Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid

Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid

International Journal of Thermal Sciences 87 (2015) 136e145 Contents lists available at ScienceDirect International Journal of Thermal Sciences jour...

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International Journal of Thermal Sciences 87 (2015) 136e145

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid Navid Freidoonimehr a, Mohammad Mehdi Rashidi b, Shohel Mahmud c, * a

Young Researchers & Elite Club, Hamedan Branch, Islamic Azad University, Hamedan, Iran Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran c School of Engineering, University of Guelph, Guelph, Ontario N1G 2W1, Canada b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 February 2014 Received in revised form 30 July 2014 Accepted 9 August 2014 Available online

In this paper we investigate the transient MHD laminar free convection flow of nano-fluid past a vertical surface. The vertical surface is considered porous and stretched under acceleration. Four different types of water based nano-fluid are considered in this investigation where copper (Cu), copper oxide (CuO), aluminum oxide (Al2O3), and titanium dioxide (TiO2) are the nano-particles. The boundary-layer forms of the governing partial differential equations (momentum and energy equations) are transformed into highly nonlinear coupled ordinary differential equations (ODEs) using similarity technique. The ordinary differential equations are solved numerically using a fourth order Runge-Kutta method based shooting technique. For some special cases, an excellent agreement is observed between the current results and the results available in the existing literature. The effects of different parameters: the nanoparticle volume fraction (4), unsteadiness parameter (A), magnetic parameter (M), buoyancy parameter (l), suction parameter (fw) and different types of nanoparticles on the fluid velocity component ðf 0 ðhÞÞ, 1=2 temperature distribution (q(h)), the skin friction coefficient ðCf Rex Þ, and the local Nusselt number 1=2 Þ are presented graphically and discussed in details. The results illustrate that selecting Al2O3 ðNux Rex and Cu as the nanoparticle leads to the minimum and maximum amounts of skin friction coefficient absolute value, and also Cu and TiO2 nanoparticles have the largest and lowest local Nusselt number. © 2014 Elsevier Masson SAS. All rights reserved.

Keywords: Unsteady boundary-layer Vertical surface Nano-fluid MHD flow Buoyancy effect Vertical surface

1. Introduction A system having energy interaction with other systems may be influenced by the working fluids used in the system. In many cases, the performance of the working fluids depend on the thermophysical properties, for example, thermal conductivity and heat capacity. Such properties along with other physical properties have significant influence in modern thermal and manufacturing processes and designs. In particular, a low thermal conductivity is one of the influential parameters that can limit the heat transfer performance significantly. Suspending the ultrafine solid metallic particles in fluids can cause an increase in the thermal conductivity. This is one of the most recent techniques applied for increasing the coefficient of heat transfer. It is expected that the ultrafine solid particle is able to increase the thermal conductivity and heat transfer performance, since the thermal conductivity of solid

* Corresponding author. E-mail addresses: [email protected], (S. Mahmud). http://dx.doi.org/10.1016/j.ijthermalsci.2014.08.009 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.

[email protected]

metals is higher than that of base fluids. Choi and Eastman [1] utilized a mixture of nanoparticles and base fluid and characterized its properties which is now widely known as “nano-fluid”. Experimental studies have displayed that with 1e5% volume of solid metallic or metallic oxide particles, the effective thermal conductivity of the resulting mixture can be increased by 20% compared to that of the base fluid [2e4]. In addition, there exist meaning relations between the thermal conductivity and the nanoparticle particle sizes' as well as the temperature of the based fluid [5]. The thermal conductivity of nanoparticleefluid mixtures enhances with decreasing the particle size [6]. Khanafer and Vafai [7] discussed the thermo-physical properties of nano-fluids and their importance in biomedical applications and heat-transfer enhancement. Their investigations illustrated that the results of the effective thermal conductivity and viscosity of nano-fluids can be determined at room temperature using the classical equations at low-volume fractions. However, these models cannot predict the thermal conductivity at other temperatures. Hosseini et al. [8] did a review of relations for physical properties of nano-fluids. They reached that the temperature difference in fluid layers made gradient in fluid density and this created gradient cause the

