Powder Technology 258 (2014) 125–133

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A comprehensive analysis of the flow and heat transfer for a nanofluidover an unsteady stretching flat plate

A.R. Ahmadi a, A. Zahmatkesh a, M. Hatami b,⁎, D.D. Ganji c

a Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iranb Esfarayen University, Department of Mechanical Engineering, Esfarayen, North Khorasan, Iranc Babol University of Technology, Department of Mechanical Engineering, Babol, Mazandaran, Iran

⁎ Corresponding author. Tel.: +98 9157186374.E-mail address: M.Hatami2010@gmail.com (M. Hatam

http://dx.doi.org/10.1016/j.powtec.2014.03.0210032-5910/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 28 December 2013Received in revised form 9 February 2014Accepted 1 March 2014Available online 11 March 2014

Keywords:NanoparticleBase fluidUnsteady stretching flat plateHeat transfer

In the present paper, the unsteady flow and the related heat transfer of a nanofluid caused by linear motion of ahorizontal flat plate has been analyzed and the nonlinear differential equations governing on the presentedsystem have been reduced to a set of ordinary differential equations and obtained equations have been solvedby Differential Transformation Method. Comparisons have been made between the obtained results andRunge–Kutta Numerical Solution, and the outcomes have been revealed that DTM is applicable in this casestudy. Furthermore, the relevant errors of the achieved solutions have been depicted graphically and these resultsindicate that the obtained analysis is trustable and applicable. It is necessary to mention that the effects of un-steadiness parameter S, Prandtl Number and finally volume fraction of the nanoparticles, ϕ on the temperatureand velocity profile have been investigated. As a main outcome, the velocity profile is increased by enlargingthe S parameter. Also, by increasing the ϕ, temperature profile has risen a lot which means the heat transferproperties can be enhanced by selecting appropriate nanoparticles sizes.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Nanofluid, which is a mixture of nano-sized particles suspended in abase fluid, is used to enhance the rate of heat transfer via its higherthermal conductivity compared to the base fluid. Recently, nanofluidshave been used in different industries and applications such asmicrochannel cooling, floor heating, and heat recovery systems. Sincethe thermal conductivity of solids is usually higher than liquids, theidea of adding particleswhich can bemade frommetallic or nonmetallicmaterials to a conventional fluid for instance water, mineral oils andethylene glycol in order to improve its heat transfer characteristicscreated. Amongst all dimensions of particles such as macro, micro, andnano due to some obstacles in the pressure drop through the systemor keeping the mixture homogenous, nano-scaled particles haveattracted more attention. These tiny particles are fairly close in size tothe molecules of the base fluid, and therefore can realize extremelystable suspensions with slight gravitational settling over long periodsof time. It is necessary to mention that the convective heat transfercharacteristic of nanofluids depends on the thermo-physical propertiesof the base fluid and the suspended particles, the flow pattern and flowstructure, the volume fraction of the suspended particles, the dimen-sions and the shape of these particles.

i).

To explain more, a quick look to Table 1 which describes thethermo-physical properties of four different kinds of nanoparticles(copper, silver, alumina and titanium dioxide) and a common basefluid (pure water) is recommended: The utility of a particularnanofluid for a heat transfer application can be established by suit-ably modeling the convective transport in the nanofluid [1]. Theterm “nanofluid” for the first time was presented by Choi [2] toshow engineered colloids which are composed of nanoparticles dis-persed in a base fluid. Afterwards, Masuda et al. [3] followed thestudy of this concept and then a considerable amount of researchup to now in this field has significantly risen. Recently, nanofluidshave been utilized in different industries and motivated the re-searchers in different fields of study by analytical and numericalmethods [4–8]. Take for example; Sheikholeslami et al. [9] investi-gated the flow of nanofluid and heat transfer characteristics betweentwo horizontal plates in a rotating system. Their results show thatfor suction and injection, the heat transfer rate at the surface is in-creased by enhancing the nanoparticle volume fraction. Hatami andGanji [8] investigated the effect of Cu-water nanofluid flow incooling of microchannel heat sink (MCHS) by analytical least squaremethod (LSM). This analytical method is also applied in other heattransfer problems [10–13] due to its simplicity and high accuracy.Hatami and Ganji [13] investigated the effect of variable magneticfield on the heat transfer and nanofluid flow between two paralleldisks. Also, Domairry and Hatami [14] studied the effect of squeezingfilm of Cu-water nanofluid between parallel plates using another

