in Fig. 3. It is illustrated that the rise in Re make a decrease in both velocity and temperature profiles for stretching/shrinking divergent channels as well as for higher values of Re the flow moves reversely and the probability of the backflow phenomenon, especially in the stretching case, increases, so if we will not to observe the backflow phenomenon, in the divergent channel, it is better to use shrinking case. Another notable point about Fig. 3 is that for stretch-
Fig. 8. Nusselt number (Nu) for different values of radiation parameter (N) and stretching/shrinking parameter (C) in (a) divergent /(b) convergent channel (CuO– water).
ing/shrinking convergent channels, the results were completely reversed and also the variation of velocity and temperature profiles is more gradual than when Re is at a higher value. The effect of the opening angle (α ) on the velocity and temperature profiles in stretching/shrinking divergent/convergent channel is shown in Fig. 4. In the stretching/shrinking convergent channel, the fluid velocity becomes flat and thickness of boundary layer reduces. In the stretching/shrinking divergent channel, as the opening angle (α ) rises, the magnitude of fluid velocity decreases. Hence, by rising the opening angle in stretching/shrinking divergent channel, the probability of the backflow phenomenon, especially in the stretching case, rises. Also, it can be seen that the temperature profile increases with the increasing of the absolute
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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variation of the heat source parameter can be viewed in Fig. 6. But since, the presence of radiation parameter leads to thinning of the thermal layer consequently, the rising value of radiation parameter leads to a decrease the temperature profile of the nanofluid. Fig. 7 shows the variation of the velocity and temperature profiles for two different structures of nanofluid (See Table 2). It can be concluded that when water-based nanofluid includes Al2 O3 (as nanoparticles) in its structure, f and values are greater than the other structure. However, this is true for the velocity profile in a divergent channel. For convergent channel this behavior of nanofluids structure is completely vice versa. Also, it is observed that in the stretching divergent channel the probability of the backflow phenomenon for CuO–water nanofluid is more than waterbased nanofluid includes Al2 O3 (as nanoparticles) in its structure. Fig. 8 displays the effect of the stretching/shrinking parameter (C) and the radiation parameter (N) on the Nusselt number (Nu). It is observed that the magnitude of the Nusselt number increases with the increasing of stretching parameter and radiation parameter, but this treatment of Nusselt number is completely vice versa for shrinking parameter. Finally, Fig. 9 shows the effect of the heat source parameter (Hs) and the Reynolds number (Re) on the Nusselt number (Nu) in stretching divergent/convergent channel. It is observed that the magnitude of the Nusselt number increases with the increasing of the heat source parameter. While, the opposite behavior is observed in a variation of the Nusselt number with Reynolds number in convergent/divergent channel. As already noted, Tables 3–6 confirm that the approach used has high accuracy for different values of the effective parameters. In the ADM, for given Re, Hs, α , N and φ , we must solve f (1) = 0 and (0) = 0 to find f (0) and (0). If we change the values of Re, we must again evaluate the values of f (0) and (0). In our work (Duan–Rach Approach), the obtained solutions for f(η) (Eq. (50)) and (η) (Eq. (52)) are purely analytical approach and we do not need any other calculations if we change any of the parameters of the problem and the final solution does not contain undetermined coefficients. 5. Conclusions In this paper, the nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion is investigated analytically using Duan–Rach Approach (DRA). The comparison between DRA and numerical results confirms the validity of this approach. The results indicate that with the increasing of stretching parameter, the fluid flow moves faster near the walls. Also, by increasing of stretching parameter the system becomes warmer. On the other hand, the temperature profile and Nusselt number increase with the increasing of heat source parameter. Fig. 9. Nusselt number (Nu) for different values of heat source parameter (Hs) and Reynolds number (Re) in stretching (a) divergent/(b) convergent channel (CuO– water).
value of the opening angle for both stretching/shrinking divergent/convergent channels. The effect of the nanoparticles volume fraction (φ ) on the velocity profile, when the other parameters are constant, is shown in Fig. 5. It can be concluded that rising nanoparticles volume fraction in divergent channel causes a decrease in velocity profile and the backflow phenomenon, especially in the stretching case, may be initiated at high Reynolds numbers. Fig. 6 shows the effect of the heat source (Hs) and radiation (N) parameters on the temperature profile. Due to the heat source parameter causes heat generation in the system, so it is expected that the temperature profile increases with the increasing of the heat source parameter. This treatment of the temperature profile with a
References [1] Jeffery GB. The two-dimensional steady motion of a viscous fluid. Philos Mag Ser 1915;29(6)):455–65. [2] Hamel G. Spiralförmige Bewgungen Zäher Flüssigkeiten Jahresbericht der. Deutschen Mathematiker-Vereinigung 1916;25:34–60. [3] Rahimi-Gorji M, Pourmehran O, Gorji-Bandpy M, Ganji DD. An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by collocation method. Ain Shams Eng J 2015;6:531–40. [4] Zhou JK. Differential transformation method and its application for electrical circuits. Wuhan, China: Hauzhang University Press; 1986. [5] Ganji DD, Dogonchi AS. Analytical investigation of convective heat transfer of a longitudinal fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation. Int J Phys Sci 2014;9:466–74. [6] Dogonchi AS, Hatami M, Domairry G. Motion analysis of a spherical solid particle in plane Couette Newtonian fluid flow. Powder Technol 2015;274:186–92. [7] Dogonchi AS, Hatami M, Hosseinzadeh K, Domairry G. Non-spherical particles sedimentation in an incompressible Newtonian medium by Padé approximation. Powder Technol 2015;278:248–56. [8] Dogonchi AS, Ganji DD. Convection-radiation heat transfer study of moving fin with temperature dependent thermal conductivity, heat transfer coefficient and heat generation. Appl Therm Eng 2016;103:705–12.
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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ARTICLE IN PRESS
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A.S. Dogonchi, D.D. Ganji / Journal of the Taiwan Institute of Chemical Engineers 000 (2016) 1–13 [9] Rashidi MM, Erfani E. Analytical method for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating. Eng Comput 2012;29:562–79. [10] Sheikholeslami M, Ganji DD, Rashidi MM. Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model. J Magn Magn Mater 2016;416:164–73. [11] Abolbashari MH, Freidoonimehr N, Nazari F, Rashidi MM. Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano-fluid. Powder Technol 2014;267:256–67. [12] Abbasbandy S, Hayat T, Alsaedi A, Rashidi MM. Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid. Int J Numer Methods Heat Fluid Flow 2014;24:390–401. [13] Rashidi MM, Ali M, Freidoonimehr N, Rostami B, Hossain MA. Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation. Adv Mech Eng 2014;6:735939. [14] Sheikholeslami M, Ellahi R, Ashorynejad HR, Domairry G, Hayat T. Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. J Comput Theor Nanosci 2014;11:486–96. [15] Adomian G. Nonlinear stochastic operator equations. New York: Academic Press; 1986. [16] Sheikholeslami M, Ganji DD, Ashorynejad HR. Investigation of squeezing unsteady nanofluid flow using ADM. Powder Technol 2013;239:259–65. [17] Sheikholeslami M, Ganji DD, Ashorynejad HR, Rokni HB. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Appl Math Mech 2012;33:25–36. [18] Sheikholeslami M, Ganji DD. Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technol 2013;235:873–9. [19] Sheikholeslami M, Ashorynejad HR, Ganji DD, Kolahdooz A. Investigation of rotating MHD viscous flow and heat transfer between stretching and porous surfaces using analytical method. Math Probl Eng 2011;2011 Article ID: 258734, 17 pages. doi:10.1155/2011/258734. [20] Sheikholeslami M, Ashorynejad HR, Ganji DD, Yıldırım A. Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk. Sci Iran 2012;19:437–42. [21] M Sheikholeslami M, Ganji DD. Magnetohydrodynamic flow in a permeable channel filled with nanofluid. Sci Iran Trans B, Mech Eng 2014;21:203–12. [22] Barzegar Gerdroodbary M, Rahimi Takami M, Ganji DD. Investigation of thermal radiation on traditional Jeffery–Hamel flow to stretchable convergent/divergent channels. Case Stud Therm Eng 2015;6:28–39. [23] Turkyilmazoglu M. Extending the traditional Jeffery-Hamel flow to stretchable convergent/divergent channels. Comput Fluids 2014;100:196–203. [24] Rashidi MM, Vishnu Ganesh N, Abdul Hakeem AK, Ganga B. Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation. J Mol Liq 2014;198:234–8. [25] Valiallah Mousavi S, Sheikholeslami M, Bandpy MG, Barzegar Gerdroodbary M. The influence of magnetic field on heat transfer of magnetic nanofluid in a sinusoidal double pipe heat exchanger. Chem Eng Res Des 2016;113:112–24. [26] Kandelousi MS. Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Eur Phys J Plus 2014;129:1–12. [27] Sheikholeslami M, Rashidi MM, Hayat T, Ganji DD. Free convection of magnetic nanofluid considering MFD viscosity effect. J Mol Liq 2016;218:393–9. [28] Safarnia H, Sheikholeslami M, Ganji DD. Electrohydrodynamic nanofluid flow and forced convective heat transfer in a channel. Eur Phys J Plus 2016;131:1–14. [29] Sheikholeslami M, Mustafa MT, Ganji DD. Nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation. Iran J Sci Technol 2015;39:433–40. [30] Sheikholeslami M, Soleimani S, Ganji DD. Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. J Mol Liq 2016;213:153–61. [31] Sheikholeslami M, Rashidi MM. Non-uniform magnetic field effect on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects. J Braz Soc Mech Sci Eng 2016;38:1171–84. [32] Sheikholeslami M, Rashidi MM, Ganji DD. Numerical investigation of magnetic nanofluid forced convective heat transfer in existence of variable magnetic field using two phase model. J Mol Liq 2015;212:117–26. [33] Sheikholeslami M, Rashidi MM, Ganji DD. Effect of non-uniform magnetic field on forced convection heat transfer of Fe3O4-water nanofluid. Comput Methods Appl Mech Eng 2015;294:299–312. [34] Sheikholeslami M, Rashidi MM. Effect of space dependent magnetic field on free convection of Fe3O4-water nanofluid. J Taiwan Inst Chem Eng 2015;56:6–15. [35] Sheikholeslami M, Mustafa MT, Ganji DD. Effect of Lorentz forces on forced-convection nanofluid flow over a stretched surface. Particuology 2016;26:108–13. [36] Sheikholeslami M, Ganji DD, Rashidi MM. Ferrofluid flow and heat transfer in a semi annulus enclosure in the presence of magnetic source considering thermal radiation. J Taiwan Inst Chem Eng 2015;47:6–17. [37] Sheikholeslami M, Ganji DD. Entropy generation of nanofluid in presence of magnetic field using Lattice Boltzmann Method. Phys A: Stat Mech Appl 2015;417:273–86. [38] Sheikholeslami M, Ganji DD. Nanofluid convective heat transfer using semi analytical and numerical approaches: a review. J Taiwan Inst Chem Eng 2016;65:43–77. [39] Sheikholeslami M, Rashidi MM. Ferrofluid heat transfer treatment in the presence of variable magnetic field. Eur Phys J Plus 2015;130:1–12.
13
[40] Sheikholeslami M. Effect of uniform suction on nanofluid flow and heat transfer over a cylinder. J Braz Soc Mech Sci Eng 2015;37:1623–33. [41] Sheikholeslami M, Ganji DD. Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer. Energy 2014;75:400–10. [42] Pourmehran O, Rahimi-Gorji M, Ganji DD. Heat transfer and flow analysis of nanofluid flow induced by a stretching sheet in the presence of an external magnetic field. J Taiwan Inst Chem Eng 2016;65:162–71. [43] Rahimi-Gorji M, Pourmehran O, Hatami M, Ganji DD. Statistical optimization of microchannel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis. Eur Phys J Plus 2015;130:1–21. [44] Pourmehran O, Rahimi-Gorji M, Gorji-Bandpy M, Ganji DD. Analytical investigation of squeezing unsteady nanofluid flow between parallel plates by LSM and CM. Alexandria Eng J 2015;54:17–26. [45] Rahimi-Gorji M, Pourmehran O, Gorji-Bandpy M, Ganji DD. Unsteady squeezing nanofluid simulation and investigation of its effect on important heat transfer parameters in presence of magnetic field. J Taiwan Inst Chem Eng 2016;67:467–75. [46] Hashim I. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. J Comput Appl Math 2006;193:658–64. [47] Esmaili Q, Ramiar A, Alizadeh E, Ganji DD. An approximation of the analytical solution of the Jeffery–Hamel flow by decomposition method. Phys Lett A 2008;372:3434–9. [48] Sheikholeslami M, Ganji DD, Ashorynejad HR, Rokni HB. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Appl Math Mech-Engl Ed 2012;33:25–36. [49] Jin C, Liu M. A new modification of Adomian decomposition method for solving a kind of evolution equation. Appl Math Comput 2005;169:953–62. [50] Jafari H, Daftardar-Gejji V. Revised Adomian decomposition method for solving a system of nonlinear equations. Appl Math Comput 2006;175:1–7. [51] Duan J-S, Rach R. A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations.. Appl Math Comput 2011;218:4090–118. [52] Dogonchi AS, Ganji DD. Investigation of heat transfer for cooling turbine disks with a non-Newtonian fluid flow using DRA. Case Stud Therm Eng 2015;6:40–51. [53] Dogonchi AS, Ganji DD. Investigation of MHD nanofluid flow and heat transfer in a stretching/shrinking convergent/divergent channel considering thermal radiation. J Mol Liq 2016;220:592–603. [54] Dogonchi AS, Divsalar K, Ganji DD. Flow and heat transfer of MHD nanofluid between parallel plates in the presence of thermal radiation. Comput Methods Appl Mech Eng 2016;310:58–76. [55] Dib A, Haiahem A, Bou-said B. An analytical solution of the MHD Jeffery–Hamel flow by the modified Adomian decomposition method. Comput Fluids 2014;102:111–15. [56] Dib A, Haiahem A, Bou-said B. Approximate analytical solution of squeezing unsteady nanofluid flow. Powder Technol 2015;269:193–9. [57] Xuan Y, Li Q, Hu W. Aggregation structure and thermal conductivity of nanofluids. AIChE J 2003;49:1038–43. [58] Chon CH, Kihm KD, Lee SP, Choi SUS. Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement. Appl Phys Lett 2005;87:153107. [59] Koo J, Kleinstreuer C. A new thermal conductivity model for nanofluids. J Nanopart Res 2004;6:577–88. [60] Vafai K. A critical synthesis of thermophysical characteristics of nanofluids. Int J Heat Mass Tran 2011;54:4410–28. [61] Koo J, Kleinstreuer C. Viscous dissipation effects in micro tubes and micro channels. Int J Heat Mass Tran 2004;47:3159–69. [62] Koo J. Computational nanofluid flow and heat transfer analyses applied to microsystems Ph.D Thesis. Raleigh, NC, The United States: NC State University; 2004. [63] Li J. Computational analysis of nanofluid flow in micro channels with applications to micro-heat sinks and bio-MEMS PhD Thesis. Raleigh, NC, The United States: NC State University; 2008. [64] Koo J, Kleinstreuer C. Laminar nanofluid flow in microheat-sinks. Int J Heat Mass Tran 2005;48:2652–61. [65] Prasher RS, Bhattacharya P, Phelan PE. Thermal conductivity of nano scale colloidal solution. Phys Rev Lett 2005;94:025901–1-4. [66] Jang SP, Choi SUS. The role of Brownian motion in the enhanced thermal conductivity of nanofluids. Appl Phys Lett 2004;84:4316–18. [67] Pourmehran O, Rahimi-Gorji M, Hatami M, Sahebi SAR, Domairry G. Numerical optimization of microchannel heat sink (MCHS) performance cooled by KKL based nanofluids in saturated porous medium. J Taiwan Inst Chem E 2015;55:49–68. [68] Sheikholeslami Kandelousi M. KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel. Phys Lett A 2014;378:3331–9. [69] Sheikholeslami M, Ganji DD. Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM. Comput Methods Appl Mech Eng 2015;283:651–63. [70] Dogonchi AS, Ganji DD. Thermal radiation effect on the nano-fluid buoyancy flow and heat transfer over a stretching sheet considering Brownian motion. J Mol Liq 2016;223:521–7. [71] Adomian G, Rach R. Inversion of nonlinear stochastic operators. J Math Anal Appl 1983;91:39–46.
