NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFER MHD FLOW WITH SORET EFFECT AND VISCOUS DISSIPATION IN THE PRESENCE OF THERMAL RADIATION AND HEAT SOURCE/SINK

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 7, Issue 3, May–June 2016, pp.170–181, Article ID: IJMET_07_03_016
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NUMERICAL INVESTIGATION OF AN
UNSTEADY MIXED CONVECTIVE MASS
AND HEAT TRANSFER MHD FLOW WITH
SORET EFFECT AND VISCOUS
DISSIPATION IN THE PRESENCE OF
THERMAL RADIATION AND HEAT
SOURCE/SINK
Suresh
Dept. of Mathematics, Gulbarga University,
Kalaburagi-585106, Karnataka, India
P.H. Veena
Dept. of Mathematics, Smt. V. G. Womens College,
Kalaburagi-585102, Karnataka, India
V. K. Pravin
Dept. of Mechanical Engineering,
P. D. A. College of Engineering, Kalaburagi-585102,
Karnataka, India
ABSTRACT
In the present paper an analysis of mixed convection of an unsteady
magneto hydrodynamic (MHD) flow of an incompressible viscous fluid
through porous media due to a vertical porous stretching sheet in the presence
of viscous dissipation, thermal radiation and heat source /sink has been
carried out. The fluid considered is viscous and incompressible. The
governing partial differential equations of the flow, mass and heat transfer are
highly non linear hence are converted into a system of ordinary differential
equations using suitable similarity transformations. These ordinary
differential equations are further converted into 7 first order ordinary
differential equations and are solved numerically by Matlab ode-45 solver via
shooting method. The effects of various physical parameters such as magnetic
parameter, porous parameter, mixed convection parameter, Eckert number,
heat source/sink parameter, unsteady parameter, Soret number, the Prandtl

Study of Performance Evaluation of Domestic Refrigerator working with Mixture of Propane,
butane and isobutene Refrigerant (LPG)
number of the flow, mass and heat transfer characteristics are analyzed and
illustrated through graphs.
Cite this Article: Suresh, P.H. Veena and V. K. Pravin, Study of Performance
Evaluation of Domestic Refrigerator working with Mixture of Propane, butane
and isobutene Refrigerant (LPG). International Journal of Mechanical
Engineering and Technology, 7(3), 2016, pp. 161–169.
http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=3
NOMENCLATURE
A, b, c constants
Externally imposed transverse magnetic field strength
C concentration of the species
Free stream concentration
Local skin-friction coefficient
Specific heat at constant pressure
Coefficient of mass diffusivity
Thermal diffusivity
Ec Eckert number
Dimensionless suction velocity
g acceleration due to gravity
k thermal conductivity
k* mass absorption co-efficient
Porous parameter
R thermal radiation parameter
Nusselt number
Pr Prandtl number
radiative heat flux in the y- direction
Local Reynolds number
Sc Schmidt number
Sr Soret number
Sherwood number
T fluid temperature
Stretching sheet temperature
Temperature far away from the stretching sheet
u,v velocity components in the x- and y-directions
t time variable
Heat source/sink parameter
x,y flow directional coordinate and normal to the stretching sheet

