Available online at www.ispacs.com/cna

Volume 2015, Issue 1, Year 2015 Article ID cna-00228, 21 Pages doi:10.5899/2015/cna-00228

Research Article

### Mixed convection stagnation-point flow of nanofluids over a

### stretching/shrinking sheet in a porous medium with internal

### heat generation/absorption

Dulal Pal1∗_{, Gopinath Mandal}2_{, Kuppalapalle Vajravalu}3

(1)*Department of Mathematics, Visva-Bharati University, Siksha-Bhavana, Santiniketan, West Bengal-731 235, India*
(2)*Siksha-Satra, Visva-Bharati University, Sriniketan, West Bengal-731 236, India*

(3)*Department of Mathematics, University of Central Florida, Orlando, FL32816, USA*

Copyright 2015 c⃝Dulal Pal, Gopinath Mandal and Kuppalapalle Vajravalu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we analyzed the buoyancy-driven radiative non-isothermal heat transfer in a nanofluid stagnation-point flow over a stretching/shrinking sheet embedded in a porous medium.The effects of thermal radiation and internal heat generation/absorption along with suction/injection at the boundary are also considered. Three different types of nanofluids, namely the Copper-water, the Alumina-water and the Titanium dioxide water are considered. The result-ing coupled nonlinear differential equations are solved numerically by a fifth-order Runge-Kutta-Fehlberg integration scheme with a shooting technique. A good agreement is found between the present numerical results and the available results in the literature for some special cases. The effects of the physical parameters on the flow and temperature characteristics are presented through tables and graphs, and the salient features are discussed. The results obtained reveal many interesting behaviors that warrant further study on the heat transfer enhancement due to the nanofluids.

Keywords:Nanofluids; porous medium; stagnation-point flow; mixed convection; thermal radiation; heat source.

1 Introduction

In recent years, a great deal of interest has been evinced in the study of mixed convective heat and mass transfer
in nanofluids as it has many industrial importance, specially, in nanotechnology. Nanofluid is a suspension of solid
nano particles or fibers of diameter 1-100 nm in a basic fluids such as water, oil and ethylene glycol. Nanoparticles
are made from various materials, such as*Cu*,*Ag*,*Au*,*Fe*,*Hg*,*Ti*etc. metals and non metallic*Al*2*O*3,*CuO*,*TiO*2,*SiO*2
etc. (Choi et al. [1]). These nanofluids exhibit poor heat transfer rates as the thermal conductivities of such fluids are
important in calculating the heat transfer coefficient. Due to better performance of heat exchange, nanofluids can be
used in several industrial applications such as in transportation, chemical production, automotive, power generation in
power plant and in nuclear system. A comprehensive survey of convective heat transfer characteristics of nanofluids
was made by Daungthong et al. [2], Das et al. [3], Kaˇkac and Pramuanjaroenkij [4], Wang and Mujumdar [5], and
Saidur et al. [6] in their books and review papers.

The boundary layer flow and heat transfer due to nanofluids over a stretching/shrinking sheets embedded in porous

medium have a broad spectrum of applications in science and engineering. Thus, it is important to study the heat
transfer characteristics of the stretching and shrinking sheets to get the desired quality of the finished product. The
unsteady boundary layer flow of a nanofluid over a permeable stretching/shrinking sheet is studied by Bachok et
al. [7]. Later, Bachok et al. [8] studied the steady two-dimensional stagnation-point flow of a nanofluid over a
stretching/shrinking sheet. Recently, Hayat et al. [9] analyzed the boundary layer flow of second grade nanofluid past
a stretching surface with thermal radiation and heat source/sink. Makinde and Aziz [10] studied the boundary layer
flow of a nanofluid over a stretching sheet with convective boundary condition. Vajravelu et al. [11] analyzed the
convective heat transfer of nanofluids over a stretching surface using*Ag*_{−}water or*Cu*_{−}water. Hamad and Ferdows
[12] studied boundary layer stagnation point flow towards a heated porous stretching sheet filled with a nanofluid with
heat absorption/generation and suction/blowing using Lie group analysis.

Uddin et al. [13] studied free convective boundary layer flow of a nanofluid over a permeable upward horizontal plate in a porous medium with thermal convective boundary condition. Rana and Bhargava [14] studied the heat transfer enhancement in mixed convection flow along a vertical plate with heat source/sink utilizing nanofluids. Chamkha and Aly [15] focused on steady convection boundary layer flow of a nanofluid along a permeable vertical plate in the presence of magnetic field, heat generation/absorption and suction/injection effects. Recently, Rahman et al. [16] investigated hydromagnetic slip flow of water based nanofluid over a wedge with convective surface in the presence of heat generation or absorption. A similarity solution of the steady boundary layer near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid in the presence of heat generation/absorption is studied by Hamad and Pop [17]. An important numerical investigation on the convective heat transfer performance of nanofluids over a permeable stretching surface in the presence of partial slip, thermal buoyancy and internal heat generation or absorption was presented by Das [18].