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Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion A.S. Dogonchi∗, D.D. Ganji Mechanical Engineering Department, Babol Noshirvani University of Technology, P.O. Box 484, Babol, Iran
a r t i c l e
i n f o
Article history: Received 16 June 2016 Revised 3 September 2016 Accepted 23 September 2016 Available online xxx Keywords: Nanofluid Brownian motion Stretchable/shrinkable walls Duan–Rach approach (DRA) KKL
a b s t r a c t In this paper, the nanofluid flow and heat transfer between non-parallel stretching walls with Brownian motion effect is investigated. The governing radial momentum and energy equations are solved by Duan–Rach Approach (DRA). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The effective viscosity and thermal conductivity of nanofluid are calculated via KKL (Koo–Kleinstreuer–Li) correlation in which influence of Brownian motion on the thermal conductivity is considered. The effects of various parameters such as the stretching/shrinking parameter, the radiation parameter, Reynolds number, the opening angle and the heat source parameter are investigated on the velocity and temperature. Also, the value of the Nusselt number is calculated and presented through figures. The results show that the fluid velocity, temperature profile and Nusselt number increase with the increasing of stretching parameter. The results also reveal that the temperature profile increases with the increasing of the heat source parameter and it decreases with the rising of radiation parameter for both divergent and convergent channel. In addition, the results were compared with the previous works and found proposed method has high accuracy to solve this nonlinear problem. © 2016 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1. Introduction The fluid flow and heat transfer between parallel/non-parallel plates have received notable consideration in recent years as they are most applicable in the mechanical engineering and industrial applications. One of the first studies of this type of flow can be traced back to Jeffery [1] and Hamel [2]. There are newly developed methods such as Collocation Method [3], Differential Transformation Method [4–10], Homotopy Analysis Method [11–14], Adomian Decomposition Method [15–17], Homotopy Perturbation Method [18–20] and Optimal Homotopy Analysis Method [21] to solve the equation of Jeffery–Hamel fluid flow and many nonlinear problems in the engineering field. The unsteady motion of a rigid spherical particle in a quiescent shear-thinning power-law fluid was studied by Rahimi-Gorji et al. [3]. Their results show that the time of reaching the terminal velocity in a falling procedure significantly increases with growing of the particle size. The falling of a spherical particle in a plane Couette Newtonian fluid flow was
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (A.S. Dogonchi), [email protected] (D.D. Ganji).
studied by Dogonchi et al. [6]. They applied the DTM-Padé to solve governing equations. They concluded that the horizontal and vertical velocities of spherical particles in water fluid are higher than the ethylene-glycol and glycerin fluids. Dogonchi et al. [7] studied the motion of a vertically falling non-spherical solid particle in an incompressible Newtonian fluid. They applied the DTM-Padé to solve governing equations. They observed that the velocity of the gold particle is higher than the aluminum and copper particles. The entropy generation for the MHD nanofluid flow past a stretching permeable surface was analyzed by Abolbashari et al. [11]. Their results show that the rising of the unsteadiness parameter, nanoparticle volume fraction parameter, magnetic parameter, Reynolds number, suction parameter, Hartmann number and Brinkman number lead to an increase of the entropy generation number. Rashidi and co-workers [12] studied the heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation effect. They found that the heat transfer to the fluid increases with the increasing value of the wedge angle. The effect of thermal radiation on the classical Jeffery–Hamel flow from a source or sink in convergent/divergent channels was studied by Barzegar Gerdroodbary et al. [22]. Their results show that the temperature profile increases with the rising of thermal radiation parameter. The classical Jeffery–Hamel flow from a source or sink in
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Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Nomenclature A1 , A2 , A3 , A4 C Ec Cp f
P Re Pr s qrad . T Tw T∞ r,θ u uc uw Hs N Nu Cf DRA ADM k k∗ Rf dp
constant parameter stretching/shrinking parameter Eckert number specific heat (J/kg K) dimensionless velocity dimensionless temperature pressure term Reynolds number Prandtl number stretching/shrinking rate (m2 /s) radiative heat flux temperature (K) surface temperature of channels (K) free stream temperature cylindrical coordinates velocity component in the radial direction (m/s) rate of movement in the radial direction (m2 /s) wall velocity component in the radial direction (m/s) heat source parameter radiation parameter Nusselt number skin fraction coefficient Duan–Rach approach Adomian decomposition method thermal conductivity mean absorption coefficient thermal interfacial resistance particle size
Greek symbols α angle of the channel η dimensionless angle σ electrical conductivity θ any angle ρ density μ dynamic viscosity ν kinematic viscosity φ solid volume fraction σ∗ Stefan–Boltzmann constant Subscripts nf nanofluid f base fluid p nano-solid-particles
convergent/divergent channels with stretchable walls was investigated by Turkyilmazoglu [23]. He observed that the temperature profile increases with the rising of stretching parameter for the convergent channel. The primary obstacle to enhance the heat transfer in engineering systems is the low thermal conductivity of customary fluids such as water, air, oil, and ethylene glycol mixture. Solid usually has a higher thermal conductivity than liquids. Addition of Nanoparticles to the conventional fluid, the so called “Nano-fluid”, can improve the thermal conductivity of the mixture. Recently, several authors studied nanofluid flow and heat transfer [24–45]. Rashidi et al. [24] studied heat transfer of nanofluid flow over a stretching sheet in the presence of transverse magnetic field considering thermal radiation and buoyancy effects. They concluded that the increasing of buoyancy parameter increases the velocity profile and decreases the temperature profile of the nanofluid. The
effect of magnetic field on the nanofluid flow inside a sinusoidal two-tube heat exchanger was investigated numerically by Valiallah Mousavi et al. [25]. They applied finite volume method to solve governing equations. Their results show that the sinusoidal formation of the internal tube significantly increases the Nusselt number inside a two-tube heat exchanger. Ferrofluid flow and heat transfer in the presence of an external variable magnetic field were studied by Kandelousi [26]. His results indicate that the Nusselt number is an increasing function of Magnetic number, Rayleigh number and nanoparticle volume fraction while it is a decreasing function of the Hartmann number. The influence of magnetic field dependent (MFD) viscosity on free convection heat transfer of nanofluid in an enclosure was studied by Sheikholeslami et al. [27]. They applied Control Volume based Finite Element Method to solve governing equations. They observed that the Nusselt number is an increasing function of Rayleigh number and volume fraction of nanoparticle while it is a decreasing function of viscosity parameter and Hartmann number. The effect of an electric field on Fe3 O4 –water nanofluid flow and heat transfer in a channel was studied by Safarnia et al. [28]. They found that the Nusselt number has a direct relationship with the Reynolds number and voltage supply. Nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation was investigated by Sheikholeslami et al. [29]. They proved that skin friction coefficient increases with increase of Reynolds number and suction parameter, but it decreases with increase of the nanoparticle volume fraction. Effect of electric field on the hydrothermal behavior of nanofluid in a complex geometry was studied by Sheikholeslami et al. [30]. Their results show that supplied voltage can change the flow shape. Impact of non-uniform magnetic field on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects was investigated by Sheikholeslami and Rashidi [31]. They reported that the Nusselt number has a direct relationship with Rayleigh number, buoyancy ratio number and Lewis number