Applied Mathematics, 2011, 2, 475-481
doi:10.4236/am.2011.24061 Published Online April 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Boundary Layer Flow and Heat Transfer of a Dusty
Fluid over a Stretching Vertical Surface
Bijjanal Jayanna Gireesha, Gosikere Kenchappa Ramesh, Hatti Jayappa Lokesh,
Channabasappa Shanthappa Bagewadi
Department of St u die s an d Re searc h in Mat h emat i cs , Kuvempu University, Shimoga, India
E-mail: bjgireesu@rediffmail.com
Received February 20, 2011; revised March 7, 2011; accepted March 9, 2011
Abstract
This paper presents the study of convective heat transfer characteristics of an incompressible dusty fluid past
a vertical stretching sheet. The governing partial differential equations are reduced to nonlinear ordinary dif-
ferential equations by using similarity transformation. The transformed equations are solved numerically by
applying Runge Kutta Fehlberg fourth-fifth order method (RKF45 Method). Here obtained non-dimensional
velocity and temperature profiles has been carried out to study the effect of different physical parameters
such as fluid-particle interaction parameter, Grashof number, Prandtl number, Eckert number. Comparison of
the obtained numerical results is made with previously published results.
Keywords: Boundary Layer Flow, Dusty Fluid, Stretching Sheet, Grashof Number, Fluid-Particle Interaction
Parameter, Numerical Solution
1. Introduction
The behavior of boundary layer flow due to a moving
flat surface immersed in a quiescent fluid was first stu-
died by Sakiadis [1], who investigated it theoretically by
both exact and approximate methods. Crane [2] presented
a closed form exponential solution for the planar viscous
flow of linear stretching case. Later this problem has been
extended to various aspects by considering non-Newton-
ian fluids, more general stretching velocity, magnetohy-
drodynamic (MHD) effect, porous sheet, porous media
and heat or mass transfer. Andreson et al. [3] extended
the work of crane [2] to non-Newtonian power law fluid
over a linear stretching sheet. Chakrabarti and Gupta [4]
have discussed the hydromagnetic flow and heat transfer
over a stretching sheet. Grubka and Bobba [5] analyzed
heat transfer studies by considering the power law varia-
tion of surface temperature. Cortell [6] studied the mag-
netohydrodynamics flow of a power-law fluid over a
stretching sheet. Chen [7] analyzed mixed convection of
a power law fluid past a stretching surface in the pres-
ence of thermal radiation and magnetic field.
Power law model has some limitations as, it does not
exhibit any elastic properties such as normal stress diffe-
rences in shear flow. In certain polymer processing appli-
cations, flow of a viscoelastic fluid over a stretching sheet
is important. For that reason Cortell [8] studied the effe-
cts of viscous dissipation and work done by deformation
on the MHD flow and heat transfer of a viscoelastic fluid
over a stretching sheet. Abel et al. ([9,10]) extended the
work and studied the viscoelastic MHD flow and heat
transfer over a stretching sheet with viscous and ohmic
dissipation, non-uniform heat source and radiation. Tsai
et al. [11] studied an unsteady flow over a stretching sur-
face with non-uniform heat source. Ishak et al. [12] ob-
tained the solution to unsteady laminar boundary layer
over a continuously stretching permeable surface.
These investigations deal with the flow and heat trans-
fer induced by a horizontal stretching sheet, but there ari-
se some situations where the stretching sheet moves ver-
tically in the cooling liquid. Under such circumstances the
fluid flow and the heat transfer characteristics are deter-
mined by two mechanisms, namely the motion of stret-
ching sheet and the buoyant force. The thermal buoyancy
resulting from heating/cooling of a vertically moving str-
etching sheet has a large impact on the flow and heat
transfer characteristics. Effects of thermal buoyancy on
the flow and heat transfer under various physical situa-
tions have been reported by many investigators (see [13-
16]).
To study the two-phase flows, in which solid spherical
B. J. GIREESHA ET AL.
Copyright © 2011 SciRes. AM
476
particles are distributed in a fluid are of interest in a wide
range of technical problems, such as flow through packed
beds, sedimentation, environmental pollution, centrifugal
separation of particles, and blood rheology. The study of
the boundary layer of fluid-particle suspension flow is
important in determining the particle accumulation and
impingement of the particle on the surface. In view of
these applications, Chakrabarti [17] analyzed the boun-
dary layer in a dusty gas. Datta and Mishra [18] have in-
vestigated boundary layer flow of a dusty fluid over a
semi-infinite flat plate. Further, researches in these fields
have been studied by many investigators such as Evgeny
and Sergei [19], XIE Ming-liang et al. [20], Palani et al.
[21], Agranat [22] and Vajravelu et al. [23]. Abdul Aziz
[24] obtained the numerical solution for laminar thermal
boundary over a flat plate with a convective surface
boundary condition using the symbolic algebra software
Maple.
Motivated by the above investigations, we present in
this paper the boundary layer flow and heat transfer of a
dusty fluid over a stretching vertical surface. The govern-
ing partial differential equations are transformed into or-
dinary differential equations using similarity transforma-
tion, before they are solved numerically by Runge Kutta
Fehlberg fourth-fifth order method. Comparison of the
results are to be in excellent agreement those reported by
Grubka and Abel. The RKF45 algorithm in Maple has
been well tested for its accuracy and robustness. The
analysis of obtained results showed that the fluid particle
interaction parameter, Grashof number, Prandtl number
and Eckert number have significant influence on the flow
and heat transfer.
2. Flow Analysis of the Problem
Consider a steady two dimensional laminar boundary
layer flow of an incompressible viscous dusty fluid over
a vertical stretching sheet. The flow is generated by the
action of two equal and opposite forces along the x-axis
and y-axis being normal to the flow. The sheet being
stretched with the velocity

