Steady MHD boundary layer flow past a shrinking sheet in
the presence of chemical reaction and suction with prescribed heat and mass
fluxes
J.
Wilfred Samuel Raj1* and S.P. Anjali Devi2
Department of Applied
Mathematics, Bharathiar University, Coimbatore - 46,
Tamil Nadu, India.
*Corresponding Author: wilfred_dphd@yahoo.com;
anjalidevi_s_p@yahoo.co.in
ABSTRACT:
A steady laminar hydromagnetic flow of a viscous, incompressible
electrically conducting fluid with heat and mass transfer over a shrinking
sheet with prescribed heat and mass fluxes in the presence of chemical reaction
is investigated. The sheet is subjected
to suction and the flow is caused due to linear shrinking of the sheet. The governing nonlinear partial differential
equations reduce to nonlinear ordinary differential equations by using
similarity transformation. Exact
solution of the momentum equation is found. The analytical solution for the
boundary layer energy and concentration equation is found using Confluent hypergeometric function.
Analytical expression for skin friction coefficient, wall temperature
and wall concentration are also obtained. In order to get the physical insight
of the problem the numerical values of the solution are obtained by fixing the
values for the physical parameters involved in the problem namely Suction
parameter, Magnetic parameter, Prandtl number, Heat
flux parameter, Schmidt number, Chemical reaction parameter and Mass flux
parameter. The results agree with the previous investigation under some special
cases.
KEY WORDS: Boundary layer flow, Magnetohydrodynamics, Shrinking sheet, heat and mass
transfer, Confluent hypergeometric function.
INTRODUCTION:
The laminar boundary layer behaviour over a moving continuous solid surface is an
important type of flow occurring in several engineering processes. The heat transfer due to a continuously
moving stretching surface through an ambient fluid is one of the thrust areas
of current research. Such investigations
find their application over a broad spectrum of science and engineering
disciplines, especially in the field of chemical engineering. Many chemical engineering processes like
metallurgical process, polymer extrusion process involve cooling of a molten
liquid being stretched into a cooling system.
Sakiadis (1961) investigated
the boundary layer flow on a continuous semi infinite sheet moving steadily
through a quiescent fluid environment. The two dimensional flow over a linearly
stretching sheet was studied by Crane (1970). Chakrabarti
and Gupta (1979) examined the hydromagnetic flow and
heat transfer over a stretching sheet.
Chen and Char (1988) investigated the heat transfer of a continuous
stretching surface with suction and blowing.
Thermal boundary layer on a power law stretched surface with suction or
injection was presented by Ali (1995).
Elbashbeshy (1998) examined heat
transfer over a stretching surface with variable surface heat flux. Anjali Devi and Ganga (2009) have
examined viscous dissipation effects on nonlinear MHD flow in a porous medium
over a stretching surface subjected to suction.
Shrinking sheet is a surface which decreases
in size to a certain area due to an imposed suction or external heat. For this
flow configuration, the sheet is shrunk towards a slot and the flow is
physically different from the stretching out case and so the velocity on the
boundary is towards the origin. It is
also shown that mass suction is required to maintain the steady flow over a
shrinking sheet. Literature survey indicates that the flow induced by a
shrinking sheet recently gains attention of modern researchers for its
interesting characteristics.
Miklavcic and Wang (2006) analysed the existence and uniqueness of solution for
steady viscous flow due to a shrinking sheet.
Hayat et al. (2009) investigated three dimensional
rotating flow induced by shrinking sheet for suction. The flow induced by a shrinking sheet with
constant or power-law velocity distribution was investigated recently. It is very important to control the drag and
the heat flux for better product quality. Very recently, an analytical solution
for thermal boundary layer flow over a shrinking sheet considering prescribed
wall temperature and prescribed wall heat flux cases was examined by Fang and
Zhang (2010). The flow of an
electrically conducting fluid past a porous plate under the effect of magnetic
field has attracted the attention of many investigators in view of its
applications in many engineering problems. Sajid et
al. (2008) have studied MHD rotating flow of a viscous fluid over a shrinking
surface. Muhaimin
et al. (2008) examined the effects of heat and mass transfer on MHD boundary
layer flow past a shrinking sheet subject to suction for the case of constant
surface temperature. Closed-form exact solution of MHD viscous flow over a
shrinking sheet was examined by Fang and Zhang (2009) without considering the
heat transfer. Bhattacharyya (2011) investigated the effects of heat
source/sink on MHD flow and heat transfer over a shrinking sheet with mass
suction.
So far no contribution has been made on steady,
laminar, nonlinear, two dimensional boundary layer MHD flow of a viscous,
incompressible electrically conducting fluid with heat and mass transfer over a
linearly shrinking sheet prescribed with variable heat and mass fluxes in the
presence of magnetic field and chemical reaction. The present investigation
extends the work of Fang and Zhang (2010) by considering Lorentz force and mass
transfer with chemical reaction for the case of prescribed heat and mass
fluxes. From the present results we
infer that the presence of magnetic field alters the flow, heat and mass
transfer characteristics significantly.