Suresh, P.H. Veena and V. K. Pravin
Greek Symbols
Coefficient of thermal expansion
Coefficient of expansion with concentration
Similarity variable
Dynamic viscosity
Kinematic viscosity
Density of the fluid
Stream function
Electrical conductivity of the fluid
Stephan–Boltzmann constant
1. INTRODUCTION
The analysis of viscous incompressible mass and heat transfer of magneto
hydrodynamic mixed convection flow through porous medium has received
considerable attention with numerous industrial applications in hydrodynamics viz
chromatography, crystal magnetic damping control, chemical catalytic reactors,
geophysics, energy related engineering problems including polymer sheets and metal
sheets. It also includes in the aerodynamic extrusion of polymer sheets, nano fluid
power plants, heat exchange between soil and atmosphere, packed sphere beds
migration of moisture through air contained in fibrous insulation, granular insulation
materials of high performance insulation buildings, transpiration cooling, packed bed
chemical reactors and continuous filament extrusion from a dye. These industrial
observations explain, how existence of practical applications of flow, mass and heat
transfer has drawn in many areas in the world.
Das et al.[1] investigated the Numerical solution of mass transfer effects on
unsteady flow past accelerated vertical porous plate with suction and solved
numerically by finite difference scheme. Prasad et al.[2] have studied the Radiation
and mass transfer effects of two-dimensional flow past an impulsively started infinite
vertical plate and solved governing equations by finite difference method. Mohamed
Abd El-Aziz [3] has interpreted the results of Radiation effect on the flow and heat
transfer over an unsteady stretching sheet using fifth order Runge-Kutta Fehlberg
integration scheme to solve differential equations via shooting technique.
Hari Rani and Chang Kim [4] worked on the numerical study of Dufour and Soret
effects on unsteady natural convection flow past an isothermal vertical cylinder by
applying a Crank-Nicolson type of implicit finite difference method with a tri-
diagonal matrix manipulation. Hayat et al [5] investigated an Unsteady flow with heat
and mass transfer of a third grade fluid over a stretching surface in the presence of
chemical reaction, and solved the system of equations by means of homotopy analysis
method (HAM). They discussed the Convergence of derived series solutions
explicitly. Vempati and Gari[6] discussed about the Soret and Dufour effects on
unsteady MHD flow past an infinite vertical porous plate with thermal radiation and
solved governing PDEs by Finite Element Method . Zaman and Ayub [7] investigated
the Series solution of an unsteady free convective flow with mass transfer along an
accelerated vertical porous plate with suction. Dulal Pal and Mondal [8] have studied
the numerical effects of Soret, Dufour, chemical reaction and thermal radiation on
MHD non-Darcy unsteady mixed convective heat and mass transfer flow over a
stretching sheet via shooting algorithm with Runge–Kutta Fehlberg integration

scheme. Chamkha and Ahmed [9] made an investigation on the Unsteady MHD Heat
and Mass Transfer by Mixed Convection Flow in the Forward Stagnation Region of a
Rotating Sphere in the Presence of Chemical Reaction and Heat Source. Singh and
Kumar [10] have interpreted the results of Fluctuating Heat and Mass transfer on
unsteady MHD free convection flow of radiating and reacting fluid past a vertical
porous plate in slip-flow regime. Chamkha et al [11] investigated an Unsteady MHD
natural convection flow from a heated vertical porous plate in a micropolar fluid with
Joule heating, chemical reaction and radiation effects. Hayat et al [12] discussed about
Mass transfer effects on the unsteady flow of UCM fluid over a stretching sheet and
Homotopy analysis method is used for the development of series solution of the
arising nonlinear problem. Husnain et al [13] discussed about the Heat and Mass
transfer analysis in unsteady boundary layer flow through porous media with variable
viscosity and thermal diffusivity solved analytically and numerically by using the
homotopy analysis method and the Runge–Kutta method via shooting technique.
Alamgir et al [14] studied the Effects of Thermophorosis on Unsteady MHD free
convective heat and mass Transfer along an Inclined porous plate with heat generation
in presence of Magnetic Field and by employing Nachtsheim-Swigert shooting
iteration technique along with sixth order Runge-Kutta integration scheme. AL-
ODAT, GHAMDI[15] studied Numerical investigation of Dufour and Soret effects on
unsteady MHD natural convection flow past a vertical plate embedded in non-Darcy
porous medium and used implicit finite difference scheme of the Crank-Nicolson type
with tri diagonal matrix manipulation method to solve governing non linear
dimensionless equations. Mustafa et al.[16] presented his work On heat and mass
transfer in the unsteady squeezing flow between parallel plates and used Homotopy
Analysis Method(HAM) to construct the series solution of the problem. Husnain et
al.[17] discussed heat and mass transfer analysis in unsteady boundary layer flow
through porous media with variable viscosity and thermal diffusivity and solved
analytically by Homotopy analysis Method and Numerically by Runge-Kutta shooting
method. Vajravelu et al.[18] interpreted Unsteady convective boundary layer flow of
a viscous fluid at a vertical surface with variable fluid properties and solved non linear
PDEs by Second order finite difference scheme known as Kellar Box Method.
Turkyilmazoglu and Pop [19] investigated heat and mass transfer of unsteady natural
convection flow of some nanofluids past a vertical infinite flat plate with radiation
effect by exact analytical method. Madhusudhan et al.[20] studied unsteady MHD
free convective heat and mass transfer flow past a semi-infinite vertical permeable
moving plate with heat absorption, radiation, chemical reaction and Soret effects by
analytical methods. Zheng et al.[21] presented the Unsteady heat and mass transfer in
MHD flow over an oscillatory stretching surface with Soret and Dufour effects by
HAM. Shankar and Yirga[22] discussed unsteady heat and mass transfer in MHD
flow of nanofluids over stretching sheet with a non-uniform heat source/sink and
solved boundary layer equations by Kellar box method. Mohamed et al.[23] presented
the results on unsteady MHD double-diffusive convection boundary-layer flow past a
radiate hot vertical surface in porous media of chemical reaction and heat sink. The
governing equations are solved in closed form by Laplace-transform technique in his
study.
Reddy et al.[24] made a note on the thermal radiation and magnetic field effects
on unsteady mixed convection flow and mass transfer over a porous stretching surface
with heat generation and Numerical solution of governing equations are obtained by
Runge-Kutta Forth order scheme along with shooting technique.

