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652 M. Sheikholeslami, D.D. Ganji / Comput. Methods Appl. Mech. Engrg. 283 (2015) 651–663

heated enclosure. They found that the suspended nanoparticles substantially increase the heat transfer rate at any

given Grashof number. Sheikholeslami et al. [3] used Control Volume based Finite Element Method to simulate the

effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with Cu–water nanofluid.

Their results indicated that Hartmann number and the inclination angle of the enclosure can be control parameters at

different Rayleigh number. Free convection heat transfer in a concentric annulus between a cold square and heated

elliptic cylinders in presence of magnetic field was investigated by Sheikholeslami et al. [4]. They found that the

enhancement in heat transfer increases as Hartmann number increases but it decreases with increase of Rayleigh

number. Nield and Kuznetsov [5] studied the natural convection in a horizontal layer of a porous medium. The analysis

reveals that for a typical nanofluid (with large Lewis number) the prime effect of the nanofluids is via a buoyancy

effect coupled with the conservation of nanoparticles, the contribution of nanoparticles to the thermal energy equation

being a second-order effect. There have been published several recent numerical studies on the modeling of natural

convection heat transfer in nanofluids [6–25].

Magnetohydrodynamic has many industrial applications such as crystal growth, metal casting and liquid metal

cooling blankets for fusion reactors. Sheikholeslami et al. [26] studied the problem of MHD free convection in an

eccentric semi-annulus filled with nanofluid. They showed that Nusselt number decreases with increase of position of

inner cylinder at high Rayleigh number. Rashidi et al. [27] considered the analysis of the second law of thermody-

namics applied to an electrically conducting incompressible nanofluid fluid flowing over a porous rotating disk. They

concluded that using magnetic rotating disk drives has important applications in heat transfer enhancement in renew-able energy systems. Recently, several authors studied the effect of magnetic field on flow and heat transfer [28–37].

The study of heat and mass transfer for unsteady squeezing viscous flow between two parallel plates has been

regarded as important research topics due to its wide range of scientific and engineering applications such as hydrody-

namical machines, polymer processing, lubrication system, chemical processing equipment, formation and dispersion

of fog, damage of crops due to freezing, food processing and cooling towers. The first research on the squeezing flow

in lubrication system was reported by Stefan [38]. Magnetohydrodynamic squeezing flow of a viscous fluid between

parallel disks was analyzed by Domairry and Aziz [39].

Most phenomena in our world are essentially nonlinear and are described by nonlinear equations. Nonlinear

differential equations usually arise from mathematical modeling of many physical systems. One of the semi-exact

methods which does not need small parameters is the Differential Transformation Method. Therefore, same as the

HAM and the HPM, the DTM can overcome the foregoing restrictions and limitations of perturbation methods.This method constructs an analytical solution in the form of a polynomial. It is different from the traditional higher-

order Taylor series method. The Taylor series method is computationally expensive for large orders. The Differential

Transform Method is an alternative procedure for obtaining an analytic Taylor series solution of differential equations.

The main advantage of this method is that it can be applied directly to nonlinear differential equations without

requiring linearization, discretization and therefore, it is not affected by errors associated to discretization. The concept

of DTM was first introduced by Zhou [40], who solved linear and nonlinear problems in electrical circuits. Chen and

Ho [41] developed this method for partial differential equations and Ayaz [42] applied it to the system of differential

equations; this method is very powerful [43]. Jang et al. [44] applied the two-dimensional differential transform

method to the solution of partial differential equations. Finally, Hassan [45] adapted the Differential Transformation

Method to solve some problems. This method was successfully applied to various application problems [46–48]. All

of these successful applications verified the validity, effectiveness and flexibility of the DTM.

In this paper, DTM is applied to find the approximate solutions of nonlinear differential equations governing the

problem of unsteady squeezing nanofluid flow and heat transfer using Koo–Kleinstreuer–Li (KKL) model. In this

model effect of Brownian motion on the effective thermal conductivity is considered. The effects of the squeeze

number and the nanofluid volume fraction on flow and heat transfer characteristics are investigated.

2. Governing equations

We consider the flow and heat transfer analysis in the unsteady two-dimensional squeezing flow of an incompress-

ible nanofluid between the infinite parallel plates. The two plates are placed at z = ±ℓ(1−αt )1/2 = ±h(t ) (ℓ is distant

of plate at t = 0 and α is squeezed parameter). For α > 0, the two plates are squeezed until they touch t = 1/α and

for α < 0, the two plates are separated. The viscous dissipation effect, the generation of heat due to fraction caused

by shear in the flow, is neglected. Further the symmetric nature of the flow is adopted.



