Afr. Mat. DOI 10.1007/s13370-012-0124-4

Nanofluid flow towards a convectively heated stretching surface with heat source/sink: a lie group analysis Prabir Kumar Kundu · Kalidas Das · Subrata Jana

Received: 2 August 2012 / Accepted: 7 November 2012 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2012

Abstract This paper considers the steady two dimensional flow of an electrically conducting nanofluid over a vertical convectively heated permeable stretching surface with variable stream conditions in presence of a uniform transverse magnetic field and internal heat source/sink. The transport equations include the effects of Brownian motion and thermophoresis. The governing partial differential equations are converted to ordinary differential equations via Lie group analysis. We employ an extensively validated, highly efficient symbolic software MATHEMATICA using finite difference code to study the problem numerically. The influences of various relevant parameters on the temperature and nanoparticle volume fraction as well as wall heat flux and wall mass flux are elucidated through graphs and tables. Keywords Nanofluid · Lie group analysis · Heat source/sink · Convective boundary condition · Brownian motion · Thermophoresis Mathematics Subject Classification (2010)

76W05

1 Introduction Lie group analysis, also called symmetry group analysis, and developed by Lie [1–3], is the most powerful, sophisticated and systematic method to generate similarity transform and is widely used in non-linear dynamical system, specially in the range of deterministic chaos. Similarity analysis reduces the number of variables that govern partial differential equations and consequently changes it to ordinary differential equations. In case of scaling

P. K. Kundu· S. Jana Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India e-mail: [email protected] K. Das (B) Department of Mathematics, Kalyani Government Engineering College, Kalyani 741235, West Bengal, India e-mail: [email protected]

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group of transformations, the group invariant solutions of initial and boundary value problems are nothing but the well known similarity solutions. The scaling group techniques have been applied by many researchers [4–11] to study different flow phenomena over different geometries arising in fluid mechanics, chemical engineering, plasma physics, aerodynamics and other engineering branches. A special form of Lie group transformations, known as scaling group of transformation, is used in the present study to find out the full set of symmetries of the problem and then to study which of them are appropriate to provided group invariant or more specifically similarity solutions. Convective heat transfer in nanofluids is a topic of major contemporary interest both in applied sciences and engineering. Choi [12] was the first to introduce the word nanofluid that represent the fluid in which nanoscale particles (diameter <50 nm) are suspended in the base fluid. With the rapid advances in nanotechnology, many inexpensive combinations of liquid/particles are now available. The base fluids used are usually water, ethylene glycol, toluene and oil. Recent research on nanofluids showed that nanoparticles changed the fluid characteristics because thermal conductivity of these particles was higher than convectional fluids. Nanoparticles are of great scientific interest as they are effectively a bridge between bulk materials and atomic or molecular structures. The common nanoparticles that have been used are aluminum, copper, iron and titanium or their oxides. Experimental studies [13,14] show that even with the small volumetric fraction of nanoparticles (usually <5 %), the thermal conductivity of the base liquid can be enhanced by 10–20 %. The enhanced thermal conductivity of nanofluids together with the thermal conductivity of the base liquid and turbulence induced by their motion contributes to a remarkable improvement in the convective heat transfer coefficient. Various benefits of the application of nanofluids include: improved heat transfer, heat transfer system size reduction, minimal clogging, micro-channel cooling and miniaturization of the system. Therefore, research is underway to apply nanofluids in environments where higher heat flux is encountered and the convectional fluid is not capable of achieving the desired heat transfer rate. It should be noticed that there have been published several recent papers [15–17] on the mathematical and numerical modeling of convective heat transfer in nanofluids. These models have some advantages over experimental studies due to many factors that influence nanofluids properties. The boundary layer flow of a nanofluids caused by a stretching surface has drawn the attention of many researchers [18–21]. Recently Kandasamy et al. [22] Investigated MHD boundary layer flow of nanofluids over a stretching surface using scaling group transformation. It is worth mentioning that while modeling the boundary layer flow and heat transfer, the boundary conditions that are usually applied are either a specified surface temperature or a specified surface heat flux. However, there are boundary layer flow and heat transfer problems in which the surface heat transfer depends on the surface temperature. Perhaps the simplest case of this is when there is a linear relation between the surface heat transfer and surface temperature. These situations arise in conjugate heat transfer problems and when there is Newtonian heating of the convective fluid from the surface. The situation with Newtonian heating arises in what is usually termed as conjugate convective flow, where the heat is supplied to the convective fluid through a bounding surface with a finite heat capacity. This results in the heat transfer rate through the surface being proportional to the local difference in the temperature with the ambient conditions. This configuration of Newtonian heating occurs in many important engineering devices, for example, in heat exchangers, where the conduction in a solid tube wall is greatly influenced by the convection in the fluid flowing over it. On the other hand, recently, heat transfer problems for boundary layer flow concerning with a convective boundary condition were investigated by Aziz [23], Makinde and Aziz [24], Ishak [25] and Makinde and Aziz [26]. Recently, Aziz and Khan [27] considered natural