while it has a reverse relationship with Hartmann number. The effect of non-uniform magnetic field on nanofluid forced convection heat transfer in a lid driven semi-annulus considering Brownian motion was studied by Sheikholeslami et al. [32]. Force convection heat transfer in a lid driven semi annulus enclosure in presence of non-uniform magnetic field was studied by Sheikholeslami et al. [33]. They applied control volume based finite element method (CVFEM) to solve the governing equations. The effect of spatially variable magnetic field on ferrofluid flow and heat transfer was investigated by Sheikholeslami and Rashidi [34]. They applied control volume based finite element method (CVFEM) to solve the governing equations. Their results indicate that the Nusselt number is an increasing function of Magnetic number, Rayleigh number and nanoparticle volume fraction, while it is a decreasing function of Hartmann number. Effect of Lorentz forces on forced-convection nanofluid flow over a stretched surface was studied by Sheikholeslami et al. [35]. Their results illustrate that the coefficient of skin friction enhances with enhancing magnetic parameter while reduces with enhancing velocity ratio parameter. Ferrofluid flow in a semi annulus enclosure considering radiative heat transfer was studied by Sheikholeslami et al. [36]. Their results show that the Nusselt number is a rising function of Rayleigh number and Magnetic number while it is a decreasing function of radiation parameter. MHD convective nanofluid flow over a vertical stretching sheet considering the thermal radiation and buoyancy effects was studied by Pourmehran et al. [42]. Their results indicate that the reduced Nusselt number has an inverse relation with increasing the nanoparticle concentrations. An analytical investigation of the heat transfer for the microchannel heat sink (MCHS) cooled by different nanofluids (Cu, Al2 O3 , Ag, TiO2 in water and ethylene glycol as base fluids) was studied by Rahimi-Gorji et al. [43]. They reported that by increasing the nanoparticles volume fraction, the Brownian
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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movement of the particles, which carries the heat and distributes it to the surroundings, increases and, consequently, the difference between the coolant and wall temperature becomes less. The unsteady flow of a nanofluid squeezing between two parallel plates was investigated by Pourmehran et al. [44]. Their results demonstrated that when the two plates move together, the Nusselt number has a direct relationship with nanoparticle volume fraction and Eckert number while it has a reverse relationship with the squeeze number. Many of engineering problems, especially some of the heat transfer equations are nonlinear. As mentioned before, some of these non-linear equations can be solved using analytical methods such as Adomian Decomposition Method, Differential Transformation Method and Homotopy Perturbation Method. One of the analytic methods which does not need discretization or linearization is the Adomian Decomposition Method (ADM). The benefits of this method are that it can provide an approximate solution for many non-linear equations without perturbation, linearization, closure approximation, or discretization methods. Also, ADM results are more realistic because it gives an approximate solution of the problem without any simplification. Adomian Decomposition Method (ADM) was proposed by Adomian in 1986 [15]. Hashim [46] applied the Adomian Decomposition Method for solving boundary value problems for fourth-order integro-differential equations and the Blasius equation. Esmaili et al. [47] used the ADM to solve the governing equation of classical Jeffery–Hamel fluid flow. Comparison of the results illustrates that the analytical and numerical methods are in an excellent agreement with each other. The effect of nanoparticle and magnetic field on the Jeffery–Hamel fluid flow was investigated by Sheikholeslami et al. [48]. They used the Adomian Decomposition Method (ADM) to solve governing equations. They reported that when the Hartmann number increases, backflow will reduce. Many authors have tried to modify the ADM. Jin and Liu [49] modified ADM for solving a kind of evolution equation. Jafari and Daftardar-Gejji [50] modified the standard ADM for solving a system of nonlinear equations. All these methods need to find the unknown initial values of the problem that the final solution depends on the accuracy of the initial values determined by numerical method. Duan and Rach [51] have introduced a new modification of the Adomian Decomposition Method to solve many nonlinear boundary value problems (BVP). This modified method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients and in fact the final solution does not contain undetermined coefficients. The non-Newtonian fluid flow in a channel with a porous wall was studied by Dogonchi and Ganji [52]. They applied this modified method (DRA) to solve governing equations. They proved that the Nusselt number has a direct relationship with Prandtl number, Reynolds number and power law index. The heat transfer of a steady, viscous incompressible water based MHD nanofluid flow between two stretchable walls with thermal radiation effect was investigated by Dogonchi and Ganji [53]. Their results indicate that the fluid velocity and temperature profile increase with the increasing of stretching parameter. The unsteady squeezing flow and heat transfer of MHD nanofluid between the infinite parallel plates with thermal radiation effect was studied by Dogonchi et al. [54]. They used the DRA to solve governing equations. Their results show that the temperature profile and Nusselt number increase with the increasing of radiation parameter. This analytical method was successfully applied to different engineering problems [55,56]. In this paper, we have applied this modified method to solve the equation of nanofluid flow and heat transfer between nonparallel stretchable walls in the presence of thermal radiation and heat source using Koo-Kleinstreuer-Li (KKL) model. In this model the impact of Brownian motion on the effective thermal conductivity and the effective viscosity is considered. The effects of the heat
3
Fig. 1. Schematic view of problem.
source parameter, the stretching/shrinking parameter, the radiation parameter and Reynolds number on the flow and heat transfer profiles are investigated. 2. Problem description The steady flow of an incompressible conductive viscous fluid due to a source or sink between two stretchable non-parallel walls that meet at an angle 2α is considered (Fig. 1). The walls are assumed to radially stretch or shrink in accordance with [23,53]:
u = uw =
s r
(1)
with s being the stretching/shrinking rate. The walls are considered to be convergent if α < 0 and divergent if α > 0. We suppose that the velocity is purely radial and depends on r and θ and moreover there is no magnetic field in the z-direction. The reduced forms of continuity, Navier–Stokes and energy equations in polar coordinates are [23,48,53]:
ρn f ∂ (ru ) = 0, r ∂r
(2)
μn f 1 ∂P ∂u u =− + ∂r ρn f ∂ r ρn f
∂ 2u 1 ∂ u 1 ∂ 2u u + + − , ∂ r2 r ∂ r r2 ∂ θ 2 r2
2μnf ∂ u 1 ∂P − = 0. ρnf r ∂θ ρnf r2 ∂θ 2 ∂T knf ∂ T 1 ∂ T 1 ∂ 2T u = + + ∂r r ∂r r2 ∂ θ 2 (ρCp )nf ∂ r2 1 1 ∂ ∂ − r qr,rad. − qθ ,rad. 2 (ρCp )nf r ∂ r (ρCp )nf r ∂θ 2 2 ∂ u u 2 ∂r + r μnf Q0 + + T 2 ρ C ρ ( p )nf + 1 ∂ u ( Cp )nf
(3) (4)
(5)
r ∂θ
where u = u(r, θ ) is the velocity, T is the temperature of the fluid, P is the fluid pressure and qrad is the radiative heat flux. Using Rosseland approximation for radiation (see Rashidi et al. [24]) we have
∂T4 ∂r
∂T4 ∂θ
qr,rad. = − 4σ ∗ /3k∗nf
qθ ,rad. = − 4σ ∗ /3k∗nf
(6) (7)
Here σ ∗ is the Stefan–Boltzmann constant and is nanofluid mean absorption coefficient. Further we assume that temperature difference within the flow is such that T4 may be panded in a Taylor series. Hence, expanding T4 about T∞ and glecting higher order terms we get 3 4 T4 ∼ T − 3T∞ = 4T∞
k∗n f
the the exne-
(8)
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u
Therefore Eq. (5) is simplified to
∂T ∂r
=
Table 1 The coefficient values of CuO–water and Al2 O3 –water nanofluids [70].
∂ 2T 1 ∂ T 1 ∂ 2T + + ∂ r2 r ∂ r r2 ∂ θ 2 2 2 3 3 16σ ∗ T∞ 16σ ∗ T∞ ∂ T 1 ∂T ∂ T + ∗ + + 3kn f (ρC p )n f ∂ r 2 r ∂ r 3k∗n f (ρC p )n f r 2 ∂ 2 θ 2 2 2 ∂∂ur + ur μn f Q0 + + T (9) ( ρ C p ) n f + 1 ∂ u 2 (ρCp )n f kn f (ρCp )n f
r ∂θ
where ρ nf is the effective density of the nanofluid and (ρ Cp )nf is the heat capacitance of the nanofluid given as:
ρn f = (1 − φ )ρ f + φρ p , (ρCp )n f = (1 − φ )(ρCp ) f + φ (ρCp ) p .
