solar radiation effects on cu–water nanofluid flow over a stretching sheet with surface slip and...
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Arab J Sci Eng (2014) 39:9015–9023DOI 10.1007/s13369-014-1380-4
RESEARCH ARTICLE - MECHANICAL ENGINEERING
Solar Radiation Effects on Cu–Water Nanofluid Flowover a Stretching Sheet with Surface Slip and Temperature Jump
Kalidas Das · Pinaki Ranjan Duari ·Prabir Kumar Kundu
Received: 27 January 2014 / Accepted: 30 May 2014 / Published online: 16 October 2014© King Fahd University of Petroleum and Minerals 2014
Abstract In the present study, hydromagnetic convectiveflow and heat transfer of an absorbing and electrically con-ducting Cu–water nanofluid over a semi-infinite, ideallytransparent, permeable stretching sheet due to solar radiationis considered. The flow considered is under both surface andthermal slip conditions. The governing equations are trans-formed into a nonlinear ordinary differential equations usingclassical Lie group approach which are solved numericallyby means of the efficient numerical shooting technique withfourth-order Runge–Kutta scheme. The effects of involvedparameters on the velocity and temperature profiles, skin fric-tion and Nusselt number are examined and discussed throughgraphs and tables. Comparisons with previously publishedworks are performed, and excellent agreement between theresults is obtained.
Keywords Nanofluid · Lie group analysis · Solar energyradiation · Magnetic field · Slip conditions
K. Das (B)Department of Mathematics, Kalyani Government EngineeringCollege, Kalyani 741235, W.B., Indiae-mail: [email protected]
P. R. Duari · P. K. KunduDepartment of Mathematics, Jadavpur University, Kolkata 700032,W.B., Indiae-mail: [email protected]
P. K. Kundue-mail: [email protected]
1 Introduction
Solar energy is the most readily available source of energy.It is also the most important of the non-conventional sourcesof energy because it is non-polluting and, therefore, helps inlessening the greenhouse effect. Solar energy, radiant lightand heat from the sun is harnessed using a range of ever-evolving technologies such as solar heating, solar photo-voltaic cells, solar thermal electricity, solar architecture andartificial photosynthesis. Thus, the utilization of solar energyand the technologies of solar energy materials attract muchmore attention in many branches of applied physics and en-gineering. Nanomaterials whose particle size is smaller thanthe wavelength of de Broglie wave and coherent wave arenewly developed energy materials. Therefore, nanoparticlesbecome to strongly absorb (i.e., skipping intermediate heattransfer steps) and selectively absorb (i.e., high absorptionin the solar range and low emittance in the infrared) incidentradiation. Hunt [23] was the first who introduced the conceptof using nanoparticles to collect solar energy. Engineeredsuspensions of nanoparticles in liquids, known recently asnanofluid, have generated considerable interest for their po-
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tentials to enhance the heat transfer rate in engineering sys-tem. Nanofluids are made from materials, such as metals(Cu, Ag, Au), oxide ceramics Al2O3, CuO, semiconductorsTiO2,SiC, carbon nanotubes and composite materials suchas alloyed nanoparticles. The base media of nanofluids areusually water, oil, ethylene glycol, acetone, etc. Comparedwith conventional heat transfer such as oil, water and ethyl-ene glycol mixture, nanofluids have significantly higher ther-mal conductivity that consequently enhances the heat transfercharacteristics of these fluids.
Natural convection heat transfer of nanofluid has drawnattraction of many researchers in recent years [2–6,8–15,19].Natural convection heat transfer in the presence of nanofluidsdue to solar energy radiation is an important phenomenon inengineering systems due to its wide applications in buildingheating, automotive technology, solar technology, cooling ofelectronic equipment, etc [16–18]. Kandasamy et al. [33]studied Hiemenz flow of Cu-nanofluid over a porous wedgedue to solar radiation. Recently, Anbuchezhian et al. [20]have reported on the effects of transverse magnetic field onnatural convection flow of a nanofluid due to solar energy.
