mixed convection flow of nanofluids over a vertical surface

Mixed Convection Flow of Nanofluids over a Vertical Surface

Embedded in a Porous Medium with Temperature Dependent Viscosity

SAHAR M. ABDEL-GAIED AND MOHAMED R. EID 1

1 Department of Science and Mathematics, Faculty of Education, Assiut University, The New Valley

72111, Egypt.

[email protected] Abstract

A boundary layer analysis is presented for the mixed convection past a vertical plate in a porous medium

saturated with a nanofluid in the presence of variable viscosity effect. The governing partial differential

equations are transformed into a set of similar equations and solved numerically by an efficient, implicit,

iterative, finite-difference method. A parametric study illustrating the influence of various physical parameters

is performed. Numerical results for the velocity, temperature, and nanoparticles volume fraction profiles, as

well as the friction factor, surface heat and mass transfer rates have been presented for parametric variations of

the mixed convection parameter λ , the variable viscosity parameter eθ , the buoyancy ratio parameter Nr, the

Brownian motion parameter Nb, the thermophoresis parameter Nt, and the Lewis number Le. The dependency

of the friction factor, surface heat transfer rate, and mass transfer rate on these parameters has been discussed.

Keywords: Nanofluid, Mixed convection, Variable viscosity, Porous medium

Nomenclature

a constant

C nanoparticles volume fraction

wC nanoparticles volume fraction at

the vertical plate

∞C ambient nanoparticles volume

fraction attained as ∞→y

BD Brownian diffusion coefficient

TD thermophoretic diffusion

coefficient

f dimensionless stream function

g gravitational acceleration

mk effective thermal conductivity

K permeability of the porous

medium

Le Lewis number

Nr buoyancy ratio

Nb Brownian parameter

Nt thermophoresis parameter

xPe Peclet number

xRa local Rayleigh number

T temperature

wT temperature at the vertical plate

∞T ambient temperature attained as

∞→y

vu, velocity components along x-

and y-directions, respectively

∞U free stream velocity

yx, Cartesian coordinates along the

plate and normal to it,

respectively

Greek symbols

mα thermal diffusivity

β volumetric expansion

coefficient

ε porosity

γ thermal property of the fluid

η similarity variable

θ dimensionless temperature

eθ the viscosity/temperature

parameter

λ mixed convection parameter

µ effective viscosity

υ kinematic viscosity

ρ density

fρ density of the base fluid

pρ nanoparticles mass density

Recent Advances in Mathematical Methods and Computational Techniques in Modern Science

ISBN: 978-1-61804-178-4 163

( )fcρ heat capacity of the fluid

( )mcρ effective heat capacity of the

porous medium

( )pcρ effective heat capacity of the

nanoparticles material

φ Rescaled nanoparticles volume

fraction

ψ stream function

Subscripts

w condition at the wall

∞ condition at the infinity

I. Background he study of convective heat transfer in

nanofluids is gaining a lot of attention.

The nanofluids have many applications in the

industry since materials of nanometer size have

unique physical and chemical properties.

Nanofluids are solid-liquid composite

materials consisting of solid nanoparticles or

nanofibers with sizes typically of 1-100 nm

suspended in liquid. Nanofluids have attracted

great interest recently because of reports of

greatly enhanced thermal properties. For

example, a small amount (<1% volume

fraction) of Cu nanoparticles or carbon

nanotubes dispersed in ethylene glycol or oil is

reported to increase the inherently poor thermal

conductivity of the liquid by 40% and 150%,

respectively [1,2]. Conventional particle-liquid

suspensions require high concentrations

(>10%) of particles to achieve such

enhancement. However, problems of rheology

and stability are amplified at high

concentrations, precluding the widespread use

of conventional slurries as heat transfer fluids.

In some cases, the observed enhancement in

thermal conductivity of nanofluids is orders of

magnitude larger than predicted by well-

established theories. Other perplexing results

in this rapidly evolving field include a

surprisingly strong temperature dependence of

the thermal conductivity [3] and a three-fold

higher critical heat flux compared with the

base fluids [4,5]. These enhanced thermal

properties are not merely of academic interest.

If confirmed and found consistent, they would

make nanofluids promising for applications in

thermal management. Furthermore,

suspensions of metal nanoparticles are also

being developed for other purposes, such as

medical applications including cancer therapy.

The interdisciplinary nature of nanofluid

research presents a great opportunity for

exploration and discovery at the frontiers of

nanotechnology.

