Nanofluids Slip Mechanisms on Hydromagnetic Flow of ...Olanrewaju.P.O et al. (2011)[14].Similarity...

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450

© Research India Publications. http://www.ripublication.com

12440

Nanofluids Slip Mechanisms on Hydromagnetic Flow of Nanofluids over a

Nonlinearly Stretching Sheet under Nonlinear Thermal Radiation

1S.P.Anjali Devi and 2Mekala Selvaraj

1Former Professor & Head, Department of Applied Mathematics, Bharathiar University, Coimbatore-46 Tamilnadu, India. 2Research Scholar,Department of Mathematics, Bharathiar University, Coimbatore-641046 Tamilnadu, India.

Orcid Id:0000-0001-7485-5114 Abstract

In this paper, Heat transfer characteristics of two

dimensional, steady hydromagnetic boundary layer flow of

water based nanofluids containing metallic nanoparticles

such as copper (Cu) and Silver (Ag) over a nonlinearly

stretching surface taking into account the effects of

nonlinear thermal radiation and viscous dissipation has been

investigated numerically. The model used for the nanofluids

incorporates the effects of Brownian motion and

thermophoresis. The governing nonlinear partial differential

equations were transformed into nonlinear ordinary

differential equations using similarity transformations and

then are solved numerically subject to the transformed

boundary conditions by most efficient Nachtsheim- Swigert

shooting iteration scheme for satisfaction of asymptotic

boundary conditions along with fourth order Runge-Kutta

Integration method. Numerical computations are carried out

for distributions of velocity, temperature and nanoparticles

volume fraction by means of graphs for different values of

physical parameters such as magnetic interaction parameter,

nonlinear stretching parameter, Eckert number, temperature

ratio parameter, radiation parameter, Prandtl number,

Brownian motion parameter, thermophoresis parameter and

Lewis number. The numerical results of the problem are

validated by comparing with previously published results in

the literature. Numerical values of skin friction coefficient

and Nusselt number at the wall are also obtained and given

in tabular form. Sherwood number is vanished due to new

mass flux condition.

Key words: Nanofluid, Stretching Sheet, MHD, Radiation.

Nomenclature

c stretching coefficient B0 magnetic induction nanoparticle volume fraction ∞ ambient nanoparticle volume fraction DB Brownian diffusion coefficient DT thermophoretic diffusion coefficient f dimensionless stream function Ec Eckert number k* Rosseland mean absorption coefficient Le Lewis number

M2 magnetic field parameter n nonlinear stretching parameter

Nb Brownian motion parameter

Nt thermophoresis parameter

Nux local Nusselt number

Pr Prandtl number

qr radiative heat flux Rexlocal Reynolds number

T temperature of the nanofluid within the boundary layer Twtemperature at the surface of the sheet T∞temperature of the ambient nanofluid u velocity along the surface of the sheet v velocity normal to the surface of the sheet (x, y) Cartesian coordinates Greek symbols

nfthermal diffusivity of the nanofluid

ρnf density of the nanofluid (cp)nf heat capacity of the nanofluid μnf viscosity of the nanofluid υnf kinematic viscosity of the nanofluid ψ stream function η similarity variable θ dimensionless temperature θw surface wall temperature φ dimensionless rescaled nanoparticle volume fraction κnf thermal conductivity of the nanofluid

τ nanoparticle heat capacity ratio σ magnetic permeability σ* Stefan-Boltzmann constant Subscripts

w surface conditions ∞ conditions far away from the surface

Superscripts

differentiation with respect to η

mailto:[email protected]



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INTRODUCTION

Ultrahigh performance cooling is one of the most vital needs

of many industrial technologies. Nanofluids which exhibit

ultra high performance cooling are engineered by

suspending nanoparticles with average size below 100nm in

traditional heat transfer fluids such as water, oil and

ethylene glycol. Nanofluid is the term coined by Choi

(1995) [5] to describe the new class of nanotechnology

based heat transfer fluids that exhibit thermal properties

superior to those of their host fluids or conventional particle

fluid suspensions. A comprehensive study on the nanofluids

characteristics is documented by Das et al.(2007)[8]. Kaufui

V. Wong and Omar De Leon (2010)[11] presented the wide

range of applications of nanofluids in current and future

such as nuclear reactors, transportation, electronics cooling,

biomedicine and food. Ahmad et al. (2011)[2] presented a

numerical study of the Blasious and Sakiadis flows in

nanofluids under isothermal condition. Their results

revealed that solid volume fraction affects the fluid flow and

heat transfer characteristics of nanofluids. An analytical

derivation of effective thermal conductivity of nanofluids

which incorporates the contribution of interfacial layer as

well as the Brownian motion was solved by Ritu Pasrija and

Sunita Srivastava (2013)[22]. Sandeep Pal et al.(2014)[24]

has presented a review on enhanced thermal conductivity of

colloidal suspension of nanosized particles (nanofluids).The

recent literature of nanofluids was reviewed by Mohameed

Saad Kamel et al.(2016)[13].