Nomenclature

S unsteadiness parameterϕ solid volume fraction of the nanofluidPr Prandtl numberu, v velocity components along x and y directionsμn f viscosity of the nanofluidρn f density of the nanofluidαn f Thermal diffusivity of the nanofluidkn f thermal conductivity of the nanofluid(ρ Cp)n f heat capacitance of the nanofluidρf density of the base fluidμf dynamic viscosity of the base fluidRex−0.5Cfx local skin friction coefficientRex−0.5Nux local Nusselt numberkf thermal conductivity of the base fluid(ρ Cp)f heat capacitance of the base fluidρs density of the nanoparticleμs dynamic viscosity of the nanoparticleks thermal conductivity of the nanoparticle(ρ Cp)s heat capacitance of the nanoparticleη similarity variableβ dimensionless film thicknessh(t) film thicknessδ(x) boundary layer thicknessε1, ε2 constantsCf skin friction coefficient

126 A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

analytical method which is introduced in the present study. In anotherresearch, Noreen Sher Akbar and Sohail Nadeem [15] modeled themixed convective peristaltic motion of a MHD Jeffrey nanofluid inan asymmetric channel with Newtonian heating using Homotopy Per-turbation Method. Also, they [16] modeled the Phan-Thien–Tanner(PTT) nanofluid in a cylindrical coordinate system for the first timeusing the same analytical method. Nadeem et al. [17] which investigat-ed the theoretical study of steady stagnation point flowwith heat trans-fer of a second grade nano fluid towards a stretching surface byHomotopy Analysis Method (HAM), found that a boundary layer isformedwhen the stretching velocity of the surface is less than the invis-cid free-stream velocity and velocity at a point increases with theincrease in the elasticity of the fluid. More study in these criteria canbe found in Refs. [18–20].

Differential Transformation Method is a powerful analytical methodwhich does not need any small parameter like p in Homotopy Perturba-tion Method (HPM) for discretization, perturbation or linearization. Thisapproach constructs an analytical solution in the form of a polynomialand is different from the traditional higher-order Taylor series method.It is notable that the concept of DTM was firstly introduced by Zhou[21] who solved linear and nonlinear problems in electrical circuits.Afterwards, his approach was successfully applied to various prob-lems [22–25]. For instance, Keimanesh et al. [26] used themulti-step dif-ferential transform method (Ms-DTM) to find the analytical solution ofthe resulted ordinary differential equation and steady hydro-magneticconvective heat and mass transfer with slip flow from a spinning disk

Table 1Thermo physical properties of water and four different kinds of nanoparticles.

Material ρ (kg/m3) Cp (J/kg K) k (W/m∙K)

Copper (Cu) 8933 385 401Silver (Ag) 10,500 235 429Alumina (Al2O3) 3970 765 40Titanium dioxide (TiO2) 4250 686.2 8.9538Pure water 997.1 4179 0.613

with viscous dissipation and also Ohmic heating was investigated byRashidi et al. [27] using DTM-Padé. Although many analytical methodsare presented in the literature, we used DTM due to following mainadvantages.

a) Unlike perturbation techniques, DTM is independent of any small orlarge quantities. So, DTM can be applied no matter if governingequations and boundary/initial conditions of a given nonlinearproblem contain small or large quantities or not.

b) Unlike Homotopy Analysis Method (HAM), DTM does not need tocalculate auxiliary parameter ℏ1, through h-curves.

c) Unlike HAM, DTM does not need initial guesses and auxiliary linearoperator and it solves equations directly.

d) DTM provides us with great freedom to express solutions of a givennonlinear problem by means of Pade approximant and Ms-DTM.

According to the above descriptions, the main objective of this studyis to apply DTM to find the approximate solution of nonlinear differentialequations governing the problem of nano-liquid film over an unsteadystretching surface which previously was studied by Xu et al. [28] usingHAM. Also in this study, the effects of active parameters are investigatedand discussed.

