hydromagnetic flow of third grade nanofluid with viscous dissipation and flux conditions

Hydromagnetic flow of third grade nanofluid with viscous dissipation and flux conditions T.. Here we introduced the prescribed surface mass flux condition to explore the characteristics

T Hussain, S A Shehzad, T Hayat, and A Alsaedi

Citation: AIP Advances 5, 087169 (2015); doi: 10.1063/1.4929725

View online: http://dx.doi.org/10.1063/1.4929725

View Table of Contents: http://aip.scitation.org/toc/adv/5/8

Published by the American Institute of Physics

Hydromagnetic flow of third grade nanofluid with viscous dissipation and flux conditions

T Hussain,1S A Shehzad,2, aT Hayat,3,4and A Alsaedi4

1Faculty of Computing, Mohammad Ali Jinnah University, Islamabad 44000, Pakistan

2Department of Mathematics, Comsats Institute of Information Technology,

Sahiwal 57000, Pakistan

3Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

4Department of Mathematics, Faculty of Science, King Abdulaziz University,

Jeddah 21589, Saudi Arabia

(Received 21 June 2015; accepted 14 August 2015; published online 24 August 2015)

This article investigates the magnetohydrodynamic flow of third grade nanofluid with thermophoresis and Brownian motion effects Energy equation is considered in the presence of thermal radiation and viscous dissipation Rosseland’s approximation

is employed for thermal radiation The heat and concentration flux conditions are taken into account The governing nonlinear mathematical expressions of velocity, temperature and concentration are converted into dimensionless expressions via transformations Series solutions of the dimensionless velocity, temperature and concentration are developed Convergence of the constructed solutions is checked out both graphically and numerically Effects of interesting physical parameters on the temperature and concentration are plotted and discussed in detail Numerical values of skin-friction coefficient are computed for the hydrodynamic and hydro-magnetic flow cases C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4929725]

INTRODUCTION

The suspension of nanoparticles such as Al2O3, Cu or CuO in the base fluids like ethylene glycol, oil or water is known as the nanofluid The recent researchers have paid special attention

to explore the characteristics of nanofluid It is because of higher thermal performance and their potential role for high heat exchange and with zero pressure drop A combination of nanofluid with biotechnical components has been extensively used in various agricultural, biological sensors and pharmaceuticals processes Different types of nanomaterials like nanofibers, nanostructures, nanowires and nanomachines are utilized in the biotechnological applications

The nanofluids can also be involved in heat removal system and the standby safety systems The heat transfer enhancement characteristics of nanofluid are quite useful in the nuclear reactor processes and nuclear reactor safety systems In recent years, the sustainable energy generation is also a challenging issue globally The solar radiation is the most suitable candidate for the renew-able energy with less environmental impact By the implementation of solar power, heat, water and electricity can be directly obtained from nature The scientists and researchers have explored that the solar collector processes and heat transfer rate can be improved through the addition of nanoparticles in the base fluids Nanomaterials are the new energy materials because their size is similar or smaller than the coherent or de Brogile waves The nanoparticles are most suitable to strongly absorb the incident radiation Due to such properties of nanomaterials, the nanofluid in solar thermal system is a new study area for the researchers and engineers Choi1experimentally pointed out that the addition of nanoparticles enhances the thermal conductivity of the fluid twice

a Corresponding author email address: ali_qau70@yahoo.com (S.A Shehzad)

