Arab J Sci Eng (2014) 39:2251–2261 DOI 10.1007/s13369-013-0792-x

RESEARCH ARTICLE - MECHANICAL ENGINEERING

MHD Mixed Convection Stagnation-Point Flow Over a Stretching Vertical Plate in Porous Medium Filled with a Nanofluid in the Presence of Thermal Radiation Mohammad Eftekhari Yazdi · Abed Moradi · Saeed Dinarvand

Received: 11 August 2012 / Accepted: 13 July 2013 / Published online: 8 September 2013 © King Fahd University of Petroleum and Minerals 2013

Abstract This article deals with the study of the twodimensional mixed convection magnetohydrodynamic boundary layer of stagnation-point flow over a stretching vertical plate in porous medium filled with a nanofluid and in the presence of thermal radiation. The stretching velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. By means of similarity transformation, the governing partial differential equations are reduced into ordinary differential equations. The similarity equations were solved for three types of nanoparticles, namely copper, alumina and titania with water as the base fluid, to investigate the effect of the nanoparticle volume fraction parameter ϕ, the constant magnetic/porous medium parameter , the mixed convection parameter λ, the Prandtl number Pr and the radiation parameter Rd on the flow and heat transfer characteristics. The skin-friction coefficient and Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters are presented graphically and discussed. Effects of the solid volume fraction on both of assisting and opposing flows on the flow and heat transfer characteristics are thoroughly examined.It is observed that, for all three nonoparticles, the

magnitude of the skin friction coefficient and local Nusselt number increases with the nanoparticle volume fraction ϕ for both cases of buoyant assisting and opposing flows. In addition, the velocity of fluid increases in case of assisting flow by decreasing  and Pr but the opposite trend is noted in the opposing flows. A similar effect on the velocity is observed when λ and Rd increases and the temperature increase by increasing  and Rd in both cases of buoyant assisting and opposing flows. The highest values of the skin friction coefficient and the local Nusselt number was obtained for the Cu nanoparticles compared to Al2 O3 and TiO2 . Keywords MHD mixed convection · Stagnation-point flow · Stretching vertical plate · Nanofluid · Porous medium · Thermal radiation

M. E. Yazdi · A. Moradi Department of Mechanical Engineering, Islamic Azad University, Central Tehran Branch, Tehran, Iran S. Dinarvand (B) Young Researchers and Elite Club, Islamic Azad University, Central Tehran Branch, Tehran, Iran e-mail: [email protected]; [email protected]

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1 Introduction The two-dimensional flow of a fluid near a stagnation point is a classical problem in fluid mechanics. The steady flow in the neighborhood of a stagnation point was first studied by Hiemenz [1], who used a similarity transformation to reduce the Navier–Stokes equations to nonlinear ordinary differential equations. This problem has been extended by Homann [2] to the case of axisymmetric stagnation-point flow. Later the problem of stagnation point flow either in the two or threedimensional cases has been extended in numerous ways to include various physical effects. The results of these studies are of great technical importance, for example in the prediction of skin friction as well as heat/mass transfer near stagnation regions of bodies in high-speed flows and also in the design of thrust bearings and radial diffusers, drag reduction, transpiration cooling and thermal oil recovery. Mahapatra and Gupta [3] and Nazar et al. [4] studied the heat transfer in the steady two-dimensional stagnation-point flow of a viscous fluid by taking into account different aspects. Mixed convection heat transfer for thermally developing airflow is encountered in a wide range of thermal engineering applications such as cooling of electronic equipment, compact heat exchangers, solar collectors and thermal energy conversion, nuclear waste repositories, high-performance building insulation, concepts of aerodynamic heat shielding with transpiration cooling, etc. Many studies have been performed in the recent books by Nield and Bejan [5], Ingham and Pop [6,7], Vafai [8,9], Pop and Ingham [10], Ingham et al. [11] and Bejan et al. [12]. The effect of thermal radiation on flow and heat transfer processes is of major importance in the design of many advanced energy conversion systems operating at high temperature. Thermal radiation within such systems occur because of the emission by the hot walls and working fluid. Low thermal conductivity of conventional fluids such as water and oil in convection heat transfer is the main problem to increase the heat transfer rate in many engineering equipments. To overcome this problem, researchers have performed considerable efforts to increase conductivity of working fluid. An innovative way to increase conductivity

