Ain Shams Engineering Journal (2014) xxx, xxx-xxx

Ain Shams University Ain Shams Engineering Journal

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MECHANICAL ENGINEERING

Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects

Mohammad Mehdi Rashidi a,b,% Behnam Rostami c, Navid Freidoonimehr c, Saeid Abbasbandy d

a Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran b University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, People's Republic of China

c Young Researchers & Elite Club, Hamedan Branch, Islamic Azad University, Hamedan, Iran d Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran 14778, Iran

Received 28 July 2013; revised 15 January 2014; accepted 12 February 2014

KEYWORDS

Heat and mass transfer; Natural convection; HAM;

Stretching surface; MHD flow; Radiation effects

Abstract The homotopy analysis method is employed to examine free convective heat and mass transfer in a steady two-dimensional magnetohydrodynamic fluid flow over a stretching vertical surface in porous medium. In this study thermal radiation and non-uniform magnetic field are taken into consideration. The two-dimensional boundary-layer governing partial differential equations are derived with considering Boussinesq and boundary-layer approximations, and the ordinary differential nonlinear forms of momentum, energy and concentration equations, obtained by the similarity solution, are solved analytically in the presence of buoyancy forces. The effects of different involved parameters such as magnetic field parameter, suction parameter, Prandtl number, buoyancy parameter, Schmidt number, Biot number and radiation parameter on velocity, temperature and concentration profiles are plotted and discussed in the paper.

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1. Introduction

* Corresponding author at: Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran. Tel.: + 98 811 8257409; fax: +98 811 8257400.

E-mail addresses: mm.rashidi@sjtu.edu.cn, mm_rashidi@yahoo.com (M.M. Rashidi).

Peer review under responsibility of Ain Shams University.

The analysis of the flow field in a boundary-layer near a stretching sheet is an important part in fluid dynamics and heat transfer occurring in a number of engineering processes such as extrusion of plastic and rubber sheets, polymer processing and metallurgy [1]. The interaction of moving fluids with magnetic fields provides a rich variety of phenomena associated with mechanical energy conversion such as metals processing, heating and flow control, or power generation from two-phase mixtures [2], but the major use of MHD is in plasma physics.

2090-4479 © 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2014.02.007

The study of magnetohydrodynamics is also stimulus due to its vast application to the delineation of space and astrophysical plasmas.

Sakiadis [3,4] started the study of boundary-layer flow over a continuous surface. Considering suction/injection effects, Erickson et al. [5] studied heat and mass transfer over a moving surface. Pal [6] illustrated the impact of different parameters on velocity, temperature and concentration profiles for both assisting and opposing flows in stagnation-point flow over a stretching surface with thermal radiation numerically. Hamad et al. [7] investigated heat and mass transfer with hydrodynamic slip via Runge-Kutta-Fehlberg fourth-fifth order method over a moving plate in porous media. Noor et al. [8] studied MHD flow over an inclined surface with heat source/sink effects via shooting method. Turkyilmazoglu [9] studied MHD fluid flow over a rotating disk. By spectral numerical integration scheme, he considered the viscous dissipation and Joule heating terms. Chen [10] employed a numerical method to study heat and mass transfer in MHD free convective flow with Ohmic heating and viscous dissipation. The effects of chemical reaction and thermal stratification over a vertical stretching surface in a porous medium were considered by Mansour et al. [11]. Rashidi et al. [12] investigated analytically a steady, incompressible and laminar-free convective flow of a two-dimensional electrically conducting visco-elastic fluid over a moving stretching surface through a porous medium. Heat and mass transfer in MHD flow over a permeable surface in the presence of slip is investigated by Turkyilmazo-glu with two different thermal boundary conditions (PST and PHF) analytically [13]. Rashidi et al. [14] applied MHD flow in medicine science. They studied the dual control mechanisms of transverse magnetic field and porous media filtration in a buoyancy-driven blood flow regime in a vertical pipe, as a model of a blood separation configuration. Beg et al. [15] used Keller-Box implicit method to study heat and mass transfer micropolar fluid flow from an isothermal sphere. Turkyilmazo-glu [16,17] presented multiple solutions in viscoelastic MHD fluid flow and heat and mass transfer over stretching and shrinking surfaces. Heat and mass transfer characteristics of MHD natural convective flow over a permeable, inclined surface with power-law variation of both wall temperature and concentration in presence of viscous dissipation and Ohmic heating are analyzed numerically by Chen [18]. Prasad et al. [19] studied a heat and mass transfer problem numerically in a non-Darcy porous medium over a vertical plate by Kellerbox method. Hamad et al. [7] employed Runge-Kutta-Fehl-berg fourth-fifth order method to study heat and mass transfer with hydrodynamic slip over a moving plate in a porous medium.