Buongiorno2provided a mathematical model to investigate the effects of Brownian motion and ther-mophoresis on the flow of nanofluid After that the flow of nanofluids is investigated largely by the

different researchers under various types of nanoparticles and flow geometries Some investigations

on nanofluid can be seen in the Refs.3 10and many therein

Very recently, Sheikholeslami et al.11numerically examined the unsteady two-phase nanofluid flow between parallel plates in presence of magnetic field Lin et al.12 reported the unsteady pseudo-plastic nanofluid flow and heat transfer over a thin liquid film in presence of variable ther-mal conductivity and viscous dissipation Analytical solutions of single and multi-phase models of nanofluid with heat transfer are presented by Turkyilmazoglu.13In another study, Turkyilmazoglu14 discussed the convective heat transfer analysis of nanofluids in circular concentric pipes with partial slip conditions Hayat et al.15 analyzed the similarity solutions of three-dimensional viscoelastic nanofluid due to a bidirectional stretching surface Mixed convection peristaltic flow of water based nanofluids with two-different models is studied by Shehzad et al.16Mustafa and Khan17numerically analyzed the boundary layer flow of Casson nanofluid over a nonlinearly stretching sheet The fully developed mixed convection flow of nanofluid in a vertical channel is reported by Das et al.18 The magnetic nanofluid manufacturing applications are enhanced in the industrial processes The magnetic nanofluids are involved in the manufacturing process of biomaterials for wound treatment, gastric medications, sterilized devices etc The magneto nanoparticles significantly ap-peared in cancer therapy, tumor analysis, sink float separation, construction of loud speakers and many others Recently, Sheikholeslami et al.19investigated the effects of applied magnetic field and viscous dissipation on nanofluid flow between two horizontal plates in a rotating system Here they calculated the viscosity and effective thermal conductivity of nanofluid through KKL correlation The radiative hydromagnetic flow of Jeffrey nanofluid induced by an exponentially stretching sheet

is reported by Hussain et al.20 Mixed convection flow of MHD nanofluid over a linearly stretch-ing sheet in the presence of thermal radiation is examined by Rashidi et al.21 The solutions are developed by employing the shooting technique together with fourth-order Runge-Kutta integration criteria Hayat et al.22studied the MHD mixed convection peristaltic transport of nanofluid under Soret and Dufour effects They utilized the slip boundary conditions in the presence of Joule heating and viscous dissipation They presented the analysis via long wave length and low Reynolds number approximation Zhang et al.23considered the MHD flow and radiation effects in nanofluid saturating through a porous medium in presence of chemical reaction They used the variable surface heat flux condition instead of prescribed surface flux condition in this investigation Magnetohydrodynamic free convection flow of Al2O3-water nanofluid under Brownian motion and thermophoresis effects

is discussed by Sheikholeslami et al.24Shehzad et al.25carried out an analysis to examine the MHD mixed convection peristaltic flow of nanofluid in the presence of Joule heating and thermophoresis

In present analysis, we considered the magnetohydrodynamic flow of third grade nanofluid

in the presence of viscous dissipation over a stretching sheet Third grade fluid26 – 30has an ability

to describe the both shear thinning and shear thickening effects This fluid model is the gener-alization of second grade fluid which only exhibits the effects of normal stress We utilized the prescribed surface heat flux and prescribed surface mass flux conditions In the previous studies, the researchers used the constant surface temperature condition or prescribed heat flux condition There

is not a single study in nanofluid literature that dealt with both the conditions Here we introduced the prescribed surface mass flux condition to explore the characteristics of third grade nanofluid with viscous dissipation and thermal radiation Mathematical formulation is made under boundary layer assumptions The expression of thermal radiation is invoked via Rosseland’s approximation Governing nonlinear problems along with corresponding boundary conditions are solved through homotopy analysis method (HAM).31 – 38 Results of interesting physical parameters on the dimen-sionless temperature and nanoparticles concentration fields are presented graphically and examined carefully

GOVERNING PROBLEMS

We consider the two-dimensional steady incompressible flow of third grade nanofluid with thermophoresis and Brownian motion effects The flow is considered to be thermally radiative An

applied magnetic field of strength B0is applied normal to the flow direction Effects of induced magnetic field are neglected due to the low magnetic Reynolds number The prescribed surface heat flux and mass flux conditions are imposed at the boundary The governing equations of third grade nanofluid with viscous dissipation and thermal radiation after employing the boundary layer theory are expressed as follows:

∂u

∂x +

∂v

u∂u

∂x +v

∂u

∂ y = ν

∂2u

∂ y2 +α1 ρ

(

u ∂3u

∂x∂ y2 + v∂ y∂3u3 +∂u∂x∂ y∂2u2 + 3∂ y∂u∂x∂ y∂2u ) + 2α2

ρ

∂u

∂ y

∂2u

∂x∂ y +6

α3 ρ

(∂u

∂ y

)2∂2u

∂ y2 −σB2

0

ρf u,

(2)

u∂T

∂x +v

∂T

∂ y = α

∂2T

∂ y2+ τ *

,

DB∂C

∂ y

∂T

∂ y +

DT T∞

(∂T

∂ y

)2

+

-+ ν

(ρc)f

(∂u

∂ y

)2 + α1 (ρc)f

(

u∂u

∂ y

∂2u

∂x∂ y +v

∂u

∂ y

∂2u

∂ y2 )

+ 2 α3 (ρc)f

(∂u

∂ y

)4

(ρc)f

∂qr

∂ y,

(3)

u∂C

∂x +v

∂C

∂ y =DB

∂2C

∂ y2 +DT T∞

∂2T

The appropriate boundary conditions for the present flow problems are

u= uw(x) = cx, v = 0, at y = 0; u → 0, v → 0 as y → ∞ (5) The boundary conditions for the prescribed heat flux (PHF) and prescribed concentration flux (PCF) are imposed as follows:

PHF : −k

(∂T

∂ y

)

= Twat y= 0 and T → T∞when y → ∞, (6) PCF : −DB

(∂C

∂ y

)

= Cwat y= 0 and C → C∞when y → ∞, (7)

in which u and v denote the velocity components parallel to the x− and y− directions, α1, α2and

α3 are the material parameters, ν= (µ/ρ) is the kinematic viscosity, µ is the dynamic viscosity,

ρf is the density of fluid, T is the fluid temperature, α is the thermal diffusivity, τ = (ρc) p

(ρc) f is the ratio of nanoparticle heat capacity and the base fluid heat capacity, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, C is the nanoparticle concentration, Twis the temperature of the fluid at the wall, Cwis the nanoparticle concentration of the fluid at the wall and T∞and C∞are the ambient temperature and nanoparticle concentration far away from the sheet The radiative heat flux qrvia Rosseland’s approximation is given below

qr= 4σ 3k∗

∂T4

in which σ is the Stefan-Boltzmann constant and k∗is mean absorption coefficient It is assumed that the temperature difference within the flow is such that T4can be written in the linear combi-nation of temperature By expanding T4about T∞in terms of Taylor’s series and neglecting higher order terms we have

and

∂qr

∂ y =−

16σT3

∞ 3k∗

∂2T

Eq (3) now becomes

u∂T

∂x +v

∂T

∂ y = α

∂2T

∂ y2+ τ *

,

DB∂C

∂ y

∂T

∂ y +

DT T∞

(∂T

∂ y

)2

+

-+ ν

Cp

(∂u

∂ y

)2 + α1 ρCp

(

u∂u

∂ y

∂2u

∂x∂ y +v

∂u

∂ y

∂2u

∂ y2 )

+ 2ρCα3 p

(∂u

∂ y

)

(ρc)f

16σT∞3 3k∗

∂2T

∂ y2

(11)

The transformations for the considered flow problems can be put in the forms

u= cx f′

(η), v = −√cν f (η), η = y cν, T = T∞+

ν c

Tw

k θ(η),

C= C∞+

ν c

Cw

DBφ(η)

(12)

Now Eq (2) is satisfied automatically and the Eqs (3)-(6) and (11) can be expressed as follows:

f′′′+ ff′′− f′2+ β1(2 f′f′′′−ff′′′′

) + (3 β1+ 2β2) f′′2+ 6ε1ε2f′′′f′′2− M2f′= 0, (13) (

1+4

3Rd

)