2. Problem description

Consider a nano-liquid film flow and heat transfer in the neighbor-hood of a thin elastic sheet. In order to understand more, the relatedphysical model has been depicted in Fig. 1.

The Cartesian coordinate (x,y) is chosen in amanner that the x-axis ismeasured in the direction of wall stretching and the y-axis is normal tothe wall. The continuous surface at y = 0 is stretched with the velocitydefined as:

Uw ¼ bx1−αt

: ð1Þ

In the above equation b and a are constants with dimensions t−1.After that the temperature distribution on the sheet is given as follows:

Ts ¼ T0−Trbx2

2vf

!1−αtð Þ−3=2

: ð2Þ

In Eq. (2), T0 is the temperature at the slit, Tr can be considered eitheras a constant reference temperature or a constant temperature differ-ence and finally vf is the kinematic viscosity of the base fluid. Supposethat the film is uniform and stable and the gravity and end effects are

Fig. 1. The schematic diagram of the physical model.

Table 2The dimensionless film thickness (β) for various values of S and ϕwhen Pr = 1.

Types of fluid S ϕ = 0.0 ϕ = 0.1 ϕ = 0.2

Cu-water 0.6 3.31710 2.66586 2.571090.8 2.15199 1.83187 1.766761.0 1.54361 1.31399 1.267291.2 1.12778 0.96002 0.925891.4 0.82103 0.69890 0.674051.6 0.57617 0.49046 0.473031.8 0.35638 0.30337 0.29259

127A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

negligible. Therefore, according to the proposed mathematical model byTiwari andDas [29], the governing differential equations for this problemare expressed as follows:

∂u∂x þ

∂v∂y ¼ 0 ;

∂u∂t þ u

∂u∂x þ v

∂u∂y ¼ μnf

ρnf

∂2u∂y2

;∂T∂t þ u

∂T∂x þ v

∂T∂y ¼ αnf

∂2T∂y2

:

ð3Þ

In Eq. (3), u and v are the velocity components along x and y direc-tions, T is defined as the temperature, and μnf, ρnf and then αnf are theviscosity, the density and the thermal diffusivity of nanofluid whichare expressed as follows:

αnf ¼knf

ρcp� �

nf

; μnf ¼μ f

1−ϕð Þ5=2 ; ρnf ¼ 1−ϕð Þρ f þ ϕρs: ð4Þ

Afterwards:

ρcp� �

nf¼ 1−ϕð Þ ρcp

� �fþ ϕ ρcp

� �s;

knfk f

¼ks þ 2kf

� �−2ϕ kf−ks

� �ks þ 2kf

� �þ ϕ kf−ks

� � :

ð5Þ

In the afore-mentioned equationsϕ is the solid volume fraction of thenanofluid, knf and (ρcp)nf are the thermal conductivity and the heatcapacitance of the nanofluid, respectively. Then, the related boundaryconditions for the differential equation governing thementioned systemare defined as:

when y ¼ 0 → u ¼ Uw ; v ¼ 0 ; T ¼ Ts ð6Þ

Fig. 2. The result of varying β in terms of ϕ for three different kinds of nanoparticles whenS = 1.2.

and then

at y ¼ h tð Þ →∂u∂y ¼ 0 ;

∂T∂y ¼ 0 ; v ¼ dh

dt: ð7Þ

In the above equation, h(t) is the thickness of the film and x is as-sumed to be a non-negative quantity. Based on the Schlichting andGersten [30], the boundary-layer thickness δ(x) is proportional to(xvf/Uw)0.5, so that the similarity variable η is defined as follows:

η ¼ y

ffiffiffiffiffiffiffiffiUw

xvf

s¼ y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib

1−αtð Þvf

s: ð8Þ

It is necessary to mention that the above equation can also beachieved by the definitions proposed by Görtler [31]. In accordancewith his explanation and substituting U(x) = Uw, it is possible to defineξ, η and ψ(x, y) brought completely in Ref. [28]. As regards the aboveexplanations, the following new variables are introduced as [28]:

ψ ¼ βvf b

1−αt

� �1=2x f ηð Þ ð9Þ

T ¼ T0−Trbx2

2vf

!1−αtð Þ3=2θ ηð Þ ð10Þ

η ¼ 1β

bvf 1−αtð Þ

" #1=2y ð11Þ

whereψ(x, y) is the stream function explained in the usual way, take forexample u ¼ ∂ψ

∂y, v ¼ −∂ψ∂x and β N 0 is the dimensionless film thickness

defined by β = (hb/vf)(1 − αt)−1/2. Note that for the limiting casesβ = 0 and β = ∞, this transformation is no longer effective and partic-ular approaches are needed to give solutions as shown inWang [31]. Asa result, the velocity components u and v can be explicitly defined as:

u ¼ ∂ψ∂y ¼ bx

1−αt

� �f ′ ηð Þ ; v ¼ −∂ψ

∂x ¼ −βvf b

1−αt

� �1=2

f ηð Þ: ð12Þ

After substituting the obtained similarity variables from Eqs. (9)–(11)to Eq. (3), the continuity equation is automatically satisfied and themomentum and energy equations are reduced to:

ε1 f‴ þ β2 f f ″− f ′

� �2−S f ′ þ η2f ″

� �� �¼ 0 ð13Þ

ε2Pr

θ″−β2 S2

3θþ ηθ′� �

þ 2θ f ′− f θ′� �

¼ 0: ð14Þ

Table 3Some fundamental operations of the differential transformation method.

Original function Transformed function

x(t) = αf(x) ± βg(t) X(k) = αF(k) ± βG(k)

x tð Þ ¼ dm f tð Þdtm

X kð Þ ¼ kþmð Þ!F kþmð Þk!

x(t) = f(t)g(t) X kð Þ ¼ ∑k

l¼0F lð ÞG k−lð Þ

x(t) = tm X kð Þ ¼ δ k−mð Þ ¼ 1; if k ¼ m;0; if k≠m:

x(t) = exp(t) X kð Þ ¼ 1

k!

x(t) = sin(ωt + α) X kð Þ ¼ ωk

k!sin

kπ2

þ α� �

x(t) = cos(ωt + α) X kð Þ ¼ ωk

k!cos

kπ2

þ α� �

128 A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

And then, the relevant boundary conditions for this problem can beexpressed as follows:

at η ¼ 0 → f 0ð Þ ¼ 0 ; f ′ 0ð Þ ¼ 1 ; θ 0ð Þ ¼ 1

at η ¼ 1 → f ″ 1ð Þ ¼ 0 ; θ′ 1ð Þ ¼ 0 ; f 1ð Þ ¼ S2:

ð15Þ

In the afore-mentioned equations, Pr = (vf/αf) is the Prandtl num-ber, S = (α/b) is the unsteadiness parameter and finally ε1 and ε2 aretwo constants explained in the following form:

ε1 ¼ 1

1−ϕð Þ2:5 1−ϕð Þ þ ϕρs=ρ f

h i ; ε2 ¼knf =kf

� �1−ϕþ ϕ ρcp

� �s=ρcpÞ f

h i :ð16Þ

To understand more, it is better to indicate that the physical quanti-ties for this problem are the skin friction coefficient Cf and the Nusseltnumber Nu as follows:

C f ¼τw

ρ f Uwð Þ2 ; Nu ¼ qwxkf Ts−T0ð Þ : ð17Þ

In the above equation, the skin friction at the surface and the heatflux from the surface are defined in the following form:

τw ¼ μnf∂ψ∂y

� �y¼0

; qw ¼ −knf∂T∂y

� �y¼0

: ð18Þ

Therefore, after substituting Eq. (18) into Eq. (17), we will have thefinal form of the physical quantities as follows:

CfxRe−1=2x ¼ 1

β 1−ϕð Þ5=2 f ″ 0ð Þ ; NuxRe1=2x ¼ −

knfk f

1βθ′ 0ð Þ: ð19Þ

It is notable that in Eq. (19) Rex = Uwx/vf is the local Reynolds num-ber. It is clear that, for the problem of a liquid film caused by an un-steady stretching flat surface in a Newtonian fluid, Wang [21]evaluated that similarity solutions are only possible in the region 0 ≤S≤ 2. For the current nano-liquidfilmflow, it is found that the similaritysolutions are available in the same range of S. Beyond this region, no so-lutions can be found. On the other hand, it is citable that the film thick-ness β decreases monotonically as S increases for both the Newtonianfluids (ϕ = 0) and the nanofluids (ϕ ≠ 0) as shown in Table 2.