One of the most well-known and reliable techniques to solve high nonlinear problems is homotopy analysis method. The HAM was employed by Liao [20,21] to offer a general analytic method for nonlinear problems. Partial slip, thermal-diffusion and diffusion-thermo on MHD convective flow over a rotating disk with viscous dissipation and Ohmic heating were studied by Rashidi et al. [22] in a rotating disk via HAM. Mustafa et al. [23] studied the effects of Brownian motion and thermophoresis in stagnation point flow of a nano-fluid toward a stretching sheet. Abbas et al. [24] studied the mixed convection of an incompressible Maxwell fluid flow over a vertical stretching surface by HAM. Dinarvand et al. [25]

employed HAM to investigate unsteady laminar MHD flow near forward stagnation point of a rotating and translating sphere. Hayat et al. [26] illustrated the thermal-diffusion and diffusion-thermo effects on two-dimensional MHD axisym-mertric flow of a second grade fluid in the presence of Joule heating and first order chemical reaction. HAM has been successfully applied to solve many types of nonlinear fluid mechanics problems including MHD flow over a porous rotating sphere [27], unsteady two-dimensional and axisymmetric squeezing flows between parallel plates [28], mixed convection boundary-layer flow of a micropolar fluid toward a heated shrinking sheet [29] and nanofluid dynamics from a non-line-arly stretching isothermal permeable sheet [30].

Understanding MHD is strongly related to the comprehension of physical effects which take place in MHD. When a conductor moves into a magnetic field, electric current is induced in the conductor and creates its own magnetic field (Lenz's law). Since the induced magnetic field tends to eliminate the original and external supported field, the magnetic field lines will be excluded from the conductor. Conversely, when the magnetic field influences the conductor to move it out of the field, the induced field amplifies the applied field. The net result of this process is that the lines of force appear to be dragged accompanied by the conductor. In this paper the conductor is the fluid with complex motions. To understand the second key effect which is dynamical we should know that when currents are induced by a motion of a conducting fluid through a magnetic field, a Lorentz force acts on the fluid and modifies its motion. In MHD, the motion modifies the field and vice versa. This makes the theory highly non-linear [31,32].

The main goal of this paper is to find the analytic solutions using a powerful technique namely the HAM for the velocity, temperature and concentration distributions to study the steady magneto hydrodynamic fluid flow over a stretching sheet in the presence of buoyancy forces. The effects of different involved parameters such as magnetic field parameter, suction parameter, Prandtl number, buoyancy parameters due to temperature and concentration effects, Schmidt number, Biot number and radiation parameter on the fluid velocity, temperature and concentration distributions are plotted and discussed.

2. Flow analysis

In this paper we examine heat and mass transfer in a steady laminar two-dimensional boundary-layer flow of an incompressible electrically conducting fluid over a permeable stretching sheet (Fig. 1). It is assumed that the stretching velocity is in the form of uw(x) = c(x)1/3 (two equal and opposite forces are applied along the x - axis with the fixed origin) where c is a constant and the surface of the sheet is heated by convection from a hot fluid at temperature Tf. The flow will be induced through the stretching sheet. A magnetic field of non-uniform strength B(x) = B0(x)~1/3 is applied normal to the sheet. The left surface of the sheet is heated by convective heat transfer and all of the fluid properties are considered as constant properties except density. The boundary-layer governing equations and boundary conditions with the Boussinesq and the boundary-layer approximations are as follows [33]. The induced magnetic field is neglected in comparison with the applied magnetic field and the viscous dissipation is small.

