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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 Mandal2, Kuppalapalle Vajravalu3

(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 asCu,Ag,Au,Fe,Hg,Tietc. metals and non metallicAl2O3,CuO,TiO2,SiO2 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

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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 usingAgwater orCuwater. 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 (Al2O3), titanium dioxide (TiO2) 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 ofCu,Al2O3andTiO2nanoparticles 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 velocityuw(x) =cx(for stretching sheet)/uw(x) =−cx(for shrinking sheet) and velocity of the free stream flowU(x) =ax, wherea andcare constants,xis the coordinate measured along the stretching/shrinking surface. The flow takes place aty0, whereyis the coordinate measured normal to the stretching/shrinking surface. It is assumed that the temperature at the stretching/shrinking surface takes the constant valuesTw, while the temperature

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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 :

ux+

v

y=0, (2.1)

uux+v

uy=U(x)

dU(x)

dx +

µn f ρn f

∂2u

y2+

µn f ρn fK

(

U(x)−u

)

+(ρβ)n f

ρn f g(TT∞), (2.2)

uTx+v

Ty =αn f

d2T dy2 −

1

Cp)n fqr

y + Q0

Cp)n f(TT∞), (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) uU(x) =ax, v0,T T∞, as y→∞, (2.5) whereuandvare velocity components along thex- andy-directions, respectively.U(x)stands for the stagnation-point velocity in the inviscid free stream.T is the temperature of the nanofluid,Kis the permeability of a porous medium, gis the acceleration due to gravity,Q0is the heat source coefficient (Q0>0) and sink coefficient (Q0<0),a,band care positive constants,lis the characteristic length andvwis the wall mass flux withvw<0 (suctions) andvw>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,(ρCp)n f is the heat capacitance of the nanofluid. These thermophysical parameters are defined as follows (Hamad and Pop [17]):

αn f = κn f

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

(ρβ)n f = (1−φ)(ρβ)f+φ(ρβ)s, (ρCp)n f = (1−φ)(ρCp)f+φ(ρCp)s κn f

κf

=(κs+2κf)−2φ(κf−κs) (κs+2κf) +2φ(κf−κs)

, (2.6)

whereφis the solid volume fraction of the nanofluid,ρfis the reference density of the fluid fraction,ρsis the reference

density of the solid fraction,µf is the viscosity of the fluid fraction,κf is the thermal conductivity of the fluid, andκs

is the thermal conductivity of the solid fraction.

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

qr=−

4

3Kgrad(eb), (2.7)

whereK∗[m−1]is the Rosseland mean spectral absorption coefficient andeb[W m−2]is the blackbody emission power which is given in terms of the absolute temperatureT by the Stefan-Boltzmann radiation laweb=σ∗T4with the

Stefan-Boltzmann constantσ∗=5.6697·10−8W m−2K−4. It is further assumed that the termT4 due to radiation within the flow can be expressed as a linear function of temperature itself. Hence,T4can be expanded as Taylor series aboutT∞and can be approximated after neglecting the higher order terms as,

T44T3T3T4. (2.8) Using Eqs. (2.7) and (2.8), we get

qry =−

16σ∗T∞3 3K

∂2T

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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)xνfF(η), θ(η) = TTTwT

, η=

uw(x) xνf

y, (2.10)

whereνf is the kinematic viscosity of the fluid and the stream function ψ is defined in the usual way asu=∂ ψy,

v=−∂ ψx, 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′′′+K1

(a

cF

′)

+ (1−φ)2.5[(

1−φ+φ(ρβ)s (ρβ)f

)

Λθ+(

1−φ+φρs ρf

)(a2

c2+FF′′−F′ 2)]

=0, (2.11)

1 Pre f f

θ′′+{

1−φ+φ(ρCp)sCp)f

}[

Fθ′−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)