This issue refers that the constants α and b as well as the wallstretching velocity Uw have significant effects on β. For a fixed value ofb, the larger α, the smaller value of b. Moreover, it is concluded fromTable 2 that the decaying rate for β between any two prescribed values

Table 4A comparison between the obtained values by DTM and Numerical Solution and the related ePr = 1 and finally S = 1.

η f(η)

NUM DTM The error of DTM

0 0.0 0.0 0%0.1 0.09345612174 0.0935895334 0.1427%0.2 0.17425663412 0.1747190347 0.2653%0.3 0.24321770321 0.2440976516 0.3617%0.4 0.30136110613 0.3026445699 0.4258%0.5 0.34989036992 0.3514717280 0.4519%0.6 0.39016599675 0.3918663230 0.4357%0.7 0.42368002460 0.4252731100 0.3760%0.8 0.45203012617 0.4532764928 0.2757%0.9 0.47689340438 0.4775824081 0.1444%1 0.49999999999 0.4999999998 0.00000004%

of S remains almost the same for all of the considered nanofluids, takefor example the decaying rate for β between S = 0.6 and S = 0.8 is31.2838% and this rate between S = 1.0 and S = 1.8 is 76.2191%. Onthe basis of the above explanations, Hang Xu et al. [20] obtained a linearformula for evaluating the variation of β in terms of nano-particlevolume fraction (ϕ) for three different kinds of nanofluids and depictedtheir results graphically as Fig. 2.

3. Differential Transformation Method (DTM)

In this section the fundamental laws of the Differential Transforma-tion Method have been introduced. Suppose that x(t) is an analyticfunction in domain D, and the expression t = ti indicates any point inthe presented domain. The function x(t) is then represented by onepower serieswhich its center is located at ti. The Taylor series expansionof x(t) is in form of:

x tð Þ ¼X∞k¼0

t−tið Þkk!

dkx tð Þdtk

" #t¼ti

; ∀t∈D: ð20Þ

Therefore, theMaclaurin series of x(t) can be obtained by taking ti=0in the above equation as follows:

x tð Þ ¼X∞k¼0

tk

k!dkx tð Þdtk

" #t¼0

; ∀t∈D: ð21Þ

As explained in Ref. [13], the differential transformation of the func-tion x(t) is defined as follows:

X kð Þ ¼X∞k¼0

Hk

k!dkx tð Þdtk

" #t¼0

: ð22Þ

In Eq. (22), X(k) represents the transformed function and x(t) is theoriginal function. The differential spectrum of X(k) is confined withinthe interval t ∈ [0, H] where H is defined as a constant value. Afterthat the differential inverse transform of X(k) is expressed as follows:

x tð Þ ¼X∞k¼0

tH

� �k

X kð Þ: ð23Þ

Obviously, the concept of Differential Transformation Method isbased on the Taylor series expansion. The function X(k) at values of ar-gument k is referred to a discrete issue which means X(0) is known asthe zero discrete, X(1) as the first discrete and so on. The more discreteavailable, the more precise and so it is possible to restore the unknownfunction. The function x(t) consists of the T-function X(k), and its value

rrors for (f(η) and θ(η) in the specified domain when β = 0.5, ε1 = −0.332, ε2 = 8.35,

θ(η)

NUM DTM The error of DTM

0.999999999999 1 0%0.993171519127 0.9931687892 0.00027%0.987307832315 0.9873047251 0.00031%0.982331096286 0.9823314688 0.000037%0.978170169099 0.9781784958 0.0008512%0.974761540350 0.9747820742 0.0021%0.972050131446 0.9720861224 0.0037%0.969989968411 0.9700429419 0.0054%0.968544729388 0.9686138292 0.0071%0.967688168589 0.9677695631 0.0084%0.967404418106 0.9674907698 0.0089%

Fig. 4. The resulted computational error for θ(η) by DTM.