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coefficient of the fluid is to suspend solid nanoparticles in it and make a mixture called nanofluid, having larger thermal conductivity coefficient than that of the base fluid. This higher thermal conductivity enhances the rate of heat transfer in industrial applications. Many researchers have investigated different aspects of nanofluids. Thermophysical properties of nanofluids such as thermal conductivity, thermal diffusivity, and viscosity of nanofluids have been studied by Kang et al. [13], Velagapudi et al. [14], Turgut et al. [15], Rudyak et al. [16], Murugesan and Sivan [17], and Nayak et al. [18]. Bachok et al. [19] studied the flow and heat transfer in an incompressible viscous fluid near the threedimensional stagnation point of a body that is placed in a water-based nanofluid containing different types of nanoparticles: copper, alumina, and titania. They [20] also investigated two-dimensional stagnation-point flow of a nanofluid over a stretching/shrinking sheet. They drived the highest values of the skin friction coefficient and the local Nusselt number were obtained for the copper nanoparticles compared with the others. Two-dimensional boundary layer flow near the stagnation point on a permeable stretching/shrinking sheet in a water-based nanofluid containing two types of nanoparticles: copper and silver, was studied by Arifin et al. [21]. Ahmad and Pop [22] examined the mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids. Nield and Kuznetsov [23] conducted the classical problems of natural convective boundary layer flow in a porous medium saturated by a nanofluid, known as the Cheng–Minkowycz’s problem. Kuznetsov and Nield [24] investigated natural convective boundary layer flow of a viscous and incompressible fluid past a vertical semi-infinite flat plate with water-based nanofluids. The two-dimensional boundary layer flow of a nanofluid past a stretching sheet in the presence of magnetic field intensity and the thermal radiation was studied by Gbadeyan et al. [25]. Olanrewaju et al. [26] examined the boundary layer flow of nanofluids over a moving surface in a flowing fluid in the presence of radiation past a moving semi-infinite flat plate in a uniform free stream. They concluded that radiation has a greater influence on both the thermal boundary layer thickness and the nanoparticle volume fraction profiles. Akbarinia and Behzadmehr [27] studied numerically fully developed laminar mixed convection of a nanofluid in horizontal curved tubes. Mirmasoumi and Behzadmehr [28] investigated the laminar mixed convection of a nanofluid in a horizontal tube using two-phase mixture model. Abu-Nada and Chamkha [29] conducted a numerical investigation on mixed convection flow in an inclined square enclosure filled with alumina-water nanofluid. Oztop and Abu-Nada [30] studied natural convection in a rectangular enclosure filled with a nanofluid containing copper, alumina, and titania as nanoparticles. They concluded that the highest value of heat transfer is obtained using copper nanoparticles.

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The main goal of the present study is to investigate the two-dimensional mixed convection magnetohydrodynamic (MHD) boundary layer of stagnation-point flow over a stretching vertical plate in porous medium filled with a nanofluid in the presence of thermal radiation. Using a similarity transform the Navier–Stokes equations have been reduced to a set of nonlinear ordinary differential equations. The resulting nonlinear system has been solved numerically using the Runge–Kutta–Fehlberg method with a shooting technique. Finally, the results are reported for three different types of nanoparticles namely alumina, titania, and copper with water as the base fluid for both of assisting and opposing flows. 2 Problem Statement and Mathematical Formulation Consider the steady flow and heat transfer of an incompressible viscous nanofluid flow near the magnetohydrodynamic stagnation point of a stretching vertical plate in a porous medium y > 0. Two equal and opposite forces are applied along the x axis so the surface (at y = 0) is stretched by keeping the origin fixed in a MHD viscous fluid of constant ambient temperature (see Fig. 1). A uniform magnetic field of strength B0 is applied in the positive direction of y axis. The temperature Tw (x) of the stretching sheet is proportional to the distance x from the stagnation point, where Tw (x) > T∞ . The stretching velocity u w (x) and the ambient fluid velocity U (x) are assumed to vary linearly from the stagnation point as u w (x) = cx and U (x) = ax, where a and c are positive constants. Under these assumptions and following the nanofluid model proposed by Tiwari and Das [31], the governing equations for the continuity, momentum and energy in laminar incompressible boundary layer flow in a nanofluid can be written as ∂v ∂u + = 0, ∂x ∂y ∂u ∂U ∂ 2u ∂u +v =U + νnf 2 ± gβT (T − T∞ ) u ∂x ∂y ∂x ∂y 2 σnf B0 νnf (U − u), (U − u) + + ρnf K