Thermal radiation effect besides convective heat transfer effect plays an important role in controlling heat transfer in manufacturing processes where the quality of the final product depend on heat control factors. The effect of thermal radiation on magnetic convection in the boundary layer flow of a nanofluid was studied by Mat et al. [19]. Hady et al. [20] examined the thermal radiation effects on viscous flow and heat transfer of a nanofluid over a non-linear stretching sheet. Elbashbeshy and Aldawody [21] investigated the effects of thermal radiation and magnetic field on unsteady boundary layer mixed convection flow and heat transfer over a porous stretching surface. Makinde [22] examined the hydromagnetic mixed convection stagnation-point flow towards a vertical plate embedded in a porous medium with thermal radiation and internal heat generation. Abdul-Kahar et al. [23] investigated boundary layer flow of a nanofluid past a porous vertical stretching surface in the presence of chemical reaction with heat radiation by using scaling group transformation. Recently, Ibrahim and Shankar [24] analyzed magnetodyhrodynamic boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet using velocity, thermal and solutal slip boundary conditions.

In view of all the above mentioned applications, the objective of the present work is to examine mixed convection
heat transfer on stagnation-point flow over a stretching/shrinking sheet in the presence of thermal radiation and heat
source/sink in three different types of nanoparticles, namely copper (*Cu*), alumina (*Al*2*O*3), titanium dioxide (*TiO*2)
with water as the base fluid. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and
no slip occurs between them. However, the use of slip boundary conditions have been found in the literature (Wang
[25], Van Gorder et al. [26] and Akyildiz et al. [27]) for related problems. In this paper, we studied the effects
of effective Prandtl number, heat source/sink, buoyancy (or mixed convection) parameter, thermal radiation, solid
volume fraction of the nanofluid and stretching/shrinking parameter on the velocity, temperature and concentration
fields using thermophysical properties of*Cu*,*Al*2*O*3and*TiO*2nanoparticles in the base fluid (Pr=6.8 for water)(see
Table-1).

2 Formulation of the problem

We consider the steady two-dimensional stagnation-point flow of nanofluids past a stretching/shrinking sheet with
linear velocity*uw(x*) =*cx*(for stretching sheet)/*uw(x*) =−*cx*(for shrinking sheet) and velocity of the free stream
flow*U*(*x*) =*ax*, where*a* and*c*are constants,*x*is the coordinate measured along the stretching/shrinking surface.
The flow takes place at*y*_{≥}0, where*y*is the coordinate measured normal to the stretching/shrinking surface. It is
assumed that the temperature at the stretching/shrinking surface takes the constant values*Tw*, while the temperature

in thermal equilibrium and no slip occurs between them. Under these assumptions, the boundary layer equations of motion and energy in the presence of heat source or sink, viscous dissipation and thermal radiation, are as follows :

∂*u*
∂*x*+

∂*v*

∂*y*=0, (2.1)

*u*∂*u*
∂*x*+*v*

∂*u*
∂*y*=*U*(*x*)

*dU*(*x*)

*dx* +

µn f ρn f

∂2_{u}

∂*y*2+

µn f
ρn f*K*

(

*U*(*x*)−*u*

)

+(ρβ)n f

ρn f *g*(*T*−*T*∞), (2.2)

*u*∂*T*
∂*x*+*v*

∂*T*
∂*y* =αn f

*d*2*T*
*dy*2 −

1

(ρ*Cp)n f*
∂*qr*

∂*y* +
*Q*0

(ρ*Cp)n f*(*T*−*T*∞), (2.3)

subject to the boundary conditions for stretching/shrinking sheets:

*u*=*uw*(*x*) =±*cx*,*v*=*vw*, *T*=*Tw*=*T*∞+*b*(*x*/*l*) *at y*=0 (2.4)
*u*_{→}*U*(*x*) =*ax*, *v*_{→}0,*T* _{→}*T*∞, *as y*→∞, (2.5)
where*u*and*v*are velocity components along the*x*- and*y*-directions, respectively.*U*(*x*)stands for the stagnation-point
velocity in the inviscid free stream.*T* is the temperature of the nanofluid,*K*is the permeability of a porous medium,
*g*is the acceleration due to gravity,*Q*0is the heat source coefficient (*Q*0>0) and sink coefficient (*Q*0<0),*a*,*b*and
*c*are positive constants,*l*is the characteristic length and*vw*is the wall mass flux with*vw*<0 (suctions) and*vw*>0

(injection), respectively. Further,ρ is the fluid density,µn f is the coefficient of viscosity of the nanofluid,βn f is the
thermal expansion of the nanofluid,κn f is the thermal conductivity of the nanofluid,αn f is the thermal diffusivity of
the nanofluid,ρn f is the effective density of the nanofluid,(ρ*Cp)n f* is the heat capacitance of the nanofluid. These
thermophysical parameters are defined as follows (Hamad and Pop [17]):

αn f = κn f

(ρ*Cp)n f*, ρn f = (1−φ)ρ*f*+φ ρs, µn f =
µf
(1−φ)2.5,

(ρβ)n f = (1−φ)(ρβ)*f*+φ(ρβ)s, (ρ*Cp)n f* = (1−φ)(ρ*Cp)f*+φ(ρ*Cp)s*
κ*n f*

κ*f*

=(κ*s*+2κ*f*)−2φ(κ*f*−κ*s)*
(κs+2κ*f*) +2φ(κ*f*−κs)

, (2.6)

whereφis the solid volume fraction of the nanofluid,ρ*f*is the reference density of the fluid fraction,ρsis the reference

density of the solid fraction,µ*f* is the viscosity of the fluid fraction,κ*f* is the thermal conductivity of the fluid, andκs

is the thermal conductivity of the solid fraction.