The porous media heat transfer problems

have numerous thermal engineering

applications such as geothermal energy

recovery, crude oil extraction, thermal

insulation, ground water pollution, oil

extraction, thermal energy storage, thermal

insulations, and flow through filtering devices.

Excellent reviews on this topic are provided in

the literature by Nield and Bejan [6], Vafai [7],

Ingham and Pop [8] and Vadasz [9]. Recently,

Cheng and Lin [10] examined the melting

effect on mixed convective heat transfer from a

permeable vertical flat plate embedded in a

liquid-saturated a porous medium with aiding

and opposing external flows. EL-Kabeir et al.

[11] applied the group theoretical method to

solve the problem of coupled heat and mass

transfer by natural convection boundary layer

flow for water-vapor around a permeable

vertical cone embedded in a non uniform

porous medium in the presence of magnetic

field and thermal radiation effects. Rashad [12]

studied the combined effect of MHD and

thermal radiation on heat and mass transfer by

free convection over vertical flat plate

embedded in a porous medium. Ibrahim et al.

[13] studied the effect of chemical reaction on

free convection heat and mass transfer for a

non-Newtonian power law fluid over a vertical

flat plate embedded in a fluid-saturated a

porous medium has been studied in the

presence of the yield stress and the Soret

effect. Hady et al. [14] applied the effect of

yield stress on the free convective heat transfer

of dilute liquid suspensions of nanofluids

flowing on a vertical plate saturated in a porous

medium under laminar conditions is

investigated considering the nanofluid obeys

the mathematical model of power-law. Abdel-

Gaied and Eid [15] presented a numerical

analysis of the free convection coupled heat

and mass transfer is presented for non-

Newtonian power-law fluids with the yield

stress flowing over a two-dimensional or

axisymmetric body of an arbitrary shape in a

fluid-saturated a porous medium. Hady et al.

[16] focused on the natural convection

boundary-layer flow over a downward-pointing

T


ISBN: 978-1-61804-178-4 164

vertical cone in a porous medium saturated

with a non-Newtonian nanofluid in the

presence of heat generation or absorption.

There have been several studies of the effect

of temperature-dependent viscosity on

free/mixed boundary layer flow. Ling and

Dybbs [17] studied the forced convection with

variable viscosity over flat plate in a porous

medium. Kafoussias and Williams [18] and

Kafoussias et al. [19] investigated the effect of

temperature-dependent viscosity on mixed

convection flow of a non-porous fluid past a

vertical isothermal flat plate. Hady et al. [20]

presented the influences of variable viscosity

and buoyancy force on laminar boundary layer

flow and heat transfer due to a continuous flat

plate. Kumari [21] studied the effect of mixed

convection boundary layer flow over heated

flat plate embedded in a porous medium with

variable viscosity. Hossain et al. [22-23]

studied the free convection flow of a viscous

and incompressible fluid past a vertical cone

and vertical wavy surface, respectively, with

viscosity inversely proportional to the linear

function of temperature. Recently, Molla et al.

[24] considered a problem of natural

convection flow from an isothermal horizontal

circular cylinder. Kim and Choi [25] presented

theoretical analysis of thermal instability

driven by buoyancy forces under a time-

dependent temperature field of conduction is

conducted in an initially quiescent, horizontal

liquid layer. The dependency of viscosity on

temperature is considered and the propagation

theory is employed for the stability analysis.

Pantokratoras [26] considered the forced

convection boundary layer flow of a viscous

and incompressible fluid past a wedge with

temperature-dependent viscosity, while Ali

[27] considered the effect of variable viscosity

on mixed convection along a vertical moving

surface. The steady mixed convection

boundary layer flow over a vertical

impermeable surface embedded in a porous

medium when the viscosity of the fluid varies

inversely as a linear function of the

temperature is studied. Both cases of assisting

and opposing flows are considered by Chin et

al. [28].

Nanofluids have been found to possess

enhanced thermophysical properties such as

thermal conductivity, thermal diffusivity,

viscosity, and convective heat transfer

coefficients compared to those of base fluids

like oil or water. It has demonstrated great

potential applications in many fields. The aim

of the present study is to investigate the effect

of variable viscosity or temperature-dependent

viscosity on mixed convection boundary layer

flow over a vertical surface embedded in a

porous medium saturated by a nanofluid. It is

assumed that viscosity of the fluid varies

inversely as a linear function of temperature.

