newtonian heating effect in nanofluid flow by a permeable cylinder

Turkyilma-zoglu [3] examined boundary layer unsteady flow of nanofluid with heat transfer in the presence of vertical flat plate.. Hsiao [17]examined heat and mass transfer in mixed conv

Newtonian heating effect in nanofluid flow by a permeable cylinder

T Hayata,b, M Ijaz Khana,⇑, M Waqasa, A Alsaedib

a

Department of Mathematics, Quaid-i-Azam University 45320 Islamabad 44000, Pakistan

b

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P O Box 80257, Jeddah

21589, Saudi Arabia

a r t i c l e i n f o

Article history:

Received 22 August 2016

Received in revised form 19 November 2016

Accepted 20 November 2016

Available online 23 November 2016

Keywords:

Stretched cylinder

Nanofluid

Newtonian heating

Brownian motion

Thermophoresis

a b s t r a c t

Here characteristics of Newtonian heating in permeable stretched flow of viscous nanomaterial are inves-tigated Adopted nanomaterial model incorporates the phenomena of Brownian motion and ther-mophoresis Concept of boundary layer is employed for the formulation procedure Convergent homotopic solutions are established for the nonlinear systems Velocity, thermal and nanoparticles fields for nonlinear boundary value problems are computed and discussed The velocity, temperature and con-centration gradients are also evaluated It is noticed that impacts of curvature and suction/injection parameters on skin friction coefficient are qualitatively similar Moreover temperature distribution enhances for larger thermophoresis and Brownian motion parameters

Ó 2016 Published 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/)

Introduction

It is well known fact that improvement of human culture

enor-mously depends the energy sources Therefore specialists and

researchers are attempting to build up the new energy assets

and energy innovations in order to utilize the sun powered/solar

energy which is very suitable, easily available and friendly source

for the various heating processes in technological and industrial

processes (which when it reaches the earth is about

4
1015 MW) This solar energy is 2000 times bigger than the

worldwide energy utilization The term nanofluid is coined by Choi

[1] This pioneering experimental research witnessed thermal

con-ductivity enhancement of nanofluid He concluded that addition of

very small amount of nanoparticles to traditional heat transfer

liq-uids enhanced the thermal conductivity of liquid up to two times

Nanoparticles have various shapes for example spherical, rod-like

or tubular Impact of Soret and Dufour on MHD convective

radia-tive heat and mass transfer is presented by Pal et al.[2]

Turkyilma-zoglu [3] examined boundary layer unsteady flow of nanofluid

with heat transfer in the presence of vertical flat plate Hussain

et al.[4]explored magnetohydrodynamic flow of Jeffrey

nanoma-terial induced by exponentially stretched surface in the presence

of Joule heating and thermal radiation Behavior of

magnetohydro-dynamic nanofluid flow over a permeable exponentially stretched

surface was presented by Bhattacharyya and Layek [5] Rashidi

et al.[6]analyzed MHD flow of nanofluid with entropy generation due to rotating porous disk Influences of viscous dissipation and melting heat transport in MHD stretched flow of viscous nanoliq-uid is reported by Mabood and Mastroberardino[7] Hayat et al

[8]addressed Joule heating and radiation characteristics in mag-neto thixotropic nanofluid Analysis of power law nanoliquid towards stretched surface with mixed convection is presented by

Si et al.[9] Hayat et al.[10]analyzed hydromagnetic mixed con-vective flow of viscous nanomaterial subject to curved stretching surface Further relevant studies on nanofluids can be seen in

[11–15]and many studies therein

Fluid flow over a flat plate or stretching cylinder has promising uses in engineering and industrial processes such as polymer expulsion, in a melt turning forms, streamlined expulsion of plastic sheets, glass fiber creation, the cooling and drying of paper and materials, water funnels, sewer funnels, watering system channels, veins and so forth Flow caused by stretching of a sheet is initially investigated by Crane[16] Hsiao [17]examined heat and mass transfer in mixed convection MHD flow of viscoelastic fluid past

a stretched surface Lin et al [18] analyzed stretched flow of pseudo-plastic nanofluid Rosca and Pop[19]analyzed fluid flow over an unsteady curved stretching/shrinking surface Abbas

et al [20] examined radiative flow of nanofluid by a curved stretched surface with partial slip Mixed convection and variable thermal conductivity effects in viscoelastic nanofluid flow over a stretched cylinder is studied by Hayat et al.[21] Simultaneous influences of magnetic dipole and homogeneous/heterogeneous

http://dx.doi.org/10.1016/j.rinp.2016.11.047

2211-3797/Ó 2016 Published by Elsevier B.V.

