c Pleiades Publishing, Ltd., 2016. ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2016, Vol. 57, No. 5, pp. 908–915.  c M. Madhu, N. Kishan. Original Russian Text 

MHD BOUNDARY-LAYER FLOW OF A NON-NEWTONIAN NANOFLUID PAST A STRETCHING SHEET WITH A HEAT SOURCE/SINK M. Madhu and N. Kishan

UDC 536.4.033

Abstract: The goal of the present paper is to examine the magnetohydrodynamic effects on the boundary layer flow of the Jeffrey fluid model for a non-Newtonian nanofluid past a stretching sheet with considering the effects of a heat source/sink. The governing partial differential equations are reduced to a set of coupled nonlinear ordinary differential equations by using suitable similarity transformations. These equations are then solved by the variational finite element method. The profiles of the velocity, temperature, and nanoparticle volume fraction are presented graphically, and the values of the Nusselt and Sherwood numbers are tabulated. The present results are compared with previously published works and are found to be in good agreement with them. Keywords: non-Newtonian nanofluid, stretching sheet, Brownian motion, thermophoresis. DOI: 10.1134/S0021894416050199 INTRODUCTION A nanofluid is a fluid containing nanometer-sized particles, called nanoparticles, which fundamentally change conductive and convective heat transfer [1]. Nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. Nanofluids are fundamentally characterized by the fact that Brownian agitation overcomes any settling motion due to gravity. Thus, a stable nanofluid is theoretically possible as long as particles stay small enough (usually smaller than 100 nm). Nanofluids would be useful as coolants in the automobile and electronics industries. Application of nanoparticles provides an effective way of improving heat transfer characteristics of fluids. Nanophase powders have much larger relative surface areas and a great potential for heat transfer enhancement. Several ideas have been proposed to explain the enhanced heat transfer characteristics of nanofluids. For example, Pak and Cho [2] attributed the increased heat transfer coefficients observed in nanofluids to the dispersion of suspended particles. Xuan and Li [3] suggested that the heat transfer enhancement was the result of the increase in turbulence induced by nanoparticle motion. A large amount of literature is available, which deals with studying nanofluids and their applications [4–9]. The momentum and heat transfer of the boundary layer flow over a stretching sheet have been applied in many chemical engineering processes, such as metallurgical processes and polymer extrusion processes, which involve cooling of a molten liquid. Sakiadis [10] initiated studying the boundary layer flow over a stretched surface moving with a constant velocity and formulated boundary layer equations for two-dimensional and axisymmetric flows. Crane [11] investigated the flow caused by a stretching sheet. Sharidan et al. [12] analyzed similarity solutions for an unsteady boundary layer flow and heat transfer due to a stretching sheet. Carraagher and Crane [13] studied the flow and heat transfer over a stretching surface in the case with a power-law dependence of the temperature difference between the surface and the ambient fluid on the distance from a fixed point. Many researchers [14–16] extended the work of Crane [11] by including the heat and mass transfer analysis under different physical situations.

Department of Mathematics, Osmania University, Hyderabad, Telangana, 500007 India; [email protected]; kishan n@rediffmail.com. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 57, No. 5, pp. 167– 175, September–October, 2016. Original article submitted September 23, 2014. 908

c 2016 by Pleiades Publishing, Ltd. 0021-8944/16/5705-0908 

Several authors have considered various aspects of this problem and obtained similarity solutions [17–24]. Nadeem et al. [25] used the Jeffrey fluid model for a non-Newtonian nanofluid over a stretching sheet to obtain a numerical solution. In reality, most of the fluids considered in industrial applications are more non-Newtonian in nature, especially those of the viscoelastic type. Various aspects of the flow of a viscoelastic fluid over a stretching sheet were considered in [26–30]. In all the previous investigations, the effects of an internal heat source or sink on heat transfer were not studied. If there is an appreciable temperature difference between the surface and the ambient fluid, one needs to consider a temperature-dependent heat source or sink, which may exert strong influence on the heat transfer characteristics. There may be also a situation of a heat source/sink present in the boundary layer. In the present paper, we analyze the heat transfer processes in a viscoelastic fluid flow over an exponentially stretching sheet. Vajravelu and Rollins [31] and Vajravelu and Nayfeh [32] studied the effect of a uniform heat source/sink on heat transfer from a stretching sheet into a cooling liquid. Abo-Eldahab and El Aziz [33] investigated heat transfer considering a nonuniform heat source/sink. However, these studies were confined to the flow and heat transfer in Newtonian fluids. In the present work, we investigate a magnetohydrodynamic boundary layer flow of a non-Newtonian nanofluid past a stretching sheet with a heat source/sink. Using suitable similarity transformation, the governing partial differential equations are transformed to a set of coupled nonlinear ordinary differential equations, which are solved numerically by the variational finite element method. The effects of various flow parameters on the velocity, temperature, and nanoparticle volume fraction profiles are discussed.

