INTRODUCTION:
Heat and mass transfer of the boundary layer flow over
a stretching sheet has been generated an important research area due to its
wide ranging applications in various engineering processes like polymer
processing, extrusion of plastic sheets, metallurgical process, paper
production. Such flows arise either due to unsteady motion of the boundary or
the boundary temperature. Sakiadis[1] was the first person to study the
boundary layer flow over a solid surface by taking velocity as constant. Crane
[2] extended the work of Sakiadis by studying the boundary layer flow over a
stretching sheet which moves in its plane varying linearly with velocity from a
fixed point. Analytical and experimental results for the heat transfer flow in
stretching sheet were examined by Tsou et al.[3] and mass transfer flow at the
stretched sheet were studied by Erickson et al. [4]. Heat and mass transfer in
hydrodynamic fluid flow over a stretching sheet with suction effect discussed
by Gupta and Gupta [5].
Recently Haritha et al. [6] studied the unsteady heat
and mass transfer flow of a Maxwell fluid over a stretching surface with
Newtonian heating. Non-Newtonian fluids gained more importance due to less
power requirement in stretching a sheet in a viscoelastic fluid than in a
Newtonian fluid also the heat transfer for viscoelastic fluid is found to be
less than that of Newtonian fluid. Similarity solutions for the velocity
distributions for Non-Newtonian power law fluid over a stretching sheet were
investigated by Anderson and Dandapat [7]. Sarpakaya [8] was the first to
examine the flow of Non-Newtonian fluids with the effect of magnetic field.
Rajagopal et al. [9] studied special case of viscoelastic fluids known as
second order fluids. Cortell [10] studied the influence of magnetic field and
heat transfer in a second grade fluid over a stretching sheet in the presence
of suction. The study of heat source/sink on heat and mass transfer is very
important in view of several physical problems. Eldahab and Aziz [11] studied
the effect of heat source with suction. Kandasamy et al. [12] investigated the
heat and mass transfer under a chemical reaction with heat source. Abel et al.
[13] examined the heat transfer in MHD viscoelastic fluid over stretching sheet
with variable thermal conductivity in the presence of heat source and
radiation.
In view of the above studies the present paper deals
with the unsteady flow of heat and mass transfer in viscoelastic fluid over a
stretching sheet with the effect of heat source and suction parameter. The
governing equations of the flow are solved numerically and results are
discussed graphically.
FORMULATION OF
THE PROBLEM:
We consider free convective, laminar boundary layer
flow and heat and mass transfer of an unsteady incompressible and electrically
conducting visco-elastic fluid over a stretching sheet. The sheet lies in the
plane 
with the flow being confined to 
The 
is being taken along the stretching sheet and 
is normal to the surface. A uniform transvers
magnetic field of strength 
is applied normal to the surface and the chemical
reaction is taking is taken into account. Using the Rosseland approximation the
radiative heat flux is incorporated in the energy equation. It is assumed that
the induced magnetic field, the external electric field and the electric field
due to the polarization of charges are negligible. The density variation and
the effects of the buoyancy are taken into account in the momentum equation
(Boussinesq’s approximation) and the concentration of species far from the wall
is infinitesimally small and the viscous dissipation term in the energy
equation is neglected (as the fluid velocity is very low). In view of this, the
governing boundary layer equations of momentum, energy and mass under
Boussinesq approximations can be derived as follows
(1)
(3)
(4)
Boundary conditions
Where 
are velocity components, 
are respectively, the temperature and concentration
of chemical species in the fluid, 
is the kinematic viscosity, 
is the non-Newtonian visco-elastic parameter, 
is the permeability coefficient of porous medium, 
is the acceleration due to gravity, 
is the volumetric coefficient of thermal expansion, 
is the volumetric concentration coefficient, 
is the fluid density, 
is the fluid electrical conductivity, 
is the thermal conductivity, 
is the specific heat at constant pressure, 
is the mass diffusivity and 
is the chemical reaction parameter
(7)
Using (7) in equations (1), (2), (3) and (4) we obtain
the following set of ordinary differential equations:
(8)
(9)
(10)
The corresponding boundary conditions are
(11)
(12)
Where subscript ‘ denotes the differentiation with
respect to 
, 
is the
viscoelastic parameters, 
and 
are the free
convection parameters, 
is the chemical
reaction parameter ,
denote prandtl number
and Schmidt number respectively and 
is the
suction/injection parameter.
The important physical quantities of engineering
interest are the local skin friction 
, Nusselt number 
and Sherwood number 
and they are defined as:
RESULTS AND DISCUSSION:
The numerical results have been discussed for
different values of visco-elastic parameter k1, Grashof number Gr,
modified Grashof number Gc, Prandtl number Pr, unsteady parameter A, Schimdt
number Sc, chemical reaction parameter γ and Suction parameter S by using
shooting technique with Runge-Kutta procedure. Figures (1- 17) depicts the
velocity, temperature and concentration profiles for different parameters.