(1)

(2)

u

∂T knf ∂ 2 T ∂qr 1 ∂T +v = , − ∂x ∂y (ρcP )nf ∂ y 2 (ρcP )nf ∂ y

(3)

subject to the boundary conditions u = u w (x) = cx, v = 0, T = Tw (x) = T∞ + bx, at y = 0,

(4)

u = U (x) = ax, T = T∞ , as y → ∞,

(5)

here u, v are the velocity components along x and y axes, respectively, T is temperature, σnf is the electrical conductivity of the nanofluid, K is the permeability of the porous medium, g is the gravitational acceleration, βT is the thermal expansion coefficient, qr is the radiative heat flux, respectively, the “+” and “−” signs in Eq. (2), respectively, correspond to assisting buoyant flow and opposing buoyant flow, μnf is the viscosity of the nanofluid, αnf is the thermal diffusivity of the nanofluid and ρnf is the density of the nanofluid, which are given by Oztop and Abu-Nada [30] μf , (ρ)nf = (1 − ϕ)(ρ)f + ϕ(ρ)s , (1 − ϕ)2.5 (ρcP )nf = (1 − ϕ)(ρcP )f + ϕ(ρcP )s ,

μnf =

knf (ks + 2kf ) − 2ϕ(kf − ks ) μnf , νnf = = , kf (ks + 2kf ) + ϕ(kf − ks ) ρnf

(6)

where ϕ is the nanoparticle volume fraction, (ρcP )nf is the heat capacity of the nanofluid, knf is the thermal conductivity of the nanofluid, kf and ks are the thermal conductivities of the fluid and of the solid fractions, respectively, and ρf and ρs are the densities of the fluid and of the solid fractions, respectively. The radiative heat flux qr is described by Roseland approximation such that (see Gbadeyan et al. [25]) qr = −

4σ ∗ ∂ T 4 , 3k ∂ y

(7)

where σ ∗ is the Stefan–Boltzmann constant and k is the mean absorption coefficient. We assume that the temperature differences within the flow are sufficiently small so that the T 4 can be expressed as a linear function after using Taylor series to expand T 4 about the free stream temperature T∞

Fig. 1 Physical model and coordinate system

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τw , ρf u 2w

cf =

3 4 T4 ∼ T − 3T∞ . = 4T∞

where the surface shear stress τw and the surface heat flux qw are given by

(8)



From Eqs. (3), (7) and (8) one obtains u

3 ∂2T ∂T knf ∂ 2 T 16σ ∗ T∞ ∂T +v = + , 2 ∂x ∂y (ρcP )nf ∂ y 3(ρcP )nf k ∂ y 2

τw = μnf

(9)

where, ψ is the stream function defined as u = ∂ψ/∂ y and v = −∂ψ/∂ x, which identically satisfy Eq. (1). Using the nondimensional variables in Eq. (10), Eqs. (2) and (3) reduce to the following ordinary differential equations 1 f  + f f  − ( f  )2 (1 − ϕ)2.5 [1 − ϕ + ϕ(ρs /ρf )] + (ε − f ) + ε ± λθ = 0, knf /kf θ  [1 + 4/3Rd ] [1 − ϕ + ϕ(ρcP )s /(ρcP )f ] + Pr ( f θ  − f  θ ) = 0, 2

(11)

(12)

where  is the constant magnetic/porous medium parameter, which is defined as  = M 2 + λ1 = const.

