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ORIGINAL ARTICLE

On radiative-magnetoconvective heat and mass transfer of a nanofluid past a non-linear stretching surface with Ohmic heating and convective surface boundary condition

Shweta Mishraa, Dulal Palb'*, Hiranmoy Mondalc, Precious Sibandac

aSchool of Engineering and Technology, Amity University, New Town, Kolkata, West Bengal 700135, India Department of Mathematics, Institute of Science, Visva-Bharati (A Central University), Santiniketan, West Bengal 731235, India cSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa

Received 30 April 2015; accepted 23 May 2016

KEYWORDS

Nanofluid; Magnetoconvection; Thermal radiation; Non-linear stretching sheet;

Ohmic heating

Abstract In this paper magnetoconvective heat and mass transfer characteristics of a two-dimensional steady flow of a nanofluid over a non-linear stretching sheet in the presence of thermal radiation, Ohmic heating and viscous dissipation have been investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and the presence of nanoparticles in the base fluid. The governing equations are transformed into a system of nonlinear ordinary differential equations by using similarity transformation. The numerical solutions are obtained by using fifth order Runge-Kutta-Fehlberg method with shooting technique. The non-dimensional parameters on velocity, temperature and concentration profiles and also on local Nusselt number and Sherwood number are discussed. The results indicate that the local skin friction coefficient decreases as the value of the magnetic parameter increases whereas the Nusselt number and Sherwood number increase as the values of the Brownian motion parameter and magnetic parameter increase.

© 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

"Corresponding author. Tel.: (+91) 3463261029.

E-mail addresses: dulalp123@rediffmail.com (Dulal Pal), hiranmoymondal@yahoo.co.in (Hiranmoy Mondal), sibandap@ukzn.ac.in (Precious Sibanda). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

http://dx.doi.org/10.1016/j.jppr.2016.11.007

2212-540X © 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The study of flow, heat and mass transfer over a non-linear stretching surface stretching surface has been given considerable attention in recent times due to its vast applications in industrial and several technological and natural processes, such as materials manufactured by extrusion, glass-fiber production, paper production, plastic and rubber sheets production, crystals growing, cooling of metallic sheets or electronic chips, etc. Ishak et al. [1] observed the mixed convection boundary layer stagnation point flow towards a stretching sheet because of wide range of applications of nanofluids, significant research interest has been found in the recent past to study heat and mass transfer characteristics on these fluids. Heat and mass transfer due to free and mixed convection in engineering systems are quite important for its wide range of applications in electronic cooling, heat exchangers etc. Tiwari and Das [2] have studied and reported results on natural convection heat transfer in nanofluids considering various flow conditions in different geometries. Partha et al. [3] studied the effects of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface.

The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid due to stretching of the sheet is of considerable interest in modern metallurgical and metal-working processes. The effect of chemical reaction and thermal radiation absorption on unsteady MHD free convection flow past a semi-infinite vertical permeable moving surface with internal heat source/suction was analyzed by Ibrahim et al. [4]. Pal [5] studied the MHD flow and heat transfer past a semi-infinite vertical plate embedded in a porous medium of variable porosity. The study of convection boundary layer flow in porous media has received special attention in recent years due to its important roles and wide applications in geophysics and thermal sciences, such as geothermal energy technology, petroleum recovery, building thermal insulation, packed bed reactors, underground disposal of chemical and nuclear waste. Rana and Bhargava [6] analyzed the flow and heat transfer of a nanofluid over a nonlinear stretching sheet. Ferdows et al. [7] investigated the effects of MHD mixed convective boundary layer flow of a nanofluid through a porous medium due to an exponentially stretching sheet. Bidin and Nazar [8] studied numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation. Pal and Mondal [9] studied the influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium. Olanrewaju et al. [10] studied the boundary layer flow of nanofluids over a moving surface in a flowing fluid in the presence of radiation.