Fa

c, θ→0 as η→∞, (2.15)

wherePr= νf

αf is the Prandtl number,λ = Q0

Cp)f is the heat source(λ >0)or sink(λ<0)parameter. K1=

νf cK is

the porous parameter,S=−vw

cνf is the mass flux parameter (S>0 corresponds to the suction andS<0 corresponds

to injection), Λ=g(ρβf)/ρfc2 is the buoyancy or mixed convection parameter (see ref. [18]), Nr=−16σ

T3

fK∗ is

the radiation parameter,Pre f f=kn fPr k f +Nr

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

Stefan-Boltzmann constantσ∗=0, soNr=0. It is noted thatPre f f= Prin 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.eTw=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 coefficientCf and the local Nusselt Number Nux, which are defined by

Cf =

2µn f ρfu2w

(∂u

y

)

y=0

, Nux= − xκn f κf(TwT∞)

(

−∂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]:

Re1x/2Cf=

2

(1−φ)2.5|F′′(0)|, Re

−1/2

x Nux=−

κn f κf

θ′(0), (2.17)

whereRex=uw(x)xf is the local Reynolds number based on the stretching/shrinking velocityuw(x).

3 Solution algorithm of the numerical method

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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 inFand 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:

F1′=F2, F2′=F3,

F3′=−K1(a/cF2)−φ1φ2{(a/c)2+F1F3−F22} −φ1φ4ΛF4 F4′=F5

F5′=−Pre f f{φ3(F1F5−F2F4) +λF4} (3.18)

where

F1=F, F2=F′, F3=F′′, F4=θ, F5=θ′ and

φ1= (1−φ)2.5, φ2=1−φ+φ(ρs/ρf),

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

F1=S, F2= +1, F4=1, atη=0(f or stretching sheet) F1=S, F2=−1, F4=1, atη=0(f or shrinking sheet)

F2→a/c, F4→0, asη→∞. (3.19) In order to integrate (3.18) and (3.19) as an initial value problem we require the values ofF3(0)andF5(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 valueF3(0)=d10andF5(0)=d20. Letγ1andγ2be the correct values ofF3(0)andF5(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 valuesF3(0)andF5(0)atη=η∞byF3(d10,d20,η∞)and F5(d10,d20,η∞), respectively. SinceF3andF5are clearly function ofγ1andγ2, they are expanded in Taylor series aroundγ1−d10andγ2−d20, respectively by retaining only the linear terms. Thus, after solving the system of Taylor series expansions forδ γ1=γ1−d10 andδ γ2= γ2−d20, we obtain the new estimated11= d10+δd10 andd21=d20+

δd20. Next the entire process is repeated starting withF1(0),F2(0),d11,F4(0)andd21as initial conditions. Iteration of the whole outlined process is repeated with the latest estimates ofγ1andγ2until prescribed boundary conditions are satisfied.

Finallyd1n=d1(n−1)+δd1(n−1) andd2n=d2(n−1)+δd2(n−1) forn=1,2,3... are obtained which seemed to be the most

desired approximate initial values ofF3(0)andF5(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 forF′′(0)andθ′(0)with those of Hamad and Ferdows

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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,K1,Λ,λ, Pre f f,Sanda/c. Thermophysical properties of

fluids and nanoparticles (Cu,Al2O3,TiO2) which are used in this study are given in Table–1. It should be noted that the Prandtl numberProf the base fluid (water) is kept constant at 6.8, whereas Pre f fis 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 coefficientF′′(0)and surface heat flux−θ′(0)are shown in Table–4 for various values ofφ,K1,Λ,λ,Pre f fandS. It is shown in Table-4 that as the nanoparticle volume fractionφincreases, the skin-friction

coefficientF′′(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 parameterS has the tendency to decrease the values ofF′′(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 increasingS. It is also noted

that the effect of increasing the porous parameterK1is to decrease bothF′′(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 bothF′′(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 Al2O3 TiO2

Cp(J/kg K) 4179 385 765 686.2

ρ(kg/m3) 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 forF′′(0)whenK1=Λ=a/c=Nr=λ =0 forCu-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

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Table 3: Comparison of result for−θ′(0)in isothermal case whenφ=K1=Λ=λ =a/c=b= 0 for stretching sheet.

