flow and heat transfer of nanofluids over a rotating disk with uniform stretching rate in the radial direction

H O S T E D B Ya rotating disk with uniform stretching rate in the radial direction Q2 Q1 a School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 10008

H O S T E D B Y

a rotating disk with uniform stretching rate

in the radial direction

Q2

Q1

a

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

b

School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China

Received 24 August 2015; accepted 23 October 2015

KEYWORDS

Nanofluid;

Rotating disk;

Stretching;

Heat transfer

Abstract This paper studiesflow and heat transfer of nanofluids over a rotating disk with uniform stretching rate Three types of nanoparticles-Cu, Al2O3 and CuO-with water-based nanofluids are considered The governing equations are reduced by Von Karman transforma-tion and then solved by the homotopy analysis method (HAM), which is in close agreement with numerical results Results indicate that with increasing in stretching strength parameter, the skin friction and the local Nusselt number, the velocity in radial and axial directions increase, whereas the velocity in tangential direction and the thermal boundary layer thickness decrease, respectively Moreover, the effects of volume fraction and types of nanofluids on velocity and temperature fields are also analyzed

& 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

1 Introduction

The problem of fluid flow over a rotating disk is one of

the classical problems in fluid mechanics, which has both

theoretical and practical values Many researches have been

carried out on flow over a rotating disk in theoretical disciplines and due to numerous practical applications in some areas such as computer storage devices, rotating machinery, electronic devices and medical equipment, such flow is also very important in the engineering processes

Von Karman [1] originally investigated the hydrodynamic flow over an infinite rotating disk in 1921 In his work, Von Karman introduced his famous similarity transformations, which reduced the governing partial differential equations into ordinary differential equations In recent years,

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http://ppr.buaa.edu.cn/

www.sciencedirect.com Propulsion and Power Research

http://dx.doi.org/10.1016/j.jppr.2017.01.004

2212-540X & 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V This is an open access article under the

CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

n Corresponding author.

E-mail address: liancunzheng@ustb.edu.cn (Liancun Zheng).

Peer review under responsibility of National Laboratory for Aeronautics

and Astronautics, China.

Propulsion and Power Research ]]]];](]):]]]–]]]

Griffiths [2] considered the boundary-layer flow due to a

rotating disk for a number of generalized Newtonianfluid

models Dandapat and Singh[3]studied the two-layerfilm

flow over a non-uniformly rotating disk in the presence of

uniform transverse magneticfield under the assumption of

planar interface

The flow due to stretching surfaces is important in the

extrusion processes in plastic and metal industries [4–6]

The steadyflow over a rotating and stretching disk was first

studied by Fang [7] Recently, Fang and Zhang [8]