Both cases of assisting and opposing flows will

be considered. The governing partial

differential equations are transformed into

ordinary differential equations which are then

solved numerically using an efficient, implicit,

iterative, finite-difference method for some

values of the physically governing parameters.

Flow and heat transfer characteristics are

presented in some tables and figures, and a

corresponding discussion is made. Quantitative

comparison with the existing results as

reported by Chin et al. [28]. All comparisons

of the existing and present results show

excellent agreements.

II. Problem formulation

We consider the steady two-dimensional free

convection boundary layer flow past a vertical

plate placed in a nanofluid saturated a porous

medium. The coordinate system is selected

such that x-axis is in the vertical direction. Fig.

1 shows the coordinate system and flow model.

At the surface, the temperature T and the

nanoparticles fraction take constant values Tw

and wC , respectively. The ambient values

attained as y tends to infinity of T and C are

denoted by T∞ and ∞C , respectively. The

Oberbeck-Boussinesq approximation is

employed and the homogeneity and local

thermal equilibrium in the porous medium are

assumed.

We consider a porous medium whose

porosity is denoted by ε and permeability by

K . The Darcy velocity is denoted by v. The

following four field equations embody the

conservation of total mass, momentum,

thermal energy, and nanoparticles,

respectively. The field variables are the Darcy

velocity v, the temperature T and the

nanoparticles volume fraction C . The basic

steady conservation of mass, momentum and

thermal energy equations for nanofluid by


ISBN: 978-1-61804-178-4 165

using usual boundary-layer approximations can

be written in Cartesian coordinates x and y as:

,0=∂

∂+

∂

∂

y

v

x

u (1)

( ) ( )( )[ ],)(1 fpf ∞∞∞∞∞∞ −−−−−+= CCgTTCgK

Uu ρρβρµ

(2)

,

2

2

2

∂

∂+

∂

∂

∂

∂+

∂

∂=

∂

∂+

∂

∂

∞ y

T

T

D

y

T

y

CD

y

T

y

Tv

x

Tu T

Bm τα

(3)

.1

2

2

2

2

y

T

T

D

y

CD

y

Cv

x

Cu T

B∂

∂+

∂

∂=

∂

∂+

∂

∂

∞ε (4)

Where u and v are the velocity

components along x and y coordinates,

respectively, µ and β are the density,

viscosity, and volumetric volume expansion

coefficient of the fluid while pρ is the density

of the particles. The gravitational acceleration

is denoted by g . We have introduced the

effective heat capacity mc)(ρ , and the effective

thermal conductivity mk of the porous

medium. The coefficients appeared in Eqs. (3)

and (4) are the Brownian diffusion coefficient

BD and the thermophoretic diffusion

coefficient TD . The corresponding boundary

conditions are defined as follows:

,,,

,0,,0

∞→→→→

====

∞∞∞ yasCCTTUu

yatCCTTv ww

(5)

Where f)( c

kmm ρ

α = , f

p

)(

)(

c

c

ρρε

τ = . (6)

We assume that the dynamic viscosity µ has

the following form (see Chin et al. [28]).

)(

1

)(1 eTTaTT −=

−+=

∞

∞

γµ

µ , (7)

where γ and ∞µ are the thermal property of

the fluid and the ambient fluid viscosity,

respectively, which is a constant. Also, a and

eT are constants, given by:

γµγ 1

, −=−= ∞∞

TTa e . (8)

The continuity equation is automatically

satisfied by defining a stream function

),( yxψ such that:

. and x

vy

u∂∂

−=∂∂

=ψψ

(9)

The dimensionless variables are:

( )

( ).

1,

-

-)( ,)(

),(2 ,2

w

2

12

1

∞∞

∞

∞

∞

∞

∞

−−=

−

−==

−

−=

=

=

TTaTT

TT

CC

CC

TT

TT

fPePe

x

y

ww

ee

w

xmx

θηφηθ

ηαψη

(10)

Where m

xUPe

α∞= (11)

is the Peclet number for a porous medium and

eθ is the viscosity/temperature parameter and

its value is determined by the

viscosity/temperature characteristics of the

fluid and the operating temperature difference

∞−=∆ TTT w .