Electrically-conducting fluid

Stretching sheet

Figure 1 The schematic diagram of the stretching sheet problem.

du dv о

dx dy '

rB2 (x)

u @x + v @y ^ t dy2 u + g(ßr(T - Ti)+ßc(C

16r*Tl d2T

dT dT_ . dx dy dy2 3pcPKi dy2 '

udC vdC _D d2C

dx dy dy2

where u and v are velocity components in the directions of x and y along and normal to the surface respectively (as shown in Fig. 1). o is the kinematic viscosity, r is the electric conductivity, q is the fluid density, B(x) is the non-uniform magnetic field parameter, g is the acceleration due to gravity, pT is the coefficient of thermal expansion, be is the coefficient of thermal expansion with concentration, T is the fluid temperature, e is the fluid concentration, a is the thermal diffusivity, cP is the specific heat at constant pressure, r is the Stephan-Boltz-man constant, j is the Rosseland mean absorption coefficient and D is the coefficient of mass diffusivity. It should be made clear that the last term in Eq. (3) refers to the radiation parameter. In this case, the Rosseland approximation has been supposed and the radiative heat flux is given by qr — — j Using the Taylor series, T4 is defined as a linear function of temperature T4 ffi 4T^T — 3T^ [12]. The thermal radiation is quite significant and the quality of final product can be controlled by the control of cooling rate via radiation parameter. In polymer industry, the thermal radiation effect may play an important role in the control of heat transfer process if the process is directed in a thermally controlled environment. The desired quality of the final product can be reached by the knowledge of radiative heat transfer [34].

The corresponding boundary conditions are as follow [35]:

' — uw(x) ,

-к — — hf(x)(Tw - T),

Cw — Ci + bx, at y — 0,

After introducing stream function W and similarity variable g, the ordinary form of equations can be derived. It is assumed that the temperature varies in the x-direction, too

(Tw = Ti + ax).

g = ycx-1'3«-1'2, W — x2=W=2f(g), TT

U(g) —

C — C C — C

/''' + / - 3/02 - Mnf ' + (1t6 + kCu) — 0, Pr

(1 + Nr)e"+ p3- (2/0'- 3/0) — 0,

(8) (9)

U"+ ^ Sc(2fu' — 3f u) — 0,

where superscript denotes the derivative with respect to g,

Mn —is the

magnetic field parameter,

kT = gbT(Tw^= — = Gt is the buoyancy

parame-

ter, where Grx —gbT(Tw——Tl)x is the Grashof number, Rex — is the Reynolds number, kC — gbc(Cw—2Ci)x1=3 is the concentration buoyancy parameter, Nr — 163r T°° is the radia-

tion parameter, Pr — a is the Prandtl number, and Sc — d is the Schmidt number and m is exponent. The corresponding boundary conditions are as follow:

f(g)—fw, f(g) — 1, 0(g) ——Bi[1 - 0(0)], u(g) —1 at g — 0, f (g)—0, 0(g) —0, u(g)— 0 as g

V 1 1/3

where fw — — 2(X/2c1/W is the suction/injection parameter >0

for suction and fw <0 for injection) and Bi — ° = jCvt hf is the Biot number [35]. Since we have considered free convective flow, the velocity at the infinity is equal to zero (as pointed in boundary conditions (10)).

3. HAM solution

We choose the initial approximations to satisfy the boundary conditions. The appropriate initial approximations are as follows:

f0(g) — fw + 1 — e—g, Bi

00(g)—Bme—g,

U0(g) — e—g.