1. MATHEMATICAL FORMULATION Let us consider a two-dimensional steady magnetohydrodynamic (MHD) boundary layer flow of an electrically conducting non-Newtonian nanofluid from a stretching sheet in the plane (x, y). The nanofluid is assumed to consist of a single phase and to be in thermal equilibrium. The nanoparticles are assumed to have identical shapes and sizes. The sheet is stretching in the plane y = 0, and the flow moves in the domain y > 0. The sheet is stretched with a linear velocity uw (x) = ax, where a > 0. The x axis is directed along the stretching surface. The boundary layer equations of the Jeffrey fluid with nanoparticles have the form ∂u ∂v + = 0; ∂x ∂y u

 ∂3u σ 2 ∂u ν  ∂2u ∂ 3 u  ∂u ∂u ∂ 2 u ∂u ∂ 2 u u − +v = + v + λ − + B u, 1 2 2 2 3 ∂x ∂y 1 + λ ∂y ∂x ∂y ∂x ∂y ∂y ∂x ∂y ∂y ρf 0  ∂2T ∂T ∂T ∂ 2T  +v =α + ∂x ∂y ∂x2 ∂y 2   ∂C ∂T Q0 ∂C ∂T  DT  ∂T 2  ∂T 2  + τ DB + + + + (T − T∞ ), ∂x ∂x ∂y ∂y T∞ ∂x ∂y ρCf u

(1)

 ∂2C ∂C ∂C ∂ 2 C  DT  ∂ 2 T ∂2T  + , +v = DB + + ∂x ∂y ∂x2 ∂y 2 T∞ ∂x2 ∂y 2 where u and v denote the velocities in the x and y directions, respectively, ρf is the density of the base fluid, ν is the kinematic viscosity of the fluid, σ is the electrical conductivity, B0 is the strength of the magnetic field, ρ is the density of the nanofluid, λ and λ1 are the ratio of the relaxation to retardation times and the retardation time, respectively, α is the thermal diffusivity, T is the fluid temperature, C is the nanoparticle fraction, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, τ = (ρc)p /(ρc)f is the ratio between the effective heat capacity of the nanoparticle material to the heat capacity of the fluid, Cf is the volumetric expansion coefficient, ρp is the density of the particles, the term Q0 (T − T∞ ) is assumed to be the amount of heat generated or absorbed per unit volume, and Q0 as a constant, which may take either a positive or negative value. If the fluid temperature at the wall Tw exceeds the free-stream temperature T∞ , we have a heat source term at Q0 > 0 and a heat sink term at Q0 < 0. u

909

The associated boundary conditions are y = 0:

u = uw (x) = ax,

y → ∞:

uy = 0,

v = 0,

u = 0,

T = Tw ,

T = T∞ ,

C = Cw ,

C = C∞ ,

where Tw and Cw are the fluid temperature and nanoparticle fraction at the wall, respectively. Let us introduce the stream function ψ defined as u=

∂ψ , ∂y

v=−

∂ψ ∂x

and apply the following similarity transformations: ψ = (aν)1/2 xf (η),

θ(η) =

T − T∞ , Tw − T∞

ϕ(η) =

C − C∞ , Cw − C∞

 η=

a y. ν

(2)

Then, the continuity equation is identically satisfied and Eqs. (1) take the following form: f  + β(f 2 − f f  ) + (1 + λ)(f f  − f 2 − M f  ) = 0, θ + Pr (f θ + NB θ ϕ + Nt θ2 + Qθ) = 0, ϕ + Le f ϕ +

Nt  θ = 0. NB

In view of Eqs. (2), the boundary conditions become f (0) = 0, θ(0) = 1,

f  (0) = 1, θ(∞) = 0,

f  (∞) = 0, ϕ(0) = 1,

f  (∞) = 0, ϕ(∞) = 0.