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convective boundary layer flow of nanofluids over a convectively heated vertical plate. But so far, no attempt has been made to analyze the MHD boundary layer flow of a nanofluid past a convectively heated vertical permeable stretching surface in the presence of heat source/sink. In this paper, the main objective is to investigate the effect of a convective boundary condition on MHD boundary layer flow of a nanofluid over a vertical stretching surface with internal heat source/sink using Lie group analysis. The similarity solutions are derived and used to predict the heat and mass transfer characteristics of the nanofluid flow. The organization of the paper is given as follows. Section 2 deals with the mathematical formulation of the convective transport model. Section 3 contains Lie group transformations. Numerical method and validation of the code are given in Sect. 4. Results and discussions are presented in Sect. 5. The conclusions have been summarized in Sect. 6.

2 Convective transport model Consider the steady two-dimensional natural convection boundary layer flow of an electrically conducting nanofluid over a permeable vertical stretching surface with convective boundary condition. The flow is assumed to be in the x-direction which is taken along the plate and y-axis is normal to it as illustrated in Fig. 1. A uniform transverse magnetic field of strength B0 is applied parallel to the y-axis. The magnetic Reynolds number of the flow is taken to be small enough so that induced magnetic field is assumed to be negligible in comparison with applied magnetic field. The temperature and the nanoparticle volume fraction at the stretching surface are deemed to have constant values T f and Cw , respectively, while the ambient temperature and nanoparticle fraction have constant values T∞ and C∞ , respectively. It is further assumed that the base fluid and the suspended nanoparticles are in thermal equilibrium and no slip occurs between them. Under the above assumptions and applying the Oberbeck–Boussinesq approximations to simplify the Buongiorno [15] convective transport equations, we obtain the following boundary layer equations:

Fig. 1 Physical model and coordinate system

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∂u ∂v + =0 ∂x ∂y u

(1)

∂ 2 u σ B20 ∂u ∂v u+[ρ f∞ β(1−C∞ )(T−T∞ )−(ρ p − ρ f∞ )(C −C∞ )]g, +v = νf 2 − ∂x ∂y ∂y ρf     ∂C ∂T Q0 ∂T ∂T κ ∂ 2T DT ∂T 2 + (T − T∞ ) + τ DB u +v = + , ∂x ∂y ρcp ∂y2 ρcp ∂y ∂y T∞ ∂y u

∂C DT ∂ 2 T ∂C ∂ 2C +v = DB 2 + ∂x ∂y ∂y T∞ ∂y2

(2) (3) (4)

Here u, v are the velocity components along the x and y-axis, respectively, νf is the kinematic viscosity of the base fluid, σ is the electrical conductivity, ρf is the density of base fluid, ρp is the nanoparticles density, α is the thermal conductivity, τ is the ratio of the effective heat capacity of the nanoparticles material and the base fluid, β is the volumetric thermal expansion coefficient of the base fluid, g is the acceleration due to gravity, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, κ is the effective thermal conductivity of nanofluid, and the subscripts ∞ denotes the values at large values of y where the fluid is quiescent. We assume the bottom surface of the plate is heated by convection from a hot fluid at temperature Tf which provides a heat transfer coefficient hf . The boundary conditions at the plate surface and far into the cold fluid may be written as  u = U(x), v = V(x), −κ ∂T ∂y = hf (Tf − T), C = Cw at y = 0 (5) u → 0, T → T∞ , C → C∞ as y → ∞ The stream wise velocity and the suction/injection velocity are taken as U(x) = cxm , V(x) = V0 x(m−1)/2