(13)

The boundary conditions (4) and (5) become f (0) = 0,

f  (0) = 1,

f  (∞) = ε,

(14)

θ (0) = 1, θ (∞) = 0, where, primes denote differentiation with respect to η and the constant λ(≥ 0) is the buoyancy or mixed convection parameter. The velocity ratio parameter ε, the mixed convection parameter λ, the Hartman number M, the porosity parameter λ1 , the Prandtl number Pr and the radiation parameter Rd are, respectively, defined as ε = a/c λ =

Gr x , Re2x

μcP , Pr = α

M2 =

σnf B02 νnf , λ1 = , ρnf c Kc

3 4σ ∗ T∞ Rd = , knf k

(15)

here, the local Grashof number Gr x and the local Reynold number Rex are defined by Gr x =

gβT (Tw − T∞ )x 3 , 2 νnf

Rex =

uw x . νnf

(16)

The physical quantities of interest are the skin friction coefficient cf and the local Nusselt number N u x , respectively, which are defined as

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(17)

    3 ∂u ∂ T  16σ ∗ T∞ , qw = knf + .  ∂ y y=0 3k ∂y  y=0

(18)

To obtain similarity solutions for the system of Eqs. (1)–(3), we introduce the following similarity variables  √ c T − T∞ y, ψ = cνf x f (η), θ (η) = , (10) η= νf Tw − T∞



N ux =

xqw , kf (Tw − T∞ )

and neglecting higher-order terms. This result is the following approximation

Using the nondimensional variables (10), we get 1 f  (0), (1 − ϕ)2.5   knf  4 −1/2 N u x Rex = − 1 + Rd θ (0). 3 kf 1/2

Rex Cf =

(19) (20)

3 Results and Discussion Equations (11) and (12) subject to the boundary conditions (14) were solved numerically using the Runge–Kutta– Fehlberg method with a shooting technique for some values of the governing parameters. Three types of nanoparticles were considered, namely, copper (Cu), alumina (Al2 O3 ), and titania (TiO2 ). Following Oztop and Abu-Nada [30] and Bachok et al. [19], the value of the Prandtl number Pr is taken as 6.2 (for water) and the volume fraction of nanoparticles is from 0 to 0.2 (0 ≤ ϕ ≤ 0.2), in which ϕ = 0 corresponds to the regular Newtonian fluid. The thermophysical properties of the fluid and nanoparticles are given in Table 1 [30]. Figures 2 and 3, respectively, show the variations of the velocity profiles f  (η) and temperature profiles θ (η) for different nanoparticle volume fractions at assisting and opposing flows for copper–water nanofluid. It can be seen from Fig. 2 that for assisting flow the velocity components increase with increase in the nanoparticle volume fraction parameter ϕ, but a reverse behavior is observed for the opposing flow. From Fig. 3, the temperature θ increases in both cases of buoyant assisting and opposing flows as the nanoparticle volume fraction parameter ϕ increases. Figures 4 and 5 show the variation of f  (η) and θ (η) for different nanoparticles in both assisting and opposing flows when ϕ = 0.2. Figure 4 shows that the Cu nanoparticle Table 1 Thermophysical properties of the base fluid and the nanoparticles [30] Physical properties

Fluid phase (water)

Cu

Al2 O3

TiO2

Cp (J/kg K)

4,179

385

765

686.2

ρ

(kg/m3 )

997.1

8,933

3,970

4,250

k (W/m K)

0.613

400

40

8.9538

α × 10−7 (m2 /s)

1.47

1163.1

131.7

30.7

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Fig. 2 The velocity profiles f  (η) for different nanoparticle volume fractions ϕ for copper–water nanofluid

Fig. 4 The velocity profiles f  (η) for different nanoparticles, when ϕ = 0.2

Fig. 3 The temperature profiles θ(η) for different nanoparticle volume fractions ϕ for copper–water nanofluid

Fig. 5 The temperature profiles θ(η) for different nanoparticles, when ϕ = 0.2

(compared to Al2 O3 and TiO2 ) has the largest and smallest velocities in the assisting and opposing flows, respectively. From Fig. 5, it is observed that the Cu nanoparticles have the highest value of temperature distribution (compared to Al2 O3 and TiO2 ) in both cases of assisting and opposing flows. Figure 6 is prepared to present the effect of the nanoparti-