investigated the flow between two stretching disks More

recently, Turkyilmazoglu [9] studied the steady

magneto-hydrodynamic (MHD) laminar flow of an electrically

conducting fluid on a radially stretchable rotating disk in

the presence of a uniform vertical magneticfield Fang and

Tao [10] investigated the laminar unsteady flow over a

stretchable rotating disk with deceleration Rashidi et al

[11] considered the first and second law analyzes of an

electrically conducting fluid past a rotating disk in the

presence of a uniform vertical magneticfield Asghar et al

[12]studied steady three dimensionalflow and heat transfer

of viscous fluid on a rotating disk stretching in radial

direction Turkyilmazoglu [13] investigated the traditional

Bödewadt boundary layer of an incompressible viscous

fluid flow and heat transfer over a stationary disk provided

that the disk is allowed to radially stretch

The term“nanofluids” was coined by Choi[14]in 1995

at the ASME Winter Annual Meeting Nanofluid is a

colloidal mixture by adding nanoparticles (o100 nm) in a

basefluid, which can considerably change the transport and

thermal properties of the basefluid and thus may improve

thermal conductivity A list of review papers on nanofluids

can be found in Refs.[15–17] Bachok et al.[18]studied

theflow and heat transfer over a rotating porous disk in a

nanofluid Rashidi et al [19] considered the entropy

generation in steady MHD flow due to a rotating porous

disk in a nanofluid Turkyilmazoglu [20] investigated the

flow and heat transfer characteristics over a rotating disk

immersed infive distinct nanofluids

The homotopy analysis method (HAM) introduced by

Liao in 1992[21–26], is an effective mathematical method

which has been successfully employed to solve different

types of nonlinear problems Many studies have verified the

validity and effectiveness of this method In this work, we

obtain the analytical solutions by using the homotopy

analysis method

Although the problem offluid flow over a rotating disk

that is stretching in the radial direction are already

involved in some works as cited above, they have not

yet been considered for nanofluids In this paper we

investigate theflow and heat transfer of nanofluid over a

stretching rotating disk, three types of nanoparticles: Cu,

CuO and Al2O3 are considered Results show that with

increasing in stretching strength parameter, the skin

friction and the local Nusselt number, the velocity in

radial and axial directions increase, whereas the velocity in tangential direction and the thermal boundary layer thick-ness decrease, respectively

2 Formulation of the problem

Consider an incompressible, steady and axially sym-metric nanofluid flow past a rotating disk that is placed at

z¼ 0 and rotates with an angular velocity Ω The disk is further stretching at a uniform rate s in the radial direction r

Physical model of rotating disk is shown in Fig 1 [13] The governing equations of the nanofluid motion and energy in cylindrical coordinates can be presented, respec-tively, as follows

∂u

∂rþ

u

u∂u

∂r

v2

r þ w∂u∂z

þρ1

nf

∂p

∂r ¼

μnf

ρnf

∂2u

∂r2þ1 r

∂u

∂r

u

r2þ∂∂z2u2

ð2Þ

u∂v

∂rþ

uv

r þ w∂v

∂z ¼

μnf

ρnf

∂2v

∂r2þ ∂

∂r

v r

 

þ∂2v

∂z2

ð3Þ

u∂w

∂r þ w

∂w

∂zþ

1

ρnf

∂p

∂z ¼

μnf

ρnf

∂2w

∂r2 þ1 r

∂w

∂rþ

∂2w

∂z2

ð4Þ

u∂T

∂rþ w

∂T

∂z ¼ αnf

∂2T

∂r2 þ1 r

∂T

∂r þ

∂2T

∂z2

ð5Þ The boundary conditions are given by

z¼ 0 : u ¼ sr; v ¼ Ωr; w ¼ 0; T ¼ Tw ð6Þ

where T is the temperature of the nanofluid, T1 is the temperature of the ambient nanofluid, the pressure is

P and the pressure of the ambient nanofluid is P1, μnf

andαnf are the dynamic viscosity and thermal diffusivity of

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Fig 1 Physical model of rotating disk.

Chenguang Yin et al

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are respectively defined as

μnf¼ μf

1φ

ð Þ2 :5; αnf¼ knf

ρCp

nf

; ρnf¼ 1φð Þρfþ φρs

ρCp

nf ¼ 1φð Þ ρCp

fþ φ ρCp

s;

knf

kf

¼ ksþ 2kf

2φ kfks

ksþ 2kf

þ φ kfks

In which,φ is the nanoparticle volume fraction, the range

ofφ is 0 to 1 μf is the viscosity of thefluid fraction, ρf and

ρs are the densities of the fluid and of the solid fractions,

respectively The heat capacitance of the nanofluid is given

by ρCp

nf and knf is the effective thermal conductivity of the nanofluid approximated by the model given by Oztop

[27], which is confined to spherical nanoparticles only

The thermophysical properties of water and different

nanoparticles are given inTable 1 [27]

3 Nonlinear boundary value problem

By means of the Von Karman's transformations,

η ¼ Ω=υf

z; u ¼ ΩrF ηð Þ; v ¼ ΩrG ηð Þ; w ¼ Ωυf

 1=2

Hð Þ;η

pp1¼ 2μfΩp ηð Þ; θ ηð Þ ¼ T Tð 1Þ= Tð wT1Þ: ð9Þ

The system (1)–(5) can be reduced into the following

ordinary differential equations

1

1φ

ð Þ2:5 1φ þ φρs=ρf

!