(6)

where c > 0 is constant and the power-law exponent m is also constant. In this study, we take c = 1. We now introduce the following non-dimensional variables θ (η) =

T − T∞ C − C∞ , φ(η) = , T f − T∞ Cw − C∞

(7)

The stream function ψ(x,y) is defined by u = ∂ψ/∂ y and v = −∂ψ/∂ x,

(8)

so that the continuity equation (1) is identically satisfied. Using (7), (8) into Eqs. (2)–(4) we obtain σ B2 ∂ψ ∂ 3ψ ∂ψ ∂ 2 ψ ∂ψ ∂ 2 ψ −ν f =− 0 − +(1 − φ∞ )ρf∞ βgθ  θ − (ρp − ρf∞ )gφφ, 2 3 ∂y ∂x∂y ∂x ∂y ∂y ρf ∂y (9)   2  2 κ ∂ θ ∂φ ∂θ DT ∂θ ∂ψ ∂θ ∂ψ ∂θ − = + +δθ +τ DB φ θ , (10) ∂y ∂x ∂x ∂y ρcp ∂y2 ∂y ∂y T∞ ∂y ∂ 2φ ∂ψ ∂φ DT θ ∂ 2 θ ∂ψ ∂φ − = DB 2 + ∂y ∂x ∂x ∂y ∂y T∞ φ ∂y2

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(11)

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The boundary conditions become

where δ =

∂ψ ∂y

⎫ (m−1)/2 , −κ ∂θ = h (1 − θ ), φ = 1 at y = 0 ⎬ = xm , ∂ψ f ∂x = −V0 x ∂y

∂ψ ∂y

→ 0, θ → 0, φ → 0 as y → ∞

Q0 ρc p

is the heat source (δ > 0) or sink (δ < 0), θ = Tf −T∞ and φ = Cw −C∞ .

(12)

⎭

3 Lie group transformations Let us introduce the simplified form of Lie group transformations namely, the scaling group of transformations,  : x∗ = xeεα1 , y∗ = yeεα2 , ψ ∗ = ψeεα3 , u∗ = ueεα4 , v∗ = veεα5 , θ ∗ = θ eεα6 , φ ∗ = φeεα7 , (13) Here ε is the parameter of the group and α’s are arbitrary real numbers whose interrelationship will be determined by our analysis. Equation (13) may be considered as a point-transformation which transforms coordinates (x, y, ψ, u, v, θ, φ) to the coordinates (x∗ , y∗ , ψ ∗ , u∗ , v∗ , θ ∗ , φ ∗ ). Substituting (13) in (9)–(11) we get, the following relationships among the exponents, namely α1 + 2α2 − 2α3 = 3α2 − α3 = α2 − α3 = −α6 = −α7 ; α1 + α2 − α3 − α6 = 2α2 − α6 = 2α2 − α6 − α7 = 2α2 − 2α6 ;

(14)

α1 + α2 − α3 − α7 = 2α2 − α7 = 2α2 − α6 = −α7 ; These relations give α2 = α1 /4 = α3 /3, α6 = α7 = 0. Also the boundary conditions yield α4 = mα1 = α1 /2, α5 = α1 (m − 1)/2 = −α1 /4, taking m = 1/2. In view of these, the boundary conditions become: ⎫ ∗ ∂ψ ∗ ∗1/2 , ∂ψ = −V x∗(−1/4) , −κ ∂θ ∗ = h (1 − θ ∗ ), φ ∗ = 1 at y∗ = 0 ⎬ 0 f ∂y∗ = x ∂x∗ ∂y∗ (15) ⎭ ∂ψ ∗ ∗ ∗ ∗ ∂y∗ → 0, θ → 0, φ → 0 as y → ∞ Thus the set of transformations (13) reduces to  : x∗ = xeεα1 , y∗ = yeεα1 /4 , ψ ∗ = ψe3εα1 /4 , u∗ = ueεα1 /2 , v∗ = ve−εα1 /4 , θ ∗ = θ, φ ∗ = φ, (16) Expanding by Taylor’s method in powers of ε and keeping terms up to the ε we get: x∗ − x = xεα1 , y∗ − y = yεα1 /4, ψ ∗ − ψ = 3ψεα1 /4,

u∗ − u = uεα1 /2,

v∗ − v = −vεα1 /4, θ ∗ = θ, φ ∗ = φ

(17)

