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0Propulsion and Power Research 2015;4(4):230-239
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ELSEVIER
ORIGINAL ARTICLE
Nonlinear radiative heat transfer in magnetohydrodynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition
CrossMark
Wubshet Ibrahim*
Department of Mathematics, Ambo University, P.O. Box 19, Ambo, Ethiopia
Received 21 October 2014; accepted 17 July 2015 Available online 28 November 2015
KEYWORDS
Nonlinear radiative effect;
Stretching sheet; Stagnation point flow; Convective boundary condition; Passive control of nanoparticles
Abstract Two-dimensional boundary layer flow of nanofluid fluid past a stretching sheet is examined. The paper reveals the effect of non-linear radiative heat transfer on magnetohydrodynamic (MHD) stagnation point flow past a stretching sheet with convective heating. Condition of zero normal flux of nanoparticles at the wall for the stretched flow is considered. The nanoparticle fractions on the boundary are considered to be passively controlled. The solution for the velocity, temperature and nanoparticle concentration depends on parameters viz. Prandtl number Pr, velocity ratio parameter A, magnetic parameter M, Lewis number Le, Brownian motion Nb, and the thermophoresis parameter Nt. Moreover, the problem is
governed by temperature ratio parameter [Nr = j>-j and radiation parameter Rd. Similarity
transformation is used to reduce the governing non-linear boundary-value problems into coupled higher order non-linear ordinary differential equation. These equations were numerically solved using the function bvp4c from the matlab software for different values of governing parameters. Numerical results are obtained for velocity, temperature and concentration, as well as the skin friction coefficient and local Nusselt number. The results indicate that the skin friction coefficient Cf increases as the values of magnetic parameter M increase and decreases as the values of velocity ratio parameter A increase. The local Nusselt number — ö'(0) decreases as the values of thermophoresis parameter Nt and radiation parameter
"Corresponding author. Tel.: (+251) 911892494.
E-mail address: wubshetib@yahoo.com Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.
2212-540X © 2015 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/).
http://dx.doi.org/10.1016/j.jppr.2015.07.007
Nr increase and it increases as the values of both Biot number Bi and Prandtl number Pr increase. Furthermore, radiation has a positive effect on temperature and concentration profiles. © 2015 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
For many years the study of stagnation point flow of a viscous incompressible fluid past a stretching sheet has got considerable interest due to its vast application in manufacturing industries. Cooling of electronic devices by fan, cooling of nuclear reactors during emergency shutdown, solar receiver etc. is important applications. Accordingly, Mahapatra and Gupta [1], Ishak et al. [2] and Hayat and et al. [3] have examined heat transfer in stagnation point past stretching sheet. The main objective of the synthesis of nanofluid is for proliferation of the thermal conductivity of a convectional heat transfer fluids such as, water, ethylene glycol and engine oil in high technological areas. This synthesized fluid has vast applications in areas such as in nuclear energy, medicine, space exploration, etc. For the past three decades nanofluid has been tremendously studied to mitigate the problem of heat transfer in high technological industries. Many scholars have contributed a lot for the advancement of study of nanofluid in capacitating common fluids by developing models and techniques for the production of nanofluids and producing important and foremost papers in the area. For instance, Choi [4] was a pioneer in injecting the idea of nanofluid for boosting heat transfer capability of convectional fluids. Following him, Buongiorno [5] developed a model which incorporates the effect of thermophoresis and Brownian motion in the convective transport of nanofluids in laminar boundary layer flow analysis.
Furthermore, Kuznetsov and Nield [6] extended the investigation and analyzed the natural convective transport of nano-fluid past a vertical surface where the nanoparticle is actively controlled at the boundary. Using the concept of controlled nanoparticle at the surface, Khan and Pop [7] examined the laminar boundary layer flow, heat transfer and nanoparticle fraction over a stretching surface in a nanofluid. The published research papers dealing with laminar boundary layer flow past a surface in nanofluid are numerous. Some of them are available in the references ([8-15]).