The influence of the nanoparticle volume fraction ϕ on −1/2 for different types the local Nusselt number N u x Rex of nanofluids in both cases of the assisting and opposing flows is shown in Fig. 7. It is observed that the local Nusselt number increases with the nanoparticle volume fraction ϕ. Moreover, it is noted that the lowest heat transfer rate is obtained for the nanoparticles TiO2 due to domination of conduction mode of heat transfer. This is because TiO2 has the lowest value of thermal conductivity compared to Cu and Al2 O3 , as seen in Table 1. This behavior of the local Nusselt number is similar to that reported by Bachok et al. [19]. However, the difference in the values of Cu and Al2 O3 is negligible. The thermal conductivity of Al2 O3 is

1/2

cle volume fraction ϕ on the skin friction coefficient Rex Cf for different types of nanofluids in both cases of the assisting and opposing flows. It is observed that the magnitude of skin friction coefficient increases with the nanoparticle volume fraction ϕ. In addition, it is noted that the highest skin friction coefficient is obtained for the Cu nanoparticle.

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Fig. 6 The effect of the nanoparticle volume fraction ϕ on the skin friction coefficient for different types of nanofluids

Fig. 8 The function f  (0) for different nanoparticle volume fractions ϕ for copper–water nanofluid

Fig. 7 The effect of the nanoparticle volume fraction ϕ on the local Nusselt number for different types of nanofluids

Fig. 9 The function −θ  (0) for different nanoparticle volume fractions ϕ for copper–water nanofluid

approximately one-tenth of that of Cu, as given in Table 1. A unique property of Al2 O3 is its slow thermal diffusivity. The reduced value of thermal diffusivity leads to higher temperature gradients and, therefore, higher enhancement in heat transfer. The Cu nanoparticles have high values of thermal diffusivity and, therefore, this reduces the temperature gradients, which will affect the performance of Cu nanoparticles. Figures 8 and 9, respectively, show the variations of f  (0) and −θ  (0) for different nanoparticle volume fractions at assisting and opposing flows for copper–water nanofluid. Figure 8 indicates that for assisting flow inside the boundary layer, the value of f  (0) is low with high value of ϕ, while outside the boundary layer the value of f  (0) is high

with high value of ϕ. The reverse trend is noted in the opposing flow. From Fig. 9 it can be seen that, inside the thermal boundary layer the value of −θ  (0) is low with high value of ϕ, while outside the thermal boundary layer the value is high with high value of ϕ in both cases of assisting and opposing flows. The effects of ε, λ, Pr, Rd and  on the velocity profiles f  (η) for copper–water nanofluid (ϕ = 0.2) are illustrated in Figs. 10, 11, 12, 13 and 14, respectively. Figure 10 displays that when ε > 1, the flow has a boundary layer structure and the thickness of the boundary layer decreases with increase in ε. According to Mahapatra and Gupta [3], it can be explained as follows: for a fixed value of c corresponding to the stretching of the surface, an increase in a in relation to

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Fig. 10 The effect of the velocity ratio parameter ε on the velocity profiles f  (η) for copper–water nanofluid, when ϕ = 0.2

Fig. 11 The effect of the buoyancy parameter λ on the velocity profiles f  (η) for copper–water nanofluid, when ϕ = 0.2

c (such that ε > 1) implies an increase in straining motion near the stagnation region resulting in increased acceleration of the external stream, and this leads to thinning of the boundary layer with an increase in ε. Further, it is seen from Fig. 10 that when ε < 1, the flow has an inverted boundary layer structure. It results from the fact that when ε < 1, the stretching velocity cx of the surface exceeds the velocity ax of the external stream. Figure 11 indicates the influence of λ on the velocity profiles f  (η). It is observed that for assisting the flow, the velocity increases at the beginning until it achieves a certain value, then decreases until the value becomes constant, that is unity, at outside the boundary layer. The results of velocity are noted to be more pronounced for large λ. This is because,

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Fig. 12 The effect of the Prandtl number Pr on the velocity profiles f  (η) for copper–water nanofluid, when ϕ = 0.2

Fig. 13 The effect of the radiation parameter Rd on the velocity profiles f  (η) for copper–water nanofluid, when ϕ = 0.2

large value of λ produces large buoyancy force which produces large kinetic energy. Then the energy is used to overcome a resistance along the flow. As a result, it decreases and becomes constant far away from the surface. Figure 12 indicates that the velocity of fluid decreases in case of assisting flow by increasing Pr but the reverse trend is noted in the opposing flow. Figure 13 shows that the velocity of fluid increases in case of assisting flow by increasing Rd , however, a reverse behavior for the opposing flow is clear from this figure. Figure 14 displays that for assisting flow the velocity f  (η) decreases as  increases but for the opposing flow it shows a reverse behavior. Moreover, The boundary layer thickness is decreased by increasing . The reason behind this phenom-