M. Sheikholeslami, D.D. Ganji / Comput. Methods Appl. Mech. Engrg. 283 (2015) 651–663 653

Fig. 1. Geometry of problem.

It is also assumed that the time variable magnetic field ( B = B e y , B = B0 (1 − αt )) is applied, where e y is unit

vectors in the Cartesian coordinate system. The electric current J and the electromagnetic force F are defined by

J = σ

V × B

and F = σ

V × B

× B, respectively. Also a heat source ( Q = Q0/ (1 − αt )) is applied between

two plates (see Fig. 1).

The nanofluid is a two component mixture with the following assumptions: Incompressible; No-chemical reaction;

Negligible radiative heat transfer; Nano-solid-particles and the base fluid are in thermal equilibrium and no slip occurs

between them. The thermo physical properties of the nanofluid are given in Table 1 [49].

The governing equations for conservative momentum and energy in unsteady two dimensional flow of a nanofluid

fluid are [50]:

∂u

∂ x+

∂v

∂ y= 0, (1)

ρn f

∂u

∂t + u

∂u

∂ x+ v

∂u

∂ y

= −

∂ p

∂ x+ µn f

∂2u

∂ x 2 +

∂2u

∂ y2

− σ n f B

2u, (2)

ρn f

∂v

∂t + u

∂v

∂ x+ v

∂v

∂ y

= −

∂ p

∂ y+ µn f

∂ 2v

∂ x 2 +

∂2v

∂ y2

, (3)

∂ T

∂t + u

∂ T

∂ x+ v

∂ T

∂ y=

k n f ρC p

n f

∂2 T

∂ x 2 +

∂2 T

∂ y2

+

QρC p

n f

T . (4)

Here u and v are the velocities in the x and y directions respectively, T is the temperature,

P is the pressure, effective density ρn f , the effective heat capacity ρC pn f and Electrical conductivity σ n f

of the nanofluid are defined as [3]:

ρn f = (1 − φ) ρ f + φ ρ p,

(ρ C p)n f = (1 − φ) (ρ C p) f + φ (ρ C p) p

σ n f

σ f

= 1 +3

σ pσ f

− 1

φσ pσ f

+ 2

−

σ pσ f

− 1

φ.

(5)

The Brownian motion has a significant impact on the effective thermal conductivity. Koo and Kleinstreuer [ 51]

proposed that the effective thermal conductivity is composed of the particle’s conventional static part and a Brownian

motion part. This 2-component thermal conductivity model takes into account the effects of particle size, particle

volume fraction and temperature dependence as well as types of particle and base fluid combinations.




Table 1

Thermo physical properties of water and nanoparticles at room temperature [49].

ρ (kg/m3) C p (J/kg K) k (W/m k ) d p (nm)

Pure water 997.1 4179 0.613 –

Al2O3 3970 765 25 47

CuO 6500 540 18 29

k eff = k static + k Brownian (6)

k static

k f

= 1 +3

k pk f

− 1

φk pk f

+ 2

−

k pk f

− 1

φ(7)

where, k static is the static thermal conductivity based on Maxwell classical correlation. The enhanced thermal con-

ductivity component generated by micro-scale convective heat transfer of a particle’s Brownian motion and affected

by ambient fluid motion is obtained via simulating Stokes’ flow around a sphere (nano-particle). By introducing two

empirical functions (β and f ) Koo [52] combined the interaction between nanoparticles in addition to the temperatureeffect in the model, leading to:

k Brownian = 5 × 104βφρ f c p, f

κbT

ρ pd p f (T ,φ). (8)

In recent years, there has been an increasing trend to emphasize the importance of the interfacial thermal resistance

between nanoparticles and based fluids (see for example, Prasher et al. [ 53] and Jang and Choi, [54]). The thermal

interfacial resistance (Kapitza resistance) is believed to exist in the adjacent layers of the two different materials; the

thin barrier layer plays a key role in weakening the effective thermal conductivity of the nanoparticle.