Idowu et al. [25] presented the effect of heat and mass Transfer on Unsteady
MHD Oscillatory Flow of Jeffrey fluid in a horizontal channel with chemical reaction
are evaluated using perturbation technique. Mohammed Ibrahim et al.[26] studied
radiation effect on unsteady MHD free convective heat and mass transfer flow of past
a vertical porous plate embedded in a porous medium with viscous dissipation by
employing shooting method with 4th
order RK integration scheme. Haroun et al.[27]
studied On unsteady MHD mixed convection in a nanofluid due to a
stretching/shrinking surface with suction/injection using the spectral relaxation
method. Nayak et al.[28] investigated Soret and Dufour effects on mixed convection
unsteady MHD boundary layer flow over stretching sheet in porous medium with
chemically reactive species and solved governing non linear differential equation by
RK4 method. Hunegnaw and Kishan[29] studied unsteady MHD heat and mass
transfer flow over stretching sheet in porous medium with variable properties
considering viscous dissipation and chemical reaction. Agarwal and Bhadauria[30]
studied the Unsteady heat and mass transfer in a rotating nano fluid Layer. Ferdows et
al.[31] interpreted the results on boundary layer flow and heat transfer of a nanofluid
over a permeable unsteady stretching sheet with viscous dissipation and solved the
flow equations numerically using the Nactsheim–Swigert shooting technique together
with Runge–Kutta six-order iteration scheme. Sengupta[32] made an analysis of
unsteady heat and mass transfer flow of radiative chemically reactive fluid past an
oscillating plate embedded in porous medium in presesnce of Soret effect. Das et
al.[33] presented an unsteady free convection flow past a vertical plate with heat and
mass fluxes in the presence of thermal radiation by analytical method. Ahmad and
Khan [34] investigated Unsteady heat and mass transfer magneto hydrodynamic
(MHD) nanofluid flow over a stretching sheet with heat source–sink using quasi-
linearization technique. Ravindran and Samyuktha [35] have investigated the
unsteady mixed convection flow over stretching sheet in presence of chemical
reaction and heat generation or absorption with non-uniform slot suction or injection
using the quasi linearization technique in combination with an implicit finite
difference scheme.
Thus motivated by the above analyses, the main objectives of present paper is to
study the Numerical investigation of unsteady mass and heat transfer of MHD viscous
flow with Soret and viscous dissipation in presence of thermal radiation and heat
source/sink. The problem addresses here is fundamental one that arises in many
practical situations. The non- linearity and mathematical difficulties associated in
basic equations encourages to use numerical techniques. The governing equations of
the given problem after similarity transformations are solved numerically by ode45 of
MATLAB differential equation of first order solver via shooting method. The effect
of various governing parameters on velocity, temperature and concentration are
analyzed and exhibited through graphs.
2. PROBLEM FORMULATION
Considered a typical laminar mixed convection boundary layer flow with heat and
mass transfer of a viscous incompressible electrically conducting fluid over a vertical
porous medium placed in the plane y = 0 and moving with the velocity
( , )
1
w
cx
U x t
t


with temperature distribution
 
0
2
2 1
w
T cx
T T
t 
 

where c and 
are constants, 0T is the reference temperature. X-axis is taken along the stretching
direction of the sheet and y-axis is normal to the surface of sheet. A magnetic field of

uniform strength 0B is applied in the negative direction of the y-axis. The fluid is
considered to be gray, absorbing and emitting radiation but non scattering medium
and to describe the radiative heat flux, the Rosseland approximation is considered in
Heat equation and is considered negligible along x-axis compared to the other axis.
Under the above considerations the governing boundary layer equations of the
model areas follows
0
u v
x y
 
 
  (1)
22
* 0
2 '
( ) ( )
u u v u
u v g T T g C C u u
t x y y k
 
    
           
     (2)
22
0
2
1
( )r
p p p
QqT T T k T u
u v T T
t x y c y c y c y

     
       
          (3)
2 2
2 2M T
C C C C T
u v D D
t x y y y
    
   
     (4)
The boundary conditions for the flow, heat and mass distribution fields are
, , , 0w w w wu U v V T T C C at y    
0 , ,u T T C C as y     (5)
Where t is the time variable, u and v are the velocity components along x- and y-
axis respectively.
1
w
c
V
t


 

is the velocity of suction parameter( wV > 0), 
is the kinematic viscosity,  is the volumetric coefficient of thermal expansion, g is
the acceleration due to gravity , 0B is the uniform magnetic field ,  is the Electrical
conductivity, pC is the Specific heat at constant pressure,  is the Density, T the
Temperature, T is the temperature for away from the stretching surface, k is the
coefficient of the thermal conductivity of the fluid, 0Q is the Heat generation constant,
rq is the Radiation heat flux,  is the Effective dynamic viscosity, mD is the mass
diffusivity, mD is the Thermal diffusivity, By Rosseland approximation, the radiative
heat flux is given by
4
*
4
3
s
r
T
q
k y
 


where s is the Stefan Boltzman Constant and
*
k is the absorption coefficient. Temperature difference in the flow is assumed to be
sufficiently small so that 4
T can be expressed as linear function of T using truncated
Talyer’s series about the free stream temperature T and neglecting higher order
terms we get 4 3 4
4 3T T T T   then equation (3) becomes
23 2
0
* 2
16
( )s
p p p
T QT T T k T u
u v T T
t x y c c k y c y
 

       
                   
(6)

Introducing now the following non dimensional variables, by choosing the stream
function
( , )x y Such that ,u v
y x
  
  
 
. Thus continuity equation (1) satisfies Identically
with this of ( , )x y where
 ( , )
1
c
x y xf
t

 


 ,
 1
c
y
t

 


,
( )
w
T T
T T
  



 ,
 
0 2
( )
2 1
cx
T T T
t
 
 

 
   
  
( )
w
C C
C C
  



 ,
 
0 2
( )
2 1
cx
C C C
t
 
 

 
   
  
0
p w
Q x
Q
C U

By consequence of the above similarity variables in (2), (4)and (6), following
ordinary differential equations are obtained ,
 
2''' '' ' ' '' ' '
1 2 2
1
0
2
f ff f S f f Mf K f
 
          
  (7)
 
 '' ' ' ' ''Pr
0
1 2
S
f f Q Ecf
R
     
 
          (8)
 '' ' ' ' ''
0
2
S
Sc f f Sr
 
          (9)
Boundary conditions corresponding to the above non-linear differential equations
converts to
'
(0) , (0) 1, (0) 1f fw f   
'
( ) 0, ( ) 0, ( ) 0f        (10)
Where A is the unsteady parameter, 1 and 2 are mixed convection parameters,
M is the Magnetic parameter, 2K is the Permeability parameter, Pr is the Prandtl
number, R is the thermal radiation parameter, Q is the heat source/sink parameter, 
is the thermal diffusivity, Ec is the Eckert number, Sc is the Schimidt number, Sr is
the Soret number, where
1
w w
t
f V
c



 is the dimensionless velocity which
determines the transpiration rate at the surface with wf > 0 for suction and wf < 0 for

injection and wf = 0 represents impermeable sheet, prime denotes the partial
differentiation with respect to similarity variable . Analysis of Flow (velocity) '
( )f 
, Temperature ( )  and Concentration ( )  allows us to determine the important
characteristics of engineering design problems.
III i) SKIN FRICTION
The wall shear stress may be expressed in terms of local skin friction coefficient as
0 ''
2
2
1
(0)
2
y
f p e
w
u
y
C C R f
U



 
  
  
III ii) NUSSELT NUMBER
The Local Nuselt number of the rate of heat transfer coefficient in the stretching
surface may be expressed as
0 '
(0)y u
u
w e
T
x
y N
N
T T R


 
  
   

III iii) SHREWOOD NUMBER
The local Sherwood number signifies the rate of mass transfer which is expressed as
Where w
e
U x
R

 Is the local Reynolds Number
0 '
(0)y
w e
C
x
y Sh
Sh
C C R


 
  
   

4. SOLUTION METHODOLOGY
The solution of the set of ordinary differential equations (7) to (9) corresponding to
laminar boundary conditions (10) are obtained Numerically by MATLAB ode45
solver with shooting technique. The values of '
,f and  are known at end points i.e
   and these end conditions are utilized to generate three unknown initial
conditions at 0  by using shooting method. The most difficulty of this method is to
choose suitable initial finite values. Starting with some initial guess values and solve
the boundary value equations(2),(5) and (6) to obtain ''
(0)f , '
(0) and '
(0) by
MATLAB software. The velocity, temperature and concentration field can easily to
be obtained for a particular set of physical parameters. The results are discussed in the
next section.
5. RESULTS AND DISCUSSION
To discuss the clear insight of the physical problem, we have been carried out
Numerical computation of the model by MATLAB ode45 solver along with shooting
technique for various values of governing parameters A, M, B, Pr, Q, Ec, Sc, K2, Sr

and fw. Many graphs are drawn for velocity field, temperature field and concentration
fields. Fig1(a). depicts the velocity profiles for different values Magnetic parameter M
and it is observed from the figure that as M increases the rate of velocity distribution
decreases slightly near the plate and outside there is no significant effect on velocity
profile. This is because of the fact that magnetic field has tendency to give resistive
type force known as Lorentz force which has a property to slow down the motion of
the fluid. This result agrees qualitating with expectation with the published result but
in case of temperature and concentration profiles it is different as shown in fig1.(b)(c).
Fig.2 (a)(b)(c) represents the graphs of velocity, temperature and concentration
profiles for various values of permeability parameter 2K . By observing the graphs it
is shown that velocity profile decreases with increasing values of 2K where the
temperature and concentration distribution increases with increasing values of 2K .
Physically it reveals the fact that increasing the tightness of porous medium results in
increasing the resistance against flow and thus fluid velocity decreases.
Fig3 (a) (b) (c). Displays the graphs to show the influence of suction parameter fw
on flow, temperature and concentration. One can depicts easily from the figures that
increasing in suction parameter results in decreasing concentration, temperature and
flow profiles.
Fig4 (a)(b)(c)explain the influence of unsteady parameter S on flow, heat and
mass transfer distribution respectively. One can observe from the figures that the
profile for temperature, concentration and flow decreases with increasing values of
Unsteady parameter S. Physically it means that increasing the values of unsteady
parameter S is to reduce the thickness in the boundary layer near the wall and fluid
velocity increases away from the wall and it is same in heat and concentration
profiles.
Figs 5(a)(b)(c) presents the graphs to show the effect of Schimidt number on flow,
temperature and concentration .It is observed from the figures that temperature and
concentration profiles decrease with increasing values of Sc. Here it is noted that
Schimidt number is inversely proportional to the diffusion coefficient.
Fig.6 (a)(b)(c) Shows the effect of Soret number on '
,f and  . For increasing
values of Soret number, the Velocity and Concentration increases and temperature
decreases. The effects of Prandtl number are drawn in the figures7(a)(b)(c)which
represents respectively velocity, temperature and concentration profiles and it can be
seen from the figures that distribution near the boundary layer flow are decreasing by
increasing values of Prandtl number i.e boundary layer thickness decreases or slew
rate of thermal diffusion.
Figs 8(a) (b) displays the influence of viscous dissipation parameter Ec or Eckert
number on velocity and temperature profiles. The fluid velocity and thermal boundary
layer increases with increasing values of Ec. Physically by definition of Eckert
number it is the ratio of Kinetic energy of the flow to the boundary layer Enthalpy
difference. It corporate the conversion of the kinetic energy into thermal energy by
work done against the viscous fluid stresses. Plate is cooled by the influence of
positive Eckert number. There fore for maximum values of Eckert number, thermal
boundary layer and velocity increases.
The distribution of radiation parameter R on flow, heat and concentration are
shown in figs 9(a)(b)(c). For various values of R, the temperature and concentration
profiles causing to increase of increasing values of R while velocity profile decreases.

Figs 10(a)(b)(c) show the influence of Heat source/sink parameter on three
distributions, it is clearly observed from the figures that velocity decreases with
increasing values of Q and temperature and concentration profile due increasing by
increasing values of Q.
In figs 11(a)(b)(c) and 12(a)(b) , for increasing mixed convection parameter t he
variations of fluid flow and thermal boundary layers are decreasing and increasing in
case of concentration profile are shown.
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NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFER MHD FLOW WITH SORET EFFECT AND VISCOUS DISSIPATION IN THE PRESENCE OF THERMAL RADIATION AND HEAT SOURCE/SINK

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NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFER MHD FLOW WITH SORET EFFECT AND VISCOUS DISSIPATION IN THE PRESENCE OF THERMAL RADIATION AND HEAT SOURCE/SINK