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Fig. 14 The effect of the constant magnetic/porous medium parameter  on the velocity profiles f  (η) for copper–water nanofluid, when ϕ = 0.2

Fig. 16 The effect of the Prandtl number Pr on the temperature profiles θ(η) for copper–water nanofluid, when ϕ = 0.2

Fig. 15 The effect of the constant magnetic/porous medium parameter  on the temperature profiles θ(η) for copper–water nanofluid, when ϕ = 0.2

Fig. 17 The effect of the radiation parameter Rd on the temperature profiles θ(η) for copper–water nanofluid, when ϕ = 0.2

enon is that application of magnetic field to an electrically conducting fluid gives rise to a resistive type force called the Lorentz force. This force has the tendency to slow down the motion of the fluid in the boundary layer. The effects of the , Pr and Rd on the temperature profile θ (η) for copper–water nanofluid (ϕ = 0.2), respectively, have been displayed in Figs. 15, 16 and 17. From Fig. 15, it is observed that the temperature profiles θ (η) increase in both cases of assisting and opposing flows by increasing . But this increment in θ (η) is larger in case of an opposing flow. Besides, the thermal boundary layer increases as  increase in both cases.

Figure 16 shows that the thermal boundary layer thickness increases with decrease in the value of Pr which is associated with an increase in the wall temperature gradient and, hence, produces an increase in the surface heat transfer rate in both the cases of assisting and opposing flows. Further, the temperature θ decreases with the increase in Pr. The effects of thermal buoyancy force are more pronounced on dimensionless temperature for a fluid with a small Pr . For small values of Pr ( 1), the fluid is highly conductive. Physically, if Pr increases, the thermal diffusivity decreases and this phenomenon leads to the decreasing of energy transfer ability that reduces the thermal boundary layer.

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Figure 17 displays that the temperature θ (η) increases by increasing Rd in both cases of assisting and opposing flows. It is observed that the increase of the radiation parameter Rd leads to an increase of the temperature profiles and to an increase of the boundary layer thickness. Therefore, higher values of Rd imply higher surface heat flux and thereby making the fluid become warmer. This enhances the effect of the thermal buoyancy of the driving body force due to mass density variations which are coupled to the temperature and, therefore, increasing the fluid velocity. Tables 2 and 3, respectively, show the values of the skin 1/2 friction coefficient Rex Cf and the local Nusselt number −1/2 for some values of the nanoparticle volume fracN u x Rex tion ϕ using different nanoparticles in both cases of assisting and opposing flows. It is observed that, the large values of average Nusselt number can be obtained by adding copper. From Tables 2 and 3, clearly, the behavior of the present results is in good agreement with the numerical results of Hayat et al. [32] for regular fluid (ϕ = 0). Table 4 is made to give the values of the skin friction coefficient and the local Nusselt number for different values of a/c, , λ and Rd for copper–water nanofluid. The values of the skin friction coefficient and the local Nusselt number increase when a/c increases in both cases of the assisting and opposing flows. From this table, for assisting flows, the

magnitude of the skin friction coefficient and the local Nusselt number decreases when  increases and also increases by increasing λ. Clearly, there is the reverse behaviors for the opposing flows. Moreover, the magnitude of the skin friction coefficient increases with increasing Rd in both cases of the assisting and opposing flows and the local Nusselt number decreases with increasing Rd .