The study of stagnation point flow of nanofluid past a stretching sheet was examined by Mustafa et al. [16] and Wubshet et al. [17]. The investigation indicated that when the free stream velocity exceeds the stretching velocity, the velocity boundary layer thickness is increasing. Moreover, the study shows that the skin friction coefficient decreases as the velocity of free stream velocity greater than stretching velocity. By varying the thermal boundary conditions, Wubshet and Shanker [18] and Wubshet and Makinde [19]
have examined the effects of convective boundary condition and double stratification on heat transfer of nanofluid past a vertical surface. Moreover, Wubshet and Shanker [20,21] analyzed the magnetohydrodynamic boundary layer flow and heat transfer of a nanofluid over non-isothermal stretching sheet with slip effect.
Their study showed that the surface temperature increases with an increase in Lewis number for prescribed heat flux case. So far the above literature described the examination of nanofluid past a vertical plate or stretching surface when the nanoparticle at the surface is actively controlled. However, very recently, Kuznetsov and Nield [22] revisited their pervious study model of natural convective boundary layer flow a nanofluid past a vertical plate and they assumed that the nanoparticle fraction at the boundary is passively controlled rather than actively and the nanoparticle flux at the wall is zero. In such condition the previous model is more physically realistic one. Furthermore, Khan et al. [23] applied the passive controlled model and examined triple diffusive free convection along a horizontal plate.
All the above studies discussed the boundary layer flow towards stretching sheet when nanoparticles flux at a surface is non-zero. None of them discussed the boundary condition on MHD boundary layer flow and heat transfer of nanofluid over a stretching sheet with zero nanoparticle flux at the wall. Therefore, the aim of this study is to fill this felt out knowledge gap. The nanofluid flow and heat transfer analysis have much practical applications in nuclear reactors, food technology, transportation and in electronics as well as in biomedicine fields. Specifically, this study has a good application area in MHD flow and heat transfer process which occurs in many industrial cooling applications, such as the cooling of nuclear reactor, the geothermal system, aerodynamic process, next-generation solar film collector and heat exchange technology design, etc.
2. Mathematical formulation
In this analysis the stagnation point flow of a steady two-dimensional viscous flow of a nanofluid past a stretching sheet with convective boundary condition and thermal radiation has been considered. At its lower surface, the sheet heated convectively with temperature Tf and a heat transfer coefficient hf. The uniform ambient temperature and concentration respectively, are T1 and C1. It is assumed that there is no nanoparticle flux at the surface and the effect of thermophoresis is taken into account in the boundary condition. The velocity of the stretching sheet at the surface is uw(x) — ax. Where a is a
Nomenclature T ambient temperature
u,v velocity component along x- and y-direction
A velocity ratio
Bi Biot number Greek symbols
ßo magnetic field parameter
C concentration at the surface a thermal diffusivity
Cf skin friction coefficient ß Deborah number
C ambient concentration n dimensionless similarity variable
Dß Brownian diffusion coefficient V dynamic viscosity of the fluid
DT thermophoresis diffusion coefficient u kinematic viscosity of the fluid
f dimensionless velocity stream function (P)f density of the fluid
h dimensionless magnetic stream function (pc)f heat capacity of the fluid
hf heat transfer coefficient (pc)p effective heat capacity of a nanoparticle
k thermal conductivity ¥ stream function
Le Lewis number a electrical conductivity
M magnetic parameter dimensionless concentration function at the surface
Nb Brownian motion parameter dimensionless concentration function at large
Nr temperature ratio values of y
Nt thermophoresis parameter 0 dimensionless temperature
Nux local Nusselt number Tw wall shear stress
Pr Prandtl number r parameter defined by (pc)p/(pc)f
Rd radiation parameter
Rex Shx local Reynolds number local Sherwood number Subscripts
T T A w Tf temperature of the fluid inside the boundary layer uniform temperature over the surface of the plate temperature of a hot fluid w condition at the free stream condition at the surface
constant. The flow is subjected to a constant transverse magnetic field of strength B — B0 which is assumed to be applied in the positive y-direction, normal to the surface. The induced magnetic field is assumed to be small compared to the applied magnetic field and is neglected. It is further assumed that the base fluid and the suspended nanoparticles are in thermal equilibrium. It has been chosen that the coordinate system x-axis is along the stretching sheet and y-axis is normal to the sheet.