Nanofluid consists of a base fluid containing colloidal suspension of nanoparticles, a term which was first introduced by Choi [11]. Nanoparticles are particles with a range of diameters from 1 to 100 nm. Common base fluids are water, oil and ethylene-glycol mixtures. The literature has revealed that the low thermal conductivity of these common

base fluids is a primary limitation in enhancing the performance and the compactness of many devices. When nanoparticles are added to these base fluids, a drastic increase in thermal conductivity is observed. The thermal conductivity of a nanofluid is found to be highly temperature dependent. Nanofluids also exhibit an increased boiling critical heat flux. Narasimhan et al. [12] studied forced convection through porous media in Newtonian fluid having temperature-dependent viscosity. Nanofluids can be utilized in several applications such for chemical production, power generation in a power plant, production of microelectronics and advanced nuclear systems due to better performance in heat exchange.

Thermophoresis is a phenomenon by which small sized particles suspended in a non-isothermal gas acquire a velocity relative to the gas in the direction of decreasing temperature. The velocity acquired by the particles is called thermophoretic velocity and the force experienced by the suspended particles due to the temperature gradient is known as thermophoretic force. Thermophoretic deposition of radioactive particles is considered to be one of the important factors causing accidents in nuclear reactors. Thermophoresis causes small particles to deposit on cold surfaces. Pakravan and Yaghoubi [13] investigated theoretically the combined thermophoresis, Brow-nian motion and Dufour effects on natural convection of nanofluids. Mehdi and Hosseinalipour [14] studied particle migration in nanofluids considering thermophoresis and its effect on convective heat transfer. Anbuchezhian et al. [15] investigated thermophoresis and Brownian motion effects on boundary layer flow of nanofluid in the presence of thermal stratification due to solar energy. Wubshet Ibrahim [16] observed nonlinear radiative heat transfer in magnetohydro-dynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition. A critical review on thermal convection in nanofluids can be found in a recent book by Straughan [17]. Khan and Pop [18] studied forced convective boundary layer flow of a nanofluid past a stretching surface by considering the model of Buongiorno [19] which includes Brownian diffusion and thermophoresis, whereas Hamad and Pop [20] studied the boundary layer flow near a stagnation-point on a heated permeable stretching surface in a porous medium saturated with a nanofluid in the presence of heat generation and absorption neglecting the Brownian diffusion and thermophoresis. Aziz [21] studied the Blassius flow over a flat plate with a convective thermal boundary condition and established the existence of similarity solution. Bataller [22] investigated the effect of radiation on the Blassius and Sakiadis flows with convective boundary condition. Makinde and Aziz [23] studied magnetohydrody-namic mixed convection heat and mass transfer flow along a vertical plate embedded in a porous medium with a convective boundary condition. Pal [24] analyzed radiative heat transfer flow over an unsteady stretching permeable surface in the presence of non-uniform heat source/sink. Makinde et al. [25] studied numerical study of chemically reacting hydromagnetic boundary layer flow with Soret/Dufour effects under con-vective surface boundary condition.

Nomenclature

a,b,c constants

Bo transverse magnetic field strength

C nanoparticle volume fraction

C w nanoparticle volume fraction at the surface

C ambient nanoparticle volume fraction

CP specific heat at constant pressure

Db Brownian diffusion coefficient

DT thermophoresis diffusion coefficient

Ec Eckert number

Grx local Grashof number

g acceleration due to gravity

Le Lewis number

M local magnetic field parameter

Nb Brownian motion parameter

Nt thermophoresis parameter

R conduction radiation parameter

Rex local Reynolds number

T temperature of the nanofluid

temperature of the ambient fluid

Greek letters

a thermal diffusivity of the nanofluid

PT volumetric expansion coefficient of base fluid

^ viscosity of the base fluid

n similarity variable

c magnetic permeability

v kinematic coefficient of viscosity

Q conduction radiation parameter

p local Reynolds number

Pf density of the nanofluid

Subscripts

w surface conduction

1 condition far away from the surface

It may be pointed out that under surface convective boundary condition, the study of boundary layer flow over a stretching sheet is unique, and the results are more realistic and practically useful. In certain polymeric (plastic films, artificial fibers) and metallurgical processes nonlinear stretching effects are very important from industrial point of view as the final product is strongly influenced by the stretching rate. Thus, the main focus of the analysis is to investigate how the flow and temperature fields of a nanofluid within the convective boundary layer is influenced by the nonlinearity of the sheet, magnetic field, viscous dissipation under convective boundary condition. The local similarity equations are derived and solved numerically using Runge-Kutta-Fehlberg method with shooting technique. Graphs and tables are presented to illustrate and discuss the important hydrodynamic and thermal features of the flow behavior of the nanofluids. The accompanying discussion provides physical interpretations of the numerical results.