(10)
The used effective thermal conductivity by some authors in their studies [16–19,53,54] to simulate nanofluids depends only on the thermal conductivity of the base fluid and particles and on the volume fraction of particles whereas numerous studies [57– 60] show that the temperature, the particle size, random motion and the kind of nanoparticle can also affect the effective thermal conductivity of a nanofluid. Hence, in this paper the effective viscosity and thermal conductivity of nanofluid are calculated via KKL (Koo-Kleinstreuer-Li) [61–64] correlation in which impacts of particle size, particle volume fraction and temperature dependence as well as types of particle and base fluid combinations on the thermal conductivity are considered.
ke f f = kstatic + kBrownian
(11)
μe f f = μstatic + μBrownian = μstatic + in which
μstatic =
(12)
kp kf
kp kf
+2 −
−1
μf (1 − φ )2.5
kBrownian = 5 × 10
4
φρ f c p, f
52.81348876 6.115637295 0.695574508 0.041745555 0.1769193 –298.1981908 –34.53271691 –3.922528928 –0.235432963 –0.999063481
∂u ∂T uc =0= ,u = ∂θ ∂θ r
(18)
and due to the stretching/shrinking convergent/divergent wall condition at the plates (θ = ±α )
s Tw u = uw = , T = 2 r r
(19)
with uc being the centerline rate of movement and Tw is the constant wall temperature. We should remark that in general conditions where asymmetry is also allowed, infinitely many solutions exist [23]. To retain the symmetry with respect to the centerline, we impose the same stretching/shrinking rates on both walls. Considering only radial flow, the Eq. (1) implies that
f (θ ) = ru(r, θ )
(20)
f (θ ) T , (η ) = r 2 where fmax Tw
η=
θ α
(21)
(13)
Substituting these into the governing equations and eliminating the pressure term yields the nonlinear third order ordinary differential equation for the flow from the radial momentum equation:
(14)
f (η ) + 2α Re
A1 f ( η ) f ( η ) + 4α 2 f ( η ) = 0 A2
(22)
And second order differential equation for the heat
kb T g (T , φ , d p ) ρpd p
(15)
It should be noted that by introducing a thermal interfacial resistance [65,66] (Rf = 4 × 10−8 km2 /W) the kp was replaced by a new kp,eff in the form:
Rf +
Al2 O3 –water
–26.59331085 –0.403818333 –33.3516805 –1.915825591 6.42185846658E–02 48.40336955 –9.787756683 190.24561 10.92853866 –0.720099837
The governing equations are accompanied with the boundary conditions, because of the symmetry assumption at the channel centerline (θ = 0)
f (η ) =
φ
CuO–water
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
using dimensionless parameters
kstatic =1+ kf
k 3 kp − 1 φ f
μf kBrownian × kf Pr f
Coefficient values
(1 + N ) (η ) + 2α 2 +
A3 Hs P r f ( η ) + 2 + 2N + (η ) A4 2A4
A2 EcP r 2 4α f (η )2 + f (η )2 = 0 A4 Re
(23)
Here A1 , A2 , A3 and A4 are constants given by:
dp dp = kp k p,e f f
(16)
The empirical gʹ-function for various nanoparticles and base fluids is different. In this paper for water based nanofluid includes Al2 O3 and CuO (as nanoparticles) in its structure, this function follows the format [61]:
g (T , φ , d p ) = (a1 + a2 ln (d p ) + a3 ln (φ ) + a4 ln (φ ) ln (d p ) +a5 ln (d p ) ) ln (T ) + (a6 + a7 ln (d p ) + a8 ln (φ ) 2
+a9 ln (φ ) ln (d p ) + a10 ln (d p ) ) 2
(17)
in which, the coefficients ai (i = 0…10) are based on the type of nanoparticles and furthermore with these coefficients, Al2 O3 – water nanofluids and CuO–water nanofluids have an R2 of 96% and 98%, respectively [63,67–70] (See Table 1).
A1 =
( ρ C p )n f ρn f μn f ke f f , A2 = , A3 = , A4 = ρf μf kf (ρ C p ) f
(24)
Subject to the boundary conditions
f (0 ) = 1, f (0 ) = 0, f (1 ) = C, (0 ) = 0, (1 ) = 1
(25)
where C = s/uc is the stretching (C > 0) or shrinking (C < 0) parameter, Re = α uc /ν f is the Reynolds number, Pr = uc ρC p /k f is the Prandtl number, Ec = α u2c /C p f Tw is the Eckert number, N = 3 /3k∗ k 16σ ∗ T∞ is the radiation parameter and Hs = r2 Q0 /kf is the nf nf heat source parameter. It is noted that setting C to zero at this stage leads to stationary wall condition for the traditional Jeffery–Hamel flow. The elliptic function solutions as given in the literature may be fulfilled here for nonzero C, but, it is thought that this is not very convenient.
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Fig. 2. Velocity (f) and temperature () profiles for different values of stretching/shrinking parameter (C) in divergent/convergent channel (CuO–water).
Physical quantities of interest are the skin fraction coefficient and Nusselt number which are defined as:
C ∗f
μn f = ρ f u2c
N u∗ =
r k f Tw
1 ∂u r ∂θ
kn f +
θ =α
σ
3 16 ∗ T∞ ∗ 3kn f
3.1. Description of the Duan–Rach approach (DRA)
1 ∂ T r ∂θ θ =α
Consider a third-order non-linear differential equation:
(26)
Lu = Nu + g(x ),
(28)
subject to the mixed set of Dirichlet and Neumann boundary conditions
After simplification, we obtain:
μn f f 1 ( ) μf kn f ( 1 + N ) Nu = (1 ) kf α
3. Applied method
u(x1 ) = α0 , u (x1 ) = α1 , u (x2 ) = α2 , x2 = x1
Cf =
(29)
3
(27)
where L = ddx3 is the linear differential operator to be inverted, Nu is an analytic nonlinear operator, and g(x) is the system input.
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Fig. 3. Velocity (f) and temperature () profiles for different values of Reynolds number (Re) in stretching/shrinking divergent/convergent channel (CuO–water).
We differentiate Eq. (31), then let x = x2 and solve for u (ξ ), hence
We take the inverse linear operator as
L−1 (· ) =
x
x0
x
x1
ξ
x
(· )dxdxdx,
(30)
where ξ is a prescribed value in the specified interval. Thus we have:
L−1 Lu = u(x ) − u(x0 ) − u (x1 )(x − x0 ) 1 − u (ξ )[(x − x1 )2 − (x0 − x1 )2 ] 2
1 u ( x2 ) − u ( x1 ) − x2 − x1 x2 − x1
(31)
(32)
x2
x1
ξ
x
[Nu + g]dxdx.
(33)
Substituting Eq. (33) into Eq. (32) yields,
u(x ) = u(x0 ) + u (x1 )(x − x0 ) +
Applying the inverse operator L − 1 to both sides of Eq. (28) yields:
L−1 [Nu + g] = u(x ) − u(x0 ) − u (x1 )(x − x0 ) 1 − u (ξ )[(x − x1 )2 − (x0 − x1 )2 ] 2
u (ξ ) =
1 (x − x1 )2 − (x0 − x1 )2 2
u ( x2 ) − u ( x1 ) + L−1 g + L−1 Nu x2 − x1 2 2 x2 x 1 ( x − x1 ) − ( x0 − x1 ) − gdxdx 2 x2 − x1 x1 ξ 2 2 x2 x 1 ( x − x1 ) − ( x0 − x1 ) − Nudxdx. 2 x2 − x1 x1 ξ
(34)
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Fig. 4. Velocity (f) and temperature () profiles for different values of opening angle parameter (α ) in stretching/shrinking divergent/convergent channel (CuO–water).
Thus in Eq. (34) the three known boundary values u(x0 ), u (x1 ) and u (x2 ) are included and the undetermined coefficient was replaced. Next, the solution is decomposed and the nonlinearity ∞ u (x ) = ∞ m=0 um (x ), Nu (x ) = m=0 Am (x ) where Am (x) = Am (u0 (x), u1 (x), … , um (x)) are the Adomian polynomials. From Eq. (33), the solution components are determined by the modified recursion scheme:
u0 = u(x0 ) + u (x1 )(x − x0 ) + u ( x2 ) − u ( x1 ) + L−1 g x2 − x1
1 ( x − x1 ) − ( x0 − x1 ) 2 x2 − x1 2
−
1 (x − x1 )2 − (x0 − x1 )2 2
2
x2 x1
ξ
um+1 = L−1 Am −
gdxdx,
(35)
2
2
x2 x1
x
ξ
Am dxdx.
