Home Search Collections Journals About Contact us My IOPscience

Effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over

a permeable moving flat plate

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2012 Phys. Scr. 86 045003

(http://iopscience.iop.org/1402-4896/86/4/045003)

View the table of contents for this issue, or go to the journal homepage for more

Download details:
IP Address: 137.149.3.15
The article was downloaded on 21/03/2013 at 07:11

Please note that terms and conditions apply.

IOP PUBLISHING PHYSICA SCRIPTA
Phys. Scr. 86 (2012) 045003 (8pp) doi:10.1088/0031-8949/86/04/045003

Effects of thermal radiation and viscous

dissipation on boundary layer flow of
nanofluids over a permeable moving
flat plate
T G Motsumi1 and O D Makinde2
1
Mathematics Department, University of Botswana, Private bag 0022, Gaborone, Botswana
2
Institute for Advanced Research in Mathematical Modelling and Computations, Cape Peninsula
University of Technology, PO Box 1906, Bellville 7535, South Africa
E-mail: motsumit@mopipi.ub.bw and makinded@cput.ac.za

Received 27 February 2012

Accepted for publication 22 August 2012
Published 27 September 2012
Online at stacks.iop.org/PhysScr/86/045003

Abstract
The effects of suction, viscous dissipation, thermal radiation and thermal diffusion are
numerically studied on a boundary layer flow of nanofluids over a moving flat plate. The
partial differential equations governing the motion are transformed into ordinary differential
equations using similarity solutions, and are solved using the Runge–Kutta–Fehlberg method
with the shooting technique. The effects of nanoparticle volume fraction, the type of
nanoparticles, the radiation parameter, the Brinkman number, the suction/injection parameter
and the relative motion of the plate on the nanofluids velocity, temperature, skin friction and
heat transfer characteristics are graphically presented and then discussed quantitatively. A
comparative study between the previously published and the present results in a limiting sense
reveals excellent agreement between them.

PACS numbers: 02.30.Jr, 05.70.−a

Nomenclature N radiation parameter

(u, v) velocity components fw suction/injection parameter
(x, y) coordinates k∗ the absorption coefficient
knf nanofluid thermal conductivity
Pr Prandtl number Greek symbols
Bi local Biot number 9 stream function
T∞ free stream temperature
2 dimensionless temperature
F dimensionless stream function
µnf nanofluid dynamic viscosity
U∞ free stream velocity
αnf nanofluid thermal diffusivity
T temperature
ks solid fraction thermal conductivity η similarity variable
Rex local Reynolds number ρnf nanofluid density
cp specific heat at constant pressure ρs solid fraction density
Br Brinkmann number υf base fluid kinematic viscosity
Tw plate surface temperature µf base fluid dynamic viscosity
Cf skin friction coefficient ϕ solid volume fraction parameter
Nu local Nusselt number ρf base fluid density
Uw plate uniform velocity σ the Stefan–Boltzman constant

0031-8949/12/045003+08$33.00 Printed in the UK & the USA 1 © 2012 The Royal Swedish Academy of Sciences
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

