Unsteady Magneto Hydrodynamic F low of Casson F luid ... · [email protected] Abstr act The...
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Unsteady Magneto Hydrodynamic Flow of Casson Fluid
through Parallel Plate Channel with Heat Source
C.K.Kirubhashankar
Sathyabama Institute of Science and Technology, Chennai, India
Abstract
The intention of this paper is to examine the unsteady magneto hydrodynamic flow of Casson fluid in
presence of heat transfer through parallel plate. The examination uncovers numerous essential parts of flow
and heat transfer. By employing appropriate transformations, the governing partial differential conditions
relating to the momentum and energy equations are changed into ordinary differential conditions. The flow
highlights and warmth exchange attributes for various estimations of the representing parameters viz. Casson
parameter, heat source parameter, Hartmann number, and Prandtl number are investigated and talked about in
detail. It was discovered that heat source and magnetic field transforms the flow pattern and increment the
temperature of the fluid. It helps the community working in the field of Physiological fluid dynamics yet
additionally to the medical professionals.
Key word: Parallel Plate Channel, Boundary Layer, Heat Source, Magnetic Field, Casson fluid.
1. Introduction
The investigation of Casson flow and heat transfer in a viscous fluid is of impressive premium as a result of
their consistently expanding modern applications and vital heading on a few mechanical procedures. Amid the
most recent decades broad research work has been done on the fluid dynamics of biological fluids in the
presence of magnetic field. For various reasons, utilizations of magneto hydrodynamics in physiological flow
issues are of developing interest. The flow because of extending of a flat surface was first researched by
Crane [1]. Pavlov [2] examined the impact of exterior magnetic field on the MHD flow over an extending
sheet. Andersson [3] examined the MHD flow of viscous fluid on an extending sheet and Mukhopadhyay et
al. [4] introduced the MHD flow and heat transfer over an extending sheet with variable fluid consistency.
Examples of Casson fluid incorporate jam, tomato sauce, nectar, soup and focused natural product juices, and
so on. Human blood can likewise be dealt with as Casson fluid. Due to presence of numerous substances like,
protein, fibrinogen and globulin in a fluid base plasma, human red platelets can shape a chainlike structure,
rouleaux. Several investigations were made to look at flow over an extending/contracting sheet under various
parts of MHD, suction/injection, heat and mass transfer and so on [5– 12]. In the event that the rouleaux carry
on like a plastic solid, at that point there exists a yield pressure that can be related to the steady yield stress in
Casson's fluid [13– 15]. Adhikary and Misra [16] exhibited a exact solution of the issue of oscillatory flow of
a fluid and heat transfer along a permeable oscillating channel in presence of an exterior magnetic field.
Tzirtzilakis [17] considered a scientific model of biomagnetic fluid dynamics (BFD), reasonable for the
portrayal of the Newtonian blood flow under the activity of magnetic field. This model is reliable with the standards of ferrodynamics and magneto hydrodynamics and considers both magnetization and electrical conductivity of blood. Ramamurthy and shanker [18] contemplated magneto hydrodynamic consequences for
blood flow through a permeable channel. They considered the blood a Newtonian fluid and conducting fluid.
International Journal of Pure and Applied MathematicsVolume 119 No. 15 2018, 1185-1195ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/
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Blood vessel MHD pulsatile flow of blood under periodic body acceleratiion has been considered by Das and
Saha [19]. The blood flow in extremely narrow vessels under the impact of transverse magnetic field has
been examined by Madhu et al. [20]. The detailed discussion on the stagnation-point flow over
extending/contracting sheet can be found in progress Bhattacharyya [21– 23].
Inspired by the previously mentioned examinations on flow of Newtonian and non-Newtonian fluids owing
to an extending sheet and its tremendous applications in several enterprises. In the present study, a numerical
model for the unsteady fluid flow through parallel plate channel with heat transfer and external transverse
magnetic field is exhibited. An endeavor is made in this paper to broaden the work by Islam M. Eldesoky
[22] for non-Newtonian Casson fluid and heat transfer under the conditions characterized in our model. The
primary target of the present work is to get analytical expressions for axial velocity, temperature distribution
and normal velocity using new boundary conditions and with converting the system of partial differential
equations into system of ordinary differential equations. The impacts of magnetic field (Hartmann number
(Ha)), Casson parameter, heat source parameter