129A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

is given by the sum of the T-function with (t/H)k as its coefficient. Bychoosing the right amount of H, it is observed that the larger values ofargument k, the less discrete of spectrum. The function x(t) is expressedby a finite series and therefore Eq. (23) can be rewritten as:

x tð Þ ¼Xnk¼0

tH

� �k

X kð Þ: ð24Þ

It is necessary to mention that some important mathematical oper-ations performed by Differential Transformation Method are listed inTable 3 as follows.

In accordancewith the above explanations, Differential Transforma-tion Method has been applied in order to solve the presented problemas follows:

DTM1 ¼ ε1 kþ 1ð Þ kþ 2ð Þ kþ 3ð Þ kþ 4ð ÞF kþ 4ð Þ

þβ2

" Xkl¼0

lþ 1ð ÞF lþ 1ð Þ kþ 1‐lð Þ kþ 2‐lð ÞF kþ 2‐lð Þ !

þ Xk

l¼0

F lð Þ kþ 1‐lð Þ kþ 2‐lð Þ kþ 3‐lð ÞF kþ 3‐lð Þ!

−2Xkl¼0

lþ 1ð ÞF lþ 1ð Þ kþ 1‐lð Þ kþ 2‐lð ÞF kþ 2‐lð Þ !

−S32

� ��kþ 1ð Þ kþ 2ð ÞF kþ 2ð Þ

þ12

Xkl¼0

δ l−1ð Þ kþ 1−lð Þ kþ 2−lð Þ kþ 3−lð ÞF kþ 3−lð Þ!!#

¼ 0

ð25Þ

DTM2 ¼ ε2Pr

kþ 1ð Þ kþ 2ð ÞΘ kþ 2ð Þ

−β2

S2

3Θ kð Þ þ

Xkl¼0

δ l−1ð Þ kþ 1−lð ÞΘ kþ 1−lð Þ!!

þ2

Xkl¼0

Θ lð Þ kþ 1−lð ÞF kþ 1−lð Þ!

−Xkl¼0

F lð Þ kþ 1−lð ÞΘ kþ 1−lð Þ!

¼ 0:

ð26Þ

Fig. 3. The achieved computational error for f(η) by DTM.

where F and Θ represent the DTM transformed form of f and θ, respec-tively. After that the given boundary conditions can be transformed asfollows:

F 0ð Þ ¼ 0 ; F 1ð Þ ¼ 1 ; F 2ð Þ ¼ a ; F 3ð Þ ¼ b ; Θ 0ð Þ ¼ 1 ; Θ 1ð Þ ¼ c:

ð27Þ

where a, b and c are constant coefficients that after specifying F(η)and Θ(η) and applying the three remained boundary conditionsfrom Eq. (15) on the obtained solutions can be determined. Whenour base fluid is pure water and the nanoparticles are copper andthe constant coefficients of the governing equations are assumed tobe β = 0.5, ε1 = −0.332, ε2 = 8.35, Pr = 1 and finally S = 1, wewill have:

a ¼ −0:6481199115 ; b ¼ 0:06070499715 ; c ¼ −0:07341467053:

ð28Þ

Fig. 5. A comparison between the obtained results by DTM and Numerical Solution interms of varying f(η) for different values of unsteadiness parameter (S).

Fig. 8. The result of varying temperature distributions for four different values of solid vol-ume fraction of the nanofluid (ϕ).

Fig. 6. Comparing the achieved results by DTM andNumerical Solution in terms of varyingθ(η) for different amounts of unsteadiness parameter (S).

130 A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

In this step, in order to avoid the repeated mathematical operationfor its ease of understanding, we omit the detailed DTM procedureand only mention the final solution in the following form:

f ηð Þ ¼ η−0:6481199115η2 þ 0:06070499715η3 þ 0:1016754379η4

− 0:01397202970η5−0:0002884940148η6: ð29Þ

And then

θ ηð Þ ¼ 1−0:07341467053ηþ 0:05239520958η2−0:01385237686η3

þ 0:001460996324η4 þ 0:001072271513η5−0:0001706601508η6:

ð30Þ

In order to be sure about the precision of the achieved solutions, acomparison between Numerical Method (Runge–Kutta) and DTM hasbeen presented in Table 4.