In all the above mentioned papers, investigators restrictedtheir analyses to flow and heat transfer with no-slip bound-ary condition. But no-slip assumption is not true for fluidflows at the micro and nanoscale. Investigation shows thatslip flow happens when the characteristic size of the flowsystem is small or the flow pressure is very low. To describethe phenomenon of slip, Navier [21] introduced a bound-ary condition which states that the component of the fluidvelocity tangential to the boundary walls is proportional totangential stress. Later, several researchers [22,23] extendedthe Navier boundary conditions. Numerous investigationshave been done analytically and numerically regarding theslip flow regimes over surfaces. Martin and Boyd [24] an-alyzed Blasius boundary layer problem in the presence ofslip boundary condition. These results demonstrated that theboundary layer equation can be used to study flow at the mi-cro electro mechanical system (MEMS) scale and provideuseful information to study the effects of rarefaction on theshear stress and structure of the flow. In another study [25],they have analyzed slip flow and heat transfer at constantwall temperature. The hydrodynamic flow in the presenceof partial slip over a stretching sheet with suction has beenstudied by Wang [26]. Van Gorder et al. [27] considered themodel proposed by Wang [28] describing the various flowdue to a stretching surface with both surface slip and suc-tion/injection. Slip effects on mixed convection flow of amicropolar fluid toward a shrinking vertical sheet was stud-ied by Das [29]. Recently, Das [30] analyzed the nanofluidflow over a shrinking sheet with surface slip.
Motivated by the above studies, we intend to investigatethe influence of slip on the behavior of fluid flow and ther-mal transport of an absorbing and electrically conducting
nanofluid over a stretching surface due to solar radiation andin the presence of an external applied magnetic field. Theplate is assumed non-reflecting, non-absorbing, ideally trans-parent and electrically non-conducting in the present work.The paper is organized as follows: In Sect. 2, the mathemati-cal analysis of the problem is discussed in detail. Numericalexperiment is presented in Sect. 3. Section 4 highlights theresults obtained through extensive computation. Section 5summarizes the important outcomes of the study.
2 Mathematical Analysis
2.1 Governing Equations
The steady two-dimensional boundary layer flow of a viscousincompressible electrically conducting Cu–water nanofluidover a semi-infinite, ideally transparent, permeable stretch-ing sheet embedded in a fluid saturated porous medium withsurface slip and temperature jump due to solar radiation isconsidered. The coordinate system is such that x measuresthe distance along the plate and y measures the distance nor-mally into the fluid. The schematics of the problem underconsideration and the coordinate system are shown in Fig. 1.A magnetic field of uniform strength B0 is applied in thenegative y direction at all times. Its interaction with the elec-trically conducting working nanofluid produces a resistiveforce called Lorentz force in the negative x-direction. Themagnetic Reynolds number is assumed to be small so thatthe induced magnetic field is neglected. In addition, no elec-tric field exists and the Hall Effect, viscous dissipation and
Fig. 1 Physical model and coordinate system
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Joule heating are all neglected. The stretching surface tem-perature is deemed to have constant value Tw, while at a largevalue of y, temperature have constant ambient value T∞.
It is assumed that the porous medium is transparent andin thermal equilibrium with the fluid. Also, it is assumedthat both the fluid and the porous medium are opaque forself-emitted thermal radiation. In the present study, the solarradiation is a collimated beam that is normal to the stretch-ing surface. Due to the heating of the absorbing nanofluidand the inclined permeable plate by solar radiation, heat istransferred from the plate to the surroundings. On the otherhand, one may have a non-absorbing fluid. In this case, thesolid porous medium absorbs the incident solar radiation andtransmits it to the working fluid by convection. The densityof the nanofluid is approximated by the standard Boussi-nesq model. The viscosity and thermal conductivity of thenanofluid are considered as variable properties. Under theabove assumptions, the boundary layer equations governingthe flow and temperature field can be written as
∂u
∂x+ ∂v
∂y= 0 (1)
u∂u
∂x+ v
∂u
∂y= 1
ρn f
[μn f
∂2u
∂y2 + (ρβ)n f g(T − T∞)
−(σ B2
0 + ν f
κρn f
)u]
(2)
u∂T
∂x+ v
∂T
∂y= αn f
∂2T
∂y2 − 1
(ρC)n f
∂q ′′rad
∂y
− Q0
(ρC p)n f(T − T∞) (3)
where u, v are the velocity components along the x-axis andy-axis, respectively, T is temperature,μn f is the effective dy-namic viscosity of the nanofluid, B0 is the uniform magneticfield, g is the gravitational acceleration, σ is the electricalconductivity, k is the permeability of the porous medium, C p
is the specific heat at constant pressure, ρn f is the effectivedensity of the nanofluid, αn f is the thermal diffusivity of thenanofluid, Q0 is the temperature dependent volumetric rate ofheat source/sink and q ′′
rad is the applied absorption radiationheat transfer per unit area. Using Rosseland approximationfor radiation, we can write
q ′′rad = −4σ1
3k∗∂T 4
∂y(4)
whereσ1 is the Stefan-Boltzmann constant and k∗ is the meanabsorption coefficient.