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Nomenclature a,c b B B0 Cf cp f g k Nux qw t T u v v0

constant parameters (t1) constant parameter (K/m) external uniform magnetic field (kg/s2A) constant magnetic flux density (kg/s2A) skin friction coefficient specific heat at constant pressure (J/kg K) self-similar velocity acceleration due to gravity (m/s2) thermal conductivity (W/mK) local Nusselt number surface heat flux time (s) fluid temperature (K) velocity component in the x direction (m/s) velocity component in the y direction (m/s) uniform suction/injection

Dimensionless parameters A unsteadiness parameter (c/a) M magnetic parameter ðsB20 =arf Þ Pr Prandtl number (nf/af)

convection heat transfer. Murshed et al. [9] investigated the thermal conductivity and viscosity of nano-fluids both experimentally and theoretically. They observed a linear increase in the effective thermal conductivity of nano-fluids with temperature. In addition, experimental results [10] and numerical investigations [11,12] have confirmed heat transfer increment achieved by nano-fluids in several other cases. Xuan and Li [13] stated the flow and heat transfer performance of nano-fluids under the turbulent flow in tubes. Their experimental results showed that the convective heat transfer coefficient and Nusselt number of nanofluids were enhanced by increasing the Reynolds number and volume fraction of nanoparticles. Keblinski et al. [14] summarized some of enhancing thermal conductivity explanations of nanoparticle suspensions. Khan and Pop [15] investigated the laminar fluid flow passing a stretching flat surface in a nano-fluid by applying an implicit finite-difference method. Their results showed that Nusselt number was a decreasing function of the involved parameters, i.e. Prandtl number, Lewis number, Brownian motion number and thermophoresis number. Rashidi and Erfani [16] illustrated the nano boundary-layer flows over the stretching surfaces with Navier boundary condition. Nano-fluids can be optimized during manufacture using a variety of techniques including plasma arc synthesis [17] and sheet processing [18]. In regard to the latter, many superior lubricants and thermal working fluids may be developed for applications in aerospace, energy systems, medical engineering, etc. A number of theoretical studies have appeared with regard to nano-fluid flows in industrial and medical engineering materials manufacture. Rashidi et al. [19] depicted the second law of thermodynamics analysis applied to an MHD incompressible nano-fluid flowing over a porous rotating disk. Rana et al. [20] applied a variational finite element method to investigate the mixed convection nano-fluids flow from an inclined plate in a porous material. Many researchers such as Gupta [21], Cobble [22] and Wilks and Hunt [23] investigated the influences of transversely applied magnetic field on the free convection of an electrically conducting fluid past a semi-infinite plate, because of its several applications in nuclear engineering dealing with the cooling of reactors. Realizing

Rex fw

137

local Reynolds number (Uwx/nf) pffiffiffiffiffiffiffiffi suction parameter ð v0 = nf aÞ

Greek symbols b thermal expansion (K1) h a scaled boundary-layer coordinate q self-similar temperature m dynamic viscosity (Ns/m2) n kinematic viscosity (m2/s) r density (kg/m3) s electrical conductivity (S/m) j stream function t skin friction l buoyancy parameter (gbf b/a2) 4 nanoparticle volume fraction Subscripts f fluid phase nf nano-fluid s solid phase w condition of the wall ∞ ambient condition

MHD is strongly related to the comprehension of physical effects which take place in MHD. Comprehensive description of MHD highly non-linear theory and its governing mechanisms can be found in Refs. [24,25]. In recent years, there are several studies in which MHD and its applications play important roles. Rashidi et al. [26] demonstrated the parametric analysis and optimization of entropy generation in unsteady MHD flow past a stretching rotating disk using artificial neural network (ANN), particle swarm optimization (PSO) algorithm, and HAM. Sheikholeslami et al. [27] employed Lattice Boltzmann method to investigate magnetohydrodynamic flow utilizing Cuewater nano-fluid in a concentric annulus. Hayat et al. [28] illustrated the effect of homogeneouseheterogeneous reaction in MHD flow bounded by a linearly stretched surface. Fang [29] displayed magneto-hydrodynamic flow over a nonlinearly (power-law velocity) moving surface analytically. The flow caused by stretching boundary gets up mostly in materials manufactured by extrusion, glass-fiber and paper production. Mohammadian and Gorla [30] employed a shooting method to analyze the micro-polar convection due to a stretching sheet. Rana and Bhargava [31] used the finite element and finite difference methods to study the nano-fluid heat and mass transfer past a nonlinearly stretching sheet. Rashidi and Mohimanianpour [32] applied Homotopy analysis method (HAM) to investigate the unsteady boundary-layer flow and heat transfer due to a stretching sheet. In another study, Bhargava et al. [33] used a finite element solution for the mixed convection micro-polar flow over a permeable stretching sheet with wall transpiration effects. Bachok et al. [34] studied the steady boundary-layer flow of nano-fluid past a moving semi-infinite flat plate in a uniform free stream. The current study is mainly motivated by the need to understand the unsteady MHD free convection boundary-layer flow due to a permeable stretching vertical surface in a nano-fluid. This model has important applications in heat transfer enhancement in the renewable energy systems, industrial thermal management, and also material processing. The well-known fourth order RungeKutta based shooting technique is employed to study the effects of physical flow parameters such as nanoparticle volume fraction,