The net radiation heat flux*qr[W m*−2_{]}_{is defined following the Rosseland approximation (Magyari and Pantokratoras}
[28]) applied to optically thick media by the expression

*qr*=−

4

3*K*∗*grad*(*eb)*, (2.7)

where*K*∗[*m*−1]is the Rosseland mean spectral absorption coefficient and*eb[W m*−2]is the blackbody emission power
which is given in terms of the absolute temperature*T* by the Stefan-Boltzmann radiation law*eb*=σ∗*T*4with the

Stefan-Boltzmann constantσ∗=5.6697·10−8*W m*−2*K*−4. It is further assumed that the term*T*4 due to radiation
within the flow can be expressed as a linear function of temperature itself. Hence,*T*4can be expanded as Taylor series
about*T*∞and can be approximated after neglecting the higher order terms as,

*T*4_{≈}4*T*_{∞}3*T*_{−}3*T*_{∞}4. (2.8)
Using Eqs. (2.7) and (2.8), we get

∂*qr*
∂*y* =−

16σ∗*T*∞3
3*K*∗

∂2_{T}

We now look for a similarity solution of the Eqs. (2.1)-(2.3) with boundary conditions (2.4)-(2.5) in the following form (Kameswaran et al. [32]):

ψ=√*uw(x*)*x*νf*F*(η), θ(η) = *T*−*T*∞
*Tw*−*T*∞

, η=

√

*uw(x*)
*x*ν*f*

*y*, (2.10)

whereν*f* is the kinematic viscosity of the fluid and the stream function ψ is defined in the usual way as*u*=∂ ψ_{∂}* _{y}*,

*v*=−∂ ψ_{∂}* _{x}*, which identically satisfies the Eq. (2.1) .

Substituting Eqs. (2.6) -(2.10) in Eqs. (2.2) and (2.3), we get the following nonlinear ordinary differential equa-tions :

*F*′′′+*K*1

(*a*

*c*−*F*

′)

+ (1−φ)2.5[(

1−φ+φ(ρβ)s
(ρβ)*f*

)

Λθ+(

1−φ+φρs
ρ*f*

)(*a*2

*c*2+*FF*′′−*F*′
2)]

=0, (2.11)

1
*Pre f f*

θ′′+{

1−φ+φ(ρ*Cp)s*
(ρ*Cp)f*

}[

*F*θ′−*F*′θ]

+λ θ=0, (2.12)

with the corresponding boundary condition as obtained from Eqs. (2.4) -(2.5) in the form:

*F*=*S*, *F*′=1, θ=1 *at* η=0 (*f or stretching sheet*), (2.13)
*F*=*S*, *F*′=−1, θ=1 *at* η=0 (*f or shrinking sheet*), (2.14)

*F*′_{→}*a*

*c*, θ→0 *as* η→∞, (2.15)

where*Pr*= ν*f*

α*f* is the Prandtl number,λ =
*Q*0

(ρ*Cp*)*f* is the heat source(λ >0)or sink(λ<0)parameter. *K*1=

ν*f*
*cK* is

the porous parameter,*S*=−_{√}*vw*

*c*ν*f* is the mass flux parameter (*S*>0 corresponds to the suction and*S*<0 corresponds

to injection), Λ=*g*(ρβ*f*)/ρ*fc*2 is the buoyancy or mixed convection parameter (see ref. [18]), *Nr*=−16σ

∗_{T}_{∞}3

3κ*fK*∗ is

the radiation parameter,*Pre f f*=_{kn f}Pr*k f* +N*r*

is the effective Prandtl number [28]. In the absence of radiation, the

Stefan-Boltzmann constantσ∗=0, so*Nr*=0. It is noted that*Pre f f*= *Pr*in absence of thermal radiation (*Nr*=0) and

nanoparticles (φ=0). It should also be noted thatΛ>0 aids the flow andΛ<0 opposes the flow, and whenΛ=0
i.e*Tw*=*T*∞represents the case of forced convection flow. Here prime denotes the differentiation with respect toη.
In isothermal case (*b*=0) in the absence of thermal radiation and mixed convection, the Eqs. (2.11) and (2.12) reduce
to those obtained by Hamad and Pop [17].