The linear operators Lff), L0(0) and are:

@3f @2f

Cfif)—@g3 + ,

u ! 0, T ! To, C ! C^, as y

L (h)=@2h+dh , dg2 dg '

_ , x d2u du +du ■

With the following properties

Lf(ci + C2 g + C3 e—g) = 0,

Lh(c4 + cse-g) = 0, Lu(c6 + cve~g) = 0:

(18) (19)

c\ — c7, are arbitrary constants. The nonlinear operators are:

N//(g; q), h(g; q) , u (g; q)] = + f/tg; q) /2^

+ (kr0(g; q) + kcU(g; q)) M @/(g;qA -Mndf(g;q)

3 \ dg

N h/(g; q) , h(g; q),9 (g; q)l = (Nr + 1)

d29(g; q) dg2

+ 2^g;q)dh(g; q)

-3 h(g; q)

d/(M) @g /

N ut/(g; q) , 9(g; q) ,9 (g; q)] =

@2 9 (g; q) dg2

+ 329(g;q) ^

-39 (g; q)

d/(g; qA @g y

The auxiliary functions are introduced as:

H/(g)=Hh (g) = Hu (g) = e-g:

The auxiliary linear operator and function, L and H, play an important role in the frame of the HAM [20]. The appropriate operators may be selected to reach the solution easier. Although in some high nonlinear problems we cannot choose the proper linear operators in the way to satisfy the initial approximations, we should try to find the best operators to increase the rate of convergence. The linear operator should normally be of the same order as the non-linear operator. The most important factor that influences the convergence of the solution is the type of base functions. When we express the solution as a polynomial or as a sum of exponential functions, we expect that base functions whose behavior imitates the actual solution should provide much better results than the functions with different behavior. The choice of the linear operator, auxiliary function, and initial approximation is often dependent on the base functions which are presented in a solution. Having selected a linear operator, auxiliary function, and an initial approximation, we can develop a solution series [36].

The mth order deformation equations (Eqs (25)-(27)) can be solved by the symbolic software MATHEMATICA.

L/[/m(g)— Vm/m—1 (g)] = ftW/(g)R/m(g), (24)

Lh[h„(g)- Vmhm-1(g)] = hHh( g)Rh,m (g),

LU[Um(g) — Vm Um—1 (g)] = (g)RU,m (g) ■ where h is the auxiliary nonzero parameter.

r. ^ d3/m—1 (g) m—2/m— 1— n(g^

R/,m(g) ^ -m-jr-+—d-g—J

— 1 m—1/d/,(g) d/m—1— n(g)\ — Mn d/m—1(g)

3 ¿A dg dg / dg

+ {krhm—1 (g)+ kc Um—1 (g)g ,

-0.5 -

-2.1 -1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0 h

Figure 2 The curves of /"(0) , h'(0) and u'(0)obtained by the 20th order approximation of the HAM solution when Pr = 1.0 and /w = 0.1.

\ - \ \ — f'(n) - 9(n) - - e (n) A Numerical Solution

\ \ \\

> \ \ \ \ \ \ \

, \ \ \ v \

\ \ \ " \\ \

\\ \ \ \ \

A \ - \\\

A i rt . . t i n n t •

012345678

Figure 3 The comparison of the results when B; = 2.0 and

/w = 0.1.

-5E-5 -0.0001 -0.00015 -0.0002

Figure 4 The residual error of Eq. (32) when Pr = 1.0 and

fw = 0.1.

Table 1 Comparison of values of /'(0) for several suction/

injection parameters when Mn = kT = kC = 0.0 and Bi ! 1.

fw -f '(0)

Cortell [40] Ferdows et al. [35] Present results

0.75 0.984417 0.984439 0.9844401

0.5 0.873627 0.873643 0.8736447

0 0.677647 0.677648 0.6776563

-0.5 0.518869 0.518869 0.5188901

-0.75 0.453521 0.453523 0.4535499

1 1 1 1 1 - fw = 0.0

- fw = 0.1

- \\ V

■ \\ V Mn = 0.5, 1.5, 2.5

\x\ AV^

012345678

Figure 5 The effect of Mn on velocity profile.