2.2 Thermophysical Properties of Nanofluid
In this study, Cu–water nanofluid is used as the working fluid.The thermophysical properties of the nanofluid involved inthe governing equations are calculated using the following
Table 1 Thermophysical properties of regular fluid and nanoparticles
Physical properties Regular fluid (water) Cu
CP (J/kg K) 4,179 385
ρ (kg/m3) 997.1 8,933
κ (W/mK) 0.613 400
α × 107 (m2/s) 1.47 1,163.1
β × 10−5 (1/K) 21 1.67
equations (see Maxwell [31], Oztop and Abu-Nada [32]) andit is listed in Table 1.
The effective dynamic viscosity of the nanofluid is givenby
μn f = μ f
(1 − ζ )2.5(5)
where ζ is the solid volume fraction of nanoparticles. Theeffective density ρn f , thermal diffusivity αn f and the heatcapacitance of the nanofluid (ρC p)n f are given by
ρn f = (1 − ζ )ρ f + ζρs (6)
αn f = kn f
(ρC p)n f(7)
(ρC p)n f = (1 − ζ )(ρC p) f + ζ(ρC p)s (8)
The thermal conductivity of nanofluid restricted to sphericalnanoparticles is approximated by the Maxwell [31]
κn f
κ f= κs + 2κ f − 2ζ(κ f − κs)
κs + 2κ f + 2ζ(κ f − κs)(9)
Here μ f is the viscosity of the base fluid, ρ f and ρs are thedensities of the pure fluid and nanoparticles, respectively,(ρC p) f and (ρC p)s are the specific heat parameters of thebase fluid and nanoparticles, respectively, and κ f and κs arethe thermal conductivities of the base fluid and nanoparticles,respectively.
2.3 Boundary Conditions
In the slip flow regime, slip velocity and temperature jumpboundary conditions should be applied to the momentum andenergy equations. These are
us = 2 − σv
σvλ0
(∂u
∂y
)y=0
(10)
where σv is the tangential momentum accommodation coef-ficient, λ0 is the molecular mean free path and
Ts = 2 − σT
σT
2r
1 + r
λ0
Pr
(∂T
∂y
)y=0
(11)
where σT is the thermal accommodation coefficient, Pr is thePrandtl number and r is the specific heat ratio, respectively.
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Thus, the appropriate boundary conditions for the presentproblem are
u = uw + us, v = V0, T = Tw + Ts at y = 0u → 0, T → T∞ as y → ∞
}(12)
where uw = U (=ax) is the velocity of the stretching sheet, a(>0) is the stretching rate (of the sheet) and V0 is the velocityof suction/injection.
2.4 Non-dimensionalization
Let us introduce the following non-dimensional variables:
u′ = u√aν f
, v′ = v√aν f
, x ′ = x√ν fa
, y′ = y√ν fa
,
θ = T − T∞Tw − T∞
(13)
Equations (1)–(3) take the non-dimensional form (droppingprime)
∂u
∂x+ ∂v
∂y= 0 (14)
u∂u
∂x+ v
∂u
∂y= 1(
1 − ζ + ζρsρ f
)[
1
(1 − ζ )2.5
∂2u
∂y2
+{
1 − ζ + ζ(ρβ)s
(ρβ) f
}λθ −
(M2 + K
)u
](15)
u∂θ
∂x+ v
∂θ
∂y= 1
Pr
1
1 − ζ + ζ(ρC p)s(ρC p) f
[κn f
κ f
∂2θ
∂y2
+4
3N∂
∂y
{(CT + θ)3
∂θ
∂y
}− θδ
](16)
with the boundary conditions
u = x + γ ∂u∂y , v = V0√
aν f, θ = 1 + ξ ∂θ
∂y at y = 0
u → 0, θ → 0 as y → ∞
}(17)
where M = B0
√σ
aρ fis the magnetic field parameter, λ =
g(ρβ) f (Tw−T∞)aρ f U is the natural convection parameter, K = ν f
k is
the permeability parameter of the porous media, Pr = ν fαn f
is
the Prandtl number, N = 4σ1T 3∞(ρC p) f k∗ is the radiation parameter,
CT = T∞Tw−T∞ is the temperature ratio where CT assumes
very small values by its definition as Tw − T∞ is very large
compared to T∞, δ = Q0ν2f (Tw−T∞)κ f U 2 is the heat source/sink
parameter, γ = 2−σvσv
√aν fλ0 is the slip velocity parameter
and ξ = 2−σTσT
2r1+r
λ0Pr
√aν f
is the thermal slip parameter.