F″HF0F2þ G2¼ 0

ð11Þ 1

1φ

ð Þ2 :5 1φ þ φρs=ρf

!

G″HG02FG ¼ 0 ð12Þ

1

Pr

Knf=Kf

ρcp

nf= ρcp

f

the transformed boundary conditions become

F 0ð Þ ¼ C; G 0ð Þ ¼ 1; H 0ð Þ ¼ 0; θ 0ð Þ ¼ 1

Fð Þ ¼ G 11 ð Þ ¼ θ 1ð Þ ¼ P 1ð Þ ¼ 0 ð14Þ

measuring the ratio of radial stretch to swirl such that C¼ 0 corresponds to the classical non-stretching case, the range

of C is 0 to 1 Pr is the Prandtl number

The skin friction coefficient Cf and the Nusselt number

Nu are physical quantities which are given by

Cf¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

τwr2þ τw ϕ2

p

ρfð ÞΩr2 ; Nu ¼ rqw

kfðTwT1Þ ð15Þ whereτwr andτw ϕare the radial and transversal shear stress

at the surface of the disk, respectively, and qwis the surface heatflux, which are introduced as

τwr¼ μ nfuzþ wϕ

z ¼ 0; τw ϕ¼ μnf vzþ1

rwϕ

z ¼ 0;

Substituting Eq.(8) in Eq (16)and using Eq (15), we have

Re1=2C

f¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

F0ð Þ02þ G0ð Þ0 2

q

1φ

ð Þ2 :5 ; Re1 =2Nu¼ knf

kf θ0ð Þ0

ð17Þ

Re¼ Ωr2=υf is the local Reynolds number

4 Results and discussion

The nonlinear ordinary differential Eqs.(10)–(13)subjected to the boundary conditions(14)are solved by the Homotopy analysis method[21–26] We obtain

H1ð Þ ¼ 2Ceη  ηð1 þ eηÞh

F1ð Þ ¼η 1

6e

 3ηð1 þ eηÞh 1 þ C2eηþ C2

eη

1φ

ð Þ2 :5 1φ þ φρs=ρf

!

G1ð Þ ¼η 1

6e

 3ηð1 þ eηÞh 2Cþ 2Ceη

1φ

ð Þ2:5 1φ þ φρs=ρf

!

θ1ð Þ ¼ η e 2ηð1 þ eηÞhKnf=Kf

2Pr ρcp

nf= ρcp

f

We then evaluate the errors of Eqs (10)–(13) using the following error estimation functions[28,29]

EH¼

0

H0þ 2F

E F ¼

Z1

0

1

1 φ

ð Þ 2:5 1φ þ φρ s =ρ f

!

F ″HF 0 F 2 þ G 2

! 2

d η ð19Þ

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Table 1 Thermophysical properties of water and different

nanoparticles [27]

Physical properties Pure water Cu CuO Al 2 O 3

ρ/(kg/m 3

EG¼Z 1

0

1 1φ

ð Þ 2:5 1φ þ φρ s =ρ f

!

G ″HG 0 2FG

! 2

d η ð20Þ

Eθ¼

0

1 Pr

Knf=Kf

ρcp

nf= ρcp

f

θ″Hθ0

!2

Substituting a certain order HAM solutions to Eq.(22), we can

obtain the corresponding error at that order, which are shown in

Table 2

The reliability of analytical results are verified with numerical

ones obtained by finite difference technique with Richardson

extrapolation[30,31]and results published in literatures[19]and

[20], which are shown inTable 3

Figs 2–5show the effects of stretching strength parameter C on

the velocity components in radial, tangential and axial directions

and temperature distribution It can be seen that the velocity

profiles in the radial and axial directions increase, whereas the

velocity profiles in the tangential direction and the thermal

boundary layer thickness decrease with the increasing C In order

to validate the analytical results obtained by HAM, the numerical solutions are presented inFigs 2–4, the results are in very good agreement