The characteristic equations are dx dy dψ du dv dθ dφ = = = = = = xα1 yα1 /4 3ψα1 /4 uα1 /2 −vα1 /4 0 0

(18)

Solving the above equations we get: −1/4

y∗ x∗

−3/4

= η, ψ ∗ = x∗

f(η), θ ∗ = θ (η), φ ∗ = φ(η)

(19)

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Using (19), Eqs. (9)–(11) become f +

1 (0.75ff − 0.5f2 ) + Ra(θ − Nrφ) − Mf = 0, Pr

1  θ + δθ + 0.75Prfθ  + Nbθ  φ  + Nbθ 2 = 0, Pr Nt  φ  + 0.75Lefφ  + θ = 0, Nb In view of (19), the boundary conditions (15) turn into

f = S, f = 1, θ  = −λ(1 − θ ), φ = 1 at η = 0  f → 0, θ → 0, φ → 0 as η → ∞

(20) (21) (22)

(23)

Here prime denotes differentiation with respect to η, Pr = ν/α is the Prandtl number, Ra = ((1 − φ∞ )βgθ )/ν f α is the local Rayleigh number, Le = ν/DB is the Lewis number, Nr = (ρp − ρf∞ )φ/(ρf∞ βθ (1 − φ∞ )) is the buoyancy ratio, Nb = τ DB φ is the Brownian motion parameter, Nt = (τ DT θ )/T∞ is the thermophoresis parameter, M = σ B20 U/ρf is the magnetic field parameter, S = −4V0 /3 is the suction/injection parameter and λ = (hf /κ)x1/4 is the surface convection parameter. For the energy equation to have a similarity solution, the surface convection parameter λ must be a constant and not a function of x. This condition can be met if the heat transfer coefficient hf is proportional to x−1/4 . We therefore assume hf = c1 x−1/4 where c1 is a constant. The quantities of physical interest in this problem are the local Nusselt number and the local Sherwood number which are defined as follows:   x ∂T  Nu = − = −Re1/2 (24) x θ (0) Tw − T∞ ∂y y=0 and x Sh = − Cw − C∞



 ∂C  = −Re1/2 x φ (0), ∂y y=0

(25)

respectively, where Rex = xU/ν f is the local Reynolds number. Thus the reduced Nusselt number, Nur and the reduced Sherwood number, Shr can be written in terms of the dimensionless temperature gradient at the sheet surface, θ  (0) and the dimensionless nanoparticle concentration gradient at the sheet surface, φ  (0), respectively, as Nur = Re−1/2 Nu = −θ  (0), x Shr = Re−1/2 Sh = −φ  (0), x

(26) (27)

It should be mentioned that in absence of magnetic field and internal heat source/sink, the problem reduces to those considered by Aziz and Khan [27]. Also, for infinitely large surface convection parameter characterizing the convective heating (which corresponds to the constant temperature boundary condition), the Eqs. (20)–(22) with boundary conditions (23) reduces to those obtained by Kandasamy et al. [22] in absence of internal heat source/sink.