Under the above assumptions and boundary layer approximations, the governing equation of the conservation of mass, momentum, energy and nanoparticles fraction in the presence of magnetic field past a stretching sheet as given by Wubshet et al. [17]:
du dv dx dy
du du u — + v — = u
d2u dy2
dUi dx
+ — (Ui- u) (2)
dT dT _ fcP-T dx dy \ dy2
+ r< D (3C dT\ + DT
I ß V dy dy) Ti
dC dC u — + v — = Dß
d2C dy2"
d2T ~df
The boundary conditions are
u = uw = ax, v = 0, — k — = hf (Tf — T),
dC DB dT Db — + — = 0 at y = 0 dy Ti dy
u — Ux = bx, v = 0,
T — Ti, C—Ci as y —1
x and y represent coordinate axes along the continuous surface in the direction of motion and normal to it, respectively. The velocity components along x and y-axis are u and v respectively. u is the kinematics viscosity, T is the temperature inside the boundary layer, r parameter defined by r — (pc)p/(pc)f (pc)p effective heat capacity of a nanoparticle, (pcf heat capacity of the base fluid, p is the density, Tx is ambient the temperature far away from the sheet.
By introducing the following similarity transforms and dimensionless quantities as:
n = \ß>y' w = PaUxf(n)' 0(n =
T - T1 Tf-Ta
$(n) =
where y(x, y) represent the stream function and is defined
so that Eq. (1) is satisfied automatically.
Using Rosseland approximation of radiation
4c*dT4 3k*
16c* 3dT
where k* is the mean absorption coefficient and c* is the Stefan Boltzmann constant. Eq. (8) results in a highly nonlinear energy equation in T. The Rosseland approximation can be linearized about ambient temperature TThis means simply replace T3 in Eq. (8) by T3 .
Then Eq. (3) becomes:
" 16c*T3' "
dT dT _ d dx dy dy
3(pc)k
+ T{ Db
fdC dT\ DT idT\
\iy ~dy) + T1 ~
Define the non-dimensional temperature 0(n) =
with T = T+ (Nr — 1)0) and Nr = ^, where Nr is temperature ratio parameter. The first term on the right hand side of Eq. (9) can be written as
«((f (1 + Rd(1 + (Nr- 1)0)3))
where Rd— i62*^
denotes the radiation parameter, and Rd — 0 indicates no thermal radiation effect.
The governing Eqs. (2)-(4) and Eq. (9) are reduced by using Eq. (6) and Eq. (8) as follows:
f" + ff" -f2 + A2 + M (A -f ) — 0 (10)
[(1 + Rd(1 + (Nr - 1)0) V ]' + Pr(f0 + Nb<p'9' + Nte'2) — 0
<b" + LePrfé' +— 0" = 0 Nb
(11) (12)
Table 1 Comparison of skin friction coefficient — f "(0) for
different values of velocity ratio parameter A when M = Nr = 0.
A Wubshet et al. [17] Ishak et al. [2] Present result
0.1 0.9694 0.9694 0.9694
0.2 0.9181 0.9181 0.9181
0.3 - - 0.8494
0.4 - - 0.7653
0.5 0.6673 0.6673 0.6673
0.8 - - 0.2994
1.0 - - 0.0000
2.0 2.0175 2.0175 2.0175
3.0 4.7293 4.7293 4.7293
5.0 - - 11.7520
7.0 - - 20.4979
10.0 - - 36.2574
With boundary conditions
f (0) = 0, f (0)=1, 0' (0) = Bi(0(O) — 1),
Nb< (0) + Nt0 (0) = 0, at n = 0,
f' (i)— A, 0(i) —0, <(i) —0, as n —1
where the governing parameters are defined by:
hf U v b a
Bi = -f-J- Pr = -, A =-, Le = —, k y a - a DB
(pc)pDBCi 16c* T 3, Nb — —-, Rd =
M — -
Nt —
(Pc)f u (pc)pDr (Tf — T i) (pc)fOTi '
Table 2 Comparison of local Nusselt number — 0 (0) at Nt = 0, Nb — 0, Rd = Nr = 0 for different values of Pr with previously published data.