2. Mathematical formulations

We consider the steady two-dimensional boundary layer flow of a nanofluid moving over a heated vertical stretching sheet. We consider a Cartesian coordinate system with the origin at the lower corner of the sheet. The x-axis vertically upwards along the sheet and the y-axis is horizontal and perpendicular to the plane of the sheet. The flow being confined to y>0. Two equal and opposite forces are introduced along the x-axis so that the surface is stretched keeping the origin fixed. The sheet is assumed to move with a velocity according to the power law form, u — axm where a is a dimensional constant known as the stretching rate and m is an arbitrary positive constant (i.e., not necessarily an integer) known as the stretching index. It is assumed that the left surface of the sheet is heated by

convection from a hot fluid at temperature T0 which provides a heat transfer coefficient h. A magnetic field of uniform strength B0 is applied in the y-direction, i.e. normal to the flow direction. The external electric field is assumed zero and the magnetic Reynolds number is assumed small. Hence, the induced magnetic field is neglected as it is small compared with the externally applied magnetic field.

We consider the nanofluid as a two-component mixture (i.e. base fluid and nanoparticles) with the assumption that the fluid is incompressible with no chemical reactions and negligible viscous dissipation, and nanoparticles and base fluid are locally in thermal equilibrium. Following these assumptions along with the usual boundary layer and Boussinesq approximations, the governing equations of the problem become (Kuznetsov and Nield [26]):

Y + dy = 0; CD

d2U 2/

= U —— — oBq (x)u

(1 — Ci) Pf 1 gßT (T — T i)

+ (Pp — Pfoo ) g ( C — C1)

dT dT _

dx dy dy2

d2T 1 dqr

pBq2 PfCP dy PfCp

dC dT DT ( dT

udC + vdC _D cÏC^Dt d2T dx dy dy2 T1 dy2

where u and v denote the velocity components in x- and y-directions, respectively. —f is the fluid density, while pp is the density of the nanoparticles, t is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, T1 is the temperature of the ambient fluid outside the boundary layer, while C1 its ambient value and qr is the radiative heat flux in the y-direction.

The terms inside square brackets in Eq. (2) are accounts for thermal buoyancy (1 — C1)Pf(xigfiT(T — Tdue to the thermal expansion of the base fluid and the buoyancy (pp — py ) g (C — C1) due to the difference in densities of the nanoparticles and the base fluid, respectively. The last term (right-hand side) is due to applied magnetic field. The interaction of the fluid velocity and applied magnetic field

creates a Lorentz force, J x B, where the electric current density follows the generalized Ohm's law, J — a(E + q x B), where a is electrical conductivity, q is fluid velocity vector. For an applied magnetic field

B — (0, B0 (x), 0) fluid velocity q — (u, v, 0) and in the

absence of any electric field E — 0 (since the surface is electrically non-conducting), the x-component of the Lor-entz force yields (—aB02u).

Eq. (3) states that heat can be transported in a nanofluid by convection (the left-hand side, i.e. u |T + v |y), by

conduction first term on the right-hand side, i.e. a ^4,

and also by radiation, term on right-hand side, i.e. -p1~ i|f).

The terms inside the square brackets are true representation of the additional contribution associated with the nanopar-ticle motion relative to the fluid (see Buongiorno [19]). The term tDb dyy |y is the thermal energy transport due to the

Brownian diffusion, while the term t D- (fy) is the energy transport due to thermophoretic effect. The (last second term from the right hand side) term is the viscous

dissipation due to the velocity component u in y-direction

and the (last term from the right hand side) term — —C_ u2 is

the Ohmic dissipation due to the velocity component in the x-direction of the Lorentz force aB02u.