⇑Corresponding author.

E-mail addresses: mikhan@math.qau.edu.pk (M.I Khan), mw_qau88@yahoo.

com (M Waqas).

Contents lists available atScienceDirect

Results in Physics

j o u r n a l h o m e p a g e : w w w j o u r n a l s e l s e v i e r c o m/ r e s u l t s - i n - p h y s i c s

reactions in stretched flows of ferrofluid is explored by Hayat et al.

[22,23] Alsaedi et al.[24]established homotopic solutions for

vis-cous nanomaterial considering gyrotactic microorganisms Mixed

convection and viscous dissipation characteristics in nonlinear

stretching flow of micropolar material with convective condition

is addressed by Waqas et al.[25] Hayat et al.[26]examined the

impacts of variable properties in three-dimensional mixed

convec-tive flow induced by exponential surface Waqas et al [27]and

Hayat et al.[28]explored the characteristics of non-Fourier flux

theory in flows of generalized Burgers and Jeffrey fluids Farooq

et al.[29]examined nonlinear radiation and

magnetohydrodynam-ics (MHD) T in convective flow of viscoelastic nanoliquid

In many realistic cases the heat transfer from the surface is

pro-portional to the local surface temperature Such an effect is known

as Newtonian heating effect Researchers utilized the Newtonian

heating process in their practical life applications such as conjugate

heat transfer around fins, to design heat exchanger and also in

con-vection flows setup where the surrounding bounding surfaces

absorb heat by solar radiations Merkin[30]presented the

bound-ary layer natural convection flow by a vertical surface in the

pres-ence of Newtonian heating Effect of Newtonian heating on MHD

unsteady flow with Navier slip is analyzed by Makinde[31]

She-hzad et al.[32]studied Jeffrey fluid flow in three dimensional with

Newtonian heating Hayat et al.[33]examined stagnation region

flow of carbon nanotubes with chemical reactions in the presence

of Newtonian heating Newtonian heating and heat generation/

absorption characteristics in flows of micropolar, Jeffrey and

Eyring-Powell materials are discussed by Hayat et al.[34–36]

From above mentioned investigations, it has been noticed that

the heat transfer analysis in the past has been mostly dealt with

the boundary condition either through prescribed temperature or

heat flux at the surface Few studies in this direction are made using

temperature Newtonian heating condition at the surface instead of

prescribed surface temperature or heat flux However, no attempt is

yet presented for the Newtonian heating mass condition at the

sur-face Theme of this study is to introduce Newtonian mass flux

con-dition in the literature Even such concon-dition has not been utilized

yet in flow analysis without nanoparticles We thus attempt here

the flow by a permeable stretching cylinder with Newtonian

heat-ing and mass flux conditions Series solutions are computed by

homotopy analysis method (HAM) [37–50] The behaviors of

different parameters on the physical quantities have been

examined

Governing problems

Here we assume the steady axisymmetric flow of viscous

nano-material along a stretched cylinder of radius R Heat and mass

transfer analysis is reported in presence of Brownian motion,

New-tonian heating and thermophoresis The flow here is being

assumed in the axial xð Þ direction Radial direction is normal to x

Fluid is assumed in compression Whole treatment is considered

via boundary layer assumptions Suction through permeable

cylinder is considered Stretching velocity for cylinder is linear

The governing equations are

@ðruÞ

@x þ

@ðrvÞ

u@u

@xþv@u@r¼m @2u

@r2þ1r @u@r

!

u@T

@xþv@T@r

¼qkcp

@2T

@r2þ1 r

@T

@r

!

þs DB@C

@r

@T

@rþ

DT

T1

@T

@r

 2!

; ð3Þ

u@C

@xþv@C@r¼ DB @2C

@r2þ1r@C@r

!