Fig. 1 shows the influence of visco-elastic
parameter on the velocity. It is observed that on increasing visco-elastic
parameter 
the velocity
decreases significantly throughout boundary layer region. Fig. 2 and Fig. 3
indicate that the temperature and concentration increase with increasing values
of visco-elastic parameter. Figs. 4-9 exhibits the effects of Grashof number
Gr and modified Grashof number Gc on velocity, temperature and concentration
respectively. It shows that the velocity increases as Gr and Gc increases
whereas the temperature and concentration of the fluid decreases with the
increase of Gr and Gc. Physically Gr < 0 means heating of the fluid or
cooling of the boundary surface, Gr > 0 means cooling of the fluid or
heating of the boundary surface. Figs. 10-12 depict the effect of Prandtl
number Pr with velocity, temperature and concentration. For increasing values
of Pr results reduction in velocity and temperature due to the fact increasing
values of Pr amounts to lesser thermal conductivity whereas the concentration
increases with the increasing values of Pr. Figs. 13-15 present the profiles of
velocity, temperature and concentration for values of schimdt number Sc. With
the increasing values of Sc the temperature increases but the velocity and
concentration decreases. This is due to the fact that as the molecular
diffusion increases the concentration decreases. Fig. 16 shows that the
increasing values of chemical reaction parameter γ the concentration decreases.
Fig. 17 displays the effect of unsteady parameter A with the temperature. It
shows that the increasing values of A decreases the temperature. Figs. 18-20
exhibit the effect of suction parameter S with velocity, temperature and
concentration. It shows that the increasing values of S the velocity,
temperature and concentration decreases and hence the thickness of boundary
layer decreases.
Fig.1.
The effect of on
velocity profile
Fig.
2 The effect of on
temperature profile
Fig.
3 The effect of on
Concentration profile
Fig.4
The effect of Gr on velocity profile
Fig.5.
The effect of Gr on temperature profile
Fig.6.
The effect of Gr on concentration profile
Fig.7
The effect of Gc on velocity profile
Fig.8
The effect of Gc on temperature profile
Fig.9
The effect of Gc on concentration profile
Fig.
10 The effect of Pr on velocity profile
Fig.11
The effect of Pr on temperature profile
Fig.12
The effect of Pr on concentration profile
Fig.13
The effect of Sc on velocity profile
Fig.14
The effect of Sc on temperature profile
Fig.15
The effect of Sc on concentration profile
Fig.16
The effect of γ on concentration profile
Fig.17
The effect of A on temperature profile
Fig.18
The effect of S on velocity profile
Fig.19
The effect of S on temperature profile
Fig.20
The effect of S on concentration profile
CONCLUSIONS:
The unsteady boundary layer flow of an incompressible,
visco-elastic fluid over a stretching sheet in the presence of thermal
radiation, buoyancy effects with chemical reaction and suction/injection is
analysed. The following conclusions are drawn. It is observed that the
velocity decreases with suction, magnetic field parameter, Prandtl number and
Schmidt number while an opposite trend is noted with thermal and solutal
buoyancy parameters, thermal radiation and heat source parameter. The
thickness of the thermal boundary layer increases with increasing values of
magnetic field parameter, radiation parameter, visco-elastic parameter while a
reduction is noticed with thermal and solutal buoyancy parameters and suction
parameter.
REFERENCES:
1.
B.C. Sakiadis, Boundary
layer behavior on continuous solid surfaces, Am. Inst. Chem. Eng. J.
7(1961)26-28.
2.
L.J. Crane, Flow past a
stretching plate, Z. Angew. Math. Phys. 21(1970) 645-647.
3.
F.K. Tsou, E.M. Sparrow,
R.J. Goldstein, Flow and heat transfer in the boundary layer on a continuous
moving surface, Int. J. Heat Mass Transfer 10 (1967) 219-223.
4.
L.E. Erickson, L.T. Fan,
V.G. Fox, Heat and mass transfer on a moving continuous moving surface, Ind.
Eng. Chem. Fund. 5 (1966) 19-25.
5.
P.S. Gupta, A.S. Gupta,
Heat and mass transfer on a stretching sheet with suction or blowing, Can. J.
Chem. Eng. 55 (1977) 744-746.
6.
A. Haritha, Y. devasena,
B. Vishali, Mhd heat and mass transfer of the unsteady flow of a Maxwell fluid
over a stretching surface with Navier slip and convective boundary conditions,
Global J. of pure and Applied mathematics, 6(2017), 2169-2179.
7.
Anderson HI, Dandapat
BS (1991) stab. Appl. Anal.Contin. Media (Italy). 1: 339
8.
T. Sarpakaya, Flow of
non-Newtonian fluids in a magnetic field, AICHE J. 7(1961) 324 – 328.
9.
K.R. Rajagopal, T.Y. Na,
A.S. Gupta, Flow of a viscoelasticfluid over a stretching sheet, Rheol. Acta 23
(1984) 213 – 215.
10.
R. Cortell, Similarity
solutions for flow and heat transfer of a viscoelastic fluid over a stretching
sheet, Int. J. Non-Linear Mech. 29 (1994) 155 – 161.
11.
E.M. Abo-Eldahab, M.A.
El Aziz, Blowing/Suction effect on hydro magnetic heat transfer by mixed
convection from an inclined stretching surface with internal heat generation ,
Int. J. Therm. Sci. 43 (2004) 709 – 719.
12.
R. Kandasamy, S.P.
Anjalidevi, Effect of chemical reaction, heat and mass transfer on laminar flow
along semi infinite horizontal plate, Heat and mass transfer, Vol. 35, 6(1999),
465 – 467.
13.
S. Abel, P. H. Veena, K.
Rajagopal and V. K. Pravin, Non – Newtonian magneto hydrodynamic flow over a
stretching surface with heat and mass transfer, Int. J. Non-Linear Mechanics,
Vol. 39, 2004, 1067 – 1078.