Steady boundary layer flow of incompressible fluids over a

stretching sheet has considerable bearing on various

technological processes. The flow over a stretching plate

was first considered by Crane (1970)[7] who found a closed

form analytic solution of the self-similar equation for steady

boundary layer flow of a Newtonian fluid. MHD was

initially known in the field of astrophysics and geophysics

and later becomes very important in engineering and

industrial processes. Pavlov (1974)[16] gave an exact

similarity solution of the MHD boundary layer equations for

the steady two-dimensional flow of an electrically

conducting fluid due to the stretching of a plane elastic

surface in the presence of a uniform transverse magnetic

field. Anjali Devi and Thiyagarajan (2006)[9] solved the

problem of steady nonlinear MHD flow of an

incompressible, viscous and electrically conducting fluid

with heat transfer over a surface of variable temperature

stretching with a power law velocity in the presence of

variable transverse magnetic field.

The role of thermal radiation is of major importance in some

industrial applications such as glass production, furnace

design, nuclear power plants space technology such as in

comical flight aerodynamics rocket, propulsion systems,

plasma physics and space craft reentry aerodynamics which

operates high temperatures. The effect of thermal radiation

on the boundary layer flow has been investigated by Rafael

Cortell (2008)[6].

Viscous dissipation plays an important role in changing the

temperature distribution which affects the heat transfer rates

considerably. The thermal radiation and viscous dissipation

effects on the laminar boundary layer about a flat plate in a

uniform stream of fluid (Blasius flow), and about a moving

plate in a quiescent ambient fluid (Sakiadis flow) both under

convective boundary condition is presented by

Olanrewaju.P.O et al. (2011)[14].Similarity solutions to

boundary layer flow and heat transfer of nanofluid over

nonlinearly stretching sheet with viscous dissipation effects

was studied by Hamad and M.Ferdows (2012)[10]. The

effect of variable viscosity on the flow and heat transfer of a

viscous Ag- water and Cu-water nanofluids was investigated

by Vajravelu (2012)[28].Convective-radiation effects on

stagnation point flow of nanofluids over a

stretching/shrinking surface with viscous dissipation was

studied by Pal et al.(2014)[15]. The radiating and

electrically conducting fluid over a porous stretching surface

with the effect of viscous dissipation was researched by

Sreenivasalu et al. (2016)[26].

Buongiorno (2006)[4] proposed a mathematical nanofluid

model by taking into account the Brownian motion and

thermophoresis effects on flow and heat transfer fields.In his

work he has considered seven slip mechanisms those affect

nanofluid flow such as inertia, Brownian diffusion,

thermophoresis, diffusiophoresis, Magnus effect, fluid

drainage and gravity. He indicated that of those seven only

Brownian diffusion and thermophoresis are important slip

mechanisms in nanofluids. Reza Azizian et al. (2012)[21]

has investigated the effect of nanoconvection caused by

Brownian motion on the enhancement of thermal

conductivity in nanofluids. The non-linear stretching of a

flat surface in a nanofluid with Brownian motion and

thermophoresis effects was investigated by Rana and

Bhargava (2012)[18].

The temperature-dependent thermo-physical properties on

the boundary layer flow and heat transfer of a nanofluid past

a moving semi-infinite horizontal flat plate in a uniform free

stream with the effects of Brownian motion, thermophoresis

and viscous dissipation due to frictional heating are

analyzed by Vajravelu and Prasad (2012)[29].The effects

of thermal radiation and viscous dissipation on

magnetohydrodynamic (MHD) stagnation point flow and

heat transfer of nanofluids towards a stretching sheet are

investigated by Yohannes Yirga and Bandari Shankar

(2013)[30]. The problem of laminar fluid flow which results

from a permeable stretching of a flat surface in a nanofluid

with the effects of heat radiation, magnetic field, velocity

slip, brownian motion and thermophoresis parameters and

convective boundary conditions have been examined by

Reddy (2014)[20].Sumalatha et al.(2016)[27] published the



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mixed convection flow of nanofluids past a nonlinear

stretching sheet in the existence of nanofluids important slip

mechanisms with MHD, variable surface temperature and

volume fraction.