Pr A Present Wubshet Mahapatra [1] Hayat [3]
result et al. [17]
1 0.1 0.6028 0.6022 0.603 0.602156
0.2 0.6246 0.6245 0.625 0.624467
0.5 0.6924 0.6924 0.692 0.692460
1.5 0.1 0.7768 0.7768 0.777 0.776802
0.2 0.7971 0.7971 0.797 0.797122
0.5 0.8648 0.8648 0.863 0.864771
2.0 0.1 0.9257 - - -
0.2 0.9447 - - -
0.5 1.0116 - - -
Table 3 Computed values of local Nusselt number — 0' (0) when
Nb = A = 0.5, Rd = Pr = M = 1, for different values of Le Nr, Nt
and Bi.
Le Nr Nt Bi —0' (0)
2 1.1 0.5 2 0.3372
5 0.3331
10 0.3306
15 0.3294
5 1.1 0.3331
1.5 0.2395
1.8 0.1854
2.0 0.1574
3 0.0800
1.1 0.2 0.3432
0.5 0.3331
1 0.3165
3 0.2555
5 0.2052
10 0.1259
0.5 0.1 0.0818
1 0.2900
2 0.3331
5 0.3645
10 0.3760
f ', d and $ are the dimensionless velocity, temperature and particle concentration respectively. n is the similarity variable, the prime denotes differentiation with respect to n. Pr, M, Nb, Nt, Le denote Prandtl number, a magnetic parameter, a Brownian motion parameter, a thermophoresis parameter, and a Lewis number, respectively.
The important physical quantities of interest in this problem are the skin friction coefficient Cf and local Nusselt number Nux are defined as:
Nux —
PuW k(Tf - T i)
where the wall shear stress tw and wall heat flux q given by
Tw — K d) y — 0'
^ —-k{ f)y — 0
— -k(Tf - Ti)J- [1 + Rd03w] 0'(0);
the convergence criteria were taken as An = 0.01 and 10 — 8, respectively. The asymptotic boundary conditions were given by Eq. (13) were replaced by a value of nmax = 10.
Figure 2 Velocity profile for different values of magnetic parameter M when A = Nb = Nt = 0.5, Pr= 1, Le = Bi = 5.
By using the above equations, we get
Cfffi — -f "(O);
Nux \/Rë~x
— -(1 + Rd0lNr)<0 (O)
where Rex — ^ and Nux are local Reynolds number and local Nusselt number, respectively.
3. Numerical solution and accuracy
The dimensionless ordinary differential equations Eqs. (10)—(12) subjected to the boundary conditions Eq. (13) have been solved by the function bvp4c from matlab for different values of governing parameters. The step size and
Figure 3 Temperature profile for different values of temperature ratio parameter Nr when A — Nt — Nb — 0.5, Bi — Le — 5, Rd — M — 1.
Figure 1 Velocity profile for different values of velocity ratio A when Nb — Nt — 0.5, Pr — 1, Le — Bi — 5, M — 1.
Figure 4 Temperature profile for different values of radiation parameter Rd when A — Nt — Nb — 0.5, Bi — Le — 5, M — 1, Nr — 1.1.
The choice of nmax — 10 ensure that all numerical solutions approached the asymptotic value correctly.
Furthermore, in order to evaluate the precision of the method used, comparison with previously reported data available in the literature has been made. From Table 1 it can be seen that the numerical values of the skin friction coefficient — f "(0) in this paper for different values of velocity ratio A when M — 0 are in an excellent agreement with the results published in Ishak et al. [2] and Wubshet et al. [17]. To further validate the numerical method used in the paper, comparison of local Nusselt number — 0'(0) for different values of Prandtl number Pr by ignoring the effects of Nt and Nb parameters has been shown in Table 2, which is also in excellent agreement with Refs. [1,3] and [18]. Comparison of the results of this study with literature values have shown in Tables 1 and 2 indicate an excellent agreement and this gives us a confidence to use the present code.
012345678
Figure 5 Temperature profile for different values of Nt when A = Nb = 0.5, Pr = M = 1, Bi = Le = 5.
0 1 2 3 4 5 6
Figure 6 Temperature profile for different values of Bi when A = Nt = Nb = 0.5, Pr = M = 1, Le = 5.
Furthermore, Table 3 shows the computed values of the local Nusselt number — 0 (0) for different values of the governing parameters such as Nr, Le, Nt and Bi. It is found
0 1 2 3 4 5 6
Figure 7 Temperature profile for different values of A when Nt = Nb = 0.5, Pr = M = 1, Bi = Le = 5.