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unsteadiness parameter, magnetic parameter, buoyancy parameter, suction parameter and different types of nanoparticles on the fluid velocity component, temperature distribution, skin friction coefficient and local Nusselt number. Recently, fourth order Runge-Kutta method has been employed in several studies as a powerful numerical technique (Ashorynejad et al. [35], Ferdows et al. [36] and Mushtaq et al. [37] employed this method to investigate the heat and mass transfer in the boundary-layer flow under different physical conditions). The graphs are plotted and discussed for the variations of different involved parameters in details. 2. Problem statement and mathematical formulation In this paper, we consider an unsteady MHD laminar free convection boundary-layer type flow of nano-fluid over a permeable accelerating stretching vertical surface. Different types of water based nano-fluids are considered in this paper. The schematic diagram of the problem is presented in Fig. 1. It is also considered that the base fluid and the nanoparticles are in thermal equilibrium with each other and no slip exists between them. Also for the time t < 0, the fluid and heat flows are steady. The unsteady fluid and heat flows start at t ¼ 0 and the surface being stretched with a velocity Uw(x, t) along the x-axis. The velocity of the mass transfer perpendicular to the stretching surface is vw(t). The surface temperature, Tw(x, t), has a linear variation with x but an inverse square law variation with the decreasing time. The ambient temperature of the ambient fluid is constant (T∞). The surface temperature is considered higher than the surrounding temperature (Tw > T∞), which corresponds to an assisting flow. For the assisting flow, the

stretching induced flow and the thermal buoyant flow assist each other. The magnetic Reynolds number is assumed very small. Thus, it is conceivable to neglect the induced magnetic field in comparison to the applied magnetic field. In addition, we also consider that the viscous dissipation and joule heating effects are neglected. Because these effects are of the same order as well as they are negligibly small [38,39]. The unsteady two-dimensional conservation of mass, momentum and thermal energy equations for the nano-fluids, using the above assumptions and applying the Boussinesq and boundarylayer approximations, can be written in the form of [40e42]:

vu vv þ ¼ 0; vx vy  rnf

vu vu vu þu þv vt vx vy

(1)  ¼ mnf

v2 u ±gðrbÞnf ðT  T∞ Þ  sB2 u; vy2

    vT vT vT v2 T þu þv ¼ knf 2 ; rcp nf vt vx vy vy

(2)

(3)

where u and v are the velocity components in the x and y directions, respectively, t refers to the time, rnf and mnf are the density and the dynamic viscosity of the nano-fluid, respectively, where mnf has been proposed by Brinkman [43], bnf is the thermal expansion of the nano-fluid, g is the acceleration due to gravity, T is the nanopffiffiffiffiffiffiffiffiffiffiffiffiffi fluid temperature, s is the electrical conductivity, B ¼ B0 = 1  ct ; is the magnetic field imposed along the y-axis, (rcp)nf is the heat

Fig. 1. Schematic of the flow configuration and geometrical coordinates.