Rh,m (g) = (Nr + 1)

Pr m-1 A, , , , @hm-1-n (g) ^ , , @fm-1-n(g)

+T g) —-g—3hn(g)

RU,m(g) =

@2Um-1(g)

+3 l2fn(g)—@g--3un(g)—^— )•

Table 2 Comparison of values of h (0) for several suction/injection parameters when Pr = 2.0, fw = 0.0, Mn — kT — 0.0 and Bi fi 1.

Nr m -h (0)

Cortell [40] Ferdows et al. [35] Present results

4/3 0.0 0.443088 0.443323 0.4434039

4/9 0.607982 0.608028 0.6080238

4/3 1.0 0.895249 0.895201 0.8952184

4/9 1.194156 1.194154 1.1941484

4/3 3.0 1.505809 1.505809 1.5058076

4/9 1.969957 1.969954 1.9699502

Table 3 Comparison of values of h (0) for several suction/injection parameters when Pr = = 2.0, m = 0.0, —h'(0) and Bi fi 1.

fw Nr h (0)

Cortell [40] Ferdows et al. [35] Present results

-0.5 4/3 0.2873762 0.287483 0.2877089

0 0.3989462 0.398951 0.3990842

0.0 4/3 0.4430879 0.443323 0.4434039

0 0.7643554 0.764374 0.7643525

0.5 4/3 0.6322154 0.632199 0.6322186

0 1.2307661 1.230952 1.2307912

For more information about the HAM solution please see Refs. [20,21].It is essential to adopt a proper value of the auxiliary parameter to control the convergence of the approximation series with the assistance of the valid region (nearly parallel to the horizontal axis) of so-called h— curve. In Fig. 2 S— curves of/"(0) , h'(0) , u'(0) are figured, obtained via 20th order of HAM solution. The averaged residual errors are defined as Eqs. (32)-(34) to acquire optimal values of auxiliary parameters. In all of the figures, we have considered Mn = 1.0, = 1.0 , kC = 1.0 , Nr = 0.05, Pr = 0.71, Bi = 0.1, and Sc = 0.78 unless it is mentioned.

d3/(g) 2„ ,d/(g) i/

+ {krh(g)+kcU(g)} ;

d/(g) dg

Resh = (Nr + 1)

Pr / + 2f(g)

d2h(g) dg2 d0(g) dg

- 3 dM 0(g)

Resu =

d2u(g) dg2

+ 3 Sc( 2f(g)

du(g)- 3 fri ^(g)

In order to choose the optimal value of auxiliary parameter S, , we have presented the average residual error as (see Refs. [37-39], for more details):

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

1 i i i i — fw = 0.0

- - fw = 0.1

vv\ \\\

\\\ \\\

- \\\ vk Mn = 2.5, 1.5, 0.5 XX

012345678

Figure 6 The effect of Mn on temperature profile.

1 1 - fw = 0.0

- fw = 0.1

\ \y X T = 3.5, 2.5, 1.5, 0.5

0 1 2 3 4 5 6 7

Figure 8 The effect of on velocity profile.

1 1 1 1 1 — fw = 0.0

-1 - fw = 0.1

" \x\ Mn = 2.5, 1.5, 0.5

012345678 n

Figure 7 The effect of Mn on concentration profile.

1 1 1 1 1 — fw = 0.0

- fw = 0.1

- \ \/ XT = 0.5, 1.5, 3.5

012345678 n

Figure 9 The effect of on temperature profile.

A/ m —

-F Kt0

Ah'm — kÇ i—0

AU,m — K^-

Res/1 ¿/-(¿Ax)

Ress I ^0,(iAx)

Resu ( y^Uj(iAx)

where Ax = 10/K and K = 20. For the given order of approximation m, the optimal value of h is given by the minimum values of A/m, As,m and A/,m corresponding to nonlinear algebraic equations:

;m — 0;

■ = 0.