4 Final Remarks Here, the two-dimensional mixed convection MHD boundary layer of stagnation-point flow over a stretching vertical plate in porous medium filled with a nanofluid and in the presence of thermal radiation has been investigated. The governing partial differential equations were converted to ordinary differential equations using a suitable similarity transformation and were then solved numerically using the Runge–Kutta– Fehlberg method with a shooting technique. The similarity equations were solved for three types of nanoparticles, namely copper, alumina and titania with water as the base fluid, to investigate the effect of the nanoparticle volume fraction parameter ϕ, the constant magnetic/porous medium parameter , the mixed convection parameter λ, the Prandtl number Pr and the radiation parameter Rd on the flow

Table 2 The effect of the various nanoparticle volume fractions on the skin friction coefficient for the different nanoparticles, when  = 1.5, λ = 1, ε = 1 and Rd = 0.5 ϕ

Hayat et al. [32] Assisting flow

Present study Opposing flow

Cu

Al2 O3

TiO2

Assisting flow

Opposing flow

Assisting flow

Opposing flow

Assisting flow

Opposing flow

0.00

0.206901

−0.210610

0.206901

−0.210610

0.206901

−0.210610

0.206901

−0.210610

0.05

–

–

0.291139

−0.297392

0.248736

−0.253624

0.249873

−0.254761

0.10

–

–

0.381035

−0.390144

0.294755

−0.300872

0.296838

−0.302954

0.15

–

–

0.477697

−0.490008

0.345587

−0.353094

0.348290

−0.355796

Table 3 The effect of the various nanoparticle volume fractions on the local Nusselt number for the different nanoparticles, when  = 1.5, λ = 1, ε = 1 and Rd = 0.5 ϕ

Hayat et al. [32] Assisting flow

Present study Opposing flow

Cu

Al2 O3

TiO2

Assisting flow

Opposing flow

Assisting flow

Opposing flow

Assisting flow

Opposing flow

0.00

4.06533

3.99083

4.065333

3.990833

4.065333

3.990833

4.065333

3.990833

0.05

–

–

4.363818

4.262388

4.334892

4.246303

4.289032

4.201800

0.10

–

–

4.670143

4.542975

4.610212

4.507495

4.515970

4.416244

0.15

–

–

4.985712

4.833312

4.893266

4.776360

4.744457

4.632750

0.20

–

–

5.314523

5.136752

5.184747

5.053754

4.981960

4.858364

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Table 4 The effect of the various values of ε, , λ and Rd on the skin friction coefficient and the local Nusselt number for copper–water nanofluid, when ϕ = 0.2

ε



λ

Rd

Opposing flow −1/2

Rex Cf

1/2

N u x Rex

−1/2

Rex Cf

1/2

N u x Rex

−2.37303

4.28863

−3.98754

3.58103

0.5

−1.26582

4.78237

−2.61113

4.48531

1.0

0.58295

5.31452

−0.59902

5.13875

2.0

5.50038

6.23393

4.51074

6.13995

0.1

1.0

1.0

1.0

1.5

1.0

0.5

0.64933

5.33169

−0.67728

5.11551

1.5

0.58295

5.31452

−0.59902

5.13875

3

0.53578

5.30434

−0.54679

5.15013

5

0.48984

5.29386

−0.49717

5.16177

0.0

1.5

1.5

1.0

0.0

0.5

0.00000

5.22927

0.00000

5.22927

0.5

0.29331

5.27263

−0.29733

5.18389

1.0

0.58295

5.31452

−0.59902

5.13675

1.5

0.86910

5.35497

−0.90543

5.08758

0.0

0.50748

6.83271

−0.51884

6.66308

0.5

0.58295

5.31452

−0.59902

5.13675

1.0

0.63291

4.50626

−0.65283

4.32529

1.5

0.67012

3.98284

−0.69283

3.80216

1.0

0.5

and heat transfer characteristics. Finally, from the presented analysis, the following observations are noted. • For all three nonoparticles, the magnitude of the skin friction coefficient and local Nusselt number increases with the nanoparticle volume fraction ϕ for both cases of buoyant assisting and opposing flows. • For a fixed value of the nanoparticle volume fraction ϕ, the velocity of fluid increases in case of assisting flow by decreasing  and Pr but the opposite trend is noted in the opposing flows. A similar effect on the velocity is observed when λ and Rd increases. • For a fixed value of the nanoparticle volume fraction ϕ, the temperature increases by increasing  and Rd in both cases of buoyant assisting and opposing flows. A similar effect on the temperature is observed when Pr decreases. • The type of nanofluid is a key factor for heat transfer enhancement. The highest values are obtained when using Cu nanoparticles. • The highest values of the skin friction coefficient and the local Nusselt number were obtained for the Cu nanoparticles compared to Al2 O3 and TiO2 .

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