w
Ux along the x-axis,
keeping the origin fixed in the fluid of ambient tempera-
ture T. Both the fluid and the dust particle clouds are
suppose to be static at the beginning. The dust particles
are assumed to be spherical in shape and uniform in size
and number density of the dust particle is taken as a con-
stant throughout the flow.
The momentum equations of the two dimensional
boundary layer flow in usual notation are [23]:
0
uv
xy


 (2.1)


2
*
2p
uv u
xyy
KN
uvv uugTT


 
(2.2)

p
pp
pp
uu
K
xy
uv muu


(2.3)

p
pp
pp
vv
K
xy
uv mvv


(2.4)
 
0
pp
Nu Nv
xy

 (2.5)
where
,uv and
,
p
p
uv are the velocity compo-
nents of the fluid and dust particle phases along x and y
directions respectively. u,
and N are the co-effi-
cient of viscosity of the fluid, density of the fluid, num-
ber density of the particle phase, K is the stokes’ resis-
tance (drag co-efficient), T and T are the fluid tem-
perature within the boundary layer and in the free stream
respectively. g is the acceleration due to gravity, *
is
the volumetric coefficient of thermal expansion, m is the
mass of the dust particle respectively. In deriving these
equations, the drag force is considered for the iteration
between the fluid and particle phases.
The boundary conditions are
, 0 at 0,
0, 0, , as .
w
pp
Uxv y
uu
u
vvN y



(2.6)
where
w
Ux cx
is a stretching sheet velocity, 0c
is stretching rate,
is the density ratio. To convert the
governing equations into a set of similarity equations, we
introduce the following transformation as mentioned
below,
 
  
, ,
, ,
pp r
c
ucxfvvc fy
v
u cxFvvcGH


 
 
(2.7)
which identically satisfies (2.1), and substituting (2.7)
into (2.2) to (2.5), we obtain the following non-linear
ordinary differential equations:
 
 
2
*0
fff fGr
lH Ff
 
 
 





(2.8)
 
20GFFF f
 




(2.9)

0GGf G




(2.10)

0HF HGGH


 (2.11)
where a prime denotes differentiation with respect to
and *
lmN
, mk
is the relaxation time of the
particle phase, 1c
is the fluid particle interaction
parameter,
*2
w
GrgTTc x
 is the Grashof num-
ber and rN

is the relative density.
The boundary conditions defined as in (2.6) will take
B. J. GIREESHA ET AL.
Copyright © 2011 SciRes. AM
477
the form,
 
0, 0 at 0ff


(2.12)
 