4 Method of solution The non-linear differential equations (20)–(22) with boundary conditions (23) have been solved in the symbolic computation software MATHEMATICA using finite difference code

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A lie group analysis Table 1 Comparison of the results for −θ  (0) with previous published works

Pr

Khan and Pop [19]

Kandasamy et al. [22]

Present work

0.07

0.0663

0.066129

0.066131

0.20

0.1691

0.169136

0.169136

0.70

0.4539

0.454285

0.454286

2.00

0.9113

0.911423

0.911430

7.00

1.8954

1.895264

1.895413

that implements the 3-stage Lobatto IIIa formula for partitioned Runge–Kutta method. The system cannot be solved on an infinite interval, and it would be impractical to solve it for even a very large finite interval. So, we have tried to solve a sequence of problems posed on increasingly larger intervals to verify the solution’s consistent behavior as the boundary approaches to ∞. We take infinity condition at a large but finite value of η where no considerable variation in temperature, nanoparticle concentration, etc. occur. In the absence of heat source/sink and for constant surface temperature, the present investigation coincides with that of Kandasamy et al. [22]. The constant surface temperature results were recovered by using a large values of λ, i.e., λ → ∞ in the third boundary condition in Eq. (23) which then gives the condition θ (0) = 1 (isothermal condition). To check the validity of the present code, the values of −θ  (0) have been calculated for constant surface temperature and for Ra = M = δ = 0 and for different values of Prandtl numbers Pr using symbolic software MATHEMATICA in Table 1. From table, it has been observed that the data produced by the present code and those of Khan and Pop [19] and Kandasamy et al. [22] show excellent agreement and, so justifies the use of the present numerical code for current model. 5 Numerical results and discussion In order to have an insight into the effects of the parameters on the heat and mass transfer characteristics of an electrically conducting nanofluids over a vertical convectively heated permeable stretching surface with variable stream conditions in presence of a uniform transverse magnetic field and internal heat source/sink, the numerical results have been presented graphically in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 and in Tables 2 and 3 for several sets of

Fig. 2 Temperature profiles for various values of λ

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Fig. 3 Temperature profiles for various values of M

Fig. 4 Temperature profiles for various values of Nt

Fig. 5 Temperature profiles for various values of Nb

values of the pertinent parameters such as surface convection parameter λ, Brownian motion parameter Nb, thermophoresis parameter Nt, magnetic field parameter M, suction/injection parameter S and heat source/sink parameter δ. In the simulation the default values of the

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Fig. 6 Temperature profiles for various values of S

Fig. 7 Nanoparticle concentration profiles for various values of Nb

Fig. 8 Nanoparticle concentration profiles for various values of δ (>0)

parameters are considered as Pr = 1.0, Le = 3.0, Ra = 1.0, Nr = 0.5, M = 0.5, S = 0.8 or −0.2, Nt = 1.0, Nb = 1.0, δ = 0.2 or −0.1 and λ = 1.0 unless otherwise specified.

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Fig. 9 Nanoparticle concentration profiles for various values of δ (<0)

Fig. 10 Nanoparticle concentration profiles for various values of Nt

Fig. 11 Nanoparticle concentration profiles for various values of λ

Table 2 has been prepared to illustrate the effect of surface convection parameter, magnetic field parameter and suction/injection parameter on the reduced Nusselt number and the reduced Sherwood number when the stretching sheet is heated convectively. The convection

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Fig. 12 Nanoparticle concentration profiles for various values of S

parameter λ = 0.05 represents a weak convective heating situation. The infinitely large convection parameter simulates the isothermal stretching model used in [22] as noted earlier. As expected, the stronger convection results in higher heat transfer rate, causing the thermal effect to penetrate deeper into quiescent fluid. It is observed that both the heat transfer rate and the mass transfer rate at the plate increase with increasing values of convection parameter λ in absence/presence of heat source and sink. It is also noted that rate of heat transfer in presence of heat sink is much greater than that in presence of heat source whereas the effect is opposite for rate of mass transfer. The result in table indicates that the strength of magnetic field has little impact on the reduced Nusselt number and hence the heat transfer process. In fact, the magnetic field reduces both the rate of heat transfer and the rate of mass transfer in presence/absence of heat source and sink. It is also found from table that the value of heat transfer rate in suction case is higher than that in the injection case. It is noteworthy that rate of heat transfer increases in presence of heat sink. Further, it is observed that the rate of mass transfer increases due to suction whereas it decreases due to injection for convectively heated stretching sheet. This effect is prominent in presence of heat source. Table 3 presents Table 2 Effects of λ, M and S on Nur and Shr λ