For example, in order to find the optimal values of h, the residual error which is displayed in Eq. (32), for the HAM 20th order of solution is presented in Fig. 3.

4. Results and discussion

In this paper, the natural convection of MHD steady two-dimensional heat and mass transfer flow of an incompressible fluid over a stretching vertical surface with considering the buoyancy effects is investigated. To check the validity of the solution, we have presented the comparison among some of our results with the numerical results obtained by the shooting method (Fig. 4). To assess the results of the analytical method, we have tabulated our local skin friction coefficient with the previously published papers for diverse parameters (Table. 1) and one can see a very good agreement. The values of local Nusselt number are compared in Tables 2 and 3 in the presence

Figure 10 The effect of kT on concentration profile.

Figure 12 The effect of kC on temperature profile.

1 1 — fw = 0.0

- fw = 0.1

- WW \ \ U

\ Vv\ WW y— X C = 3.5, 2.5, 1.5, 0.5

- \\V / -

/ x\ v>\ X k "

0123456

Figure 11 The effect of kC on velocity profile.

0 1 2 3 4 5 6 n

Figure 13 The effect of kC on concentration profile.

and absence of radiation, suction/injection parameters. In each case, we can see an excellent agreement.

Applying the numerical values to the problem parameters, we can examine the effects of diverse parameters on the velocity /' temperature h and concentration u distributions. Graphical illustration of the results is very helpful and practical in order to depict the effects of different parameters. The effect of magnetic parameter on the velocity, temperature and concentration profiles is plotted in Figs. 5-7. As illustrated in Introduction section, this parameter leads to diminishing the velocity boundary-layer thickness and to increasing both temperature and concentration distributions. It is obvious that in both cases of porous (suction case) and nonporous walls the same trend can be seen. The effect of is plotted in Figs. 810. Based on the definition of the thermal buoyancy parameter (the ratio of buoyancy to viscous forces in the boundary-layer), the increase in its value suggests a progressive increase in the

flow velocity [35]. 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 , as shown in Fig. 8. This parameter represents the effect of free convective flow and buoyancy force in the equations. Increasing the buoyancy parameter leads to amplifying the effects thermal variations on velocity and increases the effects of convection on the velocity control. In fact 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. The buoyancy force leads to the increase in temperature gradient and heat transfer rate and the temperature decreases (Fig. 9). As can be seen in Fig. 10, the effect of on concentration profile is not tangible, but the increase in this parameter leads to the decrease in concentration distribution. Due to

Figure 14 The effect of B' on velocity profile.

Figure 16 The effect of B on concentration profile.

Figure 15 The effect of B' on temperature profile.

Figure 17 The effect of Nr on velocity profile.

0 1 2 3 4 5 6 7 n

Figure 18 The effect of Nr on temperature profile.

1 ' i i i — fw = 0.0

- fw = 0.1

" \ y— Pr = 1.0, 3.0, 5.0

Y / N -

Figure 19 The effect of Pr on velocity profile.

the same effect of energy and concentration equations on momentum equation, the similar explanations may be presented for the concentration buoyancy parameter kC behaviors so the boundary-layer thickness in velocity profile increases and both temperature and concentration profiles decreases with the increase in concentration buoyancy parameter, as plotted in Figs. 11-13. As one can see and expect, the suction parameter decreases fluid velocity in porous wall and so the boundary-layer developing decreases. In all of the figures, the effect of suction parameter is to reduce the boundary-layer thickness. In this paper, the suction parameter has been considered, because the primary assumption in boundary-layer definition says that the boundary-layer thickness is supposed to be very thin and we are not allowed to increase it so we do not present the injection parameters that may lead to contravening the boundary-layer assumption presented by Prandtl in 1904 except in the verification tables. The effects of Biot