-0.25 -.-.-.-.-
0 1 2 3 4 5
Figure 8 Concentration profile for different values of Nb when Nt = A = 0.5, Pr = M = 1, Bi = Le = 5.
■0.5 -1-1-'-'-
0 1 2 3 4 5
Figure 9 Concentration profile for different values of Nt when Nb = A = 0.5, Pr = M = 1, Bi = Le = 5.
Figure 11 Concentration profile for different values of Bi when A = Nt = Nb = 0.5, Pr = M = 1, Le = 5.
that the local Nusselt number — 0(0) is a decreasing function of the parameter Le, Nr and Nt and an increasing function of Biot number Bi.
4. Result and discussion
The set of ordinary differential equations which are obtained from momentum, energy and concentration Eqs. (10)—(12) subjected to the boundary conditions Eq. (13) were numerically solved using bvp4c from a matlab. Velocity, temperature and concentration graphs for different values of governing parameters have been obtained. The results are displayed through figures and tables.
Figures 1 and 2 depict the graphs of the non-dimensional velocity profile f '(n) for different values of velocity ratio parameter A and magnetic parameter M. Figure 1 indicates that the hydrodynamic boundary layer thickness increases with increasing values of A (A > 1) and it decreases with decreasing values of A (A < 1). Physically, when the free
Figure 13 Concentration profile for different values of Nr when Nt — Nb — 0.5, Pr — M — 1, Bi — Le — 5.
stream velocity greater than stretching velocity, the ratio of free stream velocity to stretching velocity is greater than 1, as a result, a retarding force is diminished and the flow velocity increased. Figure 2 reveals the effect of Hartmann number called magnetic parameter M on velocity profile. The existence of magnetic field sets in a resistive force called Lorentz force, which is a retarding force on the velocity field, as a result a flow velocity is reduced.
The effects of temperature ratio parameter Nr, radiation parameter Rd, thermophoresis parameter Nt, convective parameter called Biot number Bi and velocity ratio parameter A on temperature profiles are given in Figures 3-6. Figure 3 shows the influences of temperature ratio parameter Nr on temperature profile. It is observed that the fluid temperature increases as Nr increases due to the fact that the conduction effect of the nanofluid increases with increasing temperature ratio parameter. The inclusion of temperature ratio parameter induces higher surface heat flux as a result the temperature increases with in the boundary layer region.
Figure 4 examines the behavior of radiation parameter Rd on temperature graph. Increasing thermal radiation effect
Figure 15 Variation of the skin friction coefficient -f "(0) with A for different values of M when Nb = Nt = 0.5, Le = Bi = 5, Pr = 1.
Figure 16 Variation of local Nusselt number -d(0) with Nt for different values of Bi when Nb = 0.5, Pr = M = 1, Le = 5, A = 0.3.
permits the thermal effect to penetrate deeper into the quiescent fluid and thus both temperature and thermal boundary layer thickness increase with an increases in Rd.
Figure 18 Variation of local Nusselt number -6'(0) with Nt for different values of Rd when Nb — 0.5, Pr — M — 1, Le — 5, Nr — 1.1.
Figure 5 depicts the influence thermophoresis parameter Nt on temperature profile. As the thermophoretic effect increase, the migration of nanoparticles from the hot surface to cold ambient fluid is occurred, as results the temperature increases in the boundary layer, this results in the growth of thermal boundary layer thickness. Figure 6 shows the impact of convective heating called Biot number Bi on temperature profile. Physically Biot number is the ratio of convection at the surface to conduction within the surface of a body. As the Biot number effect (convection at the surface) increases, temperature at the surface increases this result in an increase in the thermal boundary layer thickness. The influences of velocity ratio parameter A on temperature profile is given by Figure 7. As the values of velocity ratio A increase, temperature at the surface a sheet decreases; moreover, thermal boundary layer thickness diminishes.
The effects of governing parameters such as Brownian motion parameter Nb, thermophoresis parameter Nt, Prandtl number Pr, Lewis number Le, Biot number Bi and velocity ratio parameter A on concentration profiles are displayed in
Figures 8-13. Figure 8 illustrates the influences of Brow-nian motion parameter Nb on concentration profile ^(n). As Brownian motion effect increase, the concentration gradient increases as a result the Brownian force increases which boot the nanoparticles concentration at the surface. Hence the concentration profile ^(n) increases at the surface. Figure 9 shows the impact of thermophoresis on concentration profile ^(n). Since the impact of Brownian force is to counter balance the influences of thermophoretic force, as the influences of thermophoretic force increases the concentration gradient at the surface decreases, as result the concentration profile at a surface decreases, which is opposite to the case for Brownian motion effect.