M

S

Nur δ=0 0.8

Shr δ = 0.2

δ = −0.2

δ=0

δ = 0.2

δ = −0.2

0.05

0.5

0.045800

0.045178

0.046258

2.373010

2.38302

2.36570

2.0

–

–

0.290421

0.248356

0.326374

2.561620

2.64928

2.48762

∞

–

–

0.355782

0.286865

0.421681

2.716680

2.80592

2.63214

1.0

0.0

–

0.291755

0.250141

0.327365

2.571700

2.657700

2.49899

–

1.0

–

0.289252

0.246786

0.325805

2.552660

2.641800

2.47752

–

2.0

–

0.287297

0.244153

0.324065

2.537380

2.629030

2.46032

–

0.5

1.0

0.308925

0.270504

0.341965

2.888130

2.968430

2.82005

–

–

0.5

0.263038

0.215088

0.303646

2.093630

2.190710

2.01217

–

–

0.0

0.219569

0.160908

0.268452

1.387490

1.491530

1.30100

–

–

−0.5

0.180193

0.110419

0.237214

0.813895

0.908482

0.736166

–

–

−1.0

0.145160

0.065537

0.20877

0.418013

0.488616

0.360729

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P. K. Kundu et al. Table 3 Effects of Nb and Nt on Nur and Shr Nb

Nt

Nur δ=0

Shr δ = 0.2

δ = −0.2

δ=0

δ = 0.2

δ = −0.2

0.1

1.0

0.460054

0.428475

0.488698

0.937715

1.53803

0.435339

0.4

–

0.402493

0.366464

0.432886

2.284000

2.45984

2.137200

0.8

–

0.326804

0.285817

0.360900

2.515870

2.61926

2.429050

0.5

0.0

0.450025

0.423722

0.472972

2.384830

2.38636

2.383510

–

0.5

0.418489

0.387228

0.445336

2.305960

2.36615

2.254660

–

1.0

0.418489

0.345803

0.414693

2.376340

2.52368

2.253230

–

1.5

0.343758

0.299184

0.380823

2.643550

2.90929

2.409490

the influence of the Brownian motion parameter and the thermophoresis parameter on the reduced Nusselt number and the reduced Sherwood number when the stretching sheet is heated convectively. It is seen that the thermophoretic effect exerts a strong influence on the heat transfer; reducing it by almost 10 % as thermophoresis parameter changes from 0.0 to 0.5. This reduction is due to the nanoparticles of high thermal conductivity being driven away from the hot sheet to the quiescent fluid. But the effect on the reduced Sherwood number is fluctuating in nature due to presence of the Brownian motion of nanoparticles. The results in table indicate that as the Brownian motion parameter Nb increases, it influences a larger extent of the fluid, causing the thermal boundary layer to thicken, which in turn decreases the reduced Nusselt number. This effect is prominent in presence of heat sink than that in presence of heat source. It is also found from the table that an increase in Nb leads to increase in the values of the rate of mass transfer at the plate surface and rate of increase is very high in presence of heat sink. Now, the graphical results that provide additional insight into the problem under investigation are discussed as follows: 5.1 Temperature profiles The effects of various physical parameters on the nanofluid temperature are illustrated in Figs. 2, 3, 4, 5 and 6. Figure 2 demonstrates the impact of surface convection parameter λ on fluid temperature in presence of heat source/sink. It is observed from the figure that temperature θ (η) increases on increasing λ in the boundary layer region and it is maximum at the surface of the plate. The solution approaches to the solution for constant surface temperature for large values of λ, i.e., λ → ∞. From the boundary condition (23), it can be seen that θ (0) = 1 as λ → ∞. These results support the numerical results obtained in the present problem. Thus, the stronger convection results in higher surface temperature, causing the thermal effect to penetrate deeper into the quiescent fluid. It is worthy to mention that the thermal boundary layer thickness in presence of heat source is greater than in presence of heat sink, as expected. Figure 3 shows that the fluid temperature is the maximum near the boundary layer region and it decreases on increasing boundary layer coordinate η to approach free stream value. Also nanofluid temperature increases on increasing the magnetic field parameter M in the boundary layer region and, as a consequence, thickness of the thermal boundary layer increases for both heat source and sink. This result qualitatively agrees with the expectations, since magnetic field exerts retarding force on the natural convection flow and increases its temperature profiles. Figures 4 and 5 illustrate the effect of the thermophoresis