The introduction of the stream function ψ(x, y), definedby u = ∂ψ
∂y and v = − ∂ψ∂x , leads to Eqs. (15), (16) taking the
following form:
∂ψ
∂y
∂2ψ
∂x∂y− ∂ψ∂x
∂2ψ
∂y2 = 1(1 − ζ+ζ ρs
ρ f
)[
1
(1 − ζ )2.5
∂3ψ
∂y3
+{
1 − ζ + ζ(ρβ)s
(ρβ) f
}λθ −
(M2 + K
) ∂ψ∂y
](18)
∂ψ
∂y
∂θ
∂x− ∂ψ
∂x
∂θ
∂y= 1
Pr
1
1 − ζ + ζ(ρC p)s(ρC p) f
[κn f
κ f
∂2θ
∂y2
+4
3N∂
∂y
{(CT + θ)3
∂θ
∂y
}− θδ
](19)
with the boundary conditions
∂ψ
∂y= x+γ ∂
2ψ
∂y2 ,∂ψ
∂x=− V0√
aν f, θ=1+ξ ∂θ
∂yat y = 0
u → 0, θ → 0 as y → ∞
⎫⎬⎭(20)
2.5 Symmetry Groups of Equations
The symmetry groups of Eqs. (18),(19) are calculated usingclassical Lie group approach. The one-parameter infinitesi-mal Lie group of transformations leaving (18), (19) invariantis defined as
x∗ = x + εξ1(x, y, ψ, θ), y∗ = y + εξ2(x, y, ψ, θ),ψ∗ = ψ + εη1(x, y, ψ, θ), θ∗ = θ + εη2(x, y, ψ, θ)
}
(21)
where ε is the parameter of the group. Equation (21) maybe considered as a point-transformation which transformscoordinates (x, y, ψ, θ) to the coordinates (x∗, y∗, ψ∗, θ∗).
By carrying out a straight forward but tedious algebra, theform of the infinitesimals can be obtained as
ξ1 = d1x + d2, ξ2 = g(x), η1 = d3ψ + d4, η2 = d5 (22)
where d1, d2, d3, d4 and d5 are arbitrary parameters and theparameter d1 represents the scaling transformation and g(x)is an arbitrary function.
Imposing the restrictions from the boundary conditions onthe infinitesimals, one may obtain the following form for Eq.(22)
ξ1 = d1x, ξ2 = 0, η1 = d3ψ, η2 = 0 (23)
The relations (23) lead to the following characteristic equa-tions for similarity:
dx
x= dy
0= dψ
ψ= dθ
0(24)
from which the similarity variables, velocity and temperatureturn out to be of the form
η = y, ψ = x f (η) and θ = θ(η) (25)
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Substituting Eq. (25) into Eqs. (18), (19), one may finallyobtain the system of nonlinear ordinary differential equations
f ′′′ + (1 − ζ )2.5[(
1 − ζ + ζρs
ρ f
) (f f ′ − f ′2)
−(
M2 + K)
f ′ +{
1 − ζ + ζ(ρβ)s
(ρβ) f
}λθ
]= 0
(26)kn f
k fθ ′′ − δθ + 4
3N
{(CT + θ)3θ ′}′
+ Pr(
f θ ′ − f ′θ) {
1 − ζ + ζ(ρcp)s
(ρcp) f
}= 0 (27)
where prime denotes differentiation with respect to η.The corresponding boundary conditions become
f = S, f ′ = 1 + γ f ′′, θ = 1 + ξθ ′ at η = 0f ′ → 0, θ → 0 as η → ∞
}(28)
where S = − V0√aν f
, S > 0 corresponds to suction and S < 0
corresponds to injection.