(36)
3.2. Implementation of the method In our study, the Duan–Rach Approach must be modified. We do not use the prescribed value ξ . According to Eq. (28), Eqs. (22) and (23) can be written as follows:
A1 f ( η ) f ( η ) − 4α 2 f ( η ) A2
(37)
1 2α 2 A3 Pr f (η ) + 2 + 2N+ Hs (η ) A4 2A4 2 L2 (η ) = − N+1 + AA24 EcPr 4α f (η )2 + f (η )2 Re
(38)
L3 f (η ) = −2α Re x
1 ( x − x1 ) − ( x0 − x1 ) 2 x2 − x1
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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where N1 u and N2 u are introduced as:
N1 u = −2α Re
A1 f ( η ) f ( η ) − 4α 2 f ( η ) A2
(42)
1 2α 2 A3 Pr f (η ) + 2 + 2N + Hs (η ) A4 2A4 N2 u = − N+1 + AA24 EcRePr 4α 2 f (η )2 + f (η )2
(43)
Evidently, we do not have the values of f (0) and (0). In the standard Adomian Decomposition Method (ADM), we need to evaluate those unknown conditions with numerical methods. Hence, the boundary value problem (BVP) is turned into an initial value problem (IVP). The accuracy of the solution depends on the accuracy of the two unknown parameters. In our study, we use the Duan–Rach Approach [51] to find a totally analytical solution. By putting η = 1 in Eq. (40), we have:
f (0 ) = 2C − 2 where
L−1 3 N1 u
η=1
=
1 0
L−1 3 N1 u
η=1
η η 0
+1
(44)
(Nu )dηdηdη
0
(45)
Substituting Eq. (44) into Eq. (40) yields,
2 −1 f (η ) = 1 − η2 + C η2 + L−1 3 N1 u − η L3 N1 u
(46)
η=1
Thus the right hand side of Eq. (46) does not contain the undetermined parameter f (0). Finally, we have the modified recursive scheme:
f 0 (η ) = 1 − η 2 + C η 2
2 −1 fm+1 (η ) = L−1 3 Am ( η ) − η L3 Am ( η )
(47)
η=1
where Am (η) are the Adomian polynomials, which can be determined by the formula:
m 1 dm Am ( η ) = N λi Fi (η ) m! dλm
(48)
i=0
λ=0
that was first published by Adomian and Rach [71]. Applying Eq. (48), we obtain the terms of the Adomian polynomials and put them in Eq. (47), and we determine fm (η) as follows:
f 0 (η ) = 1 − η 2 + C η 2 1 (C − 1 )αη2 f 1 (η ) = − 30 A2 Fig. 5. Velocity profile (f) for different values of nanoparticles volume fraction (CuO–water).
3
where the differential operator L3 and L2 are given by L3 = ddη3 and 2 L2 = ddη2 respectively. Assume that the inverse operator L−1 and 3 L−1 exist, then we have: 2
L−1 3 (· ) =
η η η 0
0
0
(· )dηdηdη, L−1 2 (· ) =
η η 0
0
( · )d η d η
(39)
Operating with L−1 in Eq. (37) and after exerting boundary con3 ditions on it:
η f (η ) = f (0 ) + f (0 )η + f (0 ) + L−1 3 (N1 u ). 2 2
(40)
−10A2 α − 4A1 Re − A1C Re +10A2 αη2 + 5A1 Reη2 −A1 Reη4 + A1C Reη4
The functions f2 (η), f3 (η), … can be determined in a similar way from Eq. (47). For convenience, we do not represent all terms of fm (η). Using f (η ) = ∞ m=0 f m (η ) = f 0 (η ) + f 1 (η ) + f 2 (η ) + . . . , thus
f (η ) = 1 − η 2 + C η 2 −
1 (C − 1 )αη2 30 A2
−10A2 α − 4A1 Re − A1C Re +10A2 αη2 + 5A1 Reη2 −A1 Reη4 + A1C Reη4
+ −···
(50)
According to Eq. (50), the accuracy increases by increasing the number of solution terms (m). For (η), we proceed in the same manner. We get the following recursive scheme:
Operating with L−1 in Eq. (38) and after exerting boundary con2 ditions on it:
0 ( η ) = 1
−1 m+1 (η ) = − L−1 2 N2 u η=1 + L2 N2 u
(η ) = (0 ) + (0 )η + L−1 2 (N2 u ).
where [L−1 N2 u]η=1 = 2
(41)
(49)
1η 0
0
(51)
(N2 u )dηdη
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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9
Fig. 6. Temperature profile () for different values of heat source parameter (Hs) and radiation parameter (N) (CuO–water).
(η ) =
Using thus
∞
m=0
m ( η ) = 0 ( η ) + 1 ( η ) + 2 ( η ) + . . . ,
1 1 (η ) = 1 − 30(N + 1 ) A4 Re
−4A2 Ec Pr C 2 α 2 − 10A2 Ec Pr C 2 + − · · · + − · · ·
(52)
Evidently, the temperature field (η) depends on the number of terms of the velocity field f(η) found in Eq. (50).
4. Results and discussion The nanofluid flow and heat transfer between non-parallel stretching walls with Brownian motion effect is studied analytically by using the modified standard ADM called Duan–Rach Approach (DRA). The Prandtl number is fixed at 6.2. The effects of different parameters such as Reynolds number (Re), the heat source parameter (Hs), the radiation parameter (N), the stretching/shrinking parameter (C), etc. on the velocity and temperature profiles are investigated. The nanofluid thermo-physical properties have been
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Fig. 7. Velocity (f) and Temperature () profiles for different nanofluids structure.
Table 2 Thermo-physical properties of water and nanoparticles [69].
CuO Al2 O3 Pure water
ρ (kg/m3 )
Cp (J/kgK)
k(W/mK)
dp(nm)
6500 3970 997.1
540 765 4179
18 25 0.613
29 38.4 –
abridged in Table 2. To verify the analytical results, we compared our results with the previous works [23]. As seen in Tables 3–6, they are in an excellent agreement. In these tables, an error is in-
troduced as follows:
DRA Results − Numerical Results × 100. %Error = Numerical Results The effect of the stretching/shrinking parameter on the velocity and temperature profiles in convergent/divergent channel is shown in Fig. 2. It is important to note that the stretching/shrinking parameter (C) describes the movement of the walls (C > 0 corresponds to the walls are stretching, while C < 0 corresponds to the walls are shrinking). It is clear that with the increasing of stretching parameter, the fluid flow moves faster near the walls. Physically speaking, when C = 1 the fluid flow in the channel moves at
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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Table 3 Comparison between DRA and numerical results for F (1) when α = −5◦ , Re = 50 for different values of stretching/shrinking parameter.
C
DRA results ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–2 –1 0 1 2
–5.130921689 –4.652183982 –2.833915413 0 3.654305033
–5.1309222926 –4.6521591354 –2.8339514330 0 3.6697111853
0.0 0 0 01177 0.0 0 053412 0.00127102 – 0.41981919
Table 4 Comparison between DRA and numerical results for F (1) when α = 5◦ , Re = 50 for different values of stretching/shrinking parameter.
C
DRA results ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–1.5 –1 0 1 1.5
–5.082942399 –3.508090102 –1.109360533 0 –0.1423152590
–5.0829256568 –3.5081031667 –1.1093265266 0 –0.1464298835
0.0 0 032941 0.0 0 037242 0.0030655 – 2.80996228
Table 5 Comparison between DRA and numerical results for − (1)when α = −5◦ , Re = 50, Pr = 1, Ec = 0 for different values of stretching/shrinking parameter.
C
DRA results the absence of N, Hs and ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–2 –1 0 1 2
0.03157845854 0.03734696604 0.04214811723 0.04640127099 0.05052578617
0.0315761821 0.0373226368 0.0421517243 0.0464015106 0.0502423154
0.0072094 0.06518632 0.00855734 0.0 0 051638 0.5642072
Table 6 Comparison between DRA and numerical results for − (1)when α = 5◦ , Re = 50, Pr = 1, Ec = 0 for different values of stretching/shrinking parameter.