θ′′+ Pr f θ′+ Pr Nbθ′φ′+ Pr Ntθ′2 + Pr Ec f′′2+ Pr Ecε1f′f′′2− Pr Ecε1ff′′

f′′′+ 2 Pr Ecφφ1f′′4= 0, (14)

φ′′+ Pr Le f φ′+ (Nt/Nb) θ′′= 0, (15)

f = 0, f′= 1, θ′= −1, φ′= −1 at η = 0, (16)

Here β1= cα1

µ , β2= cα2

µ , ε1=cα3

µ are the material parameters for third grade fluid, ε2=c x 2

ν is the local Reynolds number, M2= σB2

0/ρfcis the Hartman number, Pr= ν/α is the Prandtl number,

Rd= 4σT3

∞/kk∗ is the radiation parameter, Ec= u2w

cpT ∞ is the Eckert number, Nb= (ρc)pDB(Cf

− C∞)/(ρc)fν is the Brownian motion parameter, Nt= (ρc)pDT/(ρc)fν is the thermophoresis parameter and Le= α/DBis the Lewis number

The dimensionless expressions of skin-friction coefficient, local Nusselt and Sherwood num-bers can be expressed as follows:

CfxRe1/2x = f′′

(0) + β1(3 f′(0) f′′(0) − f (0) f′′′(0)) + 2ε1ε2f′′3(0), NuRe−1/2x =

(

1+4

3Rd

) ( 1 θ(0)

) , ShRe−1/2

DEVELOPMENT OF SERIES SOLUTIONS

To develop the solution expressions via homotopy analysis method, the initial guesses and auxiliary linear operators are expressed in the forms (Liao (2009)):

f0(η) = 1 − exp(−η), θ0(η) = exp(−η), φ0(η) = exp(−η), (19)

Lf = f′′′− f′, Lθ= θ′′−θ, Lφ= φ′′−φ (20) The above initial guesses and auxiliary linear operators satisfy the following conditions

Lf(C1+ C2eη+ C3e−η) = 0, Lθ(C4eη+ C5e−η) = 0, Lφ(C6eη+ C7e−η), (21) where Ci(i = 1 − 7) are the arbitrary constants The problems at zeroth order deformation are given

by Hayat et al (2014b), Abbasbandy et al (2014) and Shehzad et al (2014b):

(1 − p) L  ˆf(η; p) − f0(η) = p~fN  ˆf(η; p) , (22)

(1 − p) Lθθ(η; p) − θ0ˆ (η)

= p~θNθ ˆf(η; p), ˆθ(η, p), ˆφ(η, p) , (23) (1 − p) Lφφ(η; p) − φ0(η)ˆ 

= p~φNφ ˆf(η; p), ˆθ(η, p), ˆφ(η, p) , (24) ˆ

f(0; p) = 0, ˆf′(0; p) = 1, ˆθ′(0, q) = −1, ˆφ′

(0, q) = −1, ˆ

f′(∞; p) = 0, ˆθ(∞, p) = 0, ˆφ(∞, p) = 0,

(25)

Nf[ ˆf(η, p)] = ∂3fˆ(η, p)

∂η3 + ˆf(η, p)∂2f∂ηˆ(η, p)2 −

(∂ ˆf(η, p)

∂η

)2

− M2∂ ˆf(η, p)

∂η + β1

(

2 ˆf(η, q)∂3fˆ(η, q)

∂η3 − ˆf(η, q)∂4fˆ(η, q)

∂η4 )

+ (3β1+ 2β2)

(∂2fˆ(η, p)

∂η2

) + 6ε1ε2∂3f∂ηˆ(η, p)3

(∂2fˆ(η, p)

∂η2

)2 ,

(26)