Fig. 7. The variation of velocity profile for some values of unsteadiness parameter.

4. Results and discussions

Generally, in order to understand that what percentage of errors isexisted in the obtained solution of differential equations, the procedurebelow should be followed: To start, the obtained solution of the givenset of differential equations consisting of Eqs. (29) and (30) should besubstituted into the main set of differential equations, g(η) and h(η)and then it is necessary to depict the yielded equations in Cartesian coor-dinates. Afterwards, the yielded errors of the given set of differentialequations can be observed from the obtained charts. To understandmore, reading the following lines are recommended: Consider a set ofdifferential equations in the following form:

g ηð Þ ¼ g f ηð Þ; f 0 ηð Þ;… �h ηð Þ ¼ h θ ηð Þ ; θ0 ηð Þ ;… � :

(ð31Þ

Fig. 9. Comparing the variation of f(η) in terms of different values of Prandtl Number.

Fig. 12. Comparing the variation of velocity profile in terms of four different values of (ϕ).Fig. 10. The result of varying (η) in terms of various amounts of Prandtl Number.

131A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

And the answer of the afore-mentioned set of equations is assumedto be a function of η in the form of:

f ¼ h1 ηð Þ ; θ ¼ h2 ηð Þ: ð32Þ

Thus by substituting Eq. (32) into Eq. (31), the computational errorof the obtained solution by each analytical or semi-analytical methodcan be achieved as follows:

g ηð Þ ¼ g f h1 ηð Þð Þ; f 0 h2 ηð Þð Þ;… �h ηð Þ ¼ h θ h1 ηð Þð Þ ; θ0 h2 ηð Þð Þ;… �

:ð33Þ

Eventually, with regard to the given physical values and bysubstituting the yielded solutions by DTM which are fromEqs. (29)–(30) to Eqs. (13)–(14), the computational errors of DTM aredepicted in the forms of.

The afore-mentioned figures (Figs. 3 and 4) show that theyielded solutions by DTM on the basis of the given physical valuessuch as Pr = 1 in the specified domain are appropriate approximationsfor solving the presented problem. In this case study, Cu-waternanofluid is chosen as a convenient example for illustration and

Fig. 11. The result of varying f(η) in terms of various amounts of solid volume fraction ofthe nanofluid (ϕ).

attempts have beenmade to discuss the effects of somephysical param-eters such as Pr, S and ϕ on the velocity and temperature distribution asshown in Figs. 5 and 6.

It is clear from Figs. 5 and 6 that by increasing the amount of un-steadiness parameter S, the values of f(η) increases but vice versa, thetemperature profile diminishes smoothly with η in the specified do-main. In accordance with Fig. 7, the velocity profile f′(η) decreases uni-formly with η for different amounts of unsteadiness parameter and it isnotable that the larger value of S, the higher velocity profile is. Themen-tioned variation will be clearer while increasing the amount of η.Inasmuch as the unsteadiness parameter is dependent to the α andb(S = α/b), it is observed that the wall stretching velocity is an impor-tant factor for determining the velocity profile. As regards Fig. 8 whichdescribes the variation of temperature distribution for any consideredvalue of unsteadiness parameter, θ(η) enlarges along with increasingϕ. This issue refers to the fact that nanofluid plays a vital role on theheat transfer properties. Afterwards, the effects of Prandtl numberhave been investigated on f(η) and θ(η) completely and the results areshown in Figs. 9 and 10. Regarding to the given figures, Pr does nothave any effect on f(η) and it enhances uniformly by increasing η, but

Fig. 13. The obtained graphical results for the variation of local skin friction coefficient interms of four different amounts of S.

Table 6The variation of NuxRex

12 in terms of ϕ for different values of parameter S.