The important quantities in this study are skin-friction or the shear stress coefficient*Cf* and the local Nusselt Number
*Nux*, which are defined by

*Cf* =

2µn f
ρ*fu*2*w*

(∂*u*

∂*y*

)

*y=*0

, *Nux*= −
*x*κn f
κ*f*(*Tw*−*T*∞)

(

−∂_{∂}*T*
*y*

)

*y=*0. (2.16)

Using Eqs. (2.10) and (2.16), the skin-friction coefficient and the local Nusselt number can be expressed as Hamad and Pop [17]:

*Re*1*x*/2*Cf*=

2

(1−φ)2.5|*F*′′(0)|, *Re*

−1/2

*x* *Nux*=−

κn f
κ*f*

θ′(0), (2.17)

where*Rex*=*uw(x*)*x*/ν*f* is the local Reynolds number based on the stretching/shrinking velocity*uw(x*).

3 Solution algorithm of the numerical method

similarity transformation before being solved numerically by the Runge-Kutta-Fehlberg method with shooting
tech-niques. The coupled ordinary differential equations (2.11) -(2.12) which are of third-order in*F*and second-order in

θ, are reduced to a system of five simultaneous equations of first-order having five unknowns.

The value of η∞varies from 1.5 to 7 depending upon the physical parameter that governs our problem. Thus the coupled nonlinear boundary value problem has been reduced to a system of five simultaneous equations of first-order for five unknowns as follows:

*F*_{1}′=*F*2, *F*2′=*F*3,

*F*_{3}′=−*K*1(*a*/*c*−*F*2)−φ1φ2{(*a*/*c*)2+*F*1*F*3−*F*22} −φ1φ4Λ*F*4
*F*_{4}′=*F*5

*F*_{5}′=−*Pre f f*{φ3(*F*1*F*5−*F*2*F*4) +λ*F*4} (3.18)

where

*F*1=*F*, *F*2=*F*′, *F*3=*F*′′, *F*4=θ, *F*5=θ′
and

φ1= (1−φ)2.5, φ2=1−φ+φ(ρs/ρ*f*),

φ3=1−φ+φ(ρ*Cp)s*/(ρ*Cp)f*, φ4=1−φ+φ(ρβ)s/(ρβ)f.
Here prime denotes differentiation with respect toη. The boundary condition now become

*F*1=*S*, *F*2= +1, *F*4=1, *at*η=0(*f or stretching sheet*)
*F*1=*S*, *F*2=−1, *F*4=1, *at*η=0(*f or shrinking sheet*)

*F*2→*a*/*c*, *F*4→0, *as*η→∞. (3.19)
In order to integrate (3.18) and (3.19) as an initial value problem we require the values of*F*3(0)and*F*5(0)but no such
values are prescribed at the boundary. The most important factor of shooting method is to choose the appropriate
finite value ofη∞. In order to determineη∞for the boundary value problem stated by Eqs. (3.18) -(3.19), we have
to start with the initial guess value*F*3(0)=*d*10and*F*5(0)=*d*20. Letγ1andγ2be the correct values of*F*3(0)and*F*5(0),
respectively for some particular set of physical parameters. The five ordinary differential equations are then solved
by fifth-order Runge-Kutta-Fehlberg method and denote the values*F*3(0)and*F*5(0)atη=η∞by*F*3(*d*10,*d*20,η∞)and
*F*5(*d*10,*d*20,η∞), respectively. Since*F*3and*F*5are clearly function ofγ1andγ2, they are expanded in Taylor series
aroundγ1−*d*10andγ2−*d*20, respectively by retaining only the linear terms. Thus, after solving the system of Taylor
series expansions forδ γ1=γ1−*d*10 andδ γ2= γ2−*d*20, we obtain the new estimate*d*11= *d*10+δ*d*10 and*d*21=*d*20+

δ*d*20. Next the entire process is repeated starting with*F*1(0),*F*2(0),*d*11,*F*4(0)and*d*21as initial conditions. Iteration
of the whole outlined process is repeated with the latest estimates ofγ1andγ2until prescribed boundary conditions
are satisfied.

Finally*d*1*n*=*d*1(n−1)+δ*d*1(n−1) and*d*2*n*=*d*2(n−1)+δ*d*2(n−1) for*n*=1,2,3... are obtained which seemed to be the most

desired approximate initial values of*F*3(0)and*F*5(0). In this way all the four initial conditions are determined. The
last value ofη∞is finally chosen to be the most appropriate value of the limitη∞for that particular set of parameters.
The value ofη∞may change for another set of physical parameters. Once the finite value ofη∞is determined then the
integration is carried out. The value ofη∞varies from 1.5 to 7 depending upon the physical parameter that governs
our problem. It is now possible to solve the resultant system of five simultaneous equations as initial value problem by
fifth-order Runge-Kutta-Fehlberg integration scheme for a set of physical parameters. Computed results are presented
in graphical and tabular forms. We have compared the results for*F*′′(0)andθ′_{(}_{0}_{)}_{with those of Hamad and Ferdows}