Motivated by the above discussed investigations and

applications, in this present work mainly concentrate on the

effects of nonlinear thermal radiation, viscous dissipation

and variable magnetic field on heat transfer flow of

nanofluids (Cu Water nanofluid and Ag Water nanofluid)

over a nonlinearly stretching sheet with variable surface

temperature. And also the model includes the effects of

Brownian motion and Thermophoresis effects.

MATHEMATICAL FORMULATION

Consider two-dimensional, hydromagnetic flow over a

nonlinearly stretching sheet with convective heat transfer in

water based nanofluids containing copper (Cu) and Silver

(Ag) nanoparticles and the Cartesian coordinates such as x -

axis runs along the direction of the continuous stretching

surface and the y - axis is measured normal to the surface of

the sheet. It is also considered that the sheet is stretching

with velocity Uw = cxn, where c > 0.Let us assume, the base

fluid (water) and the nanoparticles are in equilibrium and the

nanofluids is viscous and incompressible.(See Fig. i).

Figure i: Physical model of the problem

Taking into account the effects of Brownian motion and

thermophoresis and based on model developed by

Buongiorno [4]. The basic steady boundary-layer equations

in the presence of variable magnetic field, nonlinear thermal

radiation and viscous dissipation are given by

0

yv

xu

(1)

22

2

( )nf

nf

B xu u uu v ux y y

(2)

2 22

2nfnf

T rp B nfsp

D qT T T T T uc u v k c D yx y y y T y yy

(3)

2 2

2 2

TB

D Tu v Dx y y T y

(4)

The boundary conditions are given by

u = uw(x) = cxn, v =0,T = Tw (x) = T∞ + bxm ,

0TBD TD

y T y

at y = 0

u=0,T→T∞,, as y→∞ (5)

In the above boundary conditions, assume m = 2n is a

surface temperature parameter and the nanoparticle mass

flux due to the Brownian motion and thermophoresis effects

tends to zero at the boundary(y=0)[A.V.Kuznetsov and

D.A.Nield [12]].

where the symbols are as defined in the nomenclature.

The variable magnetic field B(x) = B0 x (n-1)/2 (Afzal 1993)[1]

is applied in the transverse direction. The magnetic

Reynolds number is assumed to be small so that the induced

magnetic field is negligible in comparison with the applied

magnetic field. Since the induced magnetic field is neglected

and B0 is independent of time, 0curl E . Also,

0Ediv

in the absence of surface charge density. Hence

0E

.

The Rosseland approximation [Rosseland (1936)[23],Raptis

(1998)[19], Sparrow and Cess(1978)[25], Brewster

(1992)[3]] is used to describe the radiative heat flux which

is negligible in x direction in comparison to that in y direction. Full radiation term has been taken into account.

Employing the Rosseland diffusion approximation, the

radiative heat flux is modeled as

yTT

kq *

*

r3

3

16σ

(6)

Hence

2* 22 3

* 2

163

3r

T Tq T Ty k y y

(7)

where σ* is the Stefan Boltzmann constant, k* is the Rosseland mean absorption coefficient.

The nonlinear governing equations (1) to (4) with the

boundary conditions (5) are solved by employing the

similarity transformations which are given below.

1

22

1

nfc x f

n

, 1

21

2

n

f

c ny x

,

w

T TT T

,

(8)

Where is the similarity space variable and f is the

dimensionless stream function.



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Using the Stream function

uy

and vx

The velocity components are expressed as follows

nxu c f ,

1

2

1

2

-1-

1

nc n f x nv f fn

(9)

Using the similarity transformations (9), equation of

continuity (1) is automatically satisfied and the equations

(7),(8) and (9), the nonlinear partial differential equation (2),

(3) and (4) with boundary conditions (5) are reduced to the

following nonlinear ordinary differential equations

2 2

1 1

20

1

nf b c ff f M fn

(10)

3

2

1

2

2 2

4 4 21 1 1 1 1 13

4 Pr.Pr.

1

f fw

R Rnf nf

f

nf

k kw wN k N k

k n Ecc f f f Nb Ntk n b

(11)

Pr 0NtLe fNb

(12)

Here, b1, c1 and c2 are constants whose values are given in

Appendix.

The appropriate boundary conditions are

0 0f , 0 1f , 0 1

0 0 0Nb Nt at =0,

0,f 0 0 as (13)

The nondimensional parameters appeared in Equations (10)

to (12) are defined as follows

22 02

1 fcBM

n

is the

magnetic interaction parameter, 3

*

4 *

fR

k kN

T is the

radiation parameter Prpf f

f

c

k

is the Prandtl

number ,

TTw

w is the Temperature ratio parameter,

2

w

p wf

uEc

c T T

is the Eckert number,

( )

( )

p s B

p f f

c DNb

c

is the Brownian motion parameter,

( )( )

p s T w

p f f

c D T TNt

c T

is the thermophoresis

parameter and f

B

LeD

is the Lewis number.