Substituting the expressions in Eq. (10) into

the governing Equations (1)-(5) we obtain the

following transformed equations:

( )φθθθ

λ Nrfe

−

−+=′ 11 (12)

02 =′+′′+′++′′ θφθθθ NtNbf (13)

,0Nb

Nt Le =′′+′+′′ θφff (14)

With the boundary conditions:

.0,0,0: as

,1,1,0:0at

===′∞→

====

φθηφθη

f

f (15)

In the above equations primes denote

differentiation with respect to η and λ is the

constant mixed convection parameter which is

given by:

( )( ) ( )( )

,1

,1

m

wx

x

xw xCTTgKRa

Pe

Ra

U

CTTgK

αυβ

υβ

λ∞

∞∞

∞∞

∞∞ −−==

−−=

(16)

where Rax being the local Rayleigh number for

a porous medium and where the four

parameters Nr, Nb, Nt and Le are defined by:


ISBN: 978-1-61804-178-4 166

( )( )( )

( ).,

,)(

,1

)(

f

fp

B

m

m

wT

m

wB

w

w

DLe

T

TTDNt

CCDNb

TTC

CCNr

εα

ατ

ατ

βρ

ρρ

=−

=

−=

−−

−−=

∞

∞

∞

∞∞∞

∞∞

(17)

It should be mentioned that 0>λ

corresponds to a heated plate (assisting flow),

0<λ corresponds to a cooled plate (opposing

flow) and 0=λ corresponds to the forced

convection flow. We notice that the effect of

variable viscosity can be neglected if eθ is

large (≫1), i.e., if either the constant a or

(Tw−T∞) is small. If eθ is small then either the

fluid viscosity changes with the temperature or

the temperature difference is high. It may be

noted that eθ is negative for liquids and

positive for gases. It is also noted that if

∞→eθ , Eqs. (8) and (9) reduce to those for a

constant viscosity case found by Merkin [29]

or Aly et al. [30] for an isothermal flat plate.

The quantities of physical interest to be

considered in this study are the local skin

friction coefficient )0(f ′′ , heat transfer

coefficient )0(θ ′− , mass transfer coefficient

)0(φ′− , the non-dimensional temperature

profiles )(ηθ , and the non-dimensional

concentration profiles )(ηφ for both cases of

the assisting and the opposing flows.

III. Results and discussion By using the similarity transformations and

the similarity representation of the effect of

variable viscosity on steady mixed convection

boundary layer flow over a vertical surface

embedded in a porous medium saturated a

nanofluid is obtained and it shown in equations

(12)-(15). The similarity reduction corresponds

to constant viscosity can be easy obtain if we

put ( ∞→eθ ). We have obtained the same

result as in Chin et al. [28]. Equations (12)-

(14) with the boundary layer (15) have been

solved numerically by using an efficient,

implicit, iterative, finite-difference method.

The basic idea is to introduce new variables,

one for each variable in the original problem

plus one for each of its derivatives up to one

less than the highest derivative appearing. We

used the finite-difference method (we used

MATLAB package for this purpose) for the

solution of boundary value problem of the

form:

bxayxfy ≤≤=′ ),,( (18)

subject to general nonlinear, two-point

boundary conditions:

( ) .0)().( =byayg (19)

Finite-difference equations are set up on a

mesh of points and estimated values for the

solution at the grid points are chosen. Using

these estimated values as starting values a

Newton iteration is used to solve the Finite-

difference equations. The accuracy of the

solution is then improved by deferred

corrections or the addition of points to the

mesh or a combination of both. The absolute

error tolerance for this method is 410−.

Computations are carried out for various values

of the mixed convection parameter λ , the

viscosity/temperature parameter eθ , the

buoyancy ratio parameter Nr, the Brownian

motion parameter Nb, the thermophoresis

parameter Nt, and the Lewis number Le. Both

assisting ( 0>λ ) and opposing ( 0<λ ) flows

are considered.

Tables 1 and 2 show the values of skin

friction coefficient )0(f ′′ and heat transfer

coefficient )0(θ ′− for various values of the

mixed convection parameter λ and the

viscosity/temperature parameter eθ . The

results calculated for 100,50,10,1=λ and

∞−−−−−∞= ,8,4,2,1,1,2,4,8,eθ . Also, in

these tables the comparisons of the present

work with Chin et al. [28]. It is found that the

agreement between both results is excellent.