1. Introduction y

Nanofluids can be described as a suspension of nanoparticles U∞ , T∞

(1–100 nm) in a base fluid. The nanoparticles are made
from copper, aluminium, carbon and, in general, materials v nanofluid
that are chemically stable. For the base fluids, the
u, T
commonly used fluids are water, ethylene, oils and lubricants,
and many other fluids. Nanofluids can be used for a u = U w , v=Vw(x), Tw x
wide variety of industries, ranging from transportation
to energy production and in electronics systems such as
Figure 1. Flow configuration and the coordinate system.
microprocessors, micro-electro-mechanical systems (MEMS)
and in the field of biotechnology. In recent years, the number
of companies that see the potential of nanofluids technology on the boundary layer flow of nanofluids over a moving
and focus on their specific industrial applications has been permeable plate. The nanoparticles considered are copper
increasing. Nanofluids can be used to cool automobile engines (Cu) and alumina (Al2 O3 ) with the base fluid as water. Using
and welding equipment and to cool high-heat-flux devices an appropriate similarity transformation, the well-known
such as high-power microwave tubes and high-power laser governing partial differential equations are reduced to the
diode arrays. A nanofluid coolant could flow through tiny ordinary differential equations. The resulting problems are
passages in MEMS to improve its efficiency [1, 2]. Several solved numerically using the Runge–Kutta–Fehlberg method
methods are employed to produce nanofluids in industries. with the shooting technique. Several results showing velocity
One of the processes involves production of nanoparticles and temperature profiles are presented graphically and
using gas condensation, which are then dispersed into the base explained. We also present and discuss the results on shear
fluid. Ultrasound is commonly used in this process in order stress and temperature gradient at the plate surface. A
to make sufficient amalgamation of base fluid and particles comparative study between the previously published results
[3, 4]. Another method known as vacuum evaporation on
and the present results in a limiting sense reveals excellent
running oil substrate (VEROS) involves evaporating nanofluid
agreement between them.
particles on an oil substrate. The particles then grew onto the
oil substrate in the base fluid [5]. Choi [1] pioneered the study
on the enhancement of thermal conductivity of fluids using
2. Mathematical model
nanoparticles. Thereafter, several authors, such as Kiblinski
et al [2], Wang et al [3], Eastman et al [4], Buongiorno [5], etc,
We consider the two-dimensional steady laminar boundary
have conducted theoretical and experimental investigations
on the heat transfer enhancement property of nanofluids. layer flow of water-based nanofluids containing two types
The numerical solution of the combined effects of Brownian of nanoparticles, Cu and Al2 O3 , over a flat permeable plate
motion and thermophoresis on boundary layer flows of moving with a constant velocity Uw in the same or opposite
nanofluids over a flat surface embedded in a saturated porous direction of the free stream velocity U∞ . The x-axis extends
medium was reported by Nield and Kuznetsov [6]. Kuznetsov parallel to the plate surface, while the y-axis extends normal to
and Nield [7] investigated the problem of natural convective the surface (see figure 1). It is assumed that both the base fluid
boundary layer flow of a nanofluid past a vertical semi-infinite (i.e. water) and the nanoparticles are in thermal equilibrium
flat plate. Makinde and Aziz [8] reported the similarity and no slip occurs between them.
solutions for the thermal boundary layer of a nanofluid past The governing equations for boundary layer flows of
a stretching sheet with a convective boundary condition. nanofluids and heat transfer are written as [2–5, 9, 10]
Ahmad et al [9] extended the classical forced convection
boundary layer flow past a static and a moving semi-infinite ∂u ∂v
+ = 0, (1)
flat plate to the case of a nanofluid using the model of ∂x ∂y
Tiwari and Das [10]. Moreover, an adequate understanding
of radiative heat transfer in flow processes is very important ∂u ∂u µnf ∂ 2 u
in engineering and industries, especially in the design of u +v = , (2)
∂x ∂y ρnf ∂ y 2
reliable equipments, nuclear plants, gas turbines and various
propulsion devices for aircraft, missiles, satellites and space  2
vehicles. Thermal radiation effects are extremely important ∂T ∂T ∂2T µnf ∂u
u +v = αnf 2 +
in the context of flow processes involving high temperature. ∂x ∂y ∂y (ρc p )nf ∂ y
The effects of thermal radiation on the boundary layer flow
∂qr
 
have also been considerably researched [11–13]. The effect of 1
− , (3)
thermal radiation on boundary layer flow and mass transfer (ρc p )nf ∂ y
over a moving vertical porous surface has been reported by
Makinde [14]. However, the number of studies on thermal where (u, v) are the velocity components of the nanofluid in
radiation effects on boundary layer flows using nanofluids is the (x, y) directions, respectively, T is the temperature of the
very limited. nanofluid, µnf is the dynamic viscosity of the nanofluid, ρnf
In this study, our objective is to investigate the combined is the density of the nanofluid, αnf is the thermal diffusivity
effects of suction, thermal radiation and viscous dissipation of the nanofluid and (ρc p )nf is the heat capacitance of the

2
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

nanofluid which is given as [9, 10] The following dimensionless variables,

µf s
µnf = , ρnf = (1 − ϕ)ρf + ϕρs , U yp
(1 − ϕ)2.5
p
η=y = Rex , ψ = υf xU f (η), u = U f 0 (η),
υf x x
knf µf
αnf = , υf = ,
1 U υf
r
(ρc p )nf ρf (4) T − T∞
v= ηf 0 − f , θ = ,