2. MATHEMATICAL ANALYSIS:
Consider steady two dimensional boundary
layer flow of a viscous incompressible and electrically conducting fluid past a
shrinking sheet subjected to suction.
The flow is assumed to be in x direction, which is taken along the plate
and the y axis is perpendicular to it.
Uniform strong magnetic field B0 is applied in the
y-direction. The sheet shrinking velocity
is uw = - ax and the wall suction
velocity is vw = -v0 (v0 > 0).
In the present work, the following
assumptions are made
Ř The fluid properties
are assumed to be constant.
Ř The induced magnetic
field is neglected because the magnetic Reynolds number is assumed to be small.
Ř It is further assumed
that there is no applied voltage which implies no electric field exists.
Ř The effect of viscous
and joules dissipation are assumed to be negligible in the energy equation.
Ř A first order
homogeneous chemical reaction takes place in the flow.
Under these assumptions, the usual boundary layer equations that are
based on law of mass, momentum, energy and concentration species for this
investigation can be written as
Fig. 1.
Schematic diagram of the problem
Fig. 2. Effect of suction parameter over the dimensionless
velocity
Fig. 3. Effect of suctin parameter on dimensionless
temperature
Fig. 4. Concentration profiles for
values of S
Fig 5 Skin Friction Coefficient for different values of S
Fig 6. Effect of magnetic parameter over
the dimensionless velocity
Fig 7. Effect of magnetic parameter on
dimensionless temprature
Fig 8. Concentration profiles for various
values of M2
Fig. 9. Influence of prandtl
number on dimensionless temperature
Fig 10.
Dimensionless temperature profiles for various values of m
Fig. 11.
Concentration profiles for various values of Sc
Fig 12.
Concentration profiles for various values of γ
Fig 13.
Concentration profiles for various values of wall concentration parameter
Table 1: Skin
Friction Coefficient F''(0) for different values of S with M2 = 2
|
S |
Present study |
Muhaimin et al. (2008) |
|
2 |
2.414214 |
2.414214 |
|
3 |
3.302775 |
3.302775 |
|
4 |
4.236068 |
4.236068 |
Table 2: Effect of Suction parameter and Magnetic
parameter over skin friction coefficient, wall temperature and wall
concentration
|
Parameters |
F''(0) |
θ (0) |
|
|
S = 2.0 |
2.00000 |
1.49169 |
1.34896 |
|
S = 2.5 |
2.50000 |
0.82048 |
1.19176 |
|
S = 3.0 |
3.00000 |
0.59415 |
1.06976 |
|
S = 3.5 |
3.50000 |
0.47340 |
0.97090 |
|
S = 4.0 |
4.00000 |
0.39747 |
0.88853 |
|
|
|
||
|
M2 = 0.0 |
2.00000 |
0.89508 |
1.22690 |
|
M2= 1.0 |
2.50000 |
0. 82048 |
1.19176 |
|
M2= 3.0 |
3.13746 |
0.76550 |
1.16077 |
|
M2= 5.0 |
3.60850 |
0.73865 |
1.14392 |
|
M2= 7.0 |
4.00000 |
0.72147 |
1.13253 |
Table 3: Effect of Prandtl
number and Heat flux parameter over F''(0) and θ (0)
|
Parameters |
F''(0) |
θ (0) |
|
Pr = 0.71 |
2.50000 |
0. 82048 |
|
Pr = 1.00 |
2.50000 |
0.54877 |
|
Pr = 1.50 |
2.50000 |
0.34102 |
|
Pr = 2.30 |
2.50000 |
0.20805 |
|
Pr = 7.00 |
2.50000 |
0.06107 |
|
|
|
|
|
m = -2 |
2.50000 |
0.51517 |
|
m = -1 |
2.50000 |
0.56338 |
|
m = 0 |
2.50000 |
0.62499 |
|
m = 1 |
2.50000 |
0.70668 |
|
m = 2 |
2.50000 |
0. 82048 |
Table 4: Effect of Schmidt number, Chemical reaction
parameter and Mass flux parameter over skin friction coefficient and wall concentration
|
Parameters |
F''(0) |
|
|
Sc = 0.22 |
2.50000 |
1. 19176 |
|
Sc = 0.50 |
2.50000 |
0.66362 |
|
Sc = 0.62 |
2.50000 |
0.56077 |
|
Sc = 0.78 |
2.50000 |
0.46505 |
|
Sc = 1.30 |
2.50000 |
0.29840 |
|
|
|
|
|
γ = 0 |
2.50000 |
3.11121 |
|
γ = 2 |
2.50000 |
1. 19176 |
|
γ = 4 |
2.50000 |
0.89304 |
|
γ = 6 |
2.50000 |
0.74841 |
|
γ = 7 |
2.50000 |
0.69990 |
|
|
|
|
|
n = -2 |
2.50000 |
0.93793 |
|
n = -1 |
2.50000 |
0.98922 |
|
n = 0 |
2.50000 |
1.04750 |
|
n = 1 |
2.50000 |
1.11433 |
|
n = 2 |
2.50000 |
1.19176 |
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Received on 05.01.2013 Accepted
on 12.02.2013
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