Therefore, we believe that this comparison

supports very well the validity of the present

work. The results show that for each eθ when

λ increases, the values of the magnitude of

)0(f ′′ and )0(θ ′− increase. Physically, this is

because the nanofluid velocity increases when

the buoyancy force increases and hence

increases the skin friction and it means the

temperature of the plate is increased. Tables 3-

6 indicate results for wall values for the

temperature and concentration functions which

are proportional to the Nusselt number and

Sherwood number, respectively. From Tables

3-5, we notice that as Nb and Nt increase, the

heat transfer rate (Nusselt number) )0(θ ′− and

mass transfer rate (Sherwood number) )0(φ ′−


ISBN: 978-1-61804-178-4 167

decrease for each eθ at the critical value of λ

( 354.1−=cλ ). As Nr increases, the surface

heat transfer and the surface mass transfer rates

increase for each eθ at the critical value of λ

( 354.1−=cλ ). Results from Table 6 indicate

that as Le increases, the heat transfer rate

decreases whereas the mass transfer rate

increases this for each positive value of eθ

while the negative value of eθ is different at

the critical value of cλ .

Fig. 2 indicates that, as Nr increases, the

velocity decreases, and the temperature and

concentration increase. Similar effects are

observed from Figs. 3 and 4 as Nb and Nt vary.

Fig. 5 illustrates the variation of velocity

within the boundary layer as Le increases. The

velocity increases as Le increases. As Le

increases, the temperature and concentration

within the boundary layer decrease and the

thermal and concentration boundary later

thicknesses decrease. Fig. 6 shows that as the

mixed convection parameter λ increases, the

velocity increases, whereas temperature and

concentration decrease. Fig. 7 focuses the

influence of eθ on the velocity, the

temperature and the concentration. We observe

that the velocity increases, whereas

temperature and concentration decrease and for

fixed values of λ , the values of )0(θ ′− and

)0(φ ′− for the base fluid is gas ( 0>eθ ) is

always higher than the values of )0(θ ′− and

)0(φ ′− for the base fluid is liquid ( 0<eθ ).

Even when the critical value cλ , we find the

same influence, which may vary

slightly at some values because the dual

solutions but only the first solutions have the

importance if compared with the second

solutions, which depicts in Tables 3-5.

Figs. 8, 9, 10, 11 and 12 display results for

wall values for the temperature, and

concentration functions which are proportional

to the Nusselt number, and Sherwood number,

respectively. From Figs. 8 and 10, we notice

that as Nr and Nt increase, the heat transfer rate

(Nusselt number) and mass transfer rate

(Sherwood number) decrease. As Nb increases,

the surface mass transfer rates increase

whereas the surface heat transfer rate decreases

as shown by Fig. 9 as we expect before. Fig. 11

indicates that as Le increases, the heat transfer

rate decreases whereas the mass transfer rate

increases. From Fig. 12, we observe that, as the

mixed convection parameter λ increases, the

heat and mass transfer rates increase. From

these Figs. we note that with variable viscosity,

the separation of boundary layer is delayed for

0>eθ than 0<eθ . This indicates that for

cλλ > , the boundary layer separates from the

plate surface, and therefore solutions based on

the boundary layer approximations are not

valid anymore.

It can also be summarize that decreasing λ

from 0>λ (assisting flow) to 0<λ

(opposing flow) leads to an increase of the

thermal boundary layer thickness and the

concentration boundary layer thickness. It is

also observed from the numerical results that

for fixed values of eθ , decreasing λ leads to

an increase of the thermal boundary layer and

the concentration thicknesses. It is also showed

that the changes of the thermal boundary layer

and concentration boundary layer thicknesses

with the changes of λ are much more cleared

than the changes with eθ . Therefore, this refers

that the parameter eθ has smaller effect on the

thicknesses of thermal boundary layer and

concentration boundary layer compared to that

of the mixed convection parameter λ . The

influence of nanoparticles on mixed convection

is modeled by accounting for Brownian motion

and thermophoresis as well as non-isothermal

boundary conditions. The thickness of the

boundary layer for the mass fraction is smaller

than the thermal boundary layer thickness for

large values of Lewis number Le. The

contribution of Nt to heat and mass transfer

does not depend on the value of Le. The

Brownian motion and thermophoresis of

nanoparticles increases the effective thermal

conductivity of the nanofluid. Both Brownian

diffusion and thermophoresis give rise to cross

diffusion terms that are similar to the familiar

Soret and Dufour cross diffusion terms that

arise with a binary fluid.

