Download Megnetohydrodynamic Flow of Casson Fliuds over a Moving

J. Appl. Environ. Biol. Sci., 7(4)192-200, 2017
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Megnetohydrodynamic Flow of Casson Fliuds over a Moving
Boundary Surface
Nadeem Tariq1 , S. Hussain2 , N. Yasmin3
1,3
Center for Advance Studies in Pure and Applied Mathematics, Baha Ud Din Zakariya University Multan
1Presently
2
working at Department of Mathematics, Govt College University Faisal Abad (Layyah Campus)
Punjab Higher Education Department, College Wing, Lahore, Pakistan
[email protected], [email protected], [email protected]
Received: December 23, 2016
Accepted: February 28, 2017
ABSTRACT
The numerical study for the effects of thermal radiation and chemical reaction is consider on magneto hydrodynamic
boundary layer flow of Casson fluids is considered. The fluid flows through a porous medium, near a stagnation
point and over the stretching /shrinking sheet. The physical problem has been fundamental in the form of non-linear
partial differential equations. The governing equations are then converted into ordinary differential form by using
similarity transforms. The problem is then solved numerically to reveal the physical nature of the problem through
effects of the pertinent parameter M is the magnetic parameter, K is the porosity parameter, Q is the heat source
parameter, β is the Casson parameter, c is the shrinking parameter, R is the radiation parameter, Sc is the Schmidt
number. The result have been represented and discussed through plots for concentration, velocity and temperature.
KEYWORDS: radiation, chemical reaction, Casson fluids, porous medium, stagnation point.
_____________________________________________________________________________________________
1. INTRODUCTION
Non-Newtonian fluid flow arises in many branches of chemical and material processing engineering. There are
different types of non-Newton fluids like Viscoelastic fluid, couple stress fluid, micropolar fluid and power-law
fluid etc. In addition with these, there is another non-Newtonian fluid model is known as the Casson fluid model. In
the published literature, it is sometimes claimed that for many materials, the Casson model is better than the general
visco plastic models in fitting the rheological data. So, it becomes the preferred rheological model for blood and
chocolate. The influence of thermal radiation and chemical reaction on micro polar fluid flow in a rotating frame
was discussed by Das [1] and concluded that an increase in the volume fraction of nano particles enhances the
velocity profiles. Hayat et al., [2] discussed the cross diffusion effects on MHD Casson fluid flow. Rashidi et al., [3]
analytically discussed the steady flow over a rotating disk in a porous medium by using homotopy analysis method.
The effects of radiation on unsteady free convection flow of a nanofluid past an infinite plate was discussed by
Sandeep et al., [4]. Further Sandeep and Sugunamma [5] studied the effect of inclined magnetic field on dusty
viscous fluid between two infinite flat plates. Nandy [6] studied the heat transfer characteristics of MHD Casson
fluid flow over a stretching sheet and found that an increase in the value of dimensionless thermal slip parameter
reduces the velocity profiles. Sandeep and Sugunamma [7] discussed the effects of radiation and inclined magnetic
field on natural convection flow over an impulsively moving vertical plate. The influence of chemical reaction on
MHD flow past a stretching sheet with heat generation has been investigated by Mohan krishna et al., [8].
Nandeppanavar [9] studied the flow and heat transfer analysis with two heating conditions considering the nonNewtonian Casson fluid due to linear stretching sheet. The solution they obtained is by a power series method
analystically, further Nandeppanavar [9-11] investigated the heat transfer analysis of Casson fluid due to stretching
sheet with convective heating condition both Numerical and analytical results in terms of Kummer’s function and
RungeKutta flurth order method with shooting technique. Attia and Ahmed[12] studied the transient Coutte flow
analysis of Casson fluid between parallel plates with heat transfer analysis. Bhattacharyya et.al[13] have given an
analytical solution for magnetohydrodynamic boundary layer flow of Casson fluid, they also studied the effect of
wall mass transfer analysis too. Swati[14] studied the effect of thermal radiation on the flow and heat transfer
analysis of Casson fluid over an unsteady stretching sheet with effect of suction and blowing. Shehzad et.al[15]
investigated the mass transfer of magnetohydrodyanic flow of Casson fluid with an chemical reaction. [16]
discussed the influence of non linear thermal radiation on MHD 3D Casson fluid flow with viscous dissipation. A
comparative study has been done by Sandeep et al. [17] to study the heat and mass transfer characteristics in nona
Corresponding Author: Sajjad Hussain Presently at Punjab Higher Education Department, Government Postgraduate College
Layyah, Punjab, Pakistan. +923336167923, Email: [email protected].