Li [55] revisited the model of Koo and Kleinstreuer [49] and combined β and f functions to develop a new g ′ func-

tion which captures the influences of particle diameter, temperature and volume fraction. The empirical g′-functiondepends on the type of nanofluid [51]. Also, by introducing a thermal interfacial resistance R f = 4 × 10−8 km2/W

the original k p in Eq. (7) was replaced by a new k p,eff in the form:

R f + d p

k p=

d p

k p,eff . (9)

For different based fluids and different nanoparticles, the function should be different. Only water based nanofluids

are considered in the current study. For Al2O3–water nanofluids and CuO–water nanofluids, this function follows the

format:

g′ T , φ, d p = a1 + a2 ln d p + a3 ln (φ) + a4 ln (φ) ln d p + a5 ln d p2 ln (T )

+

a6 + a7 ln

d p

+ a8 ln (φ) + a9 ln (φ) ln

d p

+ a10 ln

d p2

(10)

with the coefficients ai (i = 0 . . . 10) are based on the type of nanoparticles and also with these coefficients,

Al2O3–water nanofluids and CuO–water nanofluids have an R2 of 96% and 98%, respectively [30] (Table 2). Fi-

nally, the KKL (Koo–Kleinstreuer–Li) correlation is written as:

k Brownian = 5 × 104φρ f c p, f

κbT

ρ pd pg′(T , φ, d p). (11)

Koo and Kleinstreuer [49] further investigated laminar nanofluid flow in micro heat-sinks using the effective

nanofluid thermal conductivity model they had established (Koo and Kleinstreuer [ 49]). For the effective viscosity




Table 2

The coefficient values of Al2O3–Water nanofluids and CuO–Water nanofluids [49].

Coefficient values Al2O3–Water CuO–Water

a1 52.813488759 −26.593310846

a2 6.115637295 −0.403818333

a3

0.6955745084 −33.3516805

a4 4.17455552786E−02 −1.915825591

a5 0.176919300241 6.42185846658E−02

a6 −298.19819084 48.40336955

a7 −34.532716906 −9.787756683

a8 −3.9225289283 190.245610009

a9 −0.2354329626 10.9285386565

a10 −0.999063481 −0.72009983664

due to micro mixing in suspensions, they proposed:

µeff = µstatic + µBrownian = µstatic + k Brownian

k f

× µ f

Pr f

(12)

where µstatic = µ f

(1−φ)2.5 is viscosity of the nanofluid, as given originally by Brinkman.

The relevant boundary conditions are:

v = vw = d h/dt , T = T H at y = h (t ),

v = ∂ u/∂ y = ∂ T /∂ y = 0 at y = 0.(13)

We introduce these parameters:

η = y

[l(1 − αt )1/2], u =

α x

[2(1 − αt )] f ′(η),

v = − αl

[2(1 − αt )1/2] f (η), θ =

T

T H

.

(14)

Substituting the above variables into (2) and (3) and then eliminating the pressure gradient from the resulting equations

give:

f i v − S ( A1/ A4)

η f ′′′ + 3 f ′′ + f ′ f ′′ − f f ′′′

− Ha2 ( A5/ A4) f ′′ = 0. (15)

Using (14), Eq. (4) reduces to the following differential equations:

θ ′′ + Pr S

A2

A3

f θ ′ − ηθ ′

+

Hs

A3θ = 0. (16)

Here A1, A2, A3, A4 and A5 are dimensionless constants given by:

A1 =

ρn f

ρ f , A2 =

(ρ C p)n f

(ρ C p) f , A3 =

k eff

k f , A4 =

µeff

µ f , A5 =

σ n f

σ f . (17)

With these boundary conditions:

f (0) = 0, f ′′ (0) = 0,

f (1) = 1, f ′ (1) = 0,

θ ′ (0) = 0, θ (1) = 1.

(18)

where S is the squeeze number, Pr is the Prandtl number, Ha is the Hartmann number and Hs is the heat source

parameter which are defined as:

S = αℓ2

2υ f , Pr =

µ f ρC p f

ρ f k f , Ha = ℓ B

0 σ f

µ f , Ha =

Q0ℓ2

k f . (19)




Physical quantities of interest are the skin fraction coefficient and Nusselt number which are defined as:

Cf ∗ =µn f

∂u∂ y

y=h(t )

ρn f v2w

, Nu∗ =−lk n f

∂ T ∂ y

y=h(t )

k f T H . (20)

In terms of (17), we obtain

C f =( A1/ A4) f ′′(1)

,

Nu = A3 θ ′ (1)

.(21)

3. Differential Transform Method (DTM)

3.1. Basic of DTM

Basic definitions and operations of differential transformation are introduced as follows. Differential transformation

of the function f (η) is defined as follows:

F (k ) = 1

k !

d k f (η)

d ηk

η=η0

. (22)

In (22), f (η) is the original function and F (k ) is the transformed function which is called the T -function (it is also

called the spectrum of the f (η) at η = η0, in the k domain). The differential inverse transformation of F (k ) is defined

as:

f (η) =∞

k =0

F (k )(η − η0)k (23)

by combining (22) and (23) f (η) can be obtained:

f (η) =∞

k =0

d

k

f (η)d ηk

η=η0

(η − η0)k

k ! . (24)

Eq. (24) implies that the concept of the differential transformation is derived from Taylor’s series expansion, but the

method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by an iterative

procedure that is described by the transformed equations of the original functions. From the definitions of (22) and

(23), it is easily proven that the transformed functions comply with the basic mathematical operations shown below.