IV. Conclusions

The Nanofluids have attracted more

and more attention. The main driving force for

nanofluids research lies in a wide range of


ISBN: 978-1-61804-178-4 168

applications. The important technique to

enhance the stability of nanoparticles in fluids

is the use of surfactants. However, the

functionality of the surfactants under high

temperature is also a big concern, especially

for high-temperature applications. Two-step

method is the most economic method to

produce nanofluids in large scale, because

nanopowder synthesis techniques have already

been scaled up to industrial production levels.

Due to the high surface area and surface

activity, nanoparticles have the tendency to

aggregate.

In this study, we presented a boundary

layer analysis for the mixed convection past a

vertical surface embedded in a porous medium

saturated with a nanofluid ( two step method)

in the presence of the effect of variable

viscosity. The transformed equations are

solved numerically using the efficient, implicit,

iterative, finite-difference method. We have

quantitatively compared our present results

with those of Chin et al. [28], and the

agreement between both results is excellent.

Numerical results for surface heat transfer rate,

and mass transfer rate have been presented for

parametric variations of the mixed convection

parameter λ , the viscosity/temperature

parameter eθ , the buoyancy ratio parameter

Nr, Brownian motion parameter Nb,

thermophoresis parameter Nt, and Lewis

number Le. The numerical results are

influenced by mixed convection parameter λ

and the variable viscosity parameter which

defines the effect of variable viscosity of the

fluid eθ . It is observed that the separation of

boundary layer is delayed for 0>eθ than

0<eθ . In the assisting flow, the values of

)0(θ ′− and )0(φ ′− increase as λ increases. In

the opposing flow case, dual solutions exist

and the separation of boundary layer occurs.

The results indicate that, as Nr and Nt increase,

mass transfer rate (Sherwood number)

decrease. As Nb increases, the surface mass

transfer rates increase, whereas the surface heat

transfer rate decreases. As Le increases, the

heat transfer rate decreases, whereas the mass

transfer rate increases.

V. Competing interests This is just the theoretical study, every

experimentalist can check it experimentally

with our consent.

VI. Authors' information

S. M. Abdelgaied Assistant Professor

and

M. R. Eid mathematics lecturers.

VII.Endnotes

This is just a theoretical study;

every experimentalist can check it

experimentally with our consent.

VIII. References

[1] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu

and L.J. Thompson, Anomalously

increased effective thermal conductivities

containing copper nanoparticles, Applied

Physics Letters 78(6) (2001) 718–720.

[2] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E.

Lockwood and E.A. Grulke, Anomalous

thermal conductivity enhancement on

nanotube suspensions, Applied Physics

Letters 79(14) (2001) 2252–2254.

[3] H.E. Patel, S.K. Das, T. Sundararajan, A.

Sreekumaran, B. George and T. Pradeep,

Thermal conductivities of naked and

monolayer protected metal nanoparticle

based nanofluids: manifestation of

anomalous enhancement and chemical

Effects, Applied Physics Letters 14(83)

(2003) 2931–2933.

[4] S.M. You, J.H. Kim and K.H. Kim, Effect

of nanoparticles on critical heat flux of

water in pool boiling heat transfer, Applied

Physics Letters 83(16) (2003) 3374–3376.

[5] P. Vassallo, R. Kumar and S. D’Amico,

Pool boiling heat transfer experiments in

silica-water nanofluids, International


ISBN: 978-1-61804-178-4 169

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(2004) 407–411.

[6] D.A. Nield, and A. Bejan, Convection in

porous media (3rd

Ed.), Springer, New

York (2006).

[7] K. Vafai, Handbook of porous media (2nd

Ed.), Taylor & Francis, New York (2005).

[8] D.B. Ingham, and I. Pop, Transport

phenomena in porous media III, Elsevier,

Oxford (2005).

[9] P. Vadasz, Emerging topics in heat and

mass transfer in porous media, Springer,

New York (2008).

[10] W.T. Cheng, and C.H. Lin, Melting effect

on mixed convective heat transfer with

aiding and opposing external flows from

the vertical plate in a liquid-saturated

porous medium, Int. J. Heat Mass Transfer

50 (2007) 3026–3034.

[11] S.M.M. EL-Kabeir, M.A. EL-Hakiem and

A.M. Rashad, Group method analysis for

the effect of radiation on MHD coupled

heat and mass transfer natural convection

flow water vapor over a vertical cone

1through porous medium, Int. J. Appl.

Math. Mech. 3 (2007) 35–53.