0, 0, ,fFGf



as H


If 0
and 0Gr the analytical solution of (2.8)
with boundary condition (2.12) can be written in the
form:

1
f
e

3. Heat Transfer Analysis
The dusty boundary layer heat transport equations in the
presence of non-uniform internal heat generation/ab-
sorption for two dimensional flows are given by [25]:
 
*2
2
2
PP
pT pv
TTkTN N
xy y
uvTT uu
cc
 
 

(3.1)

pp
pp T
p
mP
uTT
xTT
yc
vc




(3.2)
where T and
P
T is the temperature of the fluid and
temperature of the dust particle,
P
c and m
c are the
specific heat of fluid and dust particles, T
is the ther-
mal equilibrium time and is time required by the dust
cloud to adjust its temperature to the fluid, v
is the
relaxation time of the of dust particle i.e., the time re-
quired by the a dust particle to adjust its velocity relative
to the fluid, *
k is the thermal conductivity.
In order to solve the (3.1) and (3.2), we consider non
dimensional temperature boundary condition as follows
2
at 0
, as
w
p
x
TT TAy
l
TTTT y


 


 
(3.3)
where A is a positive constant, v
lc
is a characteris-
tic length.
Now we define the non-dimensional fluid phase tem-
perature

and

p
dust phase temperature as
 

, p
p
ww
TT
TT
TT TT
 



(3.4)
where

2
x
TTA
l





Using (3.4) into (3.1) to (3.2), we obtain the following
non-linear ordinary differential equations

  
2
Pr 2
p
Tv
ff
NPrNPrEc Ff
c
 
 
 








(3.5)

 
2
0
pp
pp
mT
FG
c
cc

 



(3.6)
where *
p
c
Pr k
is the Prandtl number,
2
p
cl
Ec
A
c
is
the Eckert number.
The corresponding boundary conditions for
and
p
as
 
1 as 0
0, 0 as
p
 
 


(3.7)
4. Results and Discussion
The system of coupled ordinary differential Equations
(2.8) to (2.12) and (3.5) to (3.7) has been solved numeri-
cally using Runge-Kutta-Fehlberg fourth-fifth order me-
thod. To solve these equations we adopted symbolic al-
gebra software Maple which was given by Aziz [24].
Maple uses the well known Runge-Kutta-Fehlberg fourty-
fifth order (RFK45) method to generate the numerical
solution of a boundary value problem. The boundary
conditions
were replaced by those at 5
in
accordance with standard practice in the boundary layer
analysis. Numerical computation of these solutions have
been carried out to study the effect of various physical
parameters such as fluid particle interaction parameter
, Grashof number Gr , Prandtl number Pr and
Eckert number Ec are shown graphically.
In order to verify the accuracy of our present study,
the values for wall temperature
0
gradient for vari-
ous values of Prandtl number are given in Table 1,
which shows the excellent agreement with those reported
by Grubka and Bobba [5] and Abel and Mahesha [10].
Further, the Table 2 shows the results of thermal charac-
teristics at the wall for different values of influenced
physical parameters.
Figure 1 shows the effect of Grashof number
Gr
on the velocity profile. From this plot it is observed that
the effect of increasing values of Grashof number is to
increases the velocity distribution in the flow region.
Physically 0Gr means heating of the fluid or cooling
of the boundary surface, 0Gr means cooling of the
fluid or heating of the boundary surface and 0Gr
corresponds to the absence of free convection current.
From the Figure 2 shows the effect of fluid particle
B. J. GIREESHA ET AL.
Copyright © 2011 SciRes. AM
478
Figure 1. Effect of Gr on the velocity profiles for
0.5N, 2.0Ec , 1.0Pr and 0.2
.
Figure 2. Effect of
on the velocity profiles for
Gr
0.5N, 2.0Ec, 1.0Pr and 0.2
.
interaction parameter

on velocity components of
the fluid velocity

f
and particle velocity
F
i.e., if
increases we can find the decrease in the fluid
phase velocity and increase in the dust phase velocity.
Also it reveals that for the large values of
i.e., the
relaxation time of the dust particle decreases then the
velocities of both fluid and dust particles will be the
same.
Figure 3 which illustrate the effect of Prandtl number