Skin-friction coefficient

The skin friction coefficient (rate of shear stress) is defined

as

2

wf

f w

CU

, where

0

w nfy

uy

(14)

Substituting equations (8) and (9) into equation (14),

1/2

2.5

1Re = 0

1x

n fC f

Nusselt number

The Nusselt number (rate of heat transfer) is defined as

w

xf w

q xNu

k T T

, where surface heat flux is

3

0

16

3w nf

y

Tq k Tk y

(15)

Using equations (8) and (9), equation (15) can be written as

341 1 (0)

Re 3

nf fxw

f nfx R

k kNun

k k N

Here,

12

Ren

xf

c x

Due to the effects of Brownian motion and

thermophoresis at the boundary, the Sherwood number

vanishes because which characteristics the mass flux is zero

at y=0.

Numerical Solutions

In this work, steady, two dimensional, hydromagnetic

boundary layer flow of nonlinearly stretching surface over

two types of nanofluids namely Cu – Water nanofluid and

Ag – Water nanofluid in the presence of viscous dissipation

and nonlinear thermal radiation and also the effects of

Brownian motion and thermophoresis has been investigated.

The governing nonlinear partial differential equations are

converted to nonlinear ordinary differential equations by

similarity transformations incorporating the necessary

similarity variables. The resulting nonlinear ordinary

differential equations (10) to (12) along with the relevant

boundary conditions (13) constitute a nonlinear boundary

value problem which is difficult to solve analytically.

Hence, these equations are solved using the most efficient

shooting method such as the Nachtsheim-Swigert shooting

iteration scheme for satisfaction of the asymptotic boundary

conditions along with the Fourth-order Runge Kutta

integration method. The difficulty lies in guessing the values



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for ''(0),f (0) and (0) properly to get the convergence and solution. The level of accuracy for

convergence is chosen as 10-5.

RESULTS AND DISCUSSIONS

The numerical and graphical results for two types of water

based nanofluids such as Cu-water nanofluid and Ag-water

nanofluid are presented. The value of the Prandtl number for

the base fluid (water) is kept to be the constant Pr = 6.2.

In order to verify the accuracy of the present method, we

have compared our results with those of Cortell [17] and

Hamad et al.[10] for the Skin friction coefficient –f (0) and

nondimensional rate of heat transfer -0 in the absence of

nanoparticles ( = 0), Magnetic interaction parameter and

viscous dissipation parameter and without thermal radiation

parameter ( NR ) , Brownian motion and thermophoresis

which is shown in Table 2 and Table 3. It is clearly note that

our results are good agreement with that of Cortell and

Hamad et al.

Table 2: Comparison of results for −f(0) when = 0 and M2 = 0.0

n Cortell Hamad et al. Present work

0.0

0.2

0.5

1.0

3.0

10.0

20.0

0.6276

0.7668

0.8895

1.0000

1.1486

1.2349

1.2574

0.6369

0.7659

0.8897

1.0043

1.1481

1.2342

1.2574

0.6276

0.7668

0.8895

1.0000

1.1486

1.2348

1.2574

Table 3: Comparison of results for−θ(0) when = 0, Pr = 5.0,Ec = 0.0 and NR

n Cortell Hamad et al. Present work

0.75

1.5

7.0

10.0

3.1250

3.5677

4.1854

4.2560

3.1246

3.5672

4.1848

4.2560

3.1251

3.5679

4.1854

4.2558

Fig.1 to Fig.13 demonstrate the influence of

Magnetic interaction parameter, nonlinear stretching

parameter, viscous dissipation parameter, surface

temperature parameter, radiation parameter, Lewis number,

Brownian motion and thermophoresis parameter

respectively on velocity distribution, temperature

distribution and nanoparticle volume fraction of two types

of nanofluids such as copper water nanofluid and silver

water nanofluid.