Eq. (4) states that the nanoparticles can move homogeneously with the fluid (left-hand side, i.e. u dC + v dy), but they also posses a slip velocity relatively to the fluid (right-hand side), which is due to Brownian diffusion DB lyf and D^ fy_. The fluid is considered to be gray absorbing-emitting radiation but non-scattering medium. We also assume that the boundary layer is optically thick and the Rosseland approximation or diffusion approximation for radiation (Cortell [27]) is valid. Thus for an optically thick boundary layer (i.e. intensive absorption) the radiative heat (Sparrow and Cess [28]) is defined as

4c*dT4

where c*( = 5.67 x 10-8W/m2k4) is the Stephan-Boltz-mann constant and k*(m- 1) is the Rosseland mean absorption coefficient.

According to Raptis and Perdikis [29], temperature differences within the flow are assumed to be sufficiently small so that T4 may be expressed as a linear function of temperature T using a truncated Taylor series about free stream temperature TM, i.e.

T4 = T 1[1 + (Tr - 1) 9]4.

Using Eq. (5), the energy Eq. (3) becomes

dT dT _ d2T 16c* d dx dy dy2 3k*PfCp dy

T 3 dT

PfCP \àyj PfCi

cBl 2 0-u2 + T

dC dT DT

The boundary conditions are

(i) At y — 0:

u — ax™, v — 0 (no — slip and impermeable wall),

— km — — h (T0 — Tw) (convective surface) and dy

C — Cw (nanoparticle concentration at wall)

(ii) As y 1:

u — 0, T — T C — Ci, (7)

where the subscripts w and 1 refer to wall and boundary layer edge, respectively. Now we look for a similarity solution of Eq. (2)-(4) with the boundary conditions (7) of the following form:

n = (—) y; V = ( —) xf (n); & (n)= T Tl

T 0 - Ta

V (n) =

C C1 C — C ;

where the stream function ^ is defined in the usual form as

u— -T- , v = — -£-.

dy ' dx

Thus from Eq. (8) we have

u — axmf 0(n),

^m + 1

v =-(avxm -1)1/^m

■f (n) + nf'(n)

Here f is a non-dimensional stream function and the prime denotes differentiation with respect to rj.

Now using the similarity variables (8) and (9), following ordinary differential equations are obtained from Eq. (2) to

(4) as

f' '+ Ht" ff" - mf 2 + Хв + X ф - Mf — 0,

Prfв'

1 + 3R (1 + (Tr - 1) в)'

+PrNbe'q>' + PrNté2 + PrEcf"2

+PrMEcf 2 + 4 (1 + (Tr - 1) в)2в'2 = 0, R

im + 1\ . [Nt\

ф"+ (—) Lefф'+ Ы в" =

subject to the boundary conditions f — 0, f — 1, в = 1, ф — 1 at n = 0, f' — 0, в — 0, ф — 0 as n — I-

g(Pp -Pfoo )(Cw - Ci)x

Pf (axm)' parameter, M

is the nanoparticle local buoyancy

is the magnetic field parameter, Pr —

is the Prandtl number for the base fluid, Nb

_ tDb (Cw - Ci)

Cf —

(ax")2 V дУ/y — 0 and using Eqs. (8) and (9), Eq. (14) can be written as

1/2 —

f " (0),

(ii) Nusselt number:

The local Nusselt number (the rate of heat transfer) is defined as

Nux — 1 ( л

km (Tw T1)

where surface heat flux is defined as

4w —

The dimensionless parameters appeared in Eq. (10)-(13) are defined as:

2 — Cl-> Get is the thermal buoyancy parameter,

Grx — gPT (To — Tx3 ¡v 2 is the local thermal Grashof number, Rex — is the local Reynolds number,

V^y — 0

Using Eqs. (8) and (9), Eq. (16) can be written as

' Tr — 1

NuxRex

-1/2 _

Tr ^ 1

1 + 3R (1 + (Tr -1) в (0))3

the Brownian motion parameter, Nt — —T KT 0 ~ is thermo-phoresis parameter, R — 4™, kз is the conduction-radiation rate parameter (R-1 means there is no thermal radiation), Ec — C t"w-T ) is the Eckert number, Tr — Т- is the relative temperature ratio parameter, Le — is the Lewis number.