(11)
knf (ks + 2kf ) − 2ϕ(kf − ks ) 2 x Tw − T∞
= ,
kf (ks + 2kf ) + ϕ(kf − ks ) are introduced into the governing equations (2) and (7)
(ρc p )nf = (1 − ϕ)(ρc p )f + ϕ(ρc p )s . together with the boundary conditions in equation (9) and we
obtain
The thermal conductivity of the nanofluid is represented
by knf , ϕ is the nanoparticle volume fraction parameter, ρf is (1 − ϕ)2.5 (1 − ϕ + ϕρs /ρf ) 00
f 000 + f f = 0, (12)
the reference density of the fluid fraction, ρs is the reference 2
density of the solid fraction, µf is the viscosity of the fluid
fraction, υf is the kinematic viscosity of the fluid fraction,
kf Pr 1 − ϕ + ϕ(ρc p )s /(ρc p )f
   
kf is the thermal conductivity of the fluid fraction, c p is 4kf
1+ θ +
00
f θ0
the specific heat at constant pressure and ks is the thermal 3N knf 2knf
conductivity of the solid volume fraction. Using the Rosseland
kf Br
2
approximation [12, 13] for the thermal radiation, the radiative + f 00 = 0, (13)
heat flux is simplified as knf (1 − ϕ)2.5

4σ ∂ T 4 f (0) = f w , f 0 (0) = 1 − r, θ (0) = 1, (14)

qr = − , (5)
3k ∗ ∂ y

where σ is the Stephan–Boltzmann constant and k ∗ is the mass f 0 (∞) = r, θ (∞) = 0, (15)
absorption coefficient. The temperature differences within the
flow are assumed to be sufficiently small so that T 4 may where the prime symbol denotes differentiation with respect
be expressed as a linear function of temperature T using to η,
a truncated Taylor series about the free stream temperature k ∗ kf
T∞ , i.e. N= 3
(radiation parameter),
4σ T∞
T 4 ≈ 4T∞
3 4
T − 3T∞ . (6)

Using equations (5) and (6), equation (3) becomes µf U∞

2
Br = (the Brinkmann number),
kf (Tf − T∞ )
∂T ∂T ∂ T
3  2

1 16σ T∞
u +v = αnf +
∂x ∂y (ρc p )nf 3k ∗ ∂ y2 υf
Pr = (the Prandtl number),
µnf
 2
∂u αf
+ . (7)
(ρc p )nf ∂ y
U∞
r= (velocity ratio parameter),
The variable plate surface permeability function is given U
as
U υf
r
fw Ux
Vw (x) = − , (8) Rex = (the local Reynolds number).
2 x υf
where U = Uw + U∞ , f w is a constant with f w > 0 It is worth mentioning that ϕ = 0 corresponds to the
representing the transpiration (suction) rate at the plate conventional fluid scenario. The Blasius flat plate flow of the
surface, f w < 0 corresponds to injection and f w = 0 for an nanofluid problem can be recovered by putting r = 1 (i.e. for
impermeable surface. The boundary conditions at the plate Uw = 0), and for r = 0 (i.e. U∞ = 0) the Sakiadis flow of the
surface and far into the nanofluid may be written as [9, 14] nanofluid problem is obtained [9]. The nanofluid motion is
faster than the plate motion when 1 > r > 0.5, while the plate
u(x, 0) = Uw v(x, 0) = Vw (x), T (x, 0) = Tw , motion is faster than the nanofluid motion when 0 < r < 0.5.
(9)
u(x, ∞) = U∞ , T (x, ∞) = T∞ . Both the nanofluids and the plate are moving in the opposite
direction when r > 1. The quantities of practical interest in
The stream function ψ satisfies the continuity equation this study are the skin friction coefficient Cf and the local
(1) automatically with Nusselt number Nu, which are defined as
∂ψ ∂ψ τw xqw
u= and v = − . (10) Cf = , Nu = , (16)
∂y ∂x ρf U ∞
2 kf (Tw − T∞ )

3
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

Table 1. Thermophysical properties of the fluid phase (water) and

nanoparticles [5, 9, 10].
Physical properties Fluid phase (water) Cu Al2 O3
c p (J kg K )
−1
4179 385 765
ρ (kg m−3 ) 997.1 8933 3970
k (W mK−1 ) 0.613 400 40