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capacitance of nano-fluid and knf is the effective thermal conductivity of nano-fluid [44]. These nano-fluid constants are defined by:

ðrbÞnf ¼ ð1  4Þðr bÞf þ 4ðr bÞs ;     ks þ 2kf  24 kf  ks knf   ; ¼  kf ks þ 2k þ 4 k  ks f

f

u/0;

ax ; ð1  ctÞ

T/T∞ ;

as

t < c1 [46]. Also, b is a constant and has dimension temperature/ length (b ¼ 0 refers to absence of buoyancy force).

      rcp nf ¼ ð1  4Þ rcp f þ 4 rcp s ; mnf ¼

mf ð1  4Þ

2:5

;

v0 v ¼ vw ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1  ct

(4)

rnf ¼ ð1  4Þrf þ 4rs ;

where 4 is the nanoparticle volume fraction. The subscripts “f” and “s”, refer to fluid and solid fraction properties, respectively. It should be mentioned that the use of the approximation for knf is restricted to spherical nanoparticles and does not account for other nanoparticle shapes. The thermo-physical properties of the base fluid (water) and different nanoparticles are given in Table 1 [44]. The appropriate boundary conditions are introduced as.

u ¼ Uw ðx; tÞ ¼

139

T ¼ Tw ðx; tÞ ¼ T∞ þ

We now employ the following dimensionless functions f and q, and the similarity variable as.

!1=2 a y; nf ð1  c tÞ



 jðx; yÞ ¼

nf a 1  ct

1=2 xf ðhÞ; (6)

T  T∞ ; qðhÞ ¼ Tw  T∞

bx ð1  ctÞ2

;

at

y ¼ 0;

(5)

y/∞;

where a and c are constants (where a > 0 and c  0, with ct < 1), these two constants have dimension time1. We have a as the initial stretching rate a/(1c t) and it is increasing with time (positive constant). In the context of polymer extrusion, the material properties, especially the elasticity of the extruded sheet may change with time even if the sheet is being stretched by a constant force [45]. It is obvious that c shows the unsteadiness of the problem and for the c ¼ 0, the above boundary conditions can be employed for the traditional steady stretching surface problems. Further, the expressions for Uw(x, t), vw(t) and Tw(x, t) are valid are valid for time

where j(x,y) is the free stream function that satisfies the continuity equation (Eq. (1)) with



vj ax 0 ¼ f ðhÞ; vy 1  ct

  nf a 1=2 vj v¼ ¼ f ðhÞ; vx 1  ct

(7)

Substituting (6)e(7) into Eqns. (2)e(3) and (5), the following ordinary differential equations are obtained:

 

. o   .  n  00 00 1 00 02 0 hf r f f ðhÞ  1  4 þ 4 r ðhÞ  f ðhÞf ðhÞ þ A f ðhÞ þ ðhÞ  Mf 0 ðhÞ þ 1  4 þ 4 ðrbÞs ðrbÞf lqðhÞ ¼ 0; s f 2:5 2 ð1  4Þ 1

(8) . knf kf f ðhÞ 00 1   .  q ðhÞ þ 0 f ðhÞ Pr 1  4 þ 4 rcp rcp f s

  1 0 qðhÞ hq  A 2qðhÞ þ ðhÞ ¼ 0; 0 q ðhÞ 2

Table 1 Thermo-physical properties of the base fluid and different nanoparticles.[44]. Physical properties

Fluid phase (water)

Cu

CuO

Al2O3

TiO2

cp ðJ=kgKÞ rðkg=m3 Þ kðW=mKÞ b  105 ðK 1 Þ

4179 997.1 0.613 21

385 8933 401 1.67

531.8 6320 76.5 1.8

765 3970 40 0.85

686.2 4250 8.9538 0.9

(9)

The transformed boundary conditions become.

f ðhÞ ¼ fw ; f 0 ðhÞ ¼ 1; qðhÞ ¼ 1; at f 0 ðhÞ/0; qðhÞ/0; as h/∞;

h ¼ 0;

(10)

where primes denote differentiation with respect to h, A ¼ c/a is the unsteadiness parameter, M ¼ sB20 =arf is the magnetic parameter,

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Table 2 Comparison results of q0 ð0Þ for different values of unsteadiness parameter (A), buoyancy parameter (l) and Prandtl number (Pr) when M ¼ fw ¼ 0 and 4 ¼ 0.0 l

A 0.0

Pr

0.0

0.72 1.00 3.00 7.00 10.0 1.00

1.0 2.0 3.0 0.0 1.0

1.0

Table 3 00 Comparison results of  f ð0Þ and  q0 ð0Þ for different value of nanoparticle volume fraction whenM ¼ fw ¼ 0, Pr ¼ 6.5, A ¼ l ¼ 0.5 with Cu nanoparticle. 4

Ishak et al. [40]

Mahdy [42]

Grubka and Bobba [48]

Ali [47]