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Tariq and Hussain, 2017
Newtonian nanofluid past a permeable stretching surface. Raju et al., [18] discussed the effects of thermal diffusion
and diffusion thermo on the flow over a stretching surface with inclined magnetic field. This paper concludes that an
increase in the Soret number increases the friction factor but reduces the heat transfer rate. Very recently, the
researchers [19-24] investigated the heat and mass transfer characteristics of non-Newtonian and Newtonian flows
by considering various channels.
2. MATHEMATICAL ANALYSIS
The steady incompressible and two dimensional flow of caisson fluid is consider in the presence of uniform
magnetic field of strength B0 .The flow of fluid due to a horizontal sheet which is porous and stretches /shrinks. The
flow in plane y > 0 .the temperature at the surface of sheet Tw the fluid temperature is T. The temperature and
T∞ and U. The fluid flows through a porous medium of permeability
K1 in the presence of a species of concentration C. Where concentration of species in the external flow in C∞ . The
velocity in the external flow are respectively
fluid velocity components are u and v respectively in x and y directions.
Under the above assumptions the equations governing the problems are:
∂u ∂v
+
=0
∂x ∂y
2
2
σ e B0
1 ∂ u 
1 υ
∂u
∂u
dU 
u +v =U
+ 1 +  υ
+ 1 + 
U
−
u
+
(
)
(U − u )
β  ∂y 2 
β  K1
ρ
∂x
∂y
dx 
(1)
(2)
2
2
2
∂T
∂T
∂ T
Q
16α ∂ T σ e B0
µ  ∂u 
2
+v
=α 2 +
+
T
−
T
+
+
( ∞)
(U − u )


2
3βρ C p ∂y
∂y
ρ C p  ∂y 
ρC p
ρC p
∂x
∂y
2
u
u
∂C
∂C
∂ 2u
+v
= D 2 − R(C − C∞ )
∂x
∂y
∂y
Where ⍴ is density,
σ e is the electrical conductivity,
(3)
(4)
K1 is the permeability of the porous medium c is the specific
p
heat capacity at constant pressure, µ is dynamic viscosity, α is thermal diffusivity,
µ is coefficient of viscosity.
The boundary conditions are:
u = uw ( x) = bx, T = Tw , C = C w
at y=0
u = ue ( x) = ax, T = T∞ , C = C∞
as
(5)
y →∞
Using similarity transformations:
The velocity components are described in terms of the stream function Ѱ (x, y:
u=
∂ψ
∂ψ
, v=−
∂y
∂x
ψ ( x, y) = α xU f (η ) ,η = y
U
T = T∞ + (Tw − T∞ )θ (η ) , C = C∞ + (Cw − C∞ )φ (η )
αx ,
(6)
Equation of continuity (1) is identically satisfied.
Substituting the above appropriate relation in equations (2), (3) and (4) we get

 1
1
Pr 1 +  f ′′′ + 1 +  K (1 − f ') + M (1 − f ') + 1 = f ′2 − ff ′′
 β
 β
193
(7)
J. Appl. Environ. Biol. Sci., 7(4)192-200, 2017
(1 + Rn)θ ′′ + f θ ′ + Qθ + Ec Pr f ′′2 + Ec M (1 − f ′)2 = 0
φ ''+ S c ( f θ ′ − γφ ) = 0
f = 0, f ' = c,θ = 1, φ = 1 as η = 0
f ' → 1,θ → 0, φ → 0
as η → ∞
υ
U
where Pr =
is the Prandtl number, E c =
is
α
C p ( T w − T∞ )
parameter ,
M=
σ e B0 2υ Re x
ρU 2
dimensional rate of solutal. Sc =
(9)
(10)
Eckert number, K =
is the magnetic parameter, Re x =
υ
D
(8)
Ux
υ
υ
aK1
is the permeability
is the Reynolds number,
γ=
R
is nona
Schmidt number.