In real applications, the function f (η) in (24) is expressed by a finite series and can be written as:

f (η) = N

k =0

F (k )(η − η0)k . (25)

Eq. (25) implies that f (η) =

∞k = N +1

F (k )(η − η0)k

is negligibly small, where N is series size (see Table 3).

Theorems to be used in the transformation procedure, which can be evaluated from (22) and (23), are given below

(Table 1).

3.2. Solution with Differential Transformation Method

Now Differential Transformation Method has been applied into governing equations (Eqs. (15) and (16)). Taking

the differential transforms of Eqs. (15) and (16) with respect to χ and considering H = 1 gives:

(k + 1)(k + 2)(k + 3)(k + 4)F [k + 4] + S ( A1/ A4)

k m=0

∆[k − m − 1](m + 1)(m + 2)(m + 3)F [m + 3]

− 3S ( A1/ A4) (k + 1)(k + 2)F [k + 2] − S ( A1/ A4)




Table 3

Some of the basic operations of Differential Transformation Method.

Original function Transformed function

f (η) = αg (η) ± βh (η) F [k ] = α G [k ] ± β H [k ]

f (η) = d n g(η)d ηn F [k ] = (k +n)!

k ! G [k + n]

f (η) = g (η)h(η) F [k ] =

k m−0 F [m] H [k − m]

f (τ ) = sin(ϖ η + α) F [k ] = ϖ k

k ! sin( π k 2

+ α)

f (τ ) = cos(ϖ η + α) F [k ] = ϖ k

k ! cos( π k 2

+ α)

f (η) = eλη F [k ] = λk

k !

F (η) = (1 + η)m F [k ] = m(m−1)···(m−k +1)k !

f (η) = ηm F [k ] = δ (k − m) =

1, k = m

0, k = m

×

k m=0

((k − m + 1) F [k − m + 1](m + 1)(m + 2)F [m + 2])

+ S ( A1/ A4)

k m=0

(F [k − m](m + 1)(m + 2)(m + 3)F [m + 3])

− Ha2 ( A5/ A4) (k + 1)(k + 2)F [k + 2] = 0, (26)

∆[m] =

1 m = 1

0 m = 1

F [0] = 0, F [1] = a1, F [2] = 0, F [3] = a2 (27)

(k + 1)(k + 2)Θ [k + 2] + Pr .S . A2

A3 k

m=0

F [k − m](m + 1)Θ [m + 1]

− Pr .S .

A2

A3

k m=0

∆[k − m](m + 1)Θ [m + 1] + Hs

A3Θ [k ] = 0, (28)

∆[m] =

1 m = 1

0 m = 1

Θ [0] = a3, Θ [1] = 0 (29)

where F [k ] andΘ [k ] are the differential transforms of f (η) , θ (η) and a1, a2, a3 are constants which can be obtained

through boundary condition. This problem can be solved as follows:

F [0] = 0, F [1] = a1, F [2] = 0, F [3] = a2, F [4] = 0

F [5] = 3

20S a2 ( A1/ A2) +

1

20Sa1a2 ( A1/ A2) +

1

20a1a2 +

1

20 Ha2a2, . . .

(30)

Θ [0] = a3, Θ [1] = 0, Θ [2] = −1

2

Hs

A3a3, Θ [3] = 0.0,

Θ [4] = 1

12Pr S

A2

A3

Hs

A3a3a1, Θ [5] = 0, . . . .

(31)

The above process is continuous. By substituting Eqs. (30) and (31) into the main Eq. (25) based on DTM, it can

be obtained that the closed form of the solutions is:




Fig. 2. Effect of volume fraction of nanofluid on Skin friction coefficient and Nusselt number when S = 1, Hs = −1, Pr = 6.2 (CuO–Water).

Table 4

Comparison of −θ ′ (1) between the present results and

analytical results obtained by Mustafa et al. [50] for S = 0.5

and δ = 0.1.