þDT

T1

@2T

@r2þ1r@T@r

!

The boundary condition for present flow are designed in the form:

uðx;rÞ ¼ uwðxÞ ¼u0x

l ;vðx;rÞ ¼ vw;@T@r¼ h1T;@C@r¼ h2C at r¼ R;

u! 0; T ! T1; C ! C1as r! 1: ð5Þ where the velocity components in (x; r) direction are (u;v) respec-tively,mdenotes kinematic viscosity,qthe density, cp the specific heat,s¼ð Þqc p

q c

ð Þ fthe heat capacity ratio,ð Þqcpthe heat capacity of fluid,

qc

ð Þf the effective heat capacity of nanoparticles, DBthe Brownian diffusion coefficient, DT the thermophoresis diffusion coefficient,

T1the ambient temperature, T the temperature, C the nanoparticles concentration, Tð 1; C1Þ the ambient fluid temperature and concen-tration respectively, uwðxÞ the velocity of stretching surface,vwthe suction/injection velocity, h1the heat transfer coefficient and h2the mass diffusion coefficient

Using

u¼u0x

l f

0ðgÞ;v¼ R

r

ffiffiffiffiffiffiffiffi

u0m

l

r

fðgÞ; hðgÞ ¼T T1

T1 ;

/ðgÞ ¼C C1

C1 ;g¼

ffiffiffiffiffi

u0

ml

r

r2 R2 2R

!

continuity condition is indistinguishably fulfilled and Eqs.(2)–(5)

can be reduced as follows:

1þ 2ag

ð Þf000þ 2af00þ ff00 f02¼ 0; ð7Þ

1þ 2ag

ð Þh00þ 2ah0þ Prf h0þ Pr 1 þ 2ð agÞ N bh0/0þ Nth02

¼ 0; ð8Þ

1þ 2ag

ð Þ/00þ 2a/0þ Scf /0þNt

Nb

1þ 2ag

ð Þh00þ 2ah0

f¼ S; f0¼ 1; h0ð0Þ ¼ c1ð1þ h 0ð ÞÞ;

/0ð0Þ ¼ c2ð1þ / 0ð ÞÞ atg¼ 0;

f0! 0; h ! 0; / ! 0 asg! 1; ð10Þ where a ¼ ffiffiffiffiffiffiffim l

u 0 R 2

q

is the curvature parameter, Nb ¼sD B

m

  the Brownian motion parameter, Pr ¼lc p

k

  the Prandtl number,

Nt ¼sDT

T1m

the thermophoretic parameter, Sc ¼ m

D B

  the Lewis number, S the suction (S> 0) and S the injection (S < 0) and

c1 ¼ h1

ffiffiffiffiv l

u 0

q

and c2 ¼ h2

ffiffiffiffi

vl

u 0

q

are the conjugate heat and mass parameters respectively

Skin friction coefficient Cf

  Nusselt number ðNuxÞ and Sherwood number Shð xÞ are

Cf ¼2sw

qu2 w

; Nux¼ xqw

k Tð w T1Þ; Shx¼ xqm

DBðCw C1Þ; ð11Þ where

sw¼l @u

@r

 

r ¼R

; qw¼ k @T

@r

 

r ¼R

; qm¼ DB @C

@r

 

r ¼R : ð12Þ

In dimensional form

2CfRe

1 =2

x ¼ f00ð0Þ; NuxRe1=2x ¼c1 1þh 01

ð Þ

;

ShxRe1=2x ¼c2 1þ/ 01

ð Þ

where Rex¼u w ðxÞ

v ¼u 0 x 2

vl as the Reynolds number

Homotopic solutions and convergence

In 1992, Liao[37] initiated the concept of homotopy analysis

method (HAM) This method is used to solve highly nonlinear

equations the initial guesses and auxiliary linear operators along

with associate characteristic are

f0ðgÞ ¼ S þ 1  e g; h0ðgÞ ¼ c1

1c1expðgÞ; /0ðgÞ

¼ c2

Lf¼ f000 f0; Lh¼ h00 h; L/¼ /00 /; ð15Þ

LfðC1þ C2egþ C3egÞ ¼ 0; LhðC4egþ C5egÞ ¼ 0;