It is good to mention that the parameters X and X* depend on x and hence no self similar solutions (the solutions for which velocity as well as temperature profiles remain geometrically similar despite the growth of the boundary layer with distance x from the leading edge) could be found. However, since the approach preserves the mass and momentum conservation, it is still valid to study the flow dynamics within the boundary layer. Due to the x dependence of these parameters they are called local parameters and therefore we followed local similarity solutions (Kays and Crawford [30]) instead of self similar solutions. The local similarity approach implied that the dimensionless quantities (X and X*) are determined at any x-station and the upstream history of the flow will be ignored, except as far as it influences the similarity variable. It is a standard practice to follow such approach to study laminar boundary layer and viscous flow of a nanofluid and heat transfer over a non-linearly stretching sheet (Hady et al. [31]).

Following parameters are of engineering interest:

(i) Skin-friction coefficient:

The skin friction coefficient (rate of shear stress) is

defined as

0 (0); (18)

where Tr* — TTw. It is worth mentioning that we restrict ourselves to the case of heated wall T0 >Tw > T which means Tr > Tr* > 1 always for the existence of the free convection currents. Throughout the simulations the values of Tr and Tr* are considered constant as 2 and 1.5, respectively.

(iii) Sherwood number:

The Sherwood number (rate of mass transfer) is defined as

Shx —

DB (Cw - C1)

where surface mass flux is, —-DB^)y — 0-

Using Eq. (8) and (9), Eq. (19) can be written as ShxRex -1/2 =-p' (0) (21)

3. Numerical solutions

The set of ordinary differential Eqs. (10)—(12) with the boundary condition (13) are solved numerically using the Runge-Kutta-Fehlberg scheme along with shooting method. The values of f, Q, $ are known at n — 1. Thus, these three end conditions are utilized to produce three unknown initial conditions at n — 0 by using shooting technique. The most crucial factor of this scheme is to choose the appropriate finite value of nœ. Thus to est;imate the value of qœ, we start with some initial guess value and solve the boundary value problem consisting of Eqs. (10)-(12) to obtain f " (0), Q (0), $ (0). The solution process is repeated with larger values of n until two successive

Table 1 Comparison values of reduced Nusselt number

NuxRex '=2 for various values of Nt with Pr=Le = 10.0, Nb = 0.1,

M= Tr=Ec = 0.0, X = X* = 0.0, m- = 1.0.

Nt Khan and Pop Rahman and Eltayeb Present result

[18] [32]

0.1 0.9524 0.952376 0.9523761

0.2 0.6932 0.693174 0.6931742

0.3 0.5201 0.520079 0.5200790

0.4 0.4026 0.402581 0.4025808

0.5 0.3211 0.321054 0.3210543

Table 2 Comparison values of reduced ShxRex ~ '=2 for various values of Nt with Pr= M = Tr=Ec = 0.0, m = 1.0. Sherwood number ■Le = 10.0, Nb = 0.1,

Nt Khan and Pop [18] Rahman and Eltayeb [32] Present result

0.1 2.1294 0.2 2.2740 0.3 2.5286 0.4 0.7952 0.5 3.0351 2.129393 2.274020 2.528636 2.795167 3.035139 2.129389 2.274013 2.528638 2.795170 3.035142

Figure 1 Effect of non-linear stretching m on velocity distribution

f '(n).

Figure 2 Effect of X on velocity distribution f ' (n).

Figure 3 Effect of X* on velocity distribution f ' (n).

m=2.0, ¿=10.0, A'=5.0, 5=2.0, T=1.2,

\ ft=10.0,M>=JW=0.2, £c=0.01, Le=\0.0

\ —M= 0.0 -M= 0.2 .V/ 0.6 -A/=0.8 W-l .2

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Figure 4 Effect of M on velocity distribution f ' (n).

m=2.0, A= 10.0, A'=5.0, M=0.5, T=1.2,

Pr=\0.0,Nb=Nt=0.2,Ec=0.0l,Le=\0.0

— 5=0.1

% \ — 5=1.0

% \ 5=2.0

V \ -- 5=10.0

5=50.0

\\ \ ------5=100.0

Figure 5 Effect of R on velocity distribution f '(n).