4 Results and discussions

In order to study the important characteristics of the flow and heat transfer, the numerical results are presented
in Figs. 1-16 and in Table 2–4 for different values ofφ,*K*1,Λ,λ, Pr*e f f*,*S*and*a*/*c*. Thermophysical properties of

fluids and nanoparticles (*Cu*,*Al*2*O*3,*TiO*2) which are used in this study are given in Table–1. It should be noted that the
Prandtl number*Pr*of the base fluid (water) is kept constant at 6.8, whereas Pr*e f f*is the effective Prandtl number which

is appropriately computed by using the Prandtl number for water and assigned a constant value 3.5 or 3.6 depending on the thermal radiation parameter. In order to validate the present numerical method used in this paper, we have compared the present results with those obtained by Hamad [31] and Kameswaran et al. [32] for different values of

φas shown in Table-2 and excellent agreement has been obtained with their results. Therefore, we are confident that the present numerical method gives very accurate results. To be very sure about the numerical method used we have further validate the present result by comparing it with those of Khan and Pop [30], Hamad and Pop [17] and Hamad and Ferdows [12] for various values of Prandtl number Pr for Newtonian fluid whenφ = 0.0. In this case also the present result are in very good agreement with their results.

The variation of skin-friction coefficient*F*′′(0)and surface heat flux−θ′(0)are shown in Table–4 for various values
ofφ,*K*1,Λ,λ,Pr*e f f*and*S*. It is shown in Table-4 that as the nanoparticle volume fractionφincreases, the skin-friction

coefficient*F*′′(0)and surface temperature gradient−θ′(0)both decreases for stretching sheet, whereas reverse trend
occurs for shrinking sheet. Further, it is seen from this table that increasing the value of suction parameter*S* has
the tendency to decrease the values of*F*′′(0)for stretching sheet, whereas it increases for shrinking sheet. It is also
found that the value of−θ′_{(}_{0}_{)}_{increases for both stretching and shrinking sheets with increasing}_{S}_{. It is also noted}

that the effect of increasing the porous parameter*K*1is to decrease both*F*′′(0)and−θ′(0)for stretching sheet and
reverse trend is noticed for shrinking sheet. Also, it is observed that the effects of heat generation has the tendency to
increase−θ′_{(}_{0}_{)}_{for stretching sheet, whereas its effect is reverse for shrinking sheet. Further, it is shown in Table-4}

that increase in the buoyancy parameter it to increase both*F*′′(0)and−θ′_{(}_{0}_{)}_{for both stretching and shrinking sheets.}

Table 1: Thermophysical properties of fluid and nanoparticles (Rana and Bhargava [14])

Physical properties Fluid phase (water) Cu *Al*2*O*3 *TiO*2

*Cp(J*/*kg K*) 4179 385 765 686.2

ρ(*kg*/*m*3) 997.1 8933 3970 4250

κ(W/m K) 0.613 400 40 8.9538

β<10−5(*K*−1) 21 1.67 0.85 0.9

Table 2: Comparison of result for*F*′′(0)when*K*1=Λ=*a*/*c*=*Nr*=λ =0 for*Cu*-water for stretching sheet when Pr
=6.2.

φ Hamad [31] Kameswarana et al. [32] Present results 0.05 1.10892 1.108919904 1.08921

Table 3: Comparison of result for−θ′(0)in isothermal case whenφ=*K*1=Λ=λ =*a*/*c*=*b*= 0 for stretching sheet.

Khan and Hamad and Hamad and Present results

Pr*e f f*(=Pr) Pop [30] Pop [17] Ferdows [12]

0.2 0.1691 0.16909 0.16909 0.16952 0.7 0.4539 0.45391 0.45391 0.45391 2 0.9113 0.91136 0.91136 0.91135 7 1.8954 1.89540 1.89540 1.89540 20 3.3539 3.35390 3.35390 3.35390

Table 4: Values of*F*′′(0)and -θ′_{(}_{0}_{)}_{for the nanofluid (}_{Cu}_{-water) with different values of}_{φ}_{,} _{K}_{1}_{,} _{Λ}_{,} _{λ}_{,} _{a}_{/}_{c}_{,} _{Pr}_{e f f}

and *S*when*a*/*c*=1.

*Stretching* *Shrinking*

*Sheet* *Sheet*

φ *K*1 Λ λ *Pre f f* *S* *F*′′(0) −θ′(0) *F*′′(0) −θ′(0)

0.05 0.1 0.5 0.1 3.5 0.1 0.09439 2.47640 2.12923 −0.41161 0.2 0.09389 2.47631 2.22583 −0.34850 0.1 0.6 0.11317 2.47839 2.18776 −0.36926 0.5 0.2 0.09573 2.40319 2.14988 −0.64490 0.1 3.6 0.09343 2.51435 2.13008 −0.44172 3.5 0.2 0.09103 2.69172 2.27214 −0.14807 0.3 0.08771 2.91746 2.42203 0.15516 0.1 0.1 0.5 0.1 3.5 0.1 0.08018 2.46228 2.17446 −0.35436