L/ðC6egþ C7egÞ ¼ 0; ð16Þ

where Ciði ¼ 1  7Þ indicate the arbitrary constants Employing

HAM and solving the corresponding zeroth order and mth

deforma-tion problems one obtains

fmðgÞ ¼ f

mðgÞ þ C1þ C2egþ C3eg; ð17Þ

hmðgÞ ¼ h

mðgÞ þ C4egþ C5eg; ð18Þ

/mðgÞ ¼ /

mðgÞ þ C6egþ C7eg; ð19Þ

in which the fm; h

mand /mindicate the special solutions The values

of Ci¼ ði ¼ 1  7Þ are

C2¼ C4¼ C6¼ 0 and C3¼ @fmð Þg

@g

g¼0;

C1¼ C3 f

mð Þ; C0 5¼

@h 

mð Þg

@ g

g¼0þc1 h

mð Þg

g¼0

1c1

;

C7¼

@/ 

mð Þg

@ g

g¼0þc2 /

mð Þg

g¼0

Convergence analysis

Our motto here is to ensure the convergence of developed series

solutions through homotopy analysis method (HAM) Auxiliary

variableshf; hh; h/

in the developed series solutions have crucial role for such motto ThereforeFig 1outlines theh-plots for 14th

order of approximations It is noticed that the permissible values

of the auxiliary parameters hf; hh and h/ are

1:35 6 hf6 0:30; 1:50 6 hh6 0:35 and 1:70 6 h/6 0:30

Moreover the series converge in the entire region of g when

hf ¼ hh¼ h/¼ 0:6

Discussion

Here impacts of distinct variables on velocity f0ð Þ, temperatureg

hð Þ, nanoparticles concentration /ðg gÞ, skin friction 1

2CfRe1=2x

, Nusselt number NuRe1=2x 

and Sherwood number ShRe1=2x 

are addressed through graphs and Tables To achieve this motto

Figs 2–13along withTables 2–4are presented

Fig 2is displayed for the effect of S on f0ð Þ For higher S > 0g ð Þ some of the fluid particles are sucked through the cylinder which provides a resistance to the fluid flow and therefore the f0ð Þg

decreases.Fig 3disclosed the features ofaon f0ð Þ It is noticedg

that f0ð Þ decays near the stretching surface while it augments asg

one moves away from the stretching surface Physically an increase

inareduces the radius of cylinder due to which the contact area of the cylinder with fluid is reduced Therefore less resistance is pro-vided by the surface and consequently f0ð Þ enhances.g

The analysis of Pr on hð Þ is described ing Fig 4 It is found that

an increase in Pr reduces hð Þ The increase in Pr causes the thin-g

ning of thermal boundary layer which enhances the heat transfer rate As a result the temperature of fluid reduces Hence Pr can

be utilized to control the relative thickness of momentum and associated boundary layer in heat transfer phenomenon Behavior

ofc1on hð Þ is analyzed ing Fig 5 Clearly hð Þ and its associatedg

boundary layer thickness enhances Heat transfer coefficient increases for largerc1 Therefore more heat transfers from the heated surface of cylinder to the cooled surface of the fluid and

as a whole temperature of the fluid increases which transfers more heat from the cylinder to the fluid It is noticed thatc1¼ 0 relates

to insulated wall whilec! 1 represents the constant wall tem-perature Subsequentlyc1can be utilized as a cooling operator as

a part of the progressed innovative procedure.Fig 6displays the variation of Nton hð Þ It is inspected that temperature of the fluidg

Fig 1 h-curve for f ; h and /.