Present

0.8086 1.0000 1.9237 3.0723 3.7207 1.0873 1.1423 1.1853 1.6820 1.7039

0.80868 1.00000 1.92368 3.07224 3.72067 1.08727 1.14233 1.18528 1.68197 1.70390

0.8086 1.0000 1.9237 3.7207 3.7207

0.8058 0.9961 1.9144 3.7006

0.80863135 1.00000000 1.92368259 3.07225021 3.72067390 1.08727817 1.14233930 1.18529032 1.68199249 1.70391282

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

l ¼ gbfb/a2 is the buoyancy parameter, is the Prandtl number, and

00

q0 ð0Þ

f ð0Þ Mahdy [42]

Present

Mahdy [42]

Present

1.07432 1.10645 1.13585 1.16276 1.18736 1.20983 1.23031 1.24892 1.26579 1.28101 1.29466

1.07435113 1.10647599 1.13587582 1.16277898 1.18737983 1.20984564 1.23032249 1.24893588 1.26580046 1.28101581 1.29467174

3.78318 3.71711 3.65296 3.59062 3.53000 3.47099 3.41352 3.35752 3.30291 3.24962 3.19760

3.78318468 3.71711903 3.65297162 3.59063394 3.53000748 3.47100222 3.41353533 3.35753105 3.30291825 3.24963149 3.19760965

to reduce the boundary value problem to an initial value problem. In order to obtain the numerical solutions, the governing nonlinear ordinary differential Eqns. (8)e(9) and boundary conditions (10) are reduced to a set of simultaneous first order differential equa00 tions, where y1 ¼ f, y2 ¼ f 0 , y3 ¼ f , y4 ¼ q and y5 ¼ q0 as follows.

pffiffiffiffiffiffiffiffi fw ¼ v0 = nf a is the suction parameter. The skin friction coefficient Cf and the local Nusselt number Nux are the physical quantities which are given by.

y01 ¼ y2 ; y02 ¼ y3 ;

y03 ¼ ð1  4Þ2:5







  .  1 1  4 þ 4 rs rf y22  y1 y3 þ A y2 þ hy3 2

. o n  M y2 þ 1  4 þ 4 ðrbÞs ðrbÞf ly4



;

(14)

y04 ¼ y5 ;

  .   

 1  4 þ 4 rcp s rcp f 1 . y2 y4  y1 y5 þ A 2y4 þ hy5 ; y05 ¼ Pr 2 knf kf

Cf ¼

tw ; 2 rf Uw

Nux ¼

xqw ; kf ðTw  T∞ Þ

(11)

where tw is the skin friction and qw is the surface heat flux, introduced as

 tw ¼ mnf

vu vy



 ; y¼0

qw ¼ knf

vT vy

 ;

(12)

y¼0

Applying the non-dimensional transformations (6), one obtain. 1=2

Cf Rex

¼

1 ð1  4Þ

, 00

f ð0Þ; 2:5

Nux

1=2

Rex

knf ¼  q0 ð0Þ; kf

(13)

where Rex ¼ Uwx/nf is the local Reynolds number. 3. Numerical procedure and validation Equations (8)e(9) are coupled and non-linear differential equations. These non-linear differential equations, subjected to the boundary conditions given in Eq. (10), are solved by using the fourth order Runge-Kutta method based shooting technique. Different values of the physical parameters 4, A, M, l and fw are obtained from the obtained solutions. The essence of this method is

and

y1 ð0Þ ¼ fw ; y2 ð0Þ ¼ 1; y3 ð0Þ ¼ sð2Þ ; y4 ð0Þ ¼ 1; y5 ð0Þ ¼ sð4Þ ;

(15)

where s(2) and s(4) are determined for the case of y2(∞) ¼ 0 and y4(∞) ¼ 0. It is important to note that the “Infinity” in the above expression represents the edge of the boundary-layer. We apply the shooting numerical techniques to guess s(2) and s(4) until the boundary conditions y2(∞) ¼ 0 and are satisfied. A comparison is made between some of the results obtained in this paper with the results available in the literature [40,42,47,48]. Table 2 and Table 3 summarize this comparison and a very good agreement between the current results and results available in the literature is observed.