0.2 0.07978 2.46221 2.27064 −0.29531
0.1 0.6 0.09615 2.46395 2.22456 −0.32020
0.5 0.2 0.08132 2.38861 2.19143 −0.58369
0.1 3.6 0.07938 2.50004 2.17508 −0.38292
3.5 0.2 0.07746 2.67579 2.34558 _{−}0.07681
0.3 0.07477 2.89959 2.52473 0.23528
0.2 0.1 0.5 0.1 3.5 0.1 0.05775 2.43479 2.16337 −0.33687

0.2 0.05747 2.43474 2.26146 −0.27787 0.1 0.6 0.06927 2.43598 2.20054 −0.31191 0.5 0.2 0.05858 2.36022 2.17601 −0.57014 0.1 3.6 0.05718 2.47215 2.16381 −0.36490 3.5 0.2 0.05588 2.64459 2.35845 −0.04824 0.3 0.05401 2.86442 2.56186 0.26994

The velocity and temperature profiles for three different types of nanofluids (*Cu*,*Al*2*O*3,*TiO*2-water) for both
stretch-ing and shrinkstretch-ing sheets are shown in Figs. 2-3. It is seen from these figures that for both the sheets*Cu*-water has
higher velocity distribution and lower temperature distribution compared to other two nanofluids,*Al*2*O*3-water and
*TiO*2-water. The effect is found to be more significant in the*Cu*-water nanofluid than in the*Al*2*O*3-water and*TiO*2
-water nanofluids. It is also observed that velocity and temperature profiles for*Al*2*O*3-water and*TiO*2-water nearly
coincide with each other for specific values of governing parameters.

The influence ofφon the velocity and temperature fields for*Cu*-water nanofluid can be analyzed from Figs. 4-5. The
variation of velocity profiles of*Cu*–water nanofluids againstηis depicted in Fig. 4 for the stretching and shrinking
sheets for different values ofφ. From this figure it is seen that the velocity profile increases with an increase inφ

tem-perature of the nanofluid decreases which means that temtem-perature can be controlled by increasing/decreasing volume fraction of nanoparticles in the base fluid. These results are in good agreement with the results of Das [18].

Figs. 6-7 show the variation of velocity and temperature profiles for different values of porous parameter*K*1for
stretching and shrinking sheets. It is observed from these figures that velocity increases with increase in the value
of the porous parameter*K*1, whereas reverse effect is observed on the temperature profile for both stretching and
shrinking sheets. The effects of buoyancy parameterΛon velocity and temperature profiles of*Cu*-water nanofluid for
stretching and shrinking sheets are shown in Figs. 8-9. The buoyancy or mixed convection parameterΛ, represents a
measure of the effect of the buoyancy in comparison with the inertia of the external forced or free stream flow on the
heat and fluid flow. It is observed from these figures that an increase inΛ, there is increase in the velocity profiles but
decrease in the temperature profiles for both stretching and shrinking sheets. Fig. 10 has been plotted to analyze the
variation of temperature profiles for different values of heat source/sink parameterλ. It is observed that an increase
in the heat source parameterλ (>0), there is increase in the temperature profiles for*Cu*-water nanofluid for both
stretching and shrinking sheets.

The effect of*a*/*c*on velocity and temperature profiles over stretching and shrinking sheets are shown in Figs. 11-12
for*Cu*-water nanofluid. It is observed from Fig. 11 that for higher values of *a*/*c*, the velocity profile reaches its
boundary value more quickly for both type of sheets (stretching/shrinking). Further, it is seen from this figure that
when*a*/*c*=0, the velocity profile decreases alongηtill it matches the boundary condition*F*′(η)→0 asη→∞for
stretching sheet, whereas reverse trend is observed for the velocity profile for shrinking sheet. Fig. 12 shows that the
temperature profile of*Cu*-water decreases as*a*/*c*increase for stretching and shrinking sheets alongwith decrease in
the peak. Figs. 13-14 depict the effects of the effective Prandtl number Pr*e f f* on skin-friction coefficient and the local

Nusselt number. Effective Prandtl number Pr*e f f* depends on Prandtl number Pr as well as thermal radiation parameter
*Nr*and the thermal conductivity of nanoparticles. It is observed from Fig. 13 that there is not much significant effect
on skin-friction coefficient as Pr*e f f*increases for both stretching and shrinking sheets for all the three types of

nanoflu-ids consisting of*Cu*,*Al*2*O*3,*TiO*2nanoparticles. It is also noticed that*Cu*-water has significantly higher skin-friction
coefficient compared to those of*Al*2*O*3-water and*TiO*2-water nanofluids. On the other hand, the local Nusselt number
gradually increases for stretching sheet and decreases for shrinking sheet as Pr*e f f* increases for all the three types of

nanofluids.