Figs 6–9 show the effects of the solid volume fraction of nanoparticlesφ for a Cu-water nanofluid on the radial, tangential and axial velocity components and temperature distribution It is indicated that all velocity components decrease, respectively, with the increase in the value of φ The thermal conductivity of nanofluid increases and the thickness for thermal boundary layer increases as well, as the value ofφ increases

The analytical results for the skin friction coefficient Re1 =2C

f

and the local Nusselt number Re 1=2Nu, for a wide range of the nanoparticle volume fraction and three different types of nano-particles are presented inFigs 10and11 It is seen that the values

of the skin friction coefficient and the local Nusselt number are both increase nearly linearly with the nanoparticle volume fraction

Cu has the largest skin friction coefficient and heat transfer rate and Al2O3 has the lowest ones This is because of the largest thermal conductivity value of the Cu compared with other

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Table 2 Computational errors for various of C for Cu-water nano fluid with φ ¼ 0:1 and Pr ¼ 6:2 in the case of h ¼ 1.

Table 3 Comparison of the numerical solutions for F 0 ð Þ,0

G 0 ð Þ, H 10 ð Þ and θ 0 ð Þ, when φ ¼ 0, C ¼ 0 and Pr ¼ 6:2.0

Fig 2 Effects of C on radial velocity pro files F η ð Þ for Cu-water

nano fluid with φ ¼ 0:1 and Pr ¼ 6:2.

Fig 3 Effects of C on tangential velocity pro files G η ð Þ for Cu-water nano fluid with φ ¼ 0:1 and Pr ¼ 6:2.

Fig 4 Effects of C on axial velocity pro files H η ð Þ for Cu-water nano fluid with φ ¼ 0:1 and Pr ¼ 6:2.

Chenguang Yin et al

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nanoparticles Fig 10 displays that the increase of stretching

strength parameter C leads to increase the values of the skin

friction coefficient It also can be seen fromFig 11that the local

Nusselt number increases with the increasing stretching strength parameter C

5 Conclusions

In this paper we investigate theflow and heat transfer of nanofluid over a stretching rotating disk with three types of nanoparticles: Cu, CuO and Al2O3 The nonlinear govern-ing equations are transformed into ordinary differential equations by Von Karman transformations and then solved

by using homotopy analysis method (HAM) The effects of the stretching strength parameter, the solid volume fraction

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Fig 5 Effects of C on temperature pro files θ η ð Þ for Cu-water

nano fluid with φ ¼ 0:1 and Pr ¼ 6:2.

Fig 6 Effects of φ on radial velocity profiles F η ð Þ for Cu-water

nano fluid with φ ¼ 0:1, Pr ¼ 6:2 and C ¼ 0:1.

Fig 7 Effects of φ on tangential velocity profiles G η ð Þ for Cu-water

nano fluid with φ ¼ 0:1, Pr ¼ 6:2 and C ¼ 0:1.

Fig 8 Effects of φ on axial velocity profiles H η ð Þ for Cu-water

nano fluid with φ ¼ 0:1, Pr ¼ 6:2 and C ¼ 0:1.

Fig 9 Effects of φ on temperature profiles θ η ð Þ for Cu-water nano fluid with φ ¼ 0:1, Pr ¼ 6:2 and C ¼ 0:1.

Fig 10 Variation of the skin friction coef ficient with φ for different nanoparticles and C with Pr ¼ 6:2.

Fig 11 Variation of the Nusselt number with φ for different nanoparticles and C with Pr ¼ 6:2.

and the types of nanofluids on velocity and temperature

fields are graphically illustrated and analyzed

Acknowledgements

The work of the authors is

Science Foundations of China (Nos 51276014, 51476191)

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