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In this research paper, we have examines the unsteady flow of heat and mass transfer over a horizontal stretching sheet in presence of thermal radiation and chemical reaction effects by numerical method. The governing unsteady boundary layer

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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online)
Journal of Computer and Mathematical Sciences,
Vol.8(10), 512-519 October 2017
(An International Research Journal),
www.compmath-journal.org
512
Thermal Radiation and Chemical Reaction Effects on an Unsteady Stretching Sheet
Nabajyoti Dutta and B. R. Sharma
Department of Mathematics, Dibrugarh University, Dibrugarh-786004, Assam, INDIA. email: nabajyotidutta961@gmail.com, bishwaramsharma@yahoo.com. (Received on: October 6, Accepted: October 9, 2017)
ABSTRACT
In this research paper, we have examines the unsteady flow of heat and mass transfer over a horizontal stretching sheet in presence of thermal radiation and chemical reaction effects by numerical method. The governing unsteady boundary layer equations for the momentum, heat and mass transfer were transformed to a set of ordinary differential equations by using similarity transformation. Then these set of ordinary differential equations were
solved by using MATLAB’s built in solver
bvp4c. The velocity, temperature and concentration profiles were drawn for various values of parameters such as the Chemical reaction parameter (
, Thermal radiation parameter (
) and unsteadiness parameter (
) and results are discussed graphically.
Keywords:
Heat transfer and Mass transfer, Thermal radiation, Chemical reaction Unsteadiness parameter.
1. INTRODUCTION
The study of nonlinear MHD boundary layer flow and heat transfer over a stretching surfaces or flat plates has achieved great attention due to its applications in lots of engineering problems such as MHD paper production, power generators, petroleum industries, plasma studies, geothermal energy extractions etc. A large amount of research work has been done in the field of chemical reaction, heat and mass transfer. There has been renewed interest in studying hydro-magnetic flow and heat transfer of continuously stretched surfaces in the presence of a weak transverse magnetic field. This is because hydro-magnetic flow and heat transfer have become more essential industrially and in different branches of science and engineering. Most of the researchers like Vajravelu and Roper
1
, Ali and Magyari
2
, Sajid and Hayat
3
, and Ibrahim and Makinde
4,5
investigated the heat transfer problem in a stretching sheet with a linear, power-law or exponential surface velocity and a uniform or different surface
Nabajyoti Dutta,
et al.,
Comp. & Math. Sci. Vol.8 (10), 512-519 (2017)
513
temperature conditions. These type of problem was extended by Abo
–
Eldahab and Aziz
6
to include space-dependent exponentially decaying with internal heat generation or absorption. Abel
et al
.
7
and Bataller
8
analysed the effects of non-uniform heat source on viscoelastic fluid flow and heat transfer over a stretching sheets. Other researchers including Pantokratoras
9
and Mukhopadhyay and Layek
10
extended the problem to include the effects of variable fluid properties on the flow over a stretching sheet. In most of these investigations, the flow and temperature fields were considered at steady state. Some other researchers such as Dandapat
et al
.,
11,12
; Seini, Y. I.
13
recently investigated the unsteady flow over stretching surface in presence of non-uniform heat source and chemical reaction effects. Also Sharma and Nath
14
have studied the effects of heat source, chemical reaction, thermal diffusion and magnetic field on demixing of a binary fluid mixture flowing over a stretched surface.
In this paper, we extend the work of Shateyi and Motsa
15
. Here, we have studied the two dimensional unsteady nonlinear MHD boundary layer flow of an incompressible viscous electrically conducting binary mixture of fluids flowing over a horizontal porous stretching surface in presence of uniform magnetic field by taking into account the Thermal radiation and Chemical reaction effects. In previous studies finite- difference, Runga-Kutta integration
etc. schemes were used. Here, we are used bvp4c solver of MATLAB’s software which
implements a collocation method for the solution of BVPs. And this numerical method gives better approximations than most of the other numerical methods.
2. MATHEMATICAL FORMULATION
Fig 1: Schematic Diagram of the flow problem
Let us consider an unsteady two dimensional incompressible and viscous flow on a horizontal porous stretching sheet which comes from a narrow slot at the srcin. The fluid flow over the unsteady stretching sheet is composed of a reacting chemical species. The fluid motion arises due to the stretching of the elastic sheet. The continuous sheet aligned with the
axis at
= 0
moves in its own plane with a velocity
, =
−
, (where both
and
Nabajyoti Dutta,
et al.,
Comp. & Math. Sci. Vol.8 (10), 512-519 (2017)
514
are positive constants with dimension reciprocal time and
<
) in the positive
-direction. The fluid is considered to be Newtonian with constant temperature
∞
and concentration
∞
away from the surface. The temperature
,
of the sheet is different from that of the ambient medium and
,
is concentration distribution near the sheet and both vary with time
and that distance
along the sheet. The fluid is assumed to be gray, emitting and absorbing but non-scattering medium. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The radiative heat flux in the
direction is negligible in comparison with that in the
direction. It is assumed that the external electric field is zero and Hall effects are negligible. Here, the induced magnetic field is negligibly small. The fluid velocity and thermal conductivity are assumed to vary linearly with temperature. The system influenced by an external transverse magnetic field of strength
defined as
=
1
− ⁄
. It also assumed that the chemically reactive species undergo first order chemical reaction. Under the above assumptions, the governing equations of continuity, momentum, energy and concentration are given by
Equation of Continuity
+
= 0
(1)
Equation of Momentum
+
+
=
+
∞
+
∗
∞
(2)
Energy equation
+
+
=
(3)
Concentration equation
+
+
=
∞
(4) Where
and
are the velocity components in the
and
axes respectively.