Figure 1: Velocity profiles for various values of M2

Figure 2: Effect of M2 on Temperature profiles

Figure 3: Nanoparticle volume faction for various values of

M2

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f ' ()

M2 = 0.5, 1.0, 1.5, 2.0

Cu - Water

Ec = 1.0

Pr = 6.2

NR = 1.0

Le = 0.6

Nt = 0.5

Nb = 0.5

n = 10.0

Ag - Water

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M2 = 0.5, 1.0, 1.5, 2.0

Cu - Water

Ec = 1.0

Pr = 6.2

NR = 1.0

Le = 0.6

Nt = 0.5

Nb = 0.5

n = 10.0

Ag - Water

0 2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M2 = 0.5, 1.0, 1.5, 2.0

Ec = 1.0

Pr = 6.2

NR = 1.0

Le = 0.6

Nt = 0.5

Nb = 0.5

n = 10.0

Cu - Water

Ag - Water



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Figure 4: Velocity and volume fraction profiles for n

for Cu – water nanofluid

Fig.1 shows the plot of dimensionless velocity for different

values of magnetic interaction parameter. It is noted that as

magnetic interaction parameter increases, f decreases,

elucidating the fact that the effect of magnetic field is to

decelerate the velocity. This result qualitatively agrees with

the expectation since the Lorentz force which opposes the

flow field increases as M2 increases and leads to enhanced

deceleration of the flow. Further the effect of magnetic field

is to reduce the boundary layer thickness.

Fig.2 represents the graph of dimensionless temperature for

different values of magnetic interaction parameter. Increase

in M2 which enhances the dimensionless temperature

distribution. The influence of magnetic interaction

parameter on the dimensionless volume fraction is plotted in

Fig.3.The figure reveals that the volume fraction of the

nanofluids boosts for increasing values of M2.

Fig.4 and fig.5 respectively is a graphical representation of

dimensionless velocity, volume fraction for Cu water

nanofluid and temperature of both nanofluids for various

values of nonlinear stretching parameter. It is noted that as

the nonlinear stretching parameter increases, f(),and

diminishes. Consequently the effect of nonlinear

stretching parameter over momentum boundary layer

thickness becomes significantly less, for cu - water

nanofluid.

Figure 5:Dimensionless Temperature profiles for n

In Fig.6, the effect of Eckert number on temperature

distribution is displayed. It implied that the Eckert number

enhances temperature and contributes to the thickening of

thermal boundary layer thickness.

Figure 6:Dimensionless temperature distribution

at different values of Ec

0 2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

n = 1.0, 2.0, 3.0, 10.0

M2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

Nb = 0.5

Nt = 0.5

Le = 0.6

w = 0.80

f ' ()

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

n = 1.0, 2.0, 3.0, 4.0

Ag - Water

M2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

Nt = 0.5

Nb = 0.5

Le = 0.6

Cu - Water

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ec = 0.7, 0.8, 0.9, 1.0

Ag - Water

Cu - Water

M2 = 1.0

Pr = 6.2

NR = 1.0

Nt = 0.5

Nb = 0.5

Le = 0.6

n = 10.0



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Figure 7:Dimensionless Temperature Profiles for w

Figure 8: Radiation parameter effect on

Dimensionless Temperature profiles

Fig.7 depicts the effect of changing temperature ratio

parameter on temperature distribution. The thermal

boundary layer thickness increases with increasing surface

temperature. This can be explained by the statement the

effect of temperature ratio parameter is to increase the rate

of energy transport to the nanofluid and accordingly

increase the temperature. An increase in the radiation

parameter causes a decrease in the temperature and the

thermal boundary layer thickness as displayed in Fig.8.The

values of radiation parameter will cause no change in the

velocity profiles of the nanofluids because the transformed

momentum equation (10) is uncoupled from the energy

equation (12).

Fig.9 shows the effect of Lewis number on the volume

fraction profiles. It illustrates that the volume fraction

decreases as the Lewis number increases. This is because as

the values of Lewis number gets larger the molecular

diffusivity gets smaller thereby causes a decrease in the

volume fraction field.

Figure 9: Dimensionless nanoparticle volume fraction for

Lewis number

Figure 10: Temperature profiles for various values

of Brownian motion parameter

Figure 11: Effect of Brownian motion parameter

on volume fraction distribution

The effect of Brownian motion parameter on temperature

and volume fraction is shown in Fig.10 and Fig.11. The

temperature in the boundary layer has the less result due to

the influence of Brownian motion parameter whereas the

volume fraction decreases with the increasing values of

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

w = 0.8, 0.85, 0.9, 1.0

Cu - Water

Ag - Water

M2 = 1.0

Pr = 6.2

Ec = 1.0

NR = 1.0

Nt = 0.5

Nb = 0.5

Le = 0.6

n = 10.0

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

NR = 0.5, 1.0, 1.5, 2.0

Cu - Water

Ag - Water

M2 = 1.0

Pr = 6.2

Ec = 1.0

Nt = 0.5

Nb = 0.5

Le = 0.6

n = 10.0

0 2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Le = 0.3, 0.6, 0.9, 1.0

Cu - Water

Ag - Water

M2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

Nt = 0.5

Nb = 0.5

= 0.1

n = 10.0

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cu - Water

Nb = 0.2, 0.4, 0.5, 1.0

M2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

n = 10.0

Nt = 0.5

Le = 0.6

Ag - Water

0 2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Nb = 0.2, 0.4, 0.5, 1.0

Cu - Water

M2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

Nt = 0.5

Le = 0.6

= 0.1

n = 10.0

Ag - Water



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0 2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

n = 10.0

Nb = 0.5

Le = 0.6

Cu - Water

Nt = 0.2, 0.4, 0.5, 0.7

Ag - Water

Brownian motion parameter. Brownian motion serves to just

warm the boundary layer.