2.6 Physical Quantities of Interest
The quantities of physical interest in this problem are the skinfriction coefficient and the local Nusselt number which aredefined as follows:
C f = μn f
ρ f U 2
(∂u
∂y
)y=0
= − 1
(1 − ζ )2.5(Rex )
− 12 f ′′(0)
(29)
Nu = xkn f
k f (Tw − T∞)
(∂T
∂y
)y=0
− 4σ1
3k∗
(∂T 4
∂y
)y=0
= − (Rex )12
kn f
k fθ ′(0)
[1 + 4
3N {CT + θ(0)}3
](30)
where Rex = Uxν f
is the local Reynolds number. Thus, thereduced skin friction coefficient C f r and the reduced Nusseltnumber Nur can be as
C f r = Re12x C f = − 1
(1 − ζ )2.5f ′′(0) (31)
Nur = Re− 1
2x Nu = −kn f
k fθ ′(0)
[1 + 4
3N {CT + θ(0)}3
]
(32)
3 Numerical Experiment
3.1 Method of Solution
The set of Eqs. (26), (27) being highly nonlinear cannotbe solved analytically. These coupled equations have beensolved numerically by applying the Nachtsheim and Swigert
Table 2 Comparison of results for −θ ′(0) with previously publishedwork
Pr Wang [28] Anbuchezhian et al. [20] Present work
0.07 0.0656 0.0655580 0.0655578
0.20 0.1691 0.1690967 0.1690953
0.70 0.4539 0.4539134 0.4539130
2.00 0.9114 0.9113678 0.9113666
7.00 1.8954 1.8953998 1.8953999
20.0 3.3539 3.3538999 3.3538999
70.0 6.4622 6.4621997 6.4621968
(see Kafoussias and Williams [33]) shooting iteration tech-nique together with Runge–Kutta fourth-order integrationscheme. The unspecified initial conditions are assumed andthen integrated numerically as an initial value problem to agiven terminal point. Improvement is made on the values ofassumed missing initial conditions by iteratively comparingthe calculated value of the dependent variable at the terminalpoint with its given value there. A step size of�η = 0.01 hasbeen selected to be satisfactory for a convergence criterionof 10−6 in all cases.
3.2 Code Verification
As a test of the accuracy of the solution, the values of −θ ′(0)are compared with Anbuchezhian et al. [20] and Wang [28]for various values of Pr in the absence of external heatsource/sink, nanoparticles, slip velocity, thermal slip, solarradiation and magnetic field in Table 2. Table shows that thenumerical results obtained by the present code and the resultsreported by Anbuchezhian et al. [20] and Wang [28] are invery good agreement. Thus, the use of the present numericalcode for current model is justified.
4 Results and Discussions
In this section, we present our findings in tabular and graph-ical forms in order to investigate the important features ofthe solution for a range of values of the parameters affectingthe flow and heat transfer phenomena. The numerical resultsfor dimensionless velocity f ′(η) and temperature θ(η) arecomputed for the following general values: M = 0.5, K =1.0, λ = −1.2, γ = 0.1, ξ = 0.05, N = 0.2, S = 1.2 andζ = 0.05 unless otherwise specified.