3. RESULTS AND DISCUSSION
The resulting set of governing equations (7) to (10) is nonlinear in higher order. It is hard to find any analytical
solution of this system of equations. Thus a numerical treatment of the situation has been employed to obtain a
reliable solution of the problem. The higher order derivatives have been reduced to their first order form. We take
f ' = u , u ' = v , θ ' = w , φ ' = g Thus the resulting system of first order equations is as follows:
 1
 1
Pr 1 +  v '+ 1 +  K (1 − u ) + M (1 − u ) + 1 = u 2 − fv
 β
 β
(1+ Rn)w '+ fw + Qθ + Ec Pr u 2 + Ec M (1 − u)2 = 0
g '+ S c ( fw − γφ ) = 0
These equations along with the associated boundary conditions have been solved through ND solve command of
computing software Mathematica version 11. We made several computations for sufficient ranges of the pertinent
parameters that effect the physical nature of the problem some representative plots for, concentration function,
temperature function and velocity are presented to indicate the effects of the physical parameters of interest.
The curves for velocity
f ' as drawn in fig.1 demonstrate the effect of magnetic field parameters on the horizontal
f ' . It is obtained that velocity increases with magnetic field strength for stretching in the values
'
of porosity parameter caisson increase in this velocity component nearly f as indicated in fig.2. The effect of
'
velocity ratio parameter c on f is demonstrated in fig.3, here C < 0 stands for shrinking case and C > 0 is for
velocity component
stretching case.
'
Fig 4. Show the effect of Prandtl number Pr on velocity f in case of stretching sheet. It is observed that increase
in Prandtl number causes reduction in the magnitude of the velocity and increase in momentum boundary layer
thickness. But opposite effect of the casson parameter β is noticed through Fig 5., Here the velocity magnitude
increase with in β and the boundary layer thickness decrease.
The effect of velocity ratio parameter C on temperature distribution is persevered in fig 6.It is notices that
temperature function θ (η ) increase with increase in shrinking of the sheet (C <0) and decrease with increase in
stretching of the sheet (C >0). Moreover the temperature distribution is greater in magnitude for shrinking case than
the stretching case. The impact of heat sink source parameter θ (η ) on the thermal distribution is presented in fig 7.
The increase in heat sink Q < 0 parameter causes decrease in the magnetic of
heat source parameter Q < 0 causes increase in the magnetic of
number Pr on temperature function
θ (η ) but the increase in the value of
θ (η ) .Fig 8 demonstrate the effect of Prandtl
θ (η ) . It is obtained that temperature distribution increases with increase in the
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Tariq and Hussain, 2017
value of Pr .similarity, the fig.9 indicate the effect of radiation parameter Rn on θ (η ) . The increase in the magnitude
of θ (η ) as shown in fig.10.
γ on concentration function θ (η ) is shown in fig 11. The increasing value of
γ , increase the function θ (η ) significantly. But opposite effect is observed for the Schmidt number on θ (η ) as
The impact of solutal parameter
plotted in fig 12.
Table 1- Numerical values of
f '' (0) for different values of c , Pr , K , M
c
Pr
K
M
Santosh
result
Presents
result
Casson
Parameters
-0.5
-0.2
0.2
0.5
-0.1
1.0
0.1
0.1
1.0
2.0
5.0
1.0
1.0
0.1
0.1
1.0
0.1
0.1
1.0
3.0
5.0
1.640077
1.474584
1.109737
0.747083
1.397476
0.988166
0.625275
1.742807
1.742807
2.335104
2.805457
1.64059
1.47483
1.10983
0.747124
1.39767
0.991977
0.657446
1.74285
1.74285
2.33503
2.805346
0.814506
1.23925
1.71998
2.00645
1.36908
2.18489
0.116453
1.57134
1,57134
1.93736
2.23716
-0.1
-0.1
Table 2- Numerical values of
θ ' (0)
c
Pr
K
M
Ec
-0.5
-0.2
0.2
0.5
-0.1
1.0
0.1
0.1
0.1
1.0
2.0
1.0
1.0
0.1
0.1
0.1
1.0
0.1
0.1
1.0
0.1
0.1
0.3
0.5
0.7
-0.1
-0.1
for different values of
Santosh
result
0.314390
0.437600
0.583240
0.676620
0.476115
0.401936
0.484310
0.457348
0.326960
0.177807
0.028650
c , Pr , K , M , E c
Presents
Result
0.328191
0.444742
0.586568
0.679181
0.481873
0.411585
0.489061
0.448122
0.329751
0.177628
0.0255061
Casson
Parameters
1.25446
1.15866
1.00725
0.876803
1.12328
1.04534
1.12809
1.0964
1.00525
0.887224
0.769196
1.0
0.8
0.6
0.1, 1, 2, 3
f'
M
0.4
0.2
0.0
0
1
2
3
4
5
Fig.1: The plot for curves of f ′ under the effect of magnetic parameter M
when c=−0.1, Pr=0.5, and K=0.1
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J. Appl. Environ. Biol. Sci., 7(4)192-200, 2017
1.0
0.8
0.6
K
f'
0.1, 1, 2, 3
0.4
0.2
0.0
0
1
2
3
4
5
f ′ under the effect of porosity K
Fig.2: The plot for curves of
when, c=−0.1, Pr=0.5.