number on velocity, temperature and concentration profiles are illustrated in Figs. 14-16. With the increase in the amount of Biot number, both velocity and temperature distributions enhance, but the concentration profile diminishes not very sensitive. When Bi = 0, the left side of the plate with hot fluid is insulated. It should be stated that the internal thermal resistance of the plate is extremely high due to the Biot number definition (the ratio of the internal thermal resistance of a solid to the boundary-layer thermal resistance) and no convective heat transfer happens to the cold fluid. As the Biot number increases, the plate thermal resistance reduces and the velocity increase significantly. In addition, the fluid temperature on the right side of the plate increases, as can be seen in Fig. 15, because the thermal resistance of the plate decreases and convective heat transfer to the fluid on the right side of the plate increases, while the reverse trend is seen for concentration distribution [35]. The effect of Nr on the velocity and

0.02 -

012345678 n

Figure 20 The effect of Pr on temperature profile.

0 1 2 3 4 5 6 7 n

Figure 21 The effect of Sc on velocity profile.

temperature distributions is illustrated in Figs. 17 and 18. The effect of radiation parameter is to increase the heat transfer from the sheet to the fluid and boundary-layer thickness so with the increase in Nr the fluid temperature increases as one can see in Fig. 17. On the other hand, the increase in radiation parameter leads to overcoming the effect of convective heat transfer and so the buoyancy force increases that causes the fluid to accelerate. We can observe that the velocity boundary-layer thickness increases in Fig. 18. It is worth mentioning that the effect of Nr on the mass distribution is negligible. With the increase in Prandtl number, kinematic viscosity increases and velocity decreases (Fig. 19). The Prandtl number based on its definition is the ratio of momentum diffusion to thermal diffusion and with the increase in Pr the thermal diffusion decreases and so the thermal boundary-layer becomes thinner. In other words, we can say that a fluid with larger Pr and therefore with higher heat capacity increases the rate of heat

Figure 22 The effect of Sc on temperature profile.

Figure 23 The effect of Sc on concentration profile.

transfer and so the non-dimensional temperature decreases (Fig. 20). In order to be realistic the values of Schmidt number are chosen at the temperature 25 0C and one atmospheric pressure for hydrogen (Sc = 0.22), water vapor (Sc = 0.6) and ammonia (Sc = 0.78). With the increase in Schmidt number, both velocity and concentration profiles decrease (Figs. 21 and 23) and the temperature increases (Fig. 22). With the increase in Schmidt number, the kinematic viscosity increases that this leads the velocity to decrease. It is obvious (based on the definition) that the effect of the Schmidt number on temperature is opposite with the effect of Prandtl number and with the increase in the amount of Sc the thermal boundary-layer thickness increases. Figs. 24-26 are presented to compare the effects of Biot number and magnetic parameter on friction factor and heat and mass transfer rate. In these figures,

Figure 24 The variation of—f(0) for several Biot number and magnetic parameter.

Figure 25 The variation of —h'(0) for several Biot number and magnetic parameter.

Figure 26 The variation of —u'(0) for several Biot number and magnetic parameter.

the amount of -/'(0) , —h'(0) and —u'(0) are plotted when

/w = 0.1.

5. Conclusion

In the present investigation, an analysis in order to study the heat and mass transfer in a steady two-dimensional magneto-hydrodynamic boundary-layer flow of an incompressible electrically conducting fluid over a vertical stretching sheet is carried out. In this study, the density is dependent on both temperature and concentration. The governing equations (continuity, momentum, and energy and concentration equations) are transformed to the high non-linear ordinary differential equations by the use of a similarity transformation and are solved analytically using homotopy analysis method (HAM). In Fig. 2, the S— curves of /''(0), h'(0) , u'(0) are presented. The physical effects of different parameters such as magnetic field parameter, suction parameter, Prandtl number, Grashof number, Schmidt number, Biot number and radiation parameter on velocity, temperature and concentration profiles are depicted and discussed in this paper.

Acknowledgments

We express our gratitude to the anonymous referees for their constructive reviews of the manuscript and for helpful comments.

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