Figure 12: Temperature profiles for different values

of Thermophoresis parameter

Thermophoresis parameter plays an important key role in

temperature distribution and nanoparticle volume fraction

which is demonstrated through Fig.12 & Fig.13

respectively. It is noticed that the dimensionless temperature

as well as the dimensionless volume fraction increases by

the increase of the values of the thermophoresis parameter.

Increase in Nt causes the increment in the thermophoresis

force which tends to move nanoparticles from hot to cold

areas and consequently it enhances the magnitude for

temperature and nanoparticle volume fraction profiles.

Figure 13: Effect of Thermophoresis parameter on

volume fraction distribution

The numerical results of the skin friction co

efficient and nondimensional rate of heat transfer are

presented in table 4 and table 5 for both cu - water nanofluid

and silver water nanofluid. In Table 4, skin friction

coefficient increases due to the influence of magnetic

interaction parameter and nonlinear stretching parameter in

magnitude. Table 5 illustrates the effect of all the physical

parameters on nondimensional rate of heat transfer. For

increasing values of nonlinear stretching parameter and

surface temperature ratio parameter, the nondimensional rate

of heat transfer enhances meanwhile the physical parameters

such as magnetic interaction parameter, radiation parameter,

thermophoresis parameter, Brownian motion parameter and

Eckert number diminishes the nondimensional rate of heat

transfer.

Table 4: Skin friction coefficient for different values of M2

and n

0 1 2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Nt = 0.2, 0.4, 0.5, 0.7

Cu - WaterM

2 = 1.0

Pr = 6.2

NR = 1.0

Ec = 1.0

n = 10.0

Nb = 0.5

Le = 0.6

Ag - Water

n

M2

Cu - water Ag - Water

2.5

10

1

n f

2.5

10

1

n f

0.1

10.0

0.0

0.5

1.0

1.5

-6.26091

-6.81681

-7.32783

-7.80406

-6.52906

-7.06412

-7.55865

-8.02142

1.0

2.0

3.0

10.0

1.0 -2.69755

-3.52561

-4.19368

-7.32783

-2.77233

-3.62968

-4.32062

-7.55865



12448

Table 5: Nondimensional Heat transfer rate for different values of

M2, n, NR, Nb , Nt, Ec and w when = 0.1, Le = 0.6 and Pr = 6.2

M2

n

NR

Nb

Nt

Ec

w

Cu – Water Ag – Water

341 1 (0)

3

nf fw

f nfR

k kn

k k N

341 1 (0)

3

nf fw

f nfR

k kn

k k N

0.0

0.5

1.0

1.5

10.0 1.0 0.5 0.5 1.0 1.1 11.11955

8.83978

7.48352

6.19943

11.22666

9.74846

8.36668

7.06079

1.0 1.0

2.0

3.0

10.0

1.0 0.5 0.5 1.0 1.1 1.51643

2.77657

3.68201

7.48352

1.82964

3.19403

4.18242

8.36668

1.0 10.0 0.5

1.0

1.5

2.0

0.5 0.5 1.0 1.1 10.02533

7.48352

6.39860

5.79354

11.07347

8.36668

7.21558

6.57398

1.0 10.0 1.0 0.2

0.4

0.5

0.7

0.5 1.0 1.1 7.48446

7.48368

7.48352

7.48342

8.36734

8.36690

8.36668

8.36654

1.0 10.0 1.0 0.5 0.2

0.4

0.5

0.7

1.0 1.1 7.56653

7.51081

7.48352

7.42892

8.44682

8.39184

8.36668

8.35694

1.0 10.0 1.0 0.5 0.5 0.7

0.8

0.9

1.0

1.1 10.55091

9.59959

8.54185

7.48352

11.28963

10.38284

9.37520

8.36668

1.0 10.0 1.0 0.5 0.5 1.0 0.8

0.85

0.9

1.1

5.76352

6.01590

6.28266

7.48352

6.54953

6.81637

7.09819

8.36668

CONCLUSION

A role of Brownian motion and thermophoresis effects on

hydromagnetic flow of nanofluids past a nonlinearly

stretching sheet under consideration of viscous dissipation

and nonlinear thermal radiation have been investigated in

this work for two types of nanofluid Cu water nanofluid and

silver water nanofluid. Using similarity transformations the

governing equations of the problem are transformed into

nonlinear ordinary differential equations and solved

numerically by using most efficient Nachtsheim- Swigert

shooting iteration scheme for satisfaction of asymptotic

boundary conditions along with fourth order Runge-Kutta

Integration method (FORTRAN package). Numerical

solutions of the problem are obtained for various physical

parameters.