Khan and Hamad and Hamad and Present results

Pre 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 ofF′′(0)and -θ′(0)for the nanofluid (Cu-water) with different values ofφ, K1, Λ, λ, a/c, Pre f f

and Swhena/c=1.

Stretching Shrinking

Sheet Sheet

φ K1 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,Al2O3,TiO2-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 sheetsCu-water has higher velocity distribution and lower temperature distribution compared to other two nanofluids,Al2O3-water and TiO2-water. The effect is found to be more significant in theCu-water nanofluid than in theAl2O3-water andTiO2 -water nanofluids. It is also observed that velocity and temperature profiles forAl2O3-water andTiO2-water nearly coincide with each other for specific values of governing parameters.

The influence ofφon the velocity and temperature fields forCu-water nanofluid can be analyzed from Figs. 4-5. The variation of velocity profiles ofCu–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

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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 parameterK1for stretching and shrinking sheets. It is observed from these figures that velocity increases with increase in the value of the porous parameterK1, whereas reverse effect is observed on the temperature profile for both stretching and shrinking sheets. The effects of buoyancy parameterΛon velocity and temperature profiles ofCu-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 forCu-water nanofluid for both stretching and shrinking sheets.

The effect ofa/con velocity and temperature profiles over stretching and shrinking sheets are shown in Figs. 11-12 forCu-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 whena/c=0, the velocity profile decreases alongηtill it matches the boundary conditionF′(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 ofCu-water decreases asa/cincrease for stretching and shrinking sheets alongwith decrease in the peak. Figs. 13-14 depict the effects of the effective Prandtl number Pre f f on skin-friction coefficient and the local

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

nanoflu-ids consisting ofCu,Al2O3,TiO2nanoparticles. It is also noticed thatCu-water has significantly higher skin-friction coefficient compared to those ofAl2O3-water andTiO2-water nanofluids. On the other hand, the local Nusselt number gradually increases for stretching sheet and decreases for shrinking sheet as Pre f f increases for all the three types of

nanofluids.

Figs. 15-16 depict the effects of the suction/injection parameterS(−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 ofSfor stretching/shrinking sheet for three types of nanofluids consisting ofCu,Al2O3,TiO2nanoparticles. It is also noticed from Fig. 15 thatCu-water has significantly higher skin-friction coefficient compared to those ofAl2O3andTiO2-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, namelyCu,Al2O3,TiO2in 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 parameterK1, buoyancy or mixed convection parameterΛ, heat generation/absorption parameterλ, suction/injection parameterS, effective Prandtl number Pre f f

and the parametera/c. Some of the following conclusions are drawn from figures and tables:

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(ii) An increase in the values ofφ,K1andΛresults in an increase in the velocity profile forCu-water nanofluid for both stretching and shrinking sheets but these effects are opposite on temperature profiles.

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

(iv) Effective Prandtl number Pre 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 Sincreases 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 K1 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

Q0 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 inx-direction uw stretching/shrinking sheet velocity U free stream velocity of the nanofluid v velocity component iny-direction

(10)

Greek symbols

αn f effective thermal diffusivity of the nanofluid

αf fluid thermal diffusivity βn f thermal expansion of nanofluid

βf 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

νf 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

κf thermal conductivity of the fluid

Superscripts

differentiation with respect toy

Subscripts

n f nanofluid

f fluid

(11)

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

(12)

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

(13)

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 variousφforCu-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

'

(

)

(14)

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 variousK1forCu-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

'

(

)

(15)

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 variousΛforCu-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

(

)

(16)

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 variousa/cforCu-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

(17)

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 withPre f ffor 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

(18)

-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 withSfor 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

(19)

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Referências

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