C
DRA results the absence of N, Hs and ϕ = 0% (Present)
Numerical results (Turkyilmazoglu) [23]
Error (%)
–1.5 –1 0 1 1.5
0.03244718254 0.03475109986 0.03998210836 0.04640127099 0.05019038094
0.0324575970 0.0347758169 0.0399820121 0.0464015106 0.0504570638
0.03208635 0.07107537 0.0 0 02408 0.0 0 051638 0.52853424
the same speed with the centerline velocity (uc ) while when C > 1 the fluid flow in the channel moves faster than the centerline velocity. This means that the fluid velocity increases near the wall with the increasing of stretching parameter. Also, it is observed that stretching warm up the system by increasing the thickness of thermal layer while shrinking cools down the system by thinning of the thermal boundary layer. Hence, where the cooling is necessary, wall shrinking is advisable. The variation of the velocity and temperature profiles for different values of the Reynolds number (Re) is shown in Fig. 3. It is illustrated that the rise in Re make a decrease in both velocity and temperature profiles for stretching/shrinking divergent channels as well as for higher values of Re the flow moves reversely and the probability of the backflow phenomenon, especially in the stretching case, increases, so if we will not to observe the backflow phenomenon, in the divergent channel, it is better to use shrinking case. Another notable point about Fig. 3 is that for stretch-
Fig. 8. Nusselt number (Nu) for different values of radiation parameter (N) and stretching/shrinking parameter (C) in (a) divergent /(b) convergent channel (CuO– water).
ing/shrinking convergent channels, the results were completely reversed and also the variation of velocity and temperature profiles is more gradual than when Re is at a higher value. The effect of the opening angle (α ) on the velocity and temperature profiles in stretching/shrinking divergent/convergent channel is shown in Fig. 4. In the stretching/shrinking convergent channel, the fluid velocity becomes flat and thickness of boundary layer reduces. In the stretching/shrinking divergent channel, as the opening angle (α ) rises, the magnitude of fluid velocity decreases. Hence, by rising the opening angle in stretching/shrinking divergent channel, the probability of the backflow phenomenon, especially in the stretching case, rises. Also, it can be seen that the temperature profile increases with the increasing of the absolute
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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variation of the heat source parameter can be viewed in Fig. 6. But since, the presence of radiation parameter leads to thinning of the thermal layer consequently, the rising value of radiation parameter leads to a decrease the temperature profile of the nanofluid. Fig. 7 shows the variation of the velocity and temperature profiles for two different structures of nanofluid (See Table 2). It can be concluded that when water-based nanofluid includes Al2 O3 (as nanoparticles) in its structure, f and values are greater than the other structure. However, this is true for the velocity profile in a divergent channel. For convergent channel this behavior of nanofluids structure is completely vice versa. Also, it is observed that in the stretching divergent channel the probability of the backflow phenomenon for CuO–water nanofluid is more than waterbased nanofluid includes Al2 O3 (as nanoparticles) in its structure. Fig. 8 displays the effect of the stretching/shrinking parameter (C) and the radiation parameter (N) on the Nusselt number (Nu). It is observed that the magnitude of the Nusselt number increases with the increasing of stretching parameter and radiation parameter, but this treatment of Nusselt number is completely vice versa for shrinking parameter. Finally, Fig. 9 shows the effect of the heat source parameter (Hs) and the Reynolds number (Re) on the Nusselt number (Nu) in stretching divergent/convergent channel. It is observed that the magnitude of the Nusselt number increases with the increasing of the heat source parameter. While, the opposite behavior is observed in a variation of the Nusselt number with Reynolds number in convergent/divergent channel. As already noted, Tables 3–6 confirm that the approach used has high accuracy for different values of the effective parameters. In the ADM, for given Re, Hs, α , N and φ , we must solve f (1) = 0 and (0) = 0 to find f (0) and (0). If we change the values of Re, we must again evaluate the values of f (0) and (0). In our work (Duan–Rach Approach), the obtained solutions for f(η) (Eq. (50)) and (η) (Eq. (52)) are purely analytical approach and we do not need any other calculations if we change any of the parameters of the problem and the final solution does not contain undetermined coefficients. 5. Conclusions In this paper, the nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion is investigated analytically using Duan–Rach Approach (DRA). The comparison between DRA and numerical results confirms the validity of this approach. The results indicate that with the increasing of stretching parameter, the fluid flow moves faster near the walls. Also, by increasing of stretching parameter the system becomes warmer. On the other hand, the temperature profile and Nusselt number increase with the increasing of heat source parameter. Fig. 9. Nusselt number (Nu) for different values of heat source parameter (Hs) and Reynolds number (Re) in stretching (a) divergent/(b) convergent channel (CuO– water).
value of the opening angle for both stretching/shrinking divergent/convergent channels. The effect of the nanoparticles volume fraction (φ ) on the velocity profile, when the other parameters are constant, is shown in Fig. 5. It can be concluded that rising nanoparticles volume fraction in divergent channel causes a decrease in velocity profile and the backflow phenomenon, especially in the stretching case, may be initiated at high Reynolds numbers. Fig. 6 shows the effect of the heat source (Hs) and radiation (N) parameters on the temperature profile. Due to the heat source parameter causes heat generation in the system, so it is expected that the temperature profile increases with the increasing of the heat source parameter. This treatment of the temperature profile with a
References [1] Jeffery GB. The two-dimensional steady motion of a viscous fluid. Philos Mag Ser 1915;29(6)):455–65. [2] Hamel G. Spiralförmige Bewgungen Zäher Flüssigkeiten Jahresbericht der. Deutschen Mathematiker-Vereinigung 1916;25:34–60. [3] Rahimi-Gorji M, Pourmehran O, Gorji-Bandpy M, Ganji DD. An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by collocation method. Ain Shams Eng J 2015;6:531–40. [4] Zhou JK. Differential transformation method and its application for electrical circuits. Wuhan, China: Hauzhang University Press; 1986. [5] Ganji DD, Dogonchi AS. Analytical investigation of convective heat transfer of a longitudinal fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation. Int J Phys Sci 2014;9:466–74. [6] Dogonchi AS, Hatami M, Domairry G. Motion analysis of a spherical solid particle in plane Couette Newtonian fluid flow. Powder Technol 2015;274:186–92. [7] Dogonchi AS, Hatami M, Hosseinzadeh K, Domairry G. Non-spherical particles sedimentation in an incompressible Newtonian medium by Padé approximation. Powder Technol 2015;278:248–56. [8] Dogonchi AS, Ganji DD. Convection-radiation heat transfer study of moving fin with temperature dependent thermal conductivity, heat transfer coefficient and heat generation. Appl Therm Eng 2016;103:705–12.
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029