From the obtained numerical results and discussion

presented in the previous section, the following conclusions

are drawn

An increase in magnetic interaction parameter and

nonlinear stretching parameter decreases the nanofluid

velocity but opposite trend is occurred in skin friction

coefficient.

A rise in the magnetic interaction parameter,

thermophoresis parameter, temperature ratio parameter

and viscous dissipation parameter raises the temperature

distribution. In the mean while nonlinear stretching

parameter and radiation parameter decreases the

temperature distribution. Also the temperature has very

less effect due to Brownian motion parameter.

Nanoparticle volume fraction decelerates with an

increase in the values nonlinear stretching parameter,

Brownian motion parameter and Lewis number.



12449

Nanoparticle volume fraction accelerates for the

increasing values of thermophoresis parameter and

magnetic interaction parameter.

Nondimensional heat transfer rate enhances by means

of rise in the values of nonlinear stretching parameter

and surface temperature parameter but the

nondimensional rate of heat transfer decelerates with an

increasing value of magnetic interaction parameter and

radiation parameter, thermophoresis parameter,

Brownian motion parameter and Eckert number.

Sherwood number vanishes for nanofluids two phase

model with new type of boundary condition.

Finally, the numerical values of nondimensional rate of

heat transfer of Ag - water nanofluid is higher than the

Cu - water nanofluid.

Appendix

The expressions for the physical quantities,,nf nf nfk ,

nf ,and p nfc

are given through the following lines

[Ahmad et al. (2011)],

1nf f s ,

2.51

fnf

,

2 2

2

s f f snf f

s f f s

k k k kk k

k k k k

,

nfnf

nf

,

1p nf p pf sc c c

The constants values are as follows,

1

2.51

11 ,s

f

bc

,

2

1p

p

s

f

cc

c

Table 1:Thermo-physical properties of fluid and

nanoparticles at 25C

Physical properties Water fluid Cu Ag

CP 4179 385 235

997.1 8933 10500

K 0.613 400 429

REFERENCES

[1] N.Afzal,“Heat transfer from a stretching surface”, Int. J. Heat Mass Transfer, vol.36,pp.1128-1131, 1993.

[2] S.Ahmad,A.M.Rohni, I.Pop,“Blasious and Sakiadis

problems in nanofluids”,Acta Mechanica, vol.218,pp.195-204, 2011.

[3] M.Q.Brewster,“Thermal Radiative Transfer and

Properties”John Wiley and sons Inc, 1992.

[4] J.Buongiorna, “Convective transport in nanofluids”, Jl. of heat transfer,vol. 128(3), pp.240-250,2006.

[5] S.Choi,“Enhancing thermal conductivity of fluids with

nanoparticles. I sidiner DA, Wang HP (eds)

Developments and applications of non-Newtonian

flows”,ASMEFED, 231/MD, pp.99-105,1995.

[6] R.Cortell,“Effects of viscous dissipation on and

radiation on the thermal boundary layer over a

nonlinearly stretching sheet”, Physics Letters A, vol.372(5), pp.631-636, 2008.

[7] L.J.Crane,”Flow past a stretching plate”Zeitschrift für Angewandte Mathematik und Physik, vol.21(4),pp.645–647, 1970.

[8] S.K.Das,S.Choi,W.Yu&T.Pradet,“Nanofluids: science

and Technology,” Wiley, New Jersey, 2007.

[9] S.P.A.Devi, M.Thiyagarajan, “Steady nonlinear

hydromagnetic flow and heat transfer over a stretching

surface of variable temperature”,Heat Mass Transfer,vol.42,pp.671–677,2006.

[10] M.A.A.Hamad,M.Ferdows, “Similarity solutions to

viscous flow and heat transfer of nanofluid over

nonlinearly stretching sheet”, Appl. Math. Mech. Engl. Ed., vol.33(7), pp. 923–930, 2012.

[11] V.Kaufui, Wong and Omar De Leon, “Applications of

nanofluids:Current and Future”, Advances in mechanical engineering,pp.1-11,2010.