Nθ[ ˆθ(η, p), ˆf(η, p), ˆφ(η, p)] =

(

1+4

3Rd

) ∂2θ(η, p)ˆ

∂η2 + Pr ˆf(η, p)∂ ˆθ(η, p)∂η + Pr Nb∂ ˆθ(η, p)∂η ∂ ˆφ(η, p)∂η + Pr Nt(∂ ˆθ(η, p)∂η )

2

+ Pr Ec(∂2f∂ηˆ(η, p)2 )

2 + Pr Ecβ1∂ ˆf(η, p)∂η (∂2f∂ηˆ(η, p)2 )

2

− Pr Ec β1fˆ(η, p)∂ ˆf(η, p)

∂η

(∂2fˆ(η, p)

∂η2 )2

+ 2 Pr Ecε1ε2(∂2fˆ(η, p)

∂η2

)4 ,

(27)

Nφ[ ˆφ(η, p), ˆf(η, p), ˆθ(η, p)] = ∂2φ(η, p)ˆ∂η2 + Pr Le ˆf(η, p)∂ ˆφ(η, p)∂η + Nt

N b

∂2θ(η, p)ˆ

∂η2 (28)

In above equations, p is an embedding parameter, ~f, ~θ and ~φ are the non-zero auxiliary parameters and Nf, Nθ and Nφare the nonlinear operators For p= 0 and p = 1 we have (Rashidi

et al (2014b):

ˆ

f(η; 0) = f0(η), ˆθ(η, 0) = θ0(η), ˆφ(η, 0) = φ0(η), ˆ

f(η; 1) = f (η), ˆθ(η, 1) = θ(η), ˆφ(η, 1) = φ(η) (29) When variation of p is considered from 0 to 1 then f(η, p), θ(η, p) and φ(η, p) vary from f0(η),

θ0(η), φ0(η) to f (η), θ(η) and φ(η) Using Taylor series we can write (Hayat et al (2014c) and Shehzad et al (2014c)):

f(η, p) = f0(η) +

∞



m =1

fm(η)p m, fm(η) = 1

m!

∂mf(η; p)

∂ηm

p =0

θ(η, p) = θ0(η) +

∞



m =1

θm(η)pm, θm(η) = 1

m!

∂mθ(η; p)

∂ηm

p =0

φ(η, p) = φ0(η) +

∞



m =1

φm(η)pm, φm(η) = 1

m!

∂mφ(η; p)

∂ηm

p =0

The convergence of Eqs (30)-(32) strongly depends upon the suitable values of auxiliary parame-ters ~f, ~θ and ~φ We select that ~f, ~θ and ~φin such a manner that Eqs (30)-(32) converge for

p= 1 and thus one has

f(η) = f0(η) +

∞



m =1

θ(η) = θ0(η) +

∞



m =1

φ(η) = φ0(η) +

∞



m =1

The general solutions are represented by the following expressions:

fm(η) = f ∗

θm(η) = θ∗

φm(η) = φ∗

in which fm∗ θ∗

mand φ∗mare the special solutions

ANALYSIS AND DISCUSSION

The equations (30)-(33) depends upon the auxiliary parameters ~f, ~θ and ~φ which have vital role in controlling and adjusting the convergence regions for the development of homotopic solutions For this purpose, we plotted the ~− curves at 15th-order of approximations that give us the proper ranges of these auxiliary parameters Fig.1 demonstrates that the admissible values of

~f, ~θ, ~φare −0.90 ≤ ~f ≤ −0.20, −0.90 ≤ ~θ ≤ −0.40 and −0.90 ≤ ~φ≤ −0.40 Here the series solutions are convergent in the whole region of η for ~f = −0.6 = ~θ= ~φ