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A.S. Dogonchi, D.D. Ganji / Journal of the Taiwan Institute of Chemical Engineers 000 (2016) 1–13 [9] Rashidi MM, Erfani E. Analytical method for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating. Eng Comput 2012;29:562–79. [10] Sheikholeslami M, Ganji DD, Rashidi MM. Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model. J Magn Magn Mater 2016;416:164–73. [11] Abolbashari MH, Freidoonimehr N, Nazari F, Rashidi MM. Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano-fluid. Powder Technol 2014;267:256–67. [12] Abbasbandy S, Hayat T, Alsaedi A, Rashidi MM. Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid. Int J Numer Methods Heat Fluid Flow 2014;24:390–401. [13] Rashidi MM, Ali M, Freidoonimehr N, Rostami B, Hossain MA. Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation. Adv Mech Eng 2014;6:735939. [14] Sheikholeslami M, Ellahi R, Ashorynejad HR, Domairry G, Hayat T. Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. J Comput Theor Nanosci 2014;11:486–96. [15] Adomian G. Nonlinear stochastic operator equations. New York: Academic Press; 1986. [16] Sheikholeslami M, Ganji DD, Ashorynejad HR. Investigation of squeezing unsteady nanofluid flow using ADM. Powder Technol 2013;239:259–65. [17] Sheikholeslami M, Ganji DD, Ashorynejad HR, Rokni HB. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Appl Math Mech 2012;33:25–36. [18] Sheikholeslami M, Ganji DD. Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technol 2013;235:873–9. [19] Sheikholeslami M, Ashorynejad HR, Ganji DD, Kolahdooz A. Investigation of rotating MHD viscous flow and heat transfer between stretching and porous surfaces using analytical method. Math Probl Eng 2011;2011 Article ID: 258734, 17 pages. doi:10.1155/2011/258734. [20] Sheikholeslami M, Ashorynejad HR, Ganji DD, Yıldırım A. Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk. Sci Iran 2012;19:437–42. [21] M Sheikholeslami M, Ganji DD. Magnetohydrodynamic flow in a permeable channel filled with nanofluid. Sci Iran Trans B, Mech Eng 2014;21:203–12. [22] Barzegar Gerdroodbary M, Rahimi Takami M, Ganji DD. Investigation of thermal radiation on traditional Jeffery–Hamel flow to stretchable convergent/divergent channels. Case Stud Therm Eng 2015;6:28–39. [23] Turkyilmazoglu M. Extending the traditional Jeffery-Hamel flow to stretchable convergent/divergent channels. Comput Fluids 2014;100:196–203. [24] Rashidi MM, Vishnu Ganesh N, Abdul Hakeem AK, Ganga B. Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation. J Mol Liq 2014;198:234–8. [25] Valiallah Mousavi S, Sheikholeslami M, Bandpy MG, Barzegar Gerdroodbary M. The influence of magnetic field on heat transfer of magnetic nanofluid in a sinusoidal double pipe heat exchanger. Chem Eng Res Des 2016;113:112–24. [26] Kandelousi MS. Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Eur Phys J Plus 2014;129:1–12. [27] Sheikholeslami M, Rashidi MM, Hayat T, Ganji DD. Free convection of magnetic nanofluid considering MFD viscosity effect. J Mol Liq 2016;218:393–9. [28] Safarnia H, Sheikholeslami M, Ganji DD. Electrohydrodynamic nanofluid flow and forced convective heat transfer in a channel. Eur Phys J Plus 2016;131:1–14. [29] Sheikholeslami M, Mustafa MT, Ganji DD. Nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation. Iran J Sci Technol 2015;39:433–40. [30] Sheikholeslami M, Soleimani S, Ganji DD. Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. J Mol Liq 2016;213:153–61. [31] Sheikholeslami M, Rashidi MM. Non-uniform magnetic field effect on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects. J Braz Soc Mech Sci Eng 2016;38:1171–84. [32] Sheikholeslami M, Rashidi MM, Ganji DD. Numerical investigation of magnetic nanofluid forced convective heat transfer in existence of variable magnetic field using two phase model. J Mol Liq 2015;212:117–26. [33] Sheikholeslami M, Rashidi MM, Ganji DD. Effect of non-uniform magnetic field on forced convection heat transfer of Fe3O4-water nanofluid. Comput Methods Appl Mech Eng 2015;294:299–312. [34] Sheikholeslami M, Rashidi MM. Effect of space dependent magnetic field on free convection of Fe3O4-water nanofluid. J Taiwan Inst Chem Eng 2015;56:6–15. [35] Sheikholeslami M, Mustafa MT, Ganji DD. Effect of Lorentz forces on forced-convection nanofluid flow over a stretched surface. Particuology 2016;26:108–13. [36] Sheikholeslami M, Ganji DD, Rashidi MM. Ferrofluid flow and heat transfer in a semi annulus enclosure in the presence of magnetic source considering thermal radiation. J Taiwan Inst Chem Eng 2015;47:6–17. [37] Sheikholeslami M, Ganji DD. Entropy generation of nanofluid in presence of magnetic field using Lattice Boltzmann Method. Phys A: Stat Mech Appl 2015;417:273–86. [38] Sheikholeslami M, Ganji DD. Nanofluid convective heat transfer using semi analytical and numerical approaches: a review. J Taiwan Inst Chem Eng 2016;65:43–77. [39] Sheikholeslami M, Rashidi MM. Ferrofluid heat transfer treatment in the presence of variable magnetic field. Eur Phys J Plus 2015;130:1–12.
13
[40] Sheikholeslami M. Effect of uniform suction on nanofluid flow and heat transfer over a cylinder. J Braz Soc Mech Sci Eng 2015;37:1623–33. [41] Sheikholeslami M, Ganji DD. Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer. Energy 2014;75:400–10. [42] Pourmehran O, Rahimi-Gorji M, Ganji DD. Heat transfer and flow analysis of nanofluid flow induced by a stretching sheet in the presence of an external magnetic field. J Taiwan Inst Chem Eng 2016;65:162–71. [43] Rahimi-Gorji M, Pourmehran O, Hatami M, Ganji DD. Statistical optimization of microchannel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis. Eur Phys J Plus 2015;130:1–21. [44] Pourmehran O, Rahimi-Gorji M, Gorji-Bandpy M, Ganji DD. Analytical investigation of squeezing unsteady nanofluid flow between parallel plates by LSM and CM. Alexandria Eng J 2015;54:17–26. [45] Rahimi-Gorji M, Pourmehran O, Gorji-Bandpy M, Ganji DD. Unsteady squeezing nanofluid simulation and investigation of its effect on important heat transfer parameters in presence of magnetic field. J Taiwan Inst Chem Eng 2016;67:467–75. [46] Hashim I. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. J Comput Appl Math 2006;193:658–64. [47] Esmaili Q, Ramiar A, Alizadeh E, Ganji DD. An approximation of the analytical solution of the Jeffery–Hamel flow by decomposition method. Phys Lett A 2008;372:3434–9. [48] Sheikholeslami M, Ganji DD, Ashorynejad HR, Rokni HB. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Appl Math Mech-Engl Ed 2012;33:25–36. [49] Jin C, Liu M. A new modification of Adomian decomposition method for solving a kind of evolution equation. Appl Math Comput 2005;169:953–62. [50] Jafari H, Daftardar-Gejji V. Revised Adomian decomposition method for solving a system of nonlinear equations. Appl Math Comput 2006;175:1–7. [51] Duan J-S, Rach R. A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations.. Appl Math Comput 2011;218:4090–118. [52] Dogonchi AS, Ganji DD. Investigation of heat transfer for cooling turbine disks with a non-Newtonian fluid flow using DRA. Case Stud Therm Eng 2015;6:40–51. [53] Dogonchi AS, Ganji DD. Investigation of MHD nanofluid flow and heat transfer in a stretching/shrinking convergent/divergent channel considering thermal radiation. J Mol Liq 2016;220:592–603. [54] Dogonchi AS, Divsalar K, Ganji DD. Flow and heat transfer of MHD nanofluid between parallel plates in the presence of thermal radiation. Comput Methods Appl Mech Eng 2016;310:58–76. [55] Dib A, Haiahem A, Bou-said B. An analytical solution of the MHD Jeffery–Hamel flow by the modified Adomian decomposition method. Comput Fluids 2014;102:111–15. [56] Dib A, Haiahem A, Bou-said B. Approximate analytical solution of squeezing unsteady nanofluid flow. Powder Technol 2015;269:193–9. [57] Xuan Y, Li Q, Hu W. Aggregation structure and thermal conductivity of nanofluids. AIChE J 2003;49:1038–43. [58] Chon CH, Kihm KD, Lee SP, Choi SUS. Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement. Appl Phys Lett 2005;87:153107. [59] Koo J, Kleinstreuer C. A new thermal conductivity model for nanofluids. J Nanopart Res 2004;6:577–88. [60] Vafai K. A critical synthesis of thermophysical characteristics of nanofluids. Int J Heat Mass Tran 2011;54:4410–28. [61] Koo J, Kleinstreuer C. Viscous dissipation effects in micro tubes and micro channels. Int J Heat Mass Tran 2004;47:3159–69. [62] Koo J. Computational nanofluid flow and heat transfer analyses applied to microsystems Ph.D Thesis. Raleigh, NC, The United States: NC State University; 2004. [63] Li J. Computational analysis of nanofluid flow in micro channels with applications to micro-heat sinks and bio-MEMS PhD Thesis. Raleigh, NC, The United States: NC State University; 2008. [64] Koo J, Kleinstreuer C. Laminar nanofluid flow in microheat-sinks. Int J Heat Mass Tran 2005;48:2652–61. [65] Prasher RS, Bhattacharya P, Phelan PE. Thermal conductivity of nano scale colloidal solution. Phys Rev Lett 2005;94:025901–1-4. [66] Jang SP, Choi SUS. The role of Brownian motion in the enhanced thermal conductivity of nanofluids. Appl Phys Lett 2004;84:4316–18. [67] Pourmehran O, Rahimi-Gorji M, Hatami M, Sahebi SAR, Domairry G. Numerical optimization of microchannel heat sink (MCHS) performance cooled by KKL based nanofluids in saturated porous medium. J Taiwan Inst Chem E 2015;55:49–68. [68] Sheikholeslami Kandelousi M. KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel. Phys Lett A 2014;378:3331–9. [69] Sheikholeslami M, Ganji DD. Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM. Comput Methods Appl Mech Eng 2015;283:651–63. [70] Dogonchi AS, Ganji DD. Thermal radiation effect on the nano-fluid buoyancy flow and heat transfer over a stretching sheet considering Brownian motion. J Mol Liq 2016;223:521–7. [71] Adomian G, Rach R. Inversion of nonlinear stochastic operators. J Math Anal Appl 1983;91:39–46.
Please cite this article as: A.S. Dogonchi, D.D. Ganji, Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion, Journal of the Taiwan Institute of Chemical Engineers (2016), http://dx.doi.org/10.1016/j.jtice.2016.09.029