[12] A.V.Kuznetsov,D.A.Nield, “Natural convective

boundary layer flow of a nanofluid past a vertical

plate: A revised model,”Int. J. of thermal sciences, vol.77, 126-129,2014.

[13] Mohammed Saad Kamel, Raheem Abed

Syeal,Abdulameer Amdulhussein, “Heat transfer

enhancement using nanofluid: A review of the recent

literature,”American Jour. of Nano Research and Applications, vol.24(1), pp.1-5,2016.

[14] P.O.Olanrewaju,J.A.Gbadeyan,O.O.Agboola, and

S.O.Abah, “Radiation and viscous dissipation effects

for the Blasius and Sakiadis flows with a convective

surface boundary condition”, International Journal of Advances in Science and Technology,vol.2(4),2011.

[15] D.Pal, G.Mandal, and K.Vajravelu, “Convective-

Radiation Effects on Stagnation Point Flow of

Nanofluids Over a Stretching/Shrinking Surface with

Viscous Dissipation”,Journal of Mechanics, vol.8,pp.1-9,2014.



12450

[16] K.B.Pavlov,“Magnetohydrodynamic flow of an

incompressible viscous fluid caused by the

deformation of a plane surface”. Magnytnaya Gidrodinamika, vol.4, pp.146–147, 1974.

[17] Rafael Cortell, “Viscous flow and heat transfer over a

nonlinearly stretching sheet”, Applied Mathematics and Computation,vol.184,pp.864–873,2007.

[18] P.Rana,R.Bhargava, “Flow and heat transfer of a

nanofluid over a nonlinearly stretching sheet: A

numerical study”,Commun Nonlinear Sci Numer Simulat,vol.17,212–pp.226,2012.

[19] A.Raptis.“Flow of a micripolarfluid past a

continuously moving plate by the presence of

Radiation”, Int. Jour. of Heat & Mass

transfer,vol.41(18),pp.2865-2866,1998.

[20] M.G.Reddy,“Influence of Magnetohydrodynamic and

Thermal Radiation Boundary Layer Flow of a

Nanofluid Past a Stretching Sheet”,J. Sci. Res.Vol.6(2), pp.257-272,2014.

[21] Reza Azizian, Elham Doroodchi, and Behdad

Moghtaderi,“Effect of nanoconvection caused by

brownian motion on the enhancement of thermal

conductivity in nanofluids”, Ind. Eng. Chem. Res.,vol.51, pp.1782–1789,2012.

[22] Ritu Pasrija and Sunita Srivastava, “On the Effective

Thermal Conductivity of metallic and oxide

Nanofluids”,Int. Jour. of NanoScience and Nanotechnology,vol.4,pp.131-143,2013.

[23] S.Rosseland,“Theoretical Astrophysics”, Clarendon Press, Oxford, 1936.

[24] Sandeep Pal, Tikamchand Soni Akriti Agrawala and

Deepak Sharma, “Review on Enhanced Thermal

Conductivity of Colloidal Suspension of Nanosized

Particles (Nanofluids)”,Int. Jour of Advanced Mechanical Engineering,vol.4,pp.199-214,2014.

[25] E,M,Sparrow, R.D.Cess,“Radiation heat transfer

hemisphere”, Washington(Chaps. 7 & 10), 1978.

[26] P.Sreenivasulu,T.Poornima, Bhaskar

N.Reddy.,“Thermal radian effects on MHD boundary

layer slip flow past a permeable exponential stretching

sheet in the presence of Joule heating and viscous

dissipation”, JAFM, vol.9(1), pp.267-278,2016.

[27] Sumalatha, Chenna Shanker, Bandari,“MHD Mixed

Convection Flow of a Nanofluid Over a Nonlinear

Stretching Sheet with Variable Wall Temperature and

Volume fraction”,Journal of

Nanofluids,vol.5(5), pp.707-712,2016.

[28] K.Vajravelu, “The effect of variable viscosity on the

flow and heat transfer of a viscous Ag- water and Cu-

water nanofluids”, Journal of Hydrodynamics,vol.25, pp.1-9,2012.

[29] K.Vajravelu and K.V.Prasad,“Heat transfer

phenomena in a moving nanofluid over a horizontal

surface”, Journal of Mechanics, vol.28, pp.579-588,2012.

[30] Yohannes Yirga and Bandari Shankar, “Effects of

thermal radiation and viscous dissipation on

magnetohydrodynamic stagnation point flow and heat

transfer of nanofluid towards a stretching

sheet,”Journal of Nanofluids, vol.2, pp.283–291, 2013.
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