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parameter Nt and the Brownian motion parameter Nb on the thermal boundary layer. As both Nb and Nt increase, the thickness of the thermal boundary layer increases as the curves become less steep indicating a diminution of the Nusselt number. As seen in Fig. 6, the impact of suction/injection parameter S on the temperature profiles is noticeable only in a region close to the sheet as the curves tend to merge at large distances from the sheet. It can easily be seen from the figure that the temperature distribution across the boundary layer decreases with increasing values of S for suction and hence the thickness of thermal boundary layer decreases. But the effects are totally opposite for injection. It is also noted that these profiles satisfy the far field boundary conditions asymptotically, which support the numerical results obtained. 5.2 Nanoparticle concentration (volume fraction) profiles Figures 7, 8, 9, 10, 11 and 12 depict nanoparticle concentration profiles against η for various values of thermophysical parameters in the boundary layer region. Generally, whenever the nanoparticle concentration at the plate surface is higher than that of the free stream value, a gradual decrease in nanoparticle concentration towards the free stream value is observed. The trend is reversed whenever the nanoparticle volume fraction at the plate surface is lower than the free stream. Figure 7 presents typical profiles for nanoparticle concentration for different values of the Brownian motion parameter Nb. As seen in figure, the effect of Nb on nanoparticle concentration is noticeable only in a region close to the sheet as the curves tend to merge at larger distances from the sheet and the thickness of concentration boundary layer decreases as Nb increases. From Fig. 8 it is observed that the concentration distribution decreases with the increase in the heat source parameter δ (>0) at all points of the flow field near the stretching surface, i.e., for η < 0.7 (not precisely determined) whereas, reverse effect occurs for η > 0.7 (not precisely determined). It is noticed from Fig. 9 that the effect of heat sink parameter δ (<0) on the nanoparticle concentration is not significant near the stretching surface, i.e., for η < 0.4 (not precisely determined) but it increases with increase of δ(<0) for η > 0.4 (not precisely determined). Figure 10 illustrates the influence of the thermophoretic parameter on the nanoparticle concentration profiles. It is seen that nanoparticle concentration of the fluid increases with the increase in the thermophoretic parameter Nt. So, thermophoretic parameter is expected to alter the nanoparticle concentration boundary layer significantly. It is observed in Fig. 11 that as the convective heating of the sheet is enhanced, i.e., λ increases, the nanoparticle concentration distribution decreases across the boundary layer. The reason for this trend is that the nanoparticle concentration boundary layer becomes thin for larger values of surface convection parameter λ. Figure 12 depicts the variation of concentration distribution across the boundary layer for different values of suction/injection parameter S. These graphs reveal that the increase in the values of the suction parameter S (>0) results in the decrease of the concentration distribution in the boundary layer region but the effect is opposite for the injection parameter S (<0). It is worthy to mention that the concentration boundary layer thickness in presence of heat sink is greater than that in presence of heat source.

6 Conclusions Using group-theoretical methods, we have obtained the similarity solutions of MHD boundary layer flow of an electrically conducting nanofluid over a vertical stretching surface with variable stream conditions in presence of internal heat source/sink. The use of a convective heating boundary condition instead of a constant temperature or a constant heat flux makes

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this study more general and novel. From the numerical results, it is predicted that the effect of increasing surface convection parameter, magnetic field parameter and thermophoretic parameter leads to increase in the thermal boundary layer thickness and so the rate of heat transfer at the plate is reduced. It is also observed that as the Brownian motion intensifies, it affects a larger extent of the fluid, causing the thermal boundary layer to thicken, which in turn decreases the reduced Nusselt number. As expected, the thermal boundary layer thickness in presence of heat source is greater than that in presence of heat sink. The nanoparticle concentration boundary layer becomes thin for larger values of surface convection parameter. It is interesting to note that the impact of thermophoresis particle deposition with the Brownian motion in presence of heat source/sink plays an important role on the concentration boundary layer. The Sherwood number at the wall increases with the increase in the surface convection parameter, the Brownian motion parameter and the thermophoretic parameter whereas the effects are reverse for the magnetic field parameter and the suction/injection parameter. Acknowledgments The authors wish to express their very sincere thanks to the honorable referees for the valuable comments and suggestions to improve the quality of the paper.