0

rises as value of Ntincreases It is because for larger Nt, the

differ-ence between wall temperature and referdiffer-ence temperature

aug-ments.Fig 7pointed out the impact of Nbon hð Þ It is examinedg

that hð Þ and associated thickness of boundary layer are enhancedg

for larger Nb Physically an increase in Nb, the random movement of

molecules increases which results in an enhancement of hð Þ.g

Characteristics ofaon hð Þ are addressed throughg Fig 8 Clearly

hð Þ reduces by augmentingg a near the stretching surface but it

demonstrates increasing behavior as we move far away from the

surface From physical point of view largeraenhance the thickness

of thermal boundary layer due to which the heat transport rate reduces and thus hð Þ increases.g

Fig 9is sketched to see the impact of Nton /ðgÞ It is examined that an increase in Ntenhances /ðgÞ Physically more nanoparticles are pressed away from the hot surface Consequently the volume fraction distribution /ð ðgÞÞ boosts.Fig 10shows that larger Sch-midt number Scð Þ corresponds to a decrease in /ðgÞ Since weaker Brownian diffusion coefficient rises for higher Sc that retards /ðgÞ and thickness of boundary layer Influence of Nb on /ðgÞ is

por-trayed inFig 11 It is noticed for larger N , the random movement

Fig 3 Effects ofaon f0.

Fig 4 Effects of Pr on h.

Fig 5 Effects ofc1on h.

Fig 6 Effects of N t on h.

Fig 7 Effects of N b on h.

Fig 8 Effects ofaon h.

and also the collision of macroscopic particles of the liquid

enhances which decays /ðgÞ.Fig 12describes the influence ofa

on /ðgÞ It is exposed that /ðgÞ decays near the stretching cylinder

however it enhances as one moves away from the cylinder The

thickness of concentration boundary layer also enhances by

increasing the value of a Behavior of c2 on /ðgÞ is disclosed

throughFig 13 As expected /ðgÞ and corresponding thickness of

boundary layer are enhanced via largerc2

Table 1interprets the convergence analysis of series solutions

It is depicted that 15thorder of approximations are enough for

Table 2 Skin friction coefficient ð 1 C f Re1=2x Þ via a and S when N t ¼ N b ¼ 0:1;c1 ¼ 0:4;

c2 ¼ 0:2; Pr ¼ 1:0 and Sc ¼ 0:6.

Fig 9 Effects of N t on /.

Fig 10 Effects of Sc on /.

Fig 11 Effects of N b on /.

Fig 12 Effects ofaon /.

Fig 13 Effects ofc2 on /.

Table 1 HAM solutions Convergence whena¼ N t ¼ N b ¼ 0:1;c1 ¼ 0:4;c2 ¼ 0:2; Pr ¼ S ¼ 1:0 and Sc ¼ 0:6.

Order of approximations f 00 ð0Þ h 0 ð0Þ / 0 ð0Þ

the convergence of f and h while 35thorder is sufficient for the

con-vergence of /.Table 2indicates the numerical values of skin

fric-tion coefficient for different values of S and a As expected the

skin friction coefficient enhances for larger S and a Impacts of

emerging variables on Nusselt and Sherwood numbers are

demon-strated in Tables 3 and 4 It is examined that Nusselt number

shows increasing behavior for highera; Pr and Sc whereas reverse

behavior is noticed forc1;c2; Ntand Nb(for detail seeTable 3) We

further analyzed that Sherwood number enhances for larger

a; Pr;c2; Nband Sc whereas it reduces viac1and Nt(for detail see

Table 4)

Conclusions

Newtonian heating characteristics in the boundary layer flow of

viscous nanofluid are explored Major highlights of presented

anal-ysis are given below

 Larger suction parameter (S > 0) reduces velocity distribution

f0ð Þ.g

 Temperature distribution hð Þ boosts through larger Ng tand Nb

 Influences ofaand S on skin friction coefficient1CfRe1=2x 

are qualitatively similar

 Nusselt number NuRe1=2

x

diminishes via larger Nt and Nb while Sherwood numberShRe1=2x 

has reverse effects for Nt and Nb

 Effects of Schmidt number Sc on heat and mass transfer rates are similar

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[48] Hayat T, Ali S, Awais M, Alhuthali MS Newtonian heating in stagnation point flow of Burgers fluid Appl Math Mech 2015;36:61–8

[49] Shehzad SA, Hayat T, Alsaedi A, Chen B A useful model for solar radiation Energy Ecol Environ 2016;1:30–8

[50] Shehzad SA, Waqas M, Alsaedi A, Hayat T Flow and heat transfer over an unsteady stretching sheet in a micropolar fluid with convective boundary condition J Appl Fluid Mech 2016;9:1437–45