is the kinematic coefficient of viscosity,
is the acceleration due to gravity,
is the thermal expansion coefficient,
∗
is the concentration expansion coefficient,
is the electrical conductivity,
is the uniform magnetic field,
is the thermal conductivity of the fluid,
is the temperature of the fluid mixture,
is the fluid concentration,
∞
is the temperature far away from the sheet,
∞
is the species concentration far away from the sheet,
is the thermal diffusivity of the fluid mixture,
is the density of the fluid mixture,
is the specific heat at constant pressure,
is the molecular diffusion coefficient,
is the chemical reaction coefficient. The radiative heat flux term is simplified by using the Rosseland approximation as
Nabajyoti Dutta,
et al.,
Comp. & Math. Sci. Vol.8 (10), 512-519 (2017)
515
=
∗
∗
(5) where
∗
and
∗
are the Stefan-Boltzmann constant and the Mean absorption coefficient, respectively. We assume that the temperature differences within the flow are sufficiently small so that
can be expressed as a linear function of temperature, we expand
in a Taylor’s
series about
∞
as follows:
=
∞
+4
∞
∞
+6
∞
∞
+ …….
And neglecting higher order terms beyond the first degree in
∞
we get
≅ 4
∞
3
∞
(6) In view of equation (5) and (6), we obtain
=
∗
∞
∗
(7) So from equation (3), we have
+
+
=
+
∗
∞
(8) The initial and boundary conditions are
,0 =
,
,
,0 = 0
,
,0 =
,
,
,0 =
,
,
,
∞
⟶ 0
,
,
∞
⟶
∞
,
,
∞
⟶
∞
(9) We assume that both the surface temperature
,
and surface concentration
,
of the stretching sheet to vary with the distance
along the sheet and time in the following form:
, =
∞
+
[
]1
⁄
(10)
, =
∞
+
[
]1
⁄
(11) Where
is a heating or cooling reference temperature and
is a positive concentration reference. The equation for the temperature increases (reduces) if
is positive (negative) from
at the leading edge in proportion to
and such that the amount of temperature increase (reduction) along the sheet increases with time. Similarly, it is same for the concentration equation. In order to reduce
14
into a set of ordinary differential equations, we introduce the similarity variable
and the dimensionless variables
, and
.
=
⁄
1
⁄
,
=
⁄
1
⁄
,
=
∞
+
[
]1
⁄
,
=
∞
+
[
]1
⁄
(12) Where
,
is the physical stream function which automatically satisfies the continuity equation. The velocity components are then derived from the stream function expression and obtained as
Nabajyoti Dutta,
et al.,
Comp. & Math. Sci. Vol.8 (10), 512-519 (2017)
516
=
=
′
,
=
=
⁄
1
⁄
(13) Using (12) in equations
24
, the governing equations are reduces to
′′′
+
′′
′
′′
+ +
′
+ + = 0
(14)
3 +4
′′
+3[
′
2
′
(3 +
′
)] = 0
(15)
′′
+[
′
2
′
(3
′
)
] = 0
(16) Boundary conditions are given below
0 = 0,
′
0 = 1, 0 = 0 = 1
′
∞
⟶ 1,
∞
⟶ 0,
∞
⟶ 0
(17) Using Parameters are
=
= Unsteadiness parameter,
=
−
= Magnetic parameter,
=
−
= Permeability parameter,
=
−
∞
−
= Thermal Grashof number,
=
∗
−
∞
−
= Mass Grashof number,
=
∗
∞
= Radiation parameter,
=
= Prandtl number,
=
= Schmidt number,
=
−
= Instantaneous reaction rate parameter.
3. NUMERICAL ANALYSIS AND DISCUSSIONS
Solutions of non-linear coupled ordinary differential equations (14) to (16) under boundary condition (17) cannot be obtained in closed form. Hence these equations are solved numer
ically by using MATLAB’s built in solver bvp4c. Graphical representations of these
solutions are shown below for various values of parameters like unsteadiness parameter (
), Chemical reaction parameter (
and Thermal radiation parameter (
).
Figure 2: Effects of Thermal radiation (
) on Figure 3: Effects of Thermal radiation (
) on velocity profiles. temperature profiles.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91
f '(
)
R=1R=5R=100 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91
(
)
R=1R=5R=10

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