4. Results and discussion The non-linear ordinary differential equations (8)e(9) subject to the boundary conditions (10) are solved numerically for some values of the nanoparticle volume fraction (4), unsteadiness

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141

Fig. 2. Effect of nanoparticle volume fraction on the a) velocity profile and b) temperature distribution when A ¼ M ¼ l ¼ fw ¼ 1.

parameter (A), magnetic parameter (M), buoyancy parameter (l) and suction parameter (fw). We considered four different types of nanoparticles; copper (Cu), copper oxide (CuO), aluminum oxide (Al2O3), titanium dioxide (TiO2) and water is considered as the base fluid. Note that the copper nanoparticle is used in all figures in this section except those which focus on the influence of the type of applied nanoparticles on the engineering parameters such as skin friction coefficient and local Nusselt number. For the present investigation, we consider the value of the Prandtl number (Pr) as 6.2 (for water), that pointed out by Oztop and Abu-Nada [44]. In addition, the value of the nanoparticle volume fraction (4) varies from 0 (regular Newtonian fluid) to 0.2. Fig. 2 illustrates the effect of the nanoparticle volume fraction on the velocity profile and temperature distribution. The velocity profile decreases as the value of the nanoparticle volume fraction increases. This is because the presence of solid nanoparticles leads to further thinning of the velocity boundary-layer thickness. In addition, the thermal conductivity enhances and consequently the thermal boundary-layer thickness increases, as the nanoparticle volume fraction increases. This issue is in compliance with the primary proposes of employing nano-fluids [49,50]. This also agrees with the physical behavior, when the volume of nanoparticles enhances the thermal conductivity increases, and then the thermal boundary layer thickness increases. The effect of unsteadiness parameter on the fluid velocity profile and temperature distribution in the acceleration case (A > 0) is represented in Fig. 3. The velocity profile reduces for the higher

acceleration. This states an accompanying reduction of the thickness of the momentum boundary layer. This mentioned behavior changes by crossing away from the vertical surface. This means that the velocity boundary-layer thickness becomes thicker for the larger amplitude of unsteadiness parameter. In addition, the temperature distribution enhances for the lower acceleration. Fig. 4 depicts the effect of magnetic parameter on the velocity profile and temperature distribution. A drag-like force that named Lorentz force is created by the infliction of the vertical magnetic field to the electrically conducting fluid. This force has the tendency to slow down the flow over the vertical surface. Due to the above explanation, the velocity boundary-layer thickness gets depressed and the temperature distribution increases slightly with the increase in the magnetic parameter. It clearly demonstrates that the transverse magnetic field opposes the transport phenomena. It is important to mention that the large resistances on the fluid particles, which cause heat to be generated in the fluid, apply as the vertical magnetic field increases. The effect of buoyancy parameter on the fluid velocity and temperature distributions in the assisting case (l > 0) is demonstrated in Fig. 5. The buoyancy parameter represents a measure of the influence of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the heat and fluid flow [42]. The results display that as the buoyancy parameter enhances the fluid velocity profile increases while the temperature distribution decreases slightly. Due to the definition of the buoyancy parameter (the ratio of buoyancy to viscous forces in the boundary-layer), the increase in its value suggests a progressive

Fig. 3. Effect of unsteadiness parameter on the a) velocity profile and b) temperature distribution when M ¼ l ¼ fw ¼ 1 and 4 ¼ 0.1.

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Fig. 4. Effect of magnetic parameter on the a) velocity profile and b) temperature distribution when A ¼ l ¼ fw ¼ 1 and 4 ¼ 0.1.

Fig. 5. Effect of buoyancy parameter on the a) velocity profile and b) temperature distribution when A ¼ M ¼ fw ¼ 1 and 4 ¼ 0.1.

increase in the flow velocity [36]. In other words, since the governing equations are coupled together only with the buoyancy parameters, the Grashof number accelerates the fluid so the velocity and the boundary-layer thickness increases with the increase in l. The buoyancy force acts like a favorable pressure gradient and accelerates the fluid, so the velocity and the boundary-layer thickness increase with the increase in Grashof number and more production occur.

Fig. 6 shows the effect of suction parameter on the velocity profile and temperature distribution. In the current investigation, the suction parameter has been applied, because the primary assumption in boundary-layer definition says that the boundarylayer thickness is supposed to be practically very thin (according to the boundary-layer assumption presented by Prandtl in 1904). Applying suction at the vertical surface causes to draw the amount of the fluid into the surface and consequently the hydrodynamic

Fig. 6. Effect of suction parameter on the a) velocity profile and b) temperature distribution when A ¼ M ¼ l ¼ 1 and 4 ¼ 0.1.