Figs. 15-16 depict the effects of the suction/injection parameter*S*(−0.5<*S*<0.5) of nanofluids on skin-friction
coef-ficient and the local Nusselt number. Figs. 15 and 16 show that skin-friction coefcoef-ficient and local Nusselt number both
increase smoothly with increasing the value of*S*for stretching/shrinking sheet for three types of nanofluids consisting
of*Cu*,*Al*2*O*3,*TiO*2nanoparticles. It is also noticed from Fig. 15 that*Cu*-water has significantly higher skin-friction
coefficient compared to those of*Al*2*O*3and*TiO*2-water nanofluids for both stretching and shrinking sheets. Further, it
is found that the skin friction coefficient profiles for shrinking sheet are higher than stretching sheet, whereas reverse
trend is observed for the local Nusselt number, which shows that the profiles for local Nusselt number for stretching
sheet is higher than those of shrinking sheet. In this paper we have discussed the stable solution of shrinking sheet for
various physical parameters and nanoparticle volume fraction. The unstable solution for shrinking sheet is beyond the
scope of this paper. Further extended may be taken for unstable solution of shrinking sheet.

5 Conclusion

In this study, we have numerically investigated the combined effects of mixed convection and thermal radiation
on stagnation-point flow over a permeable stretching/shrinking sheet considering nanofluids. The presence of three
different types of nanoparticle, namely*Cu*,*Al*2*O*3,*TiO*2in the base fluid (water) is considered. The governing partial
differential equations were transformed into a set of of nonlinear ordinary differential equation using similarity
trans-formation before being solved numerically by Runge-Kutta-Fehlberg method with shooting techniques. These results
mainly depend on the nanoparticles volume fractionφ, permeability parameter*K*1, buoyancy or mixed convection
parameterΛ, heat generation/absorption parameterλ, suction/injection parameter*S*, effective Prandtl number Pr*e f f*

and the parameter*a*/*c*. Some of the following conclusions are drawn from figures and tables:

(ii) An increase in the values ofφ,*K*1andΛresults in an increase in the velocity profile for*Cu*-water nanofluid for
both stretching and shrinking sheets but these effects are opposite on temperature profiles.

(iii) With an increase inλ, the temperature profiles for*Cu*-water increase for both stretching and shrinking sheets.

(iv) Effective Prandtl number Pr*e f f* does not have significant effect on the skin-friction coefficient, whereas the local

Nusselt number increases for stretching sheet but decreases for shrinking sheet.

(v) The values of skin-friction coefficient and the local Nusselt number increase as *S*increases from a negative to
positive value for all the three types of nanofluids for both stretching and shrinking sheets which agree very
well with the available results in the literature.

Acknowledgement

We thank the reviewer for the constructive comments that led to a definite improvement in the paper.

Nomenclature

*Cf* skin-friction coefficient

*Cp* specific heat at constant pressure
*K* permeability of the porous medium
*K*1 porous parameter

*K*∗ Rosseland mean spectral absorption coefficient
*l* characteristic length

*Nr* thermal radiation parameter
*Nux* local Nusselt number
*Pr* Prandtl number

*Pre f f* effective Prandtl number
*qr* thermal radiative heat flux

*Q*0 dimensional heat generation/absorption coefficient
*Rex* local Reynolds number

*S* suction/injection parameter
*T* temperature of the fluid
*T*∞ free stream temperature
*Tw* temperature at the wall

*u* velocity component in*x*-direction
*uw* stretching/shrinking sheet velocity
*U* free stream velocity of the nanofluid
*v* velocity component in*y*-direction

*Greek symbols*

αn f effective thermal diffusivity of the nanofluid

α*f* fluid thermal diffusivity
βn f thermal expansion of nanofluid

β*f* fluid thermal expansion coefficient

βs thermal expansion coefficient of nanoparticles

φ solid volume fraction of the nanoparticles

η similarity variable

λ heat source/sink parameter

Λ buoyancy or mixed convection parameter

µn f effective dynamic viscosity of the nanofluid

µf dynamic viscosity of the fluid

ν*f* kinematic viscosity of the fluid
ρn f effective density of the nanofluid

σ∗ _{Stefan-Boltzmann constant}

θ dimensionless temperature of the fluid

ψ stream function

κn f effective thermal conductivity of the nanofluid

κ*f* thermal conductivity of the fluid

*Superscripts*

′ _{differentiation with respect to}_{y}

*Subscripts*

*n f* nanofluid

*f* fluid

Figure 1: Flow configuration.

0.0 0.5 1.0 1.5 2.0 -1.0

-0.5 0.0 0.5 1.0 1.5 2.0

Shrinking Stretching

=0.2, a/c=2.0, K 1

=0.2, =0.5, =0.1, S=0.1, Pr

eff =3.5

F

'

(

)

Cu-W ater Al

2 O

3 -W ater TiO

2 -W ater

0.0 0.4 0.8 1.2 0.0

0.2 0.4 0.6 0.8 1.0

Cu-W ater Al

2 O

3 -W ater TiO

2 -W ater

Shrinking

Stretching

=0.2, a/c=2.0, K 1

=0.2, =0.5, =0.1, S=0.1, Pr

eff =3.5

(

)

Figure 3: Temperature profiles for three different types of nanofluids for stretching/shrinking sheet.