Pr on the temperature profiles. We infer from this
figure that the temperature decreases with an increase in
the Prandtl number, which implies viscous boundary
layer is thicker than the thermal boundary layer. From
these plots it is evident that large values of Prandtl num-
ber result in thinning of the thermal boundary layer. In
this case temperature asymptotically approaches to zero
in free stream region. This is in contrast to the effects of
other parameters, except Gr and
on heat transfer.
Figures 4 is plotted for the temperature profiles for
different values of
Ec . We observe that the effect of
increasing values of Eckert number is to enhance the
temperature at a point which is true for both the fluid
phase as well as dust phase temperatures. Physically it
means that the heat energy is stored in the fluid due to
the frictional heating.
Figure 5 depict the effect of Grashof number
Gr
versus
. It is evident from these plots that increasing
value of Gr results in thinning of the thermal boundary
layer associated with an increase in the wall temperature
gradient and hence produces an increase in the heat
transfer rate.
Figure 6, which is a graphical representation of the
temperature profiles for different values of
versus
Figure 3. Effect of
r on the temperature profiles for
0.5
N, 2.0Ec
, 0.5Gr and 0.2
.
Figure 4. Effect of
E
c on the temperature profiles for
0.5
N, 0.5
Gr , 1.0Pr and 0.2
.
B. J. GIREESHA ET AL.
Copyright © 2011 SciRes. AM
479
Figure 5. Effect of Gr on the temperature profiles for
0.5
N, 2.0Ec, 1.0Pr and 0.2
.
Figure 6. Effect of
on the temperature profiles for
0.5
Gr N, 2.0Ec , 1.0Pr and 0.2
.
Table 1. Comparison of the results for the dimensionless
temperature gradient
in the case of 0
, 0
N
and 0
Gr .
Pr Grubka and
Bobba [5]
Abel and Mahesha
[10]
Present Study
0
0.72 1.0885 1.0885 1.0886
1.0 1.3333 1.3333 1.3333
10.0 4.7969 4.7968 4.7968
. We infer from these figures that temperature of the
fluid and dust particle decreases with increase in
res-
pectively. From all the graphs we can observed that fluid
phase temperature is higher than the dust phase tempera-
ture and also it indicates that the fluid particles tempera-
Table 2. Values of wall temperature gradient
for
different values of the parameters
, Gr ,
r and
E
c.
Ec
P
r Gr
0
0.2 2.0 1.0 0.5 0.07441
0.5 1.11703
1.0 1.39241
0.5 0.0 1.0 0.5 1.70879
1.0 1.41767
2.0 1.11703
0.5 2.0 1.0 0.5 1.11703
2.0 1.56421
3.0 1.89316
0.5 2.0 1.0 0.0 1.12770
0.5 1.11703
1.0 1.09157
ture is parallel to the dust particles temperature. We have
used throughout our thermal analysis the values of
0.5
Tv
and 0.2, 0.5,1
pm
cc c
.
5. Conclusions
The two-dimensional boundary layer flow and heat
transfer of a dusty fluid due to a vertical stretching sheet
has been investigated. The governing partial differential
equation is converted into ordinary differential equations
by using similarity transformations. The effect of several
parameters controlling the velocity and temperature pro-
files are shown graphically and discussed briefly. The in-
fluence of the parameter , , Gr Ec
and Pr on di-
mensionless velocity and temperature profiles were exa-
mined.
Some of the important findings of our analysis ob-
tained by the graphical representation are listed below.
The effect of Gr is to increase the momentum
boundary layer thickness and to decrease the ther-
mal boundary layer thickness.
Ec has significant effect on the boundary layer
growth.
The boundary layers are highly influenced by the
Prandtl number. The effect of Pr is to decreases
the thermal boundary layer thickness.
The rate of heat transfer