\\ m=2.0,A=10.0,A*= =5.0, M=0.5, T =1.2,

V \ Pr=\Q.0,Nb=Nt=0.2,Ec=0.0\,Le=\0.G

I , \ -5=0.1 — 5=1.0 5=2.0 - 5=5.0 5=10.0 5=50.0 5=100.0

values of/"(0), if {()), <//(()) differ only after desired significant digit The value of may vary depending upon the physical parameters such as Prandtl number, thermal radiation parameter, Figure 6 Variation of temperature profile 0 (n) for different values of R.

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 1

Figure 7 Effect of Pr on velocity distribution f (n).

Figure 8 Variation of temperature profile 0 (n) for different values of Pr.

Figure 10 Variation of concentration profile </>(n) for different values of Nb.

Figure 11 Effect of Nt on velocity distribution f (n).

Figure 9 Effect of Nb on velocity distribution f (n).

thermophoresis parameter, magnetic parameter and Schmidt number so that no numerical oscillations would occur. Thus the coupled nonlinear boundary value problem of third-order in f, second-order in 0 and has been reduced to a system of seven simultaneous equations of first-order for seven unknowns. Now it is possible to solve the resultant system of seven simultaneous equations by fifth order Runge-Kutta-Fehlberg integration scheme so that velocity, temperature, and concentration field can easily be obtained for a particular set of physical parameters. The results are provided in several tables and with discussion in the next section.

Figure 12 Variation of concentration profile ф (n) for different values of Nt.

4. Results and discussion

The numerical results are obtained by solving Eqs. (10)-(12) along with the boundary condition (13) using the method described in the previous section for various values of physical parameters to describe the physics of the problem. The numerical values are then tabulated in Tables 1 and 2 and the profiles for dimensionless velocity f (n), dimensionless temperature в(n) and dimensionless concentration ф (n) are depicted in Figures 1-24 for various

Figure 17 Nusselt number for different values of Nb and Nt.

Figure 18 Sherwood number for different values of Nb and Nt.

Figure 13 Effect of Nb and Nt on velocity distribution f (n).

/7\ m=2.0, A=10.0, A*=5.0, R=2.0, T=1.2,

/ M=0.5,Pi-=\0.0,Nb=Nl=0.2,Ec=0.0\

— Ie=1.0 -Le=2.0 Le=5.0 — Z,e=10.0

Figure 14 Effect of Le on velocity distribution f (n).

Figure 15 Variation of concentrati°n profile <p(n) for different values Figure 19 Skin function coefficient for different values of M and R. of Le.

Figure 21 Sherwood number for different values of M and R.

Figure 22 Skin function coefficient for different values of Nb and R.

Figure 23 Nusselt number for different values of Nb and R.

Figure 24 Sherwood number for different values of Nb and R.