0.0 0.5 1.0 1.5 2.0 -0.8

-0.4 0.0 0.4 0.8 1.2 1.6 2.0

Cu-W ater Stretching

Shrinking

a/c=2.0, K 1

=0.2, =0.5, =0.1, S=0.1, Pr

eff =3.5

F

'

(

)

=0.0 =0.1 =0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0

0.2 0.4 0.6 0.8 1.0

Cu-W ater

a/c=2.0, K 1

=0.2, =0.5, =0.1, S=0.1, Pr

eff =3.5 =0.0 =0.1 =0.2

Stretching Shrinking

(

)

Figure 5: Temperature profiles for variousφfor*Cu*-water for stretching/shrinking sheet.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 -0.8

-0.4 0.0 0.4 0.8 1.2 1.6 2.0

Cu-water

=0.1, a/c=2.0, =0.5, =0.1, S=0.1, Pr

eff =3.5 K

1 =0.0 K

1 =0.5 K

1 =1.0 Shrinking

Stretching

F

'

(

)

0.00 0.25 0.50 0.75 1.00 1.25 0.0

0.2 0.4 0.6 0.8 1.0

=0.1, a/c=2.0, =0.5, =0.1, S=0.1, Pr

eff =3.5 Cu-water

K 1

=0.0 K

1 =0.5 K

1 =1.0

Stretching Shrinking

(

)

Figure 7: Temperature profiles for various*K*1for*Cu*-water for stretching/shrinking sheet.

0.0 0.4 0.8 1.2 -0.8

-0.4 0.0 0.4 0.8 1.2 1.6 2.0

Cu-W ater =0.0 =1.0 =2.0

=0.1, a/c=2.0, K 1

=0.2, =0.1, S=0.1, Pr

eff =3.5 Stretching

Shrinking

F

'

(

)

0.00 0.25 0.50 0.75 1.00 1.25 0.0

0.2 0.4 0.6 0.8 1.0

=0.1, a/c=2.0, K 1

=0.2, =0.1, S=0.1, Pr

eff =3.5 Cu-W ater

=0.0 =1.0 =2.0

Shrinking

Stretching

(

)

Figure 9: Temperature profiles for variousΛfor*Cu*-water for stretching/shrinking sheet.

0.0 0.5 1.0 1.5 0.0

0.2 0.4 0.6 0.8 1.0

=0.1, a/c=2.0, K 1

=0.2, =0.5, S=0.1, Pr

eff =3.5 Cu-water

Shrinking Stretching

=0.0 =0.1 =0.5

(

)

0 1 2 3 4 5 6 -1.0

-0.5 0.0 0.5 1.0 1.5 2.0

Cu-water Stretching

a/c=0.0 a/c=1.0 a/c=2.0

=0.1, K 1

=0.2, =0.5, =0.1, S=0.1, Pr

eff =3.5 Shrinking

F

'

(

)

Figure 11: Velocity profiles for various*a*/*c*for*Cu*-water for stretching/shrinking sheet.

0 1 2 3 4 5 6 0.0

0.5 1.0 1.5 2.0 2.5

=0.1, K 1

=0.2, =0.5, =0.1, S=0.1, Pr

eff =3.5 Cu-water

Shrinking

Stretching

(

)

a/c=0.0 a/c=1.0 a/c=2.0

3.5 4.0 4.5 5.0 5.5 6.0 6.5 8

10 12 14 16 18 20

Shrinking

Stretching

=0.2, a/c=2.0, K 1

=0.2, =0.5, =0.1, S=0.1

R

e

x

1

/2

C

f

Pr eff

Cu-W ater Al

2 O

3 -W ater TiO

2 -W ater

Figure 13: Variation of the skin-friction with*Pre f f*for different types of nanofluids.

3.5 4.0 4.5 5.0 5.5 6.0 6.5 2

3 4 5 6 7 8 9

Shrinking Stretching

Cu-W ater Al

2 O

3 -W ater TiO

2 -W ater

=0.2, a/c=2.0, K 1

=0.2, =0.5, =0.1, S=0.1

R

e

x

-1

/2

N

u

x

Pr eff

-0.50 -0.25 0.00 0.25 0.50 6 8 10 12 14 16 18 20 22 24 Cu-W ater Al 2 O 3 -W ater TiO 2 -W ater Shrinking Stretching

=0.2, a/c=2.0, K 1

=0.2, =0.5, =0.1, Pr

eff =3.5 R e x 1 / 2 C f S

Figure 15: Variation of the skin-friction with*S*for different types of nanofluids.

-0.50 -0.25 0.00 0.25 0.50 0 1 2 3 4 5 6 7 8 9

=0.2, a/c=2.0, K 1

=0.2, =0.5, =0.1, Pr

eff =3.5 Shrinking Stretching R e x -1 / 2 N u x S Cu-W ater Al 2 O 3 -W ater TiO 2 -W ater

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