[12] A.M. Rashad, Influence of radiation on

MHD free convection from a vertical flat

plate embedded in porous medium with

thermophoretic deposition particles,

Commun Nonlinear Sci Numerical

Simulation 13 (2008) 2213–2222.

[13] F.S. Ibrahim, F.M. Hady, S.M. Abdel-

Gaied and M.R. Eid, Influence of chemical

reaction on heat and mass transfer of non-

Newtonian fluid with yield stress by free

convection from vertical surface in porous

medium considering Soret effect, Appl.

Math. Mech. -Engl. Ed. 31(6) (2010) 675–

684.

[14] F.M. Hady, F.S. Ibrahim, S.M. Abdel-

Gaied and M.R. Eid, Influence of yield

stress on free convective boundary-layer

flow of a non-Newtonian nanofluid past a

vertical plate in a porous medium, Journal

of Mechanical Science and Technology 25

(8) (2011) 2043–2050.

[15] S.M. Abdel-Gaied and M.R. Eid, Natural

convection of non-Newtonian power-law

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dimensional bodies of arbitrary shape in

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Table 1:Comparison of skin friction coefficient )0(f ′′ with 0=== NtNbNr for various values λ

and eθ :

eθ Chin et al. [28] Present study

λ 1 10 50 100 1 10 50 100

∞−

-1.0191 -21.445 -225.67 -633.26 -1.0191 -21.446 -225.67 -633.28

-8 -1.2963 -27.835 -294.22 -826.12 -1.2963 -27.833 -294.23 -826.12

-4 -1.5820 -34.592 -367.02 -1031.1 -1.5820 -34.593 -367.04 -1031.1

-2 -2.1779 -49.134 -524.56 -1474.8 -2.1779 -49.135 -524.58 -1474.8

-1 -3.4625 -81.969 -882.79 -2484.7 -3.4625 -81.969 -882.79 -2484.7

1 0.8603 12.908 120.16 330.92 0.8603 12.908 120.17 330.93

2 0 0 0 0 0 0 0 0


ISBN: 978-1-61804-178-4 171

4 -0.4911 -9.8446 -102.41 -286.91 -0.4911 -9.8446 -102.41 -286.91

8 -0.7506 -15.440 -161.63 -453.23 -0.7506 -15.440 -161.63 -453.23

∞ -1.0191 -21.445 -225.67 -633.26 -1.0191 -21.445 -225.67 -633.26

Table 2:Comparison of heat transfer coefficient )0(θ ′− with 0=== NtNbNr for various values λ

and eθ :

eθ Chin et al. [28] Present study

λ 1 10 50 100 1 10 50 100

∞−

1.0191 2.1445 4.5133 6.3326 1.0191 2.1443 4.5118 6.3322

-8 1.0370 2.2268 4.7075 6.6089 1.0370 2.2267 4.7061 6.6089

-4 1.0546 2.3061 4.8937 6.8738 1.0546 2.3060 4.8924 6.8705

-2 1.0889 2.4567 5.2456 7.3741 1.0889 2.4566 5.2448 7.3718

-1 1.1542 2.7323 5.8853 8.2822 1.1542 2.7323 5.8853 8.2826

1 0.8603 1.2908 2.4032 3.3092 0.8603 1.2905 2.4008 3.2990

2 0.9435 1.7748 3.6288 5.0705 0.9435 1.7746 3.6267 5.1429

4 0.9821 1.9689 4.0963 5.7382 0.9821 1.9687 4.0945 5.7382

8 1.0008 2.0587 4.3101 6.0431 1.0008 2.0585 4.3084 6.1593

∞ 1.0191 2.1445 4.5133 6.3326 1.0191 2.1443 4.5118 6.4934


ISBN: 978-1-61804-178-4 172

Table 3:Values related to the heat transfer and the mass transfer coefficients for different values of the

governing parameters eθ and Nr with 1.0== NtNb , 10=Le and 354.1−=cλ :

eθ Nr )0(θ ′− )0(φ ′−

-8

0.1 0.1499 0.0252

0.2 0.2737 0.3470

0.3 0.3246 0.6463

0.4 0.3626 0.8984

0.5 0.3945 1.1160

-4

0.1 0.0003 0.0000

0.2 0.2117 0.1042

0.3 0.2836 0.4118

0.4 0.3285 0.7002

0.5 0.3646 0.9525

4

0.1 0.3941 1.0080

0.2 0.4204 1.1714

0.3 0.4434 1.3158

0.4 0.4640 1.4466

0.5 0.4830 1.5670

8

0.1 0.3389 0.6932

0.2 0.3758 0.9198

0.3 0.4056 1.1095

0.4 0.4313 1.2755

0.5 0.4542 1.4247


ISBN: 978-1-61804-178-4 173


governing parameters eθ and Nb with 1.0== NtNr , 10=Le and 354.1−=cλ :