The effects of interesting physical parameters namely material parameters β1and β2, Hartman number M, Prandtl number Pr, Brownian motion parameter N b, thermophoresis parameter Nt, Eckert number Ec and thermal radiation parameter Rd on the dimensionless temperature field θ(η) are investigated in the Figs.2-9 Figs 2and3 are plotted to see the behavior of the temperature profile for different values of material parameter β1 and β2 Here we examined that an increase

in the material parameters β1and β2show a reduction in the temperature and thermal boundary layer thickness It is also seen that the temperature at the wall also decreases for the larger material parameters β1and β2 Physically, an increase in material parameters β1 and β2correspond to an enhancement in the normal stress differences due to which lower temperature and thinner thermal boundary layer thickness are achieved Fig 3 presents the variations in temperature profile for

different values of Hartman number This Fig depicts that the temperature field and thermal bound-ary layer thickness are stronger for hydromagnetic flow situation in comparison to hydrodynamic

FIG 1 ~ – curves for functions f (η), θ(η) and φ(η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nt = 0.1 = Nb,

M = 0.5 = Ec and Rd = 0.3.

FIG 2 Influence of β 1 on temperature θ (η) when ε 1 = 0.2, β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and

Rd = 0.3.

FIG 3 Influence of β 2 on temperature θ (η) when β 1 = 0.2 = ε 1 , ε 2 = 0.1, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and

Rd = 0.3.

FIG 4 Influence of M on temperature θ(η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nt = 0.1 = Nb, Ec = 0.5 and

Rd = 0.3.

FIG 5 Influence of Pr on temperature θ (η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Le = 0.7, Nt = 0.1 = Nb, M = 0.5 = Ec and

Rd = 0.3.

FIG 6 Influence of Nb on temperature θ (η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nt = 0.1, M = 0.5 = Ec and

Rd = 0.3.

FIG 7 Influence of Nt on temperature θ(η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nb = 0.1, M = 0.5 = Ec and

Rd = 0.3.

FIG 8 Influence of Ec on temperature θ(η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 and

Rd = 0.3.

flow(M = 0) The larger values of Hartman number lead to higher temperature and thicker thermal boundary layer thickness Obviously Lorentz force appears in Hartman number Hence Lorentz force resists the flow that caused to en enhancement in the temperature and thermal boundary layer thickness From Fig 5 we studied that an increase in the values of Prandtl number creates

a reduction in the temperature field and its related boundary layer thickness An enhancement in Prandtl number corresponds to weaker thermal diffusivity It is well known fact that the fluids with weaker thermal diffusivity have lower temperature Due to such weaker thermal diffusivity, lower temperature and thinner thermal boundary layer thickness is obtained in Fig.5 It is also seen that the temperature at the wall is decreased gradually for the larger Prandtl number

Figs.6and7are displayed to observe the influences of Brownian motion and thermophoresis parameters on the dimensionless temperature field θ(η) These Figs illustrate that an enhancement

in the Brownian motion and thermophoresis parameters lead to larger temperature and thicker thermal boundary layer thickness Clearly the Brownian motion and thermophoresis parameters appeared due to the presence of nanoparticles The presence of nanoparticles enhances the thermal conductivity of fluid The fluids with stronger thermal conductivity have higher temperature Due to such reasons an enhancement in the temperature and thermal boundary layer thickness is observed corresponding to the larger values of Brownian motion and thermophoresis parameters From Fig.8,

it is examined that the temperature and thermal boundary layer thickness are increased when we give rise to the values of Eckert number Ec Temperature is higher in the presence of viscous

FIG 9 Influence of Rd on temperature θ(η) when β 1 = 0.2 = ε 1 , β 2 = 0.1 = ε 2 , Pr = 0.7 = Le, Nt = 0.1 = Nb and M = 0.5

= Ec.

in the material parameters β 1and. .. convergence of Eqs (30)-(32) strongly depends upon the suitable values of auxiliary parame-ters ~f, ~θ and ~φ We select that ~f, ~θ and ~φin such a manner that Eqs (30)-(32) converge for

p= and. .. temperature profile for

different values of Hartman number This Fig depicts that the temperature field and thermal bound-ary layer thickness are stronger for hydromagnetic flow situation