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143

Fig. 7. Variation of the a) skin friction coefficient and b) Nusselt number with respect to nanoparticle volume fraction for different values of unsteadiness parameter when M ¼ l ¼ fw ¼ 1.

Fig. 8. Variation of the a) skin friction coefficient and b) Nusselt number with respect to nanoparticle volume fraction for different values of magnetic parameter when A ¼ l ¼ fw ¼ 1.

boundary-layer gets thinner and also the thermal boundary-layer gets depressed by increasing the suction parameter. Figs. 7e11 present the numerical results for the skin friction 1=2 coefficient CfRe1/2 and the local Nusselt number Nux Rex for a x wide range of the nanoparticle volume fraction and different physical parameters, i.e. unsteadiness parameter, magnetic

parameter, buoyancy parameter, and suction parameter and four types of nanoparticles. It can be observed that the value of skin friction coefficient reduces and local Nusselt number enhances almost linearly with increasing the nanoparticle volume fraction. As it is obvious from Figs. 7e10, the skin friction coefficient enhances as the unsteadiness parameter, magnetic parameter, and

Fig. 9. Variation of the a) skin friction coefficient and b) Nusselt number with respect to nanoparticle volume fraction for different values of buoyancy parameter when A ¼ M ¼ fw ¼ 1.

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Fig. 10. Variation of the a) skin friction coefficient and b) Nusselt number with respect to nanoparticle volume fraction for different values of suction parameter when A ¼ M ¼ l ¼ 1.

suction parameter get depressed or the buoyancy parameter increases. In addition, by increasing the unsteadiness parameter, buoyancy parameter, and suction parameter or depressing magnetic parameter, the local Nusselt number enhances. The effect of nanoparticle variations on the skin friction coefficient and local Nusselt number is illustrated in Fig. 11. From Table 1, Cu and Al2O3 have the maximum and minimum densities between the different considered types of nanoparticles. Therefore, selecting Al2O3 as the nanoparticle leads to the minimum amount of skin friction coefficient absolute value, while choosing the Cu gives the maximum value of it. Because of the largest thermal conductivity value, Cu has the largest local Nusselt number. In addition, it is obvious that the lowest heat transfer rate is obtained for the TiO2 nanoparticles due to domination of conduction mode of heat transfer. As can be observed from the Table 1, this is because TiO2 has the lowest thermal conductivity compared to other nanoparticles. This behavior is similar to that reported by Oztop and Abu-Nada [44] and Bachok et al. [51]. The results of this figure illustrate that the nanoparticle type is an important factor in the cooling and heating processes.

surface in the form of assisting flow. We have considered the water as the base fluid and four different types of nanoparticles; copper, copper oxide, aluminum oxide and titanium dioxide. The present numerical computations agree closely with the previous studies available in the current literature. The effects of the five key thermo-physical parameters governing the flow i.e. nanoparticle volume fraction, unsteadiness parameter, magnetic parameter, buoyancy parameter, and suction parameter on dimensionless velocity and temperature distributions, skin friction coefficient and local Nusselt number have been presented graphically and interpreted in details. The results show that by reducing the nanoparticle volume fraction, unsteadiness parameter, magnetic parameter, and suction parameter or increasing the buoyancy parameter, the skin friction coefficient enhances. Furthermore, the local Nusselt number enhances by increasing the nanoparticle volume fraction, unsteadiness parameter, buoyancy parameter, and suction parameter or depressing magnetic parameter. In addition, choosing Al2O3 and Cu as the nanoparticle leads to the minimum and maximum amounts of skin friction coefficient absolute value, respectively and also Cu and TiO2 have the largest and lowest local Nusselt number.

5. Conclusions Acknowledgment The fourth order Runge-Kutta method based shooting technique has been employed to solve the transformed differential equations describing the unsteady MHD free convection flow in a nano-fluid past a permeable accelerating stretching vertical

The authors are very grateful to the anonymous referees for carefully reading the paper and for their constructive comments and suggestions which have improved the paper.

Fig. 11. Variation of the a) skin friction coefficient and b) Nusselt number with respect to nanoparticle volume fraction for different types of nanoparticle when A ¼ M ¼ l ¼ fw ¼ 1.

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