Table 3 presents the effects for various values of the ra-diation parameter N, slip velocity parameter γ , thermal slipparameter ξ , nanoparticle volume fraction ζ and magneticfield parameter M on the reduced skin friction coefficient C f rand the reduced Nusselt number Nur. From Table, it may benoticed that the reduced skin friction coefficient at the wall
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Table 3 Effects of various parameters on C f r and Nur
ζ γ ξ M N C f r Nur
0.00 0.2 0.03 0.5 0.2 1.38397 4.04710
0.05 1.61773 4.29743
0.10 1.87390 4.55141
0.15 2.15855 4.81379
0.05 0.0 2.35023 4.67176
0.2 1.61773 4.29743
0.4 1.24842 4.05833
0.6 1.02152 3.88535
0.1 0.00 1.91814 4.96337
0.03 – 4.45961
0.06 – 4.02269
0.10 – 3.53826
0.03 0.5 1.91814 4.45961
1.0 2.01153 4.4232
2.0 2.18974 4.3591
5.0 2.60095 4.21082
0.5 0.0 1.90570 4.15859
0.2 – 4.45961
0.4 – 4.71725
0.6 – 4.94103
1.0 – 5.31626
decreases with an increase in the slip parameter γ . This isless pronounced with an increase in the value of γ . That is,as expected, for the fluid flows at nanoscales, the shear stressat the wall decreases with an increase in the slip parameterγ . One may noted that in the no-slip condition problem thehighest wall shear stress occurs. Also, the Nusselt numberdecreases with the increase in slip parameter for Cu–waternanofluid. Further, it is observed from table that an increasein thermal slip parameter ξ leads to decrease in the valuesof the rate of heat transfer at the boundary wall of the plate.Therefore, it is desirable, to reach a high heat transfer rate,less slip on the wall by a liquid with a high Prandtl number isneeded. It can be noticed that the heat transfer rate at the plateincreases with increasing values of radiation parameter N buteffect is not prominent on C f r . It is noteworthy that the solarradiation effect exerts a strong influence on the heat transferat the plate; enhancing it by near about 27 % as the radia-tion parameter changes from 0.0 to 1.0. This enhancement isdue to the nanoparticles of high thermal conductivity beingdriven away from the hot sheet to the quiescent nanofluid.The wall skin friction coefficient increases with an increasein the nanoparticle volume fraction ζ as shown in Table 3. Itis also found from this table that the presence of nanoparticlesresults in an increase of the reduced Nusselt number.
Now, the graphical results that provide additional insightinto the problem under investigation are discussed as follows:
Fig. 2 Velocity profiles for various values of γ
Fig. 3 Velocity profiles for various values of M
4.1 Computational Results for Velocity Profiles
Figure 2 draws out the influence of slip parameter γ on thefluid velocity. As slip parameter γ increases, the slip at thesurface wall of the plate increases. Thus, the fluid velocity in-creases with the increase of slip parameter γ in the presenceof solar radiation for η < 0.7 (not precisely determined).But the effect is not significant for η > 0 (not preciselydetermined). This yields an increase in the boundary layerthickness. Thus, hydrodynamic boundary layer thickness in-creases as the slip parameter γ increases for both the regular(water) and nanofluid and as a result, the local velocity alsoincreases. The physics behind this is that the increased slipparameter near the wall region decreases the velocity gradi-ent at there. In the no-slip condition, γ approaches zero, sothe slip velocity at the wall is equal to zero (i.e., us = 0),accordingly the fluid velocity adjacent to the wall is equal tothe velocity of the stretching surface (uw), then f ′(0) = 1. Itshould be noted that the effect of slip parameter is prominentfor water than Cu–water nanofluid. The effect of magneticfield parameter M on the fluid velocity is depicted in Fig. 3.It is observed from the figure that the fluid velocity decreaseswith an increase in the values of M near the boundary layer re-
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Fig. 4 Velocity profiles for various values of ζ
Fig. 5 Temperature profiles for various values of γ
gion and so decreases the thickness of momentum boundarylayer. The reason behind this phenomenon is that applica-tion of magnetic field to an electrically conducting nanofluidgives rise to a resistive type force called the Lorentz force.This force has the tendency to slow down the motion of thenanofluid in the boundary layer region. Figure 4 illustratesthe variation of the velocity distribution for various values ofthe nanoparticle volume fraction parameter ζ . It is seen fromthe figure that the velocity distribution across the boundarylayer decreases with the increase of ζ for Cu–water. Thus,the momentum boundary layer thickness decreases and tendsasymptotically to zero as the distance increases from theboundary. This sensitivity of the boundary layer thicknessto the volume fraction of nanoparticles is related to the in-creased thermal conductivity of the nanofluid. In fact, highervalues of the thermal conductivity are accompanied by highervalues of thermal diffusivity.