Pr Ec Mustafa et al. [50] Present work

1 1 3.026324 3.0263235

2 1 5.98053 5.9805303

5 1 14.43941 14.439413

1 2 6.052647 6.0526471

1 5 15.13162 15.131617

F (η) = a1η + a2η3 +

3

20S a2 ( A1/ A2) +

1

20Sa1a2 ( A1/ A2) +

1

20a1a2 +

1

20 Ha2a2

η4 + · · · (32)

θ (η) = a3 +

−

1

2

Hs

A3a3

η2 +

1

12Pr S

A2

A3

Hs

A3a3a1

η4 + · · · (33)

by substituting the boundary condition from Eq. (18) into Eqs. (32) and (33) in point η = 1 it can be obtained the

values of a1, a2, a3. By substituting obtained a1, a2, a3 into Eqs. (32) and (33), it can be obtained the expression of

F (η) and Θ (η) (see Table 4).

4. Results and discussion

In this study, unsteady flow between parallel plates in presence of variable magnetic field is investigated analytically

using Differential Transformation Method (DTM). This method is compared with previous work [50] in Fig. 2. This

companion indicates that DTM has good accuracy to solve such problem. The kind of nanoparticle is a key factor

for heat transfer enhancement. Tables 5 and 6 show a comparison among two different types of nanoparticles to find

selecting which of them leads to higher enhancement for this problem. It is also found that choosing CuO as the

nanoparticle leads to the greater amount of the Skin friction coefficient and Nusselt number. So we select CuO–water

as nanofluid. The influences of the squeeze number, Hartmann number, Heat source and the nanofluid volume fraction

on flow and heat transfer characteristics are studied. Fig. 2 shows the effect of nanofluid volume fraction on Skin

friction coefficient and Nusselt number. As nanofluid volume fraction increases Nusselt number increases while Skin

friction coefficient decreases.




Fig. 3. Effect of the squeeze number on velocity and temperature profiles when Ha = 1, Hs = −1, Pr = 6.2 (CuO–Water).

Table 5

Comparison of the Skin friction coefficient between

Al2O3–Water and CuO–Water at φ = 0.04, Pr = 6.2.

S Ha C f

CuO Al2O3

1 0 3.779263 3.629287

1 8 8.809621 8.676915

10 0 7.23826 6.897281

10 8 10.61304 10.33316

Fig. 3 shows the effect of the squeeze number on the velocity and temperature profiles. It is important to note

that the squeeze number ( S ) describes the movement of the plates (S > 0 corresponds to the plates moving apart,

while S < 0 corresponds to the plates moving together the so-called squeezing flow). Vertical velocity decreases

with increase of squeeze number while horizontal velocity has different behavior. It means that horizontal velocity

decreases with increase of S when η < 0.5 while opposite trend is observed for η > 0.5. Thermal boundary layer




Fig. 4. Effect of the Hartmann number on velocity and temperature profiles when S = 1, Hs = −1, Pr = 6.2 (CuO–Water).

Table 6

Comparison of the Nusselt number between Al2O3–Water and

CuO–Water at φ = 0.04, Pr = 6.2.

Hs S Ha Nu

CuO Al2O3

−1 1 0 0.717776 0.632233

−1 1 8 0.78598 0.700244

−1 10 0 0.291036 0.243075

−1 10 8 0.348638 0.293618

−10 10 8 3.532066 2.130418

thickness increases with increase of squeeze number. Effect of the Hartmann number on velocity and temperature

profiles is shown in Fig. 4. Effects of Hartmann number on velocity profiles are similar to that of squeeze number.

While increase in Hartmann number leads to decrease in thermal boundary layer thickness.




Fig. 5. Effects of Hartmann number and squeeze number on Skin friction coefficient and Nusselt number when φ = 0.04, Hs = −1, Pr =6.2 (CuO–Water).

Fig. 6. Effects of heat source parameter on temperature profile and Nusselt number when φ = 0.04, Hs = −1, Pr = 6.2 (CuO–Water).

Fig. 5 depicts the effects of Hartmann number and squeeze number on Skin friction coefficient and Nusselt number.This figure shows that Hartmann number has direct relationship with both of the Skin friction coefficient and Nusselt

number. While squeeze number has direct relationship with Skin friction coefficient and reverse relationship with

Nusselt number. Effects of heat source parameter on temperature profile and Nusselt number is shown in Fig. 6. As

heat source parameter increases temperature boundary layer thickness decreases and in turn Nusselt number increases.

5. Conclusion

In this study, unsteady MHD nanofluid flow and heat transfer between parallel plates are investigated. Differential

Transformation Method is used to solve the governing equations. The effect of the squeeze number and the nanofluid

volume fraction on heat and fluid flow are investigated. The results show that the higher values of heat transfer

enhancement are obtained when CuO selected as nanoparticle. Also, it can be found that Nusselt number is an




increasing function of nanoparticle volume fraction, Hartmann number and heat source parameter while it is a

decreasing function of the squeeze number.

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