values of different physical parameters. The values of the local Nusselt number and local Sherwood number for different values of thermophoresis parameter are presented in Tables 1 and 2. From Tables 1 and 2, it is observed that a comparison of the present results of local Nusselt number and Sherwood number with those obtained by Khan and Pop [18] and Rahman and Eltayeb [32], in the presence of viscous dissipation, magnetic parameter, buoyancy force, and radiation parameter show a very good agreement. It is also seen from Table 1 that the local Nusselt number decreases with increase in the thermophoresis parameter Nt but reverse effects are seen from Table 2 that local Sherwood number increases due to increase in the thermophor-esis parameter Nt. The effect of the non-linear stretching parameter m — 0.3, 1, 1.2, 2 and 3 on the velocity distribution within the boundary layer are shown in Figure 1 for fixed values of the other physical parameters X — 10, X* — 5, M—0.5, R — 2, Tr — 1.2, Pr — 10, Nb—Nt — 0.2, Ec—0.01, and Le —10. It is to be mentioned that the velocity distribution within the boundary layer decreases with increase of non-linear stretching parameter m. Figure 2 shows the effect of thermal buoyancy parameter X on the velocity profile for fixed values of other parameters. It is seen from this figure that the velocity profiles within the boundary layer increases with increase in the thermal buoyancy parameter X. This is due to the fact that when the thermal buoyancy parameter X increases the thermal state of the nanofluid also increases which in turn induce the flow rate to increase. Figure 3 explains the variation of the velocity distribution against n for various values of the nanoparticle buoyancy parameter X* at fixed values of other parameters. From this figure, we see that the velocity profile within the boundary layer increases with increase of the nanoparticle buoyancy parameter due to increase in the thickness of the thermal boundary layer with increase in the nanoparticle buoyancy parameter X*. The effect of the magnetic field parameter M on the velocity profile against n is displayed in Figure 4 for fixed values of other physical parameters. From this figure we notice that the velocity distribution decreases with increase in the value of the magnetic field parameter M. This is due to the fact that application of a magnetic field on the fluid flow domain creates a Lorentz force which acts like retarding force on the fluid motion and as a consequence the temperature of the fluid within the boundary layer increases. The thermal boundary layer thickness also increases with increase of the applied magnetic field strength. Thus the surface temperature of the sheet can be controlled by having control on the strength of the applied magnetic field.

Figures 5 and 6 represent graph of the velocity and temperature profile for different values of the conduction-radiation parameter R — 0.1, 1, 2, 5, 10, 50, etc. for fixed values of different physical parameters. From Figure 5, it is seen that the effect of increasing the strength of the conduction-radiation parameter is to decrease the velocity profile within the boundary layer but outside the boundary layer no effect on velocity profiles is seen. The effect of

radiation is inversely proportional to the conduction-radiation parameter R as can be seen from mathematical equation. Thus small value of R signifies a large radiation effect which is clearly seen from Figure 6 which shows that an increase of the conduction-radiation parameter decreases the temperature profile within the thermal boundary layer. The thickness of the thermal boundary layer increases with increase in the radiation effect, i.e. decreasing the value of R.

Figures 7 and 8 represent the graph of velocity and temperature profiles within the boundary layer for different values of Prandtl number Pr for fixed values of other parameters. It is seen that the effect of increasing the Prandtl number Pr is to decrease the velocity profiles and the temperature profile throughout the boundary layer which results in decrease in the thermal boundary layer thickness. The rate of heat transfer increases with the increasing the values of the Prandtl number due to slow rate of thermal diffusion.

The effect of Brownian diffusion parameter Nb on the velocity and concentration profiles for constant values of other parameters is displayed in Figures 9 and 10. It is seen that the effect of increasing the Brownian diffusion parameter Nb is to increase the velocity profile throughout the boundary layer as seen from Figure 9. From Figure 10, it is observed that the increase of Brownian diffusion parameter Nb, the concentration profile of the nanofluid within the boundary layer decreases for the specified conditions. Therefore, the thickness of the species boundary layer decreases with increase in the Brownian diffusion parameter Nb. It can be noticed that concentration profile decreases almost exponentially with the increase of n. Figures 11 and 12 show the velocity and concentration profiles within the boundary layer for different values of the thermophoresis parameter Nt for fixed values of other parameters. From Figure 11, it is seen that the increase of thermophoresis parameter Nt is to increase the velocity profile within the boundary layer. The thermophoresis phenomenon describes the fact that small micron sized particles suspended in a non-isothermal gas will acquire a velocity in the direction of decreasing concentration profile. An increase in thermo-phoresis parameter Nt, shows an increase of the species difference between the stretching surface and the ambient fluid. From Figure 12, we see that the increase in the values of thermophoresis parameter Nt increases the concentration profile as well as the thickness of the species boundary layer.