Here β = λ1 c is the Deborah number, Pr = ν/α is the Prandtl number, M = σB02 /(aρf ) is the magnetic field parameter, NB = (ρc)p DB C∞ /(ν(ρc)f ) represents the Brownian motion, Nt = (ρc)p DT (Tw − T∞ )/(T∞ ν(ρc)f ) is the thermophoresis parameter, Q = Q0 /(a(ρC)f ) is the heat source/sink parameter, and Le = ν/DB is the Lewis number. For practical purposes, the functions θ(η) and ϕ(η) allow us to determine the local Nusselt number Nu and local Sherwood number Sh: xqm xqw , Sh = . (3) Nu = α(Tw − T∞ ) DB (Cw − C∞ ) Here we have

 ∂T  qw = −α , ∂y y=0

qm = −DB

 ∂C  . ∂y y=0

The dimensionless relations (3) can be written as Nu = −θ (0), Re−1/2 x

Re−1/2 Sh = −ϕ (0), x

where Rex = uw (x)x/ν is the local Reynolds number. Kuznetsov and Nield [8] introduced the reduced Nusselt number Nur = −θ (0) and reduced Sherwood number Shr = −ϕ (0).

2. METHOD OF THE SOLUTION The finite element method is used to solve the resultant system of equations. The computational domain is divided into 1000 quadratic elements of equal size. Each element is three-noded; therefore, the entire domain contains 2001 nodes. At each node, four functions are to be evaluated. Hence, we obtain a system of 8004 equations. After imposing the boundary conditions, only a system of 7997 equations remains for the solution. This system is solved by the Gauss elimination method. 910

Table 1. Reduced Nusselt number −θ  (0) and reduced Sherwood number −ϕ (0) for Le = Pr = 10 and β = λ = M = Q = 0 −θ  (0)

−ϕ (0)

NB

Nt

Data of Khan and Pop [7]

Present results

Data of Khan and Pop [7]

Present results

0.1

0.1 0.2 0.3

0.9524 0.6932 0.5201

0.952 37 0.693 11 0.520 04

2.1294 2.2740 2.5286

2.129 16 2.273 84 2.528 57

0.2

0.1 0.2 0.3

0.5056 0.3654 0.2731

0.505 45 0.365 27 0.272 66

2.3819 2.5152 2.6555

2.381 75 2.515 14 2.655 38

0.3

0.1 0.2 0.3

0.2522 0.1816 0.1355

0.252 18 0.181 44 0.135 42

2.4100 2.5150 2.6088

2.409 35 2.514 31 2.608 64

Table 2. Values of Nur , Shr , NB , and Nt for β = 0.3, λ = 0.3, Pr = 7, Le = 5, M = 1, and Q = 0.2 NB = 0.1

Nt 0.1 0.2 0.3

NB = 0.2

NB = 0.3

Nur

Shr

Nur

Shr

Nur

Shr

0.5885 0.4177 0.2796

1.0568 1.1120 1.3912

0.3933 0.2446 0.1258

1.5016 1.6038 1.7605

0.1691 0.0610 −0.0162

1.5525 1.6475 1.7608

f0 1.0

f0 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

1

1 2

2 3

0.2 3 4

0

2

4 Fig. 1.

6 n

0

2

4

6 n

Fig. 2.

Fig. 1. Flow velocity versus the distance from the sheet for β = 0.2 (1), 0.4 (2), and 0.8 (3). Fig. 2. Flow velocity versus the distance from the sheet for M = 0 (1), 0.5 (2), 1.0 (3), and 1.5 (4).

3. RESULTS AND DISCUSSION The effects of the physical parameters on the velocity, temperature, and nanoparticle volume fraction are analyzed. In order to assure the accuracy of the present numerical method, the computed values of the reduced Nusselt number −θ (0) and reduced Sherwood number −ϕ (0) are compared in Table 1 with the available results of Khan and Pop [7]. It follows from Table 1 that the results are in excellent agreement. Tables 2 presents the reduced Nusselt and Sherwood numbers. It can be observed that the value of Nur decreases with increasing Nt and NB , whereas the value of Shr increases. The computational results are presented in Figs. 1–10. Figures 1 and 2 illustrate the effect of the Deborah number β and magnetic field parameter M on the velocity profile. It can be seen that the velocity profile increases 911

o,f 1.0

o,f 1.0

0.8

0.8

0.6

0.6 1

0.4 1

0.2

1

0.4

2

2 1

3

0.2

2

3 2

3

0

3

1

2

3 n

0

1

Fig. 3.