4.2 Computational Results for Temperature Profiles
The effect of slip velocity parameter γ on temperature dis-tribution is shown in Fig. 5. Figure indicates that an increaseof slip parameter tends to increase the temperature into the
Fig. 6 Temperature profiles for various values of ξ
Fig. 7 Temperature profiles for various values of M
fluid field for both Cu–water nanofluid and regular fluid, i.e.,water but effect is not significant for regular fluid, i.e., forwater in comparison to the nanofluid. Thus, by escalatingγ , thermal boundary layer thickness enhances. So, we caninterpret that the rate of heat transfer decreases with the in-crease in slip parameter γ . This phenomenon is more promi-nent in the presence of nanoparticles. Figure 6 presents theeffect of thermal slip parameter ξ on fluid temperature inthe presence of solar radiation. It is observed from the fig-ure that in the absence of thermal slip parameter, i.e., whenξ = 0, the temperature of the fluid and that of the stretch-ing surfaces is same, which in the present problem is one,i.e., Tw = 1. As the slip parameter ξ increases, the temper-ature of the fluid decreases near the boundary layer regionand hence the thickness of thermal boundary layer decreases.It is worth for mentioning that these profiles satisfy the farfield boundary conditions asymptotically, which support thenumerical results obtained. The influence of magnetic fieldon the fluid temperature is illustrated in Fig. 7. Figure showsthat the fluid temperature is the maximum near the bound-ary layer region and it decreases on increasing the boundarylayer coordinate η to approach free stream value. Also fluidtemperature increases with increasing the values of M in the
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Fig. 8 Temperature profiles for various values of N
Fig. 9 Temperature profiles for various values of ζ
boundary layer region and, as a consequence, thickness ofthe thermal boundary layer increases. The impact of radia-tion parameter N on the temperature profiles is presented inFig. 8. It can easily be seen from figure that the temperaturedecreases as the boundary layer coordinate η increases for afixed value of N but the rate of decrease is faster as value ofN goes on increasing. For a nonzero fixed value of η, temper-ature distribution across the boundary layer increases withthe increasing values of N and hence the thickness of ther-mal boundary layer increases. Another important fact is thatthe effect of N is more significant for Cu–water nanofluidthan that of regular fluid. Figure 9 demonstrates the effect ofnanoparticle volume fraction parameter ζ on nanofluid tem-perature in the presence of heat source/sink. It is observedfrom the figure that temperature enhances on increasing ζin the boundary layer region and is maximum at the surfaceof the plate. Thus, the presence of nanoparticles, namely Culeads to an increase in the thickness of the thermal boundarylayer profile and tends asymptotically to zero as the distanceincreases from the boundary.
5 Conclusions
In the present study, the influence of slip velocity and ther-mal slip on the behavior of fluid flow and thermal trans-port of an absorbing and electrically conducting nanofluidover a stretching surface due to solar radiation and in thepresence of an external applied magnetic field is analyzedusing numerical technique. Lie group transformations areapplied to the governing equations. The reduced nonlinearordinary differential equations are solved numerically by em-ploying Runge–Kutta–Fehlberg method with shooting tech-nique. The following conclusion can be drawn from thepresent investigation:
• In boundary layer region, the velocity of the fluid in-creases with the increase of slip parameter γ but effect isreverse for magnetic field parameter M and nanoparticlevolume fraction ζ .
• An increase in the slip velocity, solar radiation andnanoparticles volume fraction lead to increase the ther-mal boundary layer thickness but opposite effect occursfor thermal slip.
• The skin friction coefficient C f r increases with the in-crease in the nanoparticle volume fraction ζ and mag-netic field parameter M while it decreases with slip pa-rameter γ .
• The increasing values of γ , M and ξ are to decreasethe reduced Nusselt number; consequently, less heat iscarried out of the sheet, resulting in an increase of thethermal boundary layer thickness and hence decreasingthe heat transfer rate but the nature is opposite for solarradiation parameter N and nanoparticle volume fractionζ .
• The results demonstrate that nanofluid clearly reduceheat transfer rate compared to their own base fluid.
• The limiting case of the present results (γ = ξ = 0) isin excellent agreement with the results of Anbuchezhianet al. [20].
Acknowledgments Research supported from, UGC, Government ofIndia under the scheme of UGC-BSR research fellowship in sciencefor meritorious students, gratefully acknowledged by second author.The authors are very thankful to the reviewers for their constructivecomments which have improved significantly the quality of the paper.
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