Figure 13 depicts the variation of the velocity profile within the boundary layer for some values of Brownian diffusion parameter Nb and thermophoresis parameter Nt. From the figure we can easily see that the effect of increasing the Brownian diffusion parameter Nb and thermophoresis parameter Nt is to increase the velocity profile throughout the boundary layer. Figures 14 and 15 show the velocity and concentration profiles within the boundary layer for different values of the Lewis number Le. From Figure 14, it is seen that

the velocity decreases with increase in the Lewis number Le within the boundary layer. But from Figure 15 we have seen that the concentration profile decreases with increase in the values of the of Lewis number Le throughout the species boundary layer. Figures 16-18 show the graph of skin function coefficient, local Nusselt number and local Sherwood number for different values of the Brownian diffusion parameter Nb and the thermophoresis parameter Nt at constant values of other physical parameters. Figure 16 shows that increasing the values of Brownian diffusion Nb and thermophoresis parameter Nt, the coefficient profiles of skin function increase and we can also see that, two profiles of skin function coefficient intersect when the values of Nb = 0.2, 0.4 for Nt = 0.4. The profiles of Nusselt number increase with increase of the parameters Nb and Nt as presented in Figure 18. Similarly, the graph of Sherwood number is presented in Figure 18, which shows that the graph decreases with an increase in the values of Nb and Nt. It can be easily be seen that all the graphs intersect at a common point and after that they decreases as the values of Nb and Nt are increased.

Figures 19-21 show the graph of skin function coefficient, local Nusselt number and local Sherwood number for different values of magnetic field parameter M and the conduction-radiation parameter R for constant values of other physical parameters. As the values of the local magnetic field parameter M and conduction-radiation parameter R increase there is enhancement in both the heat and mass transfer characteristics and as a result skin function coefficient, local Nusselt number and local Sherwood number decrease, which can be seen in these Figures Figures 22-24 show the graph of skin function coefficient, local Nusselt number, and local Sherwood number for different values of the Brownian diffusion parameter Nb and the conduction-radiation parameter R at fixed values of other physical parameters. Figure 22 shows the decreasing in the profiles of the skin-friction coefficient for different values of Brownian diffusion parameter Nb and the conduction-radiation parameter R. We can see from Figure 23 that the profiles of Nusselt number increases with an increase in the values of the Brownian diffusion parameter Nb and the conduction-radiation parameter R. The profiles of the local Sherwood number decrease with increase in the Brownian diffusion parameter Nb and the conduction-radiation parameter R which can be seen from Figure 24.

5. Conclusion

The problem of two-dimensional mixed convection flow of a nanofluid due to stretching sheet in the presence of thermal radiation and Ohmic dissipation in the presence of convective boundary condition is investigated. The resulting partial differential equations, describing the problem, are transformed into ordinary differential equations by using similarity transformations. These transformed equations

were more conveniently solved numerically by using Runge-Kutta-Fehlberg method with shooting technique for the computation of flow, heat and mass transfer characteristics. The important effects on skin friction, heat transfer rate and mass transfer rate at the stretching sheet for various values of magnetic parameter, radiation parameter and thermophoresis parameter are also analyzed and discussed. The numerical results obtained for the present problem are compared with previously reported results in the open literature under some limiting cases and they are found to be in very good agreement. From the present investigation, following conclusions are drawn:

(i) Increasing the non-linear stretching parameter decreases the flow velocity and temperature in the boundary layer.

(ii) Temperature profile decreases and velocity profile increases with increase in the value of the thermal buoyancy parameter.

(iii) Velocity profiles increase and temperature profiles decrease with increase in the nanoparticle buoyancy parameter.

(iv) Increasing the magnetic parameter decreases the flow velocity but increases the temperature in the thermal boundary layer.

(v) Velocity and temperature profiles both decrease with increase in the conduction-radiation parameter.

(vi) Increasing of relative temperature ratio parameter is to increase velocity profiles as well as temperature profile.

(vii) The effect of Prandtl number is to decrease both velocity and temperature in the thermal boundary layer.

(viii) Both temperature and velocity profiles increase with increase in either Brownian motion parameter or thermophoresis parameter or both.

(ix) Increasing the value of the Eckert number increases the velocity and temperature profiles.

(x) Velocity profile decreases and temperature profile increases with increase in the Lewis number.

It is hoped that the present investigation would be useful in the study the effects of chemical reaction on MHD flow over a vertical plate embedded in non-Darcian porous medium which can be utilized as the basis for many scientific and industrial applications and studying more complex problems.

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