References 1. Lie, S.: Verallgemeinerung und neue Verwertung der Jacobischen Multiplikatortheorie. In: Christiania Forh., pp. 255–274 (1874) 2. Lie, S., Engel, F.: Theorie der Transformationsgruppen, vol. 3. Teubner, Leipzig (1888–1893) 3. Lie, S. : Gesammelte Abhandlungen, vol. 7. Teubner, Leipzig (1922–1960) 4. Hansen, A.G.: Similarity Analysis of Boundary Layer Problems in Engineering. Prentice Hall, Englewood Cliffs (1964) 5. Ames, W.F.: Nonlinear Partial Differential Equations in Engineering. Academic Press, New York (1972) 6. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989) 7. Pakdemirli, M., Yurusoy, M.: Similarity transformations for partial differential equations. SIAM Rev. 40, 96–101 (1998) 8. Oberlack, M.: Similarity in non-rotating and rotating turbulent pipe flows. J. Fluid Mech. 379, 1–22 (1999) 9. Yurusoy, M., Pakdemirli, M., Noyan, O.F.: Lie group analysis of creeping flow of a second grade fluid. Int. J. Non-Linear Mech. 36, 955–960 (2001) 10. Torrisi, M., Tracina, R.: Second order differential invariants of a family of diffusion equations. J. Phys. A Math. Theor. 38, 7519–7526 (2005) 11. Gandarias, M.L., Torrisi, M., Tracina, R.: On some differential invariants for a family of diffusion equations. J. Phys. A Math. Theor. 40, 8803–8813 (2007) 12. Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles. Dev. Appl. Non-Newtonian Flows 66, 99–105 (1995) 13. Eastman, J.A., Choi, S.L.S.S., Yu, W., Thompson, L.J.: Anomalously increased effective thermal conductivity of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett. 78(6), 718–720 (2001) 14. Xuan, Y., Li, Q.: Investigation on convective heat transfer and flow features of nanofluids. J. Heat Transf. 125, 151–155 (2003) 15. Buongiorno, J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006) 16. Kuznetsov, A.V., Nield, D.A.: Natural convective boundary layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 49, 243–247 (2010) 17. Khan, W.A., Aziz, A.: Natural convection flow of a nanofluid over a vertical plate with uniform surface heat flux. Int. J. Therm. Sci. 50, 1207–1214 (2011) 18. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces. Commun. Nonlinear Sci. Numer. Simul. 15, 1494–1500 (2010) 19. Khan, W.A., Pop, I.: Boundary layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53, 2477–2483 (2010) 20. Hassan, M., Tabar, M.M., Nemati, H., Domairry, G., Noori, F.: An analytical solution for boundary layer flow of a nanofluid past a stretching sheet. Int. J. Therm. Sci. 50, 2256–2263 (2011)

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A lie group analysis 21. Yao, S., Fang, T., Zhong, Y.: Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 16, 752–760 (2011) 22. Kandasamy, R., Loganathanb, P., Puvi Arasub, P.: Scaling group transformation for MHD boundary layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection. Nucl. Eng. Des. 241, 2053–2059 (2011) 23. Aziz, A.: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul. 14, 1064–1068 (2009) 24. Makinde, O.D., Aziz, A.: MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Int. J. Therm. Sci. 49, 1813–1820 (2010) 25. Ishak, A.: Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Appl. Math. Comput. 217, 837–842 (2010) 26. Makinde, O.D., Aziz, A.: Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int. J. Therm. Sci. 50, 1326–1332 (2011) 27. Aziz, A., Khan, W.A.: Natural convective boundary layer flow of a nanofluid past a convectively heated vertical plate. Int. J. Therm. Sci. 52, 83–90 (2012)

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