eθ Nb )0(θ ′− )0(φ ′−

-8

0.1 0.1499 0.0252

0.2 0.1749 0.0064

0.3 0.0672 0.0000

0.4 0.0039 0.0001

0.5 0.0038 0.0001

-4

0.1 0.0003 0.0000

0.2 0.0003 0.0000

0.3 0.0003 0.0000

0.4 0.0003 0.0000

0.5 0.0003 0.0000

4

0.1 0.3941 1.0080

0.2 0.3597 1.0153

0.3 0.3287 1.0067

0.4 0.3002 0.9945

0.5 0.2742 0.9812

8

0.1 0.3389 0.6932

0.2 0.3073 0.6741

0.3 0.2789 0.6480

0.4 0.2531 0.6203

0.5 0.2295 0.5921


ISBN: 978-1-61804-178-4 174


governing parameters eθ and Nt with 1.0== NbNr , 10=Le and 354.1−=cλ :

eθ Nt )0(θ ′− )0(φ ′−

-8

0.1 0.1499 0.0252

0.2 0.1284 0.0305

0.3 -0.0542 -0.0408

0.4 0.0910 0.0172

0.5 0.3597 0.2423

-4

0.1 0.0003 0.0000

0.2 0.0003 0.0001

0.3 0.1250 0.0530

0.4 -0.0011 -0.0007

0.5 0.0002 0.0001

4

0.1 0.3941 1.0080

0.2 0.3729 0.9884

0.3 0.3528 0.9862

0.4 0.3337 0.9990

0.5 0.3156 1.0246

8

0.1 0.3389 0.6932

0.2 0.3180 0.6843

0.3 0.2981 0.6897

0.4 0.2790 0.7071

0.5 0.2608 0.7340


ISBN: 978-1-61804-178-4 175


governing parameters eθ and Le with 1.0=== NtNbNr and 354.1−=cλ :

eθ Le )0(θ ′− )0(φ ′−

-8

1 0.1592 0.2146

5 0.1178 0.0546

10 0.1499 0.0252

100 0.1738 0.0024

-4

1 0.1457 0.2330

5 -0.0644 -0.2509

10 0.0003 0.0000

100 0.0973 0.0010

4

1 0.4331 0.3341

5 0.4073 0.6907

10 0.3941 1.0080

100 0.3734 2.4863

8

1 0.3770 0.3138

5 0.3513 0.5480

10 0.3389 0.6932

100 0.3225 0.8376


ISBN: 978-1-61804-178-4 176

Fig. 1 Flow model and coordinate system.

Fig. 2 Velocity, temperature and concentration profiles for various values of buoyancy ratio Nr.


ISBN: 978-1-61804-178-4 177

Fig. 3 Velocity, temperature and concentration profiles for various values of Brownian motion

parameter Nb.

Fig. 4 Velocity, temperature and concentration profiles for various values of thermophoresis

parameter Nt.


ISBN: 978-1-61804-178-4 178

Fig. 5 Velocity, temperature and concentration profiles for various values of Lewis number Le.

Fig. 6 Velocity, temperature and concentration profiles for various values of mixed convection

parameter λ .


ISBN: 978-1-61804-178-4 179

Fig. 7 Velocity, temperature and concentration profiles for various values of variable viscosity

parameter eθ .

Fig. 8 Heat transfer rate and mass transfer rate for various values of buoyancy ratio Nr.


ISBN: 978-1-61804-178-4 180

Fig. 9 Heat transfer rate and mass transfer rate for various values of Brownian motion parameter Nb.

Fig. 10 Heat transfer rate and mass transfer rate for various values of thermophoresis parameter Nt.


ISBN: 978-1-61804-178-4 181

Fig. 11 Heat transfer rate and mass transfer rate for various values of Lewis number Le.

Fig. 12 Heat transfer rate and mass transfer rate for various values of mixed convection parameter λ .


ISBN: 978-1-61804-178-4 182

mixed convection flow of nanofluids over a vertical surface

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