0
decreases with in-
creasing the Pr and
. While it increases with
increasing the Ec.
If 0Gr
we can find the results of the horizontal
stretching sheet.
B. J. GIREESHA ET AL.
Copyright © 2011 SciRes. AM
480
6. References
[1] B. C. Sakiadis, “Boundary Layer Behaviour on Conti-
nuous Solid Surface,” AIChE Journal, Vol. 7, No. 1,
1961, pp. 26-28. doi:10.1002/aic.690070108
[2] L. J. Crane, “Flow Past a Stretching Sheet,” Zeitschrift
für Angewandte Mathematik und Physik (ZAMP), Vol. 21,
No. 4, 1970, pp. 645-647. doi:10.1007/BF01587695
[3] H. I. Anderson, K. H. Bech and B. S. Dandapat, “Mag-
netohydrodynamic Flow of a Power-Law Fluid over a
Stretching Sheet,” International Journal of Non-Linear
Mechanics, Vol. 27, No. 6, 1992, pp. 929-936.
doi:10.1016/0020-7462(92)90045-9
[4] A. Chakrabarti and A. S. Gupta, “Hydromagnetic Flow
and Heat Transfer over a Stretching Sheet,” Quarterly of
Applied Mathematics, Vol. 37, No. 1, 1979, pp. 73-78.
[5] L. J. Grubka and K. M. Bobba, “Heat Transfer Characte-
ristics of a Continuous Stretching Surface with Variable
Temperature,” Journal of Heat Transfer, Vol. 107, No. 1,
1985, pp. 248-250. doi:10.1115/1.3247387
[6] R. Cortell, “A Note on Magnetohydrodynamic Flow of a
Power-Law Fluid over a Stretching Sheet,” Applied Ma-
thematics and Computation, Vol. 168, No. 1, 2005, pp.
557-56. doi:10.1016/j.amc.2004.09.046
[7] C. H. Chen, “MHD Mixed Convection of a Power Law
Fluid Past a Strtching Surface in the Presence of Thermal
Radiation and Internal Heat Generation/Absorption,” In-
ternational Journal of Nonlinear Mechanics, Vol. 44, No.
6, 2009, pp. 596-603.
[8] R. Cortell, “Effects of Viscous Dissipation and Work
Done by Deformation on the MHD Flow and Heat Trans-
fer of a Viscoelastic Fluid over a Stretching Sheet,” Phy-
sics Letters A, Vol. 357, No. 4-5, 2006, pp. 298-305.
doi:10.1016/j.physleta.2006.04.051
[9] M. S. Abel, E. Sanjayanand and M. M. Nandeppanavar,
“Viscoelastic MHD Flow and Heat Transfer over a Stre-
tching Sheet with Viscous and Ohmic Dissipation,” Com-
munications in Nonlinear Science and Numerical Simula-
tion, Vol. 13, No. 9, 2008, pp. 1808-1821.
[10] M. S. Abel and N. Mahesha, “Heat Transfer in MHD
Viscoelastic Fluid Flow over a Stretching Sheet with Va-
riable Thermal Conductivity, Non-Uniform Heat Source
and Radiation,” Applied Mathematical Modelling, Vol.
32, No. 10, 2008, pp. 1965-1983.
doi:10.1016/j.apm.2007.06.038
[11] R. Tsai, K. H. Huang and J. S. Haung, “Flow and Heat
Transfer over an Unsteady Stretching Surface with Non-
uniform Heat Source,” International Communications in
Heat and Mass Transfer, Vol. 35, No. 10, 2008, pp.
1340-1343. doi:10.1016/j.icheatmasstransfer.2008.07.001
[12] A. Ishak, R. Nazar and I. Pop, “Heat Transfer over an
Unsteady Stretching Permeable Surface with Prescribed
Wall Temperature,” Non-Linear Analysis, Real World
Applications, Vol. 10, No. 5, 2009, pp. 2909-2913.
doi:10.1016/j.nonrwa.2008.09.010
[13] A. Ishak, R. Nazar and I. Pop, “Boundary Layer Flow and
Heat Transfer over an Unsteady Stretching Vertical Sur-
face,” Meccanica, Vol. 44, No. 4, 2009, pp. 369-375.
doi:10.1007/s11012-008-9176-9
[14] F. Aman, A. Ishak and R. Nazar, “Boundary Layer Flow
and Heat Transfer Adjacent to a Stretching Vertical Sheet
with Prescribed Surface Heat Flux,” Matematika, Vol. 26,