ϕ S = 0.5 S = 1 S = 1.5 S = 1.9

0 4.1166 4.6547 5.1325 5.48350.01 4.1617 4.7158 5.2027 5.56040.02 4.2178 4.7772 5.2733 5.63750.03 4.2186 4.8388 5.3441 5.71500.04 4.3196 4.9007 5.4153 5.79280.05 4.3709 4.9630 5.4868 5.87100.06 4.4224 5.0256 5.5588 5.9496

Fig. 14. Investigating the effect of (ϕ) on the local Nusselt number in the specified domain.

132 A.R. Ahmadi et al. / Powder Technology 258 (2014) 125–133

the temperature distribution decreases significantly by growing thevalues of Prandtl number and it vanishes at the free surface (η = 1)for large amount of Prandtl number. Therefore, it is citable that the tem-perature at the free surface is equal to the ambient temperature.

Moreover, the variation of f(η) in terms of various amounts of nano-particle volume fraction has been analyzed in Fig. 11. Based on thegraphical results, it is revealed that f(η) enhances in the specified do-main of η and for the effect of ϕ, we can indicate that f(η) decreasesmonotonically by enlarging ϕ.

In this step, the effects of solid volume fraction of the nanofluid onthe velocity distribution have been presented in Fig. 12. In accordancewith the given information by increasing the values ofϕ, f′(η) decreasesin terms of η∈ {0, 0.5} but this tendency inverses for η∈ {0.5, 1} whichmeans the velocity profile enhances by increasing the values of ϕ. Even-tually, the variations of local skin friction coefficient and also the localNusselt number presented as the physical quantities of the mentionedproblem in terms of ϕ are graphically represented in Figs. 13 and 14 asfollows.

It is observed that by enlarging the unsteadiness parameter, theamount of local skin friction coefficient enhances significantly and thelocal skin friction coefficient reduces uniformly with increasing ϕ inthe specified domain. This issue proves that nanofluids are sufficientlyuseful for reducing the drag force of fluid flow. Then, the variation oflocal Nusselt numberwith solid volume fraction of the nanofluid for dif-ferent amounts of S presented in Fig. 14. It can easily be seen that thelocal Nusselt number increases continuously with ϕ defined from 0 to0.6. This item indicates that more values of ϕ, the more convectiveheat transfer of the nanofluids. Furthermore, it is necessary to mentionthat the local Nusselt number enlarges by increasing the unsteadinessparameter S. Therefore, the obtained results indicate that the suspended

Table 5The variation of CfxRex

−12 in terms of ϕ for different values of parameter S.

ϕ S = 0.5 S = 1 S = 1.5 S = 1.9

0 −1.4566 −1.1022 −0.6073 −0.12950.01 −1.5094 −1.1459 −0.6325 −0.13500.02 −1.5631 −1.1902 −0.6580 −0.14050.03 −1.6177 −1.2349 −0.6837 −0.14610.04 −1.6733 −1.2802 −0.7096 −0.15180.05 −1.7299 −1.3262 −0.7359 −0.15750.06 −1.7875 −1.3729 −0.7625 −0.1633

nanoparticles mainly enhance the heat transfer rate at any given valuesof S and Prandtl number.

To deeply understand the above procedure, the variation of localskin friction coefficient and the local Nusselt number in terms of ϕ fordifferent values of unsteadiness parameter has been presented in theform of numerical data in Tables 5 and 6.

5. Conclusions

In the current study, the flow of a nanofluid film and its heat transferdistribution nearby of an unsteady stretching flat plate has thoroughlybeen investigated by Differential Transformation Method (DTM). DTMresults are compared with Runge–Kutta numerical outcomes andthose of HAM [28] and an excellent agreement is observed which con-firms the accuracy of applied method. After that the effects of somephysical parameters such asϕ, S and Pr on the temperature and velocityprofiles have completely been investigated. It has been mentioned thatS has an important role on the velocity profile whichmeans that the ve-locity profile is increased by enlarging S. It is citable that by increasingϕ,θ(η) has risen a lot whichmeans that the heat transfer properties can beenhanced by selecting appropriate nanoparticles. Moreover, it has beenobserved that the value of θ(η) diminishes at the free surface (η=1) forlarge Prandtl numbers. Eventually, it has been found that local skin fric-tion coefficient decreases by rising ϕ but on the contrary, local Nusseltnumber increases along with enhancing the amount of solid volumefraction of the nanofluid.

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