The variation of Prandtl number on concentration profile is shown in Figure 10. It can be seen from a graph that as the influence of Pr increases, the nanoparticles diffuses out towards outside, as a result the nanoparticles concentration at the surface decreases.
The influences of Lewis number Le, Biot number Bi and velocity ratio A on concentration profile in this study are similar as shown in the Figures 11-13. The Lewis number effect on nanoparticle on concentration graph is described in Figure 11. Increasing the Lewis number corresponds to a poor Brownian diffusion coefficient which leads to short penetration depth for concentration profile ^(n). As a result the concentration at the surface decreases with increasing the influences of Lewis number. Similar effects are observed when the values of Biot number Bi, velocity ratio parameter A (Figure 12) and radiation parameter Nr (Figure 13) increase.
As the values of Nb and Le increase, the dimensionless concentration at a surface increases, but opposite effect is observed as the values of Nt, Pr and Bi increases. The graph also reveals that the concentration boundary layer thickness increases as the values of Nb and Le, but it decreases as the values of Nt, Pr and Bi increases. For all the governing parameters considered here, the nanofluid parameters overshooting the dimensionless concentration at the surface and asymptotic to zero far away from the surface.
Figures 14 and 15 show the variation of the coefficient of skin friction with respect to magnetic field parameter M for different values of velocity ration A parameter and vice versa. In both cases it is observed that the magnitude of skin friction coefficient increases as the values of magnetic parameters M increase and decreases as the values of A increase.
Figures 16-18 show the variation of local Nusselt number — 0 (0) with respect to different parameters such as thermo-phoresis parameter Nt, velocity ratio parameter A, convective parameter Bi and temperature ratio parameter Nr. The graphs show that as the values of Nt and Nr increase, the local Nusselt number — 0 (0) at the surface decreases, however, it increase as the values of A, Pr and Bi increase.
5. Conclusion
In this paper, the effects of nonlinear radiative heat transfer, magnetic field and convective heating on boundary
layer flow and heat transfer of nanofluid over a stretching sheet is discussed. The boundary layer equations governing the flow problem are reduced into a coupled high order non-linear ordinary differential equations using the similarity transformation. The differential equations are solved numerically using bvp4c from matlab software. The effects of various physical parameters such as thermal radiation parameter Rd, temperature ratio parameter Nr, magnetic parameter M, Prandtl number Pr, Brownian motion parameter Nb, thermophoresis parameter Nt and Lewis number Le on momentum, energy and concentration equation are discussed using figures and tables. Significant effects of governing parameters are plotted and justify our results through tables and graphs. The outcome of this study indicated that the flow velocity and the skin friction coefficient on stretching sheet are strongly influenced by velocity ratio A and magnetic parameter M. It is observed that the magnetic parameter M boots the growth of skin friction coefficient - f' (0), however, velocity ratio parameter A inhibited the growth of skin friction coefficient. The study also shows that the Biot number Bi, Prandtl number Pr and velocity ratio A parameters favor the local Nusselt number — 0'(0) but thermophoresis parameter Nt and temperature ratio parameter Nr act in opposite way. It is also found that the influences of Brownian motion on temperature and nanoparticle volume fraction is minimal.
The following are brief summary conclusions drawn from the analysis:
1. A and M have opposite influences on skin friction coefficient.
2. The influences of Nb and Nt on concentration profile is opposite.
3. The influence of A and Bi on the local Nusselt number is positive.
4. Prandtl number Pr and convective parameter Bi have a similar impact on local Nusselt number.
5. The thickness of thermal boundary layer are antagonistic with parameters Nt and Pr.
6. Nanoparticle volume fraction near the stretching surface is negative and it is zero far away from the surface.
7. Radiation parameter Rd favors the thermal boundary layer thickness.
8. The temperature ratio parameter Nr has a diminishing effect on local Nusselt number -0'(0).
Acknowledgements
The author wish to express his very sincere thanks to referees for their valuable comments and suggestions.
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