No. 2, 2010, pp. 197-206.
[15] S. M. Alharbi, M. A. A. Bazid and M. S. E. Gendy, “Heat
and Mass Transfer in MHD Visco-Elastic Fluid Flow
through a Porous Medium over a Stretching Sheet with
Chemical Reaction,” Applied Mathematics, Vol. 1, No. 6,
2010, pp. 446-455. doi:10.4236/am.2010.16059
[16] I. Olajuwon, “Heat and Mass Transfer in MHD Vis-
co-Elastic Fluid Flow through a Porous Medium over a
Stretching Sheet with Chemical Reaction,” International
Journal of Nonlinear Science, Vol. 7, No. 1, 2009, pp.
50-56.
[17] K. M. Chakrabarti, “Note on Boundary Layer in a Dusty
Gas,” AIAA Journal, Vol. 12, No. 8, 1974, pp. 1136-1137.
doi:10.2514/3.49427
[18] N. Datta and S. K. Mishra, “Boundary Layer Flow of a
Dusty Fluid over a Semi-Infinite Flat Plate,” Acta Me-
chanica, Vol. 42, No. 1-2, 1982, pp. 71-83.
doi:10.1007/BF01176514
[19] E. S. Asmolov and S. V. Manuilovich, “Stability of a
Dusty Gas Laminar Boundary Layer on a Flat Plate,”
Journal of Fluid Mechanics, Vol. 365, No. 1, 1998, pp.
137-170. doi:10.1017/S0022112098001256
[20] M.-L. Xie, J.-Z. Lin and F.-T. Xing, “On the Hydrody-
namic Stability of a Particleladen Flow in Growing Flat
Plate Boundary Layer,” Journal of Zhejiang University
SCIENCE A (Springer), Vol. 8, No. 2, 2007, pp. 275-284.
doi:10.1631/jzus.2007.A0275
[21] G. Palani and P. Ganesan, “Heat Transfer Effects on
Dusty Gas Flow past a Semi-Infinite Inclined Plate,”
Forsch Ingenieurwes (Springer), Vol. 71, No. 3-4, 2007,
pp. 223-230. doi:10.1007/s10010-007-0061-9
[22] V. M. Agranat, “Effect of Pressure Gradient on Friction
and Heat Transfer in a Dusty Boundary Layer,” Fluid
Dynamics, Vol. 23, No. 5, 1988, pp. 729-732.
doi:10.1007/BF02614150
[23] K. Vajravelu and J. Nayfeh, “Hydromagnetic Flow of a
Dusty Fluid over a Stretching Sheet,” International Jour-
nal of Non-Linear Mechanics, Vol. 27, No. 6, 1992, pp.
937-945. doi:10.1016/0020-7462(92)90046-A
[24] A. Aziz, “A Similarity Solution for Laminar Thermal
Boundary Layer over a Flat Plate with a Convective Sur-
face Boundary Condition,” Communications in Nonlinear
Science and Numerical, Vol. 14, No. 4, 2009, pp. 1064-
1068. doi:10.1016/j.cnsns.2008.05.003
[25] H. Schlichting, et al., “Boundary Layer Theory,”
McGraw-Hill, New York, 1968.
B. J. GIREESHA ET AL.
Copyright © 2011 SciRes. AM
481
Nomenclature
A
constant
c stretching rate
m
c specific heat of dust phase
p
c specific heat of fluid
Ec Eckert number
Gr Grashof number
K stokes resistance
*
k thermal conductivity
l characteristic length
N number density of the dust phase
Pr Prandtl number
T temperature of the fluid
p
T temperature of the dust phase
w
T temperature at the wall
T temperature at large distance from the wall

w
Ux stretching sheet velocity
,uv
velocity components of the fluid along
x
and
y directions
,
p
p
uv velocity components of the dust particle along
x
and y directions
x
coordinate along the stretching sheet
y distance normal to the stretching sheet
Greek symbols
coefficient of the viscosity of the fluid
density of the fluid
density ratio
relaxation time of the dust phase
fluid particle interaction parameter
r
relative density
T
thermal equilibrium time
v
relaxation time of the dust phase
fluid phase temperature
p
dust phase temperature
dimensionless space variable
Subscripts
w properties at the plate
free stream condition