2

3 n

Fig. 4.

Fig. 3. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for β = 0.2 (1), 0.4 (2), and 0.8 (3). Fig. 4. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for λ = 0.2 (1), 0.4 (2), and 0.8 (3).

o,f 1.0

o,f 1.0

0.8

0.8

0.6

0.6 1

0.4 0.2

0.4

2

1

3

2

1

0.2

4 3

2 3

4

0

1

2

1

2 Fig. 5.

3 n

0

1

3

2

3 n

Fig. 6.

Fig. 5. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for M = 0 (1), 0.5 (2), 1.0 (3), and 1.5 (4). Fig. 6. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for Pr = 5 (1), 7 (2), and 10 (3).

with an increase in the Deborah number β. The effect of M is to reduce the velocity. The reason can be formulated as follows: because of the application of a transverse magnetic field in an electrically conducting fluid, a resistive Lorentz force similar to a drag force is generated, leading to a decrease in the flow velocity. Figure 3 shows the effect of the Deborah number β on the temperature and nanoparticle volume fraction profiles. The effect of β is to decrease the temperature θ, as well as the nanoparticle volume fraction ϕ. The effects of the ratio of the relaxation to retardation times λ on the temperature and nanoparticle volume fraction profiles are shown in Fig. 4, where both increase with increasing λ. An increase in the magnetic field parameter M leads to an increase in the temperature and nanoparticle volume fraction profiles owing to the action of the Lorentz force 912

o,f

o, f 1.0

1.0

3

3 2

0.8

2

0.8

1

1

0.6

0.6

0.4

0.4

1 2

1

0.2

3

0.2

2 3

0

1

2

3 n

0

1

3 n

2

Fig. 7.

Fig. 8.

Fig. 7. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for NB = 0.2 (1), 0.4 (2), and 0.8 (3). Fig. 8. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for Nt = 0.2 (1), 0.4 (2), and 0.6 (3).

o,f 1.0

o,f 1.0

0.8

0.8

0.6

0.6

0.4

0.4

1

0.2

1 1

2 3

3 4

0

2

2

3

3 2

1

0.2 4

2 Fig. 9.

1

3 n

0

0.4

0.8

1.2

1.6

2.0 n

Fig. 10.

Fig. 9. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for Q = −0.4 (1), −0.2 (2), 0 (3), and 0.2 (4). Fig. 10. Temperature (solid curves) and nanoparticle volume fraction (dashed curves) versus the distance from the sheet for Le = 10 (1), 20 (2), and 30 (3).

decelerating the flow (Fig. 5). Thus, increasing the magnetic field, it is possible to increase the temperature and the thermal boundary layer thickness. It follows from Fig. 6 that the temperature decreases with an increase in the Prandtl number, whereas the nanoparticle volume fraction profile increases in the vicinity of the boundary layer and decreases far away from the sheet. The Prandtl number is the ratio of the kinematic viscosity to the thermal diffusivity. Therefore, the thickness of the thermal boundary layer increases as the Prandtl number decreases, as well as the temperature. 913

Figure 7 shows that the temperature increases with increasing parameter characterizing the Brownian motion NB , whereas the nanoparticle volume fraction decreases. Hypothetically, the enhanced thermal conductivity of the nanofluid is mainly due to the Brownian motion, which ensures micromixing. An increase in the thermophoresis parameter Nt leads to an increase in both the temperature and nanoparticle volume fraction (Fig. 8). Figure 9 depicts the effects of the heat source/sink parameter on the temperature and nanoparticle volume fraction profiles as functions of the distance from the sheet. In the case of the heat source (Q > 0), the temperature in the thermal boundary layer increases with increasing Q. In the case with the heat sink (Q < 0), the temperature in the thermal boundary layer decreases. The case Q = 0 represents the absence of heat transfer. In the case of the heat source, the nanoparticle volume fraction decreases near the sheet and increases far from it. The effect of the Lewis number is to reduce the temperature and nanoparticle volume fraction profiles (Fig. 10). Thus, the effects of the Deborah number β, ratio of the relaxation to retardation time λ, Prandtl number Pr , magnetic field parameter M , Brownian motion parameter NB , thermophoresis parameter Nt , heat source/sink parameter Q, and Lewis number Le on the flow velocity, nanofluid temperature, and nanoparticle volume fraction are analyzed.

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