Table 2. Computations of Rex1/2 Cf showing comparison with

Ahmad et al [9] results for Br = 0, f w = 0, N = ∞ and Pr = 6.2.
ϕ Cu–water Al2 O3 –water Cu–water Al2 O3 –water
[9] [9] (present) (present)
0 0.3321 0.3321 0.3321 0.3321
0.008 0.3459 0.3394 0.3459 0.3394
0.014 0.3563 0.3449 0.3563 0.3449
0.016 0.3597 0.3468 0.3597 0.3468
0.02 0.3667 0.3506 0.3667 0.3506
0.1 0.5076 0.4316 0.5076 0.4316 Figure 2. Velocity profiles for Pr = 6.2, Br = 0.1, f w = 0.2,
0.2 0.7066 0.5545 0.7066 0.5545 ϕ = 0.1, N = 1 and r = 1.

where τw is the skin friction and qw is the heat flux from the
plate, which are given by
∂u 

τw = µnf ,
∂ y  y=0
(17)
3 
∂ T 
 
16σ T∞
qw = − knf + .
3k ∗ ∂ y  y=0

Substituting equations (17) into (16), we obtain

1/2 1
Rex Cf = f 00 (0),
(1 − ϕ)2.5
  (18)
−1/2 knf 4
Rex N u = − + θ 0 (0).
kf 3N Figure 3. Velocity profiles with Al2 O3 –water as a working fluid for
Pr = 6.2, Br = 0.1, f w = 0.2, ϕ = 0.1 and N = 1.
The above set of equations (12) and (13) subject to the
boundary conditions (14) and (15) were solved numerically
3.1. Effects of parameters variation on the velocity profile
by the Runge–Kutta–Fehlberg method with the shooting
technique [15]. Both the velocity and temperature profiles The velocity profiles of the two water-based nanofluids (Cu
were obtained and utilized to compute the skin-friction and Al2 O3 ) are shown in figure 2 for the case when the
coefficient and the local Nusselt number in equation (18). flow over a stationary plate is driven by the free stream
velocity (r = 1). It is observed that the water–Al2 O3 nanofluid
3. Results and discussion produced a thicker momentum boundary layer than that of
water–Cu nanofluid. Figure 3 illustrates the effects of the
In this section, we consider two types of water-based velocity ratio parameter (r) on the nanofluid velocity profiles
Newtonian nanofluids containing copper (Cu) and alumina for alumina. As expected, the velocity profiles satisfied the
(Al2 O3 ). The Prandtl number of the base fluid (water) is kept prescribe velocity ratio boundary conditions. The case of r =
constant at 6.2 and the effect of solid volume fraction ϕ is 0 corresponds to the Sakiadis flow scenario with only the plate
investigated in the range of 0 6 ϕ 6 0.2. Following [9, 10], moving, while r = 1 depicts the Blasius flow over a stationary
the thermophysical properties of water and the elements Cu, flat plate. When r = 0.5, both the plate and the free stream
Al2 O3 are shown below (table 1). are moving with the same velocity and r = 1.5 corresponds to
For the special case of infinite Biot number and in the case when both the plate and the free stream are moving
the absence of viscous dissipation effect, our results agreed in opposite directions. In figure 4, it is interesting to note that
perfectly with those reported by Ahmad et al [9] as shown in the nanofluid momentum boundary layer thickness increases
table 2 and this serves as a benchmark for the accuracy of our slightly by increasing the values of the solid fraction (ϕ).
numerical procedure. In figure 5, it is observed that the velocity boundary layer
In the following subsections, we highlight the effects of thickness for alumina decreases with increasing the values
various thermophysical parameters on the nanofluid velocity of the suction parameter ( f w ). This can be attributed to a
and temperature profiles as well as the skin friction and the decrease in the fluid velocity due to increasing nanofluid
local Nusselt number on the plate surface. suction at the moving plate surface (r = 0).

4
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

ϕ =0

ϕ =0.1

ϕ =0.15

ϕ =0.2

Figure 6. Temperature profiles for Pr = 6.2, Br = 0.1, f w = 0.2,

ϕ = 0.1, N = 1 and r = 1.
Figure 4. Velocity profiles with Al2 O3 –water as a working fluid for
Pr = 6.2, Br = 0.1, f w = 0.2, r = 0 and N = 1.

____

ooo

++++

…….

Figure 7. Temperature profiles with Al2 O3 –water as a working

fluid for Pr = 6.2, Br = 0.1, f w = 0.2 and ϕ = 0.1, N = 1.

Figure 5. Velocity profiles with Al2 O3 –water as a working fluid for

Pr = 6.2, Br = 0.1, r = 0, ϕ = 0.1 and N = 1.

3.2. Effects of parameters variation on the temperature

profile
Figures 6–11 demonstrate the effects of various parameters on
the nanofluid temperature profiles. Generally, the nanofluid
temperature is highest at the plate surface and decreases
gradually to its zero free stream value far away from the plate.
In figure 6, it is observed that alumina produces a thicker
thermal boundary layer with higher temperature as compare to
the water–Cu nanofluid. From figure 7, it can be seen that the
temperature increases for alumina within the boundary layer
region as the value of the velocity ratio parameter (r) increases
from 0 to 1.4. This can be attributed to the fact that as the
value of r increases from 0 to 1.4, the plate velocity decreases
to zero and then starts moving in the opposite direction
of the free stream, leading to an increase in the internal Figure 8. Temperature profiles with Al2 O3 –water as a working
heat generation within the nanofluid due to the combined fluid for Pr = 6.2, Br = 0.1, f w = 0.2, r = 0 and N = 1.

5
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

Figure 12. Skin friction coefficient for Pr = 6.2, N = 1, r = 1,

f w = 0.2 and Br = 0.1.

Figure 9. Temperature profiles with Al2 O3 –water as a working

fluid for Pr = 6.2, Br = 0.1, r = 0, ϕ = 0.1 and N = 1.

Figure 13. Skin friction for Al2 O3 –water as a working fluid for
Pr = 6.2, Br = 0.1 and ϕ = 0.1.

Figure 10. Temperature profiles with Al2 O3 –water as a working

fluid for Pr = 6.2, Br = 0.1, r = 0, f w = 0.2 and ϕ = 0.1.

Figure 14. Nusselt number for Pr = 6.2, N = 1, r = 1, f w = 0.2

and Br = 0.1.

effects of nanoparticle action and viscous dissipation. Figure 8

shows the effect of increasing the nanoparticles solid fraction
for alumina on the temperature profiles. As expected, it is
observed that the thermal boundary layer thickness increases
with increasing the value of nanoparticle solid fraction (ϕ),
leading to an increase in the temperature. An increase in
the suction parameter ( f w ) and the radiation parameter (N)
Figure 11. Temperature profiles with Al2 O3 –water as a working for alumina causes a decrease in the temperature and the
fluid for Pr = 6.2, N = 1, r = 0, f w = 0.2 and ϕ = 0.1. thermal boundary layer thickness as displayed in figures 9

6
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

Figure 15. Nusselt number for Al2 O3 –water as a working fluid for Pr = 6.2, Br = 0.1 and ϕ = 0.1.

Figure 16. Nusselt number for Al2 O3 –water as a working fluid for Pr = 6.2, r = 0 and ϕ = 0.1.

and 10, respectively. However, it is noteworthy that the with an increase in solid fraction, with water–Cu showing a
nanofluid temperature increases with an increase in the plate surface higher heat transfer rate than alumina. Figure 15
Brinkmann number (Br) due to the action of viscous heating shows the combined effects of suction and velocity ratio
as shown in figure 11. parameter on the Nusselt number. The rate of heat transfer at
the plate surface decreases with increasing the values of r and
increases with increasing the intensity of nanofluid suction.
3.3. Effects of parameters variation on the skin friction and
Figure 16 shows that the Nusselt number (Nu) decreases with
Nusselt number
increasing the values of both the Brinkmann number (Br) and
Figures 12–16 show the effects of parameter variation on the radiation parameter (N).
the skin friction and the Nusselt number. From figure 12,
it is observed that skin friction grows with an increase in 4. Conclusions
solid fraction. Water–Cu nanofluid shows faster growth than
alumina. Figure 13 shows that skin friction grows with an This paper presents an analysis of the combined effects
increase of the suction rate ( f w ) and the friction decreases of thermal radiation, viscous dissipation, suction and solid
when the plate is moving and the free stream is stationary volume fraction on the thermal boundary layer of water-based
(r = 0) until it is zero when both the free stream and the plate nanofluids over a moving or fixed flat surface. The governing
move with same velocity (r = 0.5) and then it increases until nonlinear partial differential equations were transformed into
the plate is stationary and the free stream is moving (r = 1). ordinary differential equations using the similarity approach
In figure 14, it is observed that the Nusselt number increases and solved numerically using the Runge–Kutta–Fehlberg

7
Phys. Scr. 86 (2012) 045003 T G Motsumi and O D Makinde

method coupled with the shooting technique. Two types of particles (nanofluids) Int. J. Heat Mass Transfer
nanofluids were considered, Cu–water and Al2 O3 –water, and 42 855–63
[3] Wang X, Xu X and Choi S U S 1999 Thermal conductivity of
our results revealed, among others, the following.
nanoparticle fluid mixture J. Thermophys. Heat Transfer
• Alumina shows a thicker velocity boundary than 13 474–80
[4] Eastman J A, Choi S U S, Li S, Yu W and Thompson L J 2001
water–Cu nanofluid. The velocity boundary layer Anomalously increased effective thermal conductivity of
thickness decreases with increasing the intensity of ethylene glycol-based nanofluids containing copper
suction at the plate surface. nanoparticles Appl. Phys. Lett. 78 718–20
• Alumina shows a thicker thermal boundary than [5] Buongiorno J 2006 Convective transport in nanofluids ASME
water–Cu nanofluid. The thermal boundary layer J. Heat Transfer 128 240–50
[6] Nield D A and Kuznetsov A V 2009 The Cheng–Minkowycz
thickness increases with increasing the values of r, ϕ, Br, problem for natural convective boundary-layer flow in a
whereas it decreases with increasing the values of N, f w . porous medium saturated by a nanofluid Int. J. Heat Mass
• The skin friction increases with increasing the volume Transfer 52 5792–5
fraction ϕ; the water–Cu nanofluid shows a higher skin [7] Kuznetsov A V and Nield D A 2010 Natural convective
friction than alumina. boundary-layer flow of a nanofluid past a vertical plate Int.
J. Therm. Sci. 49 243–7
• The heat transfer rate at the plate surface decreases [8] Makinde O D and Aziz A 2011 Boundary layer flow of a
with increasing the Brinkmann number (Br), radiation nanofluid past a stretching sheet with a convective boundary
parameter (N) and velocity ratio parameter (r), whereas it condition Int. J. Therm. Sci. 50 1326–32
increases with the suction rate f w and nanoparticle solid [9] Ahmad S, Rohni A M and Pop I 2011 Blasius and Sakiadis
fraction (ϕ). problems in nanofluids Acta Mech. 218 195–204
[10] Tiwari R K and Das M K 2007 Heat transfer augmentation
in a two-sided lid-driven differentially heated square cavity
Acknowledgments utilizing nanofluids Int. J. Heat Mass Transfer
50 2002–18
TGM collaborated with Professor O D Makinde during [11] Mohammadein A A and El-Amin M F 2000 Thermal
dispersion-radiation effects on non-Darcy natural convection
his visit to the African Institute for Mathematical Sciences in a fluid saturated porous medium Transp. Porous Medium
(AIMS) and the Cape Peninsula University in Cape Town 40 153–63
South Africa and acknowledges their hospitality. [12] Cortell R 2008 Effects of viscous dissipation and radiation on
the thermal boundary layer over a nonlinearly stretching
sheet Phys. Lett. A 372 631–6
References [13] Raptis A, Perdikis C and Takhar H S 2004 Effect of thermal
radiation on MHD flow Appl. Math. Comput. 153 645–9
[1] Choi S U S 1995 Enhancing thermal conductivity of fluids [14] Makinde O D 2005 Free-convection flow with thermal
with nanoparticles Developments and Applications of radiation and mass transfer past a moving vertical porous
Non-Newtonian Flows ed D A Siginer and H P Wang plate. Int. Commun. Heat Mass Transfer 32 1411–9
(New York: American Society of Mechanical Engineers) [15] Nachtsheim P R and Swigert P 1965 Satisfaction of the
FED 231/MD 66 pp 99–105 asymptotic boundary conditions in numerical solution of the
[2] Kiblinski P, Phillpot S R, Choi S U S and Eastman J A 2002 system of nonlinear equations of boundary layer type
Mechanism of heat flow is suspensions of nano-sized NASA Technical Report Server NASA TND-3004