1.0
0.8
0.6
0.4
f'
C
0.5. 0.2, 0.2, 0.5
0.2
0.0
0.2
0.4
0
1
2
3
4
5
Fig.3: The plot for curves of f ′ under the effect of parameter c
when Κ=0.1, Μ =0.1, Pr=0.5.
1.0
0.8
0.6
0.5, 2, 3, 4
f'
Pr
0.4
0.2
0.0
0
1
Fig.4: The plot for curves of
2
3
4
5
f ′ under the effect of parameter Pr
when c=-0.1 , Κ=0.1,Μ =0.1.
196
Tariq and Hussain, 2017
1.0
0.8
0.6
f'
1, 2, 3, 4
0.4
0.2
0.0
0
1
2
Fig.5: The plot for curves of
3
4
5
f ′ under the effect of casson parameter β
when c=-0.1, Κ=0.1,Μ =0.1, Pr=0.5.
1.0
0.8
C
0.6
0.5. 0.2, 0.2, 0.5
0.4
0.2
0.0
0
1
2
3
4
5
f ′ under the effect of parameter c
Fig.6: The plot for curves of
when c=-0.1, Κ=0.1,Μ =0.1, Pr=0.5.
1.0
0.8
Q
0.25 , 0, 0.25 , 0.5
0.6
0.4
0.2
0.0
0
1
Fig.7: The plot for curves of
2
3
4
5
f ′ under the effect of heat source parameter Q
when c=-0.1, Κ=0.1,Μ =0.1, Pr=0.5,
197
J. Appl. Environ. Biol. Sci., 7(4)192-200, 2017
1.0
0.8
Pr
0.1, 1, 3, 5
0.6
0.4
0.2
0.0
0
1
2
3
4
5
Fig.8: The plot for curves of θ under the effect of Prandtl number Pr
when c=-0.1,M=0.1, Εc=0.1 and Rn=0.1.
1.0
0.8
Rn
0.1, 0.5, 1, 1.5
0.6
0.4
0.2
0.0
0
1
2
3
4
5
Fig.9: The plot for curves of θ under the effect of Radiation parameter Rn
when c=-0.1,M=0.1, Εc=0.1 and Pr=0.5
1.0
0.8
Ec
0.1, 0.2, 0.3, 0.4
0.6
0.4
0.2
0.0
0
1
2
3
4
5
Fig.10: The plot for curves of θ under the effect of Eckert number Ec
when c=-0.1,M=0.1, Pr=0.5 and Rn=0.1.
198
Tariq and Hussain, 2017
1.2
1.0
0.8
0.6
0.4
0.1, 1, 1.5, 2
0.2
0.0
0
1
2
Fig.11: The plot for curves of
3
4
5
φ under the effect of Solutal number γ
when c=-0.1,M=0.1 Pr=0.1 and Rn=0.1.
1.0
0.8
Sc
0.1, 0.3, 0.5, 0.8
0.6
0.4
0.2
0.0
0
1
2
Fig.12: The plot for curves of
3
4
5
φ under the effect of Schmidt number Sc
when c=-0.1,M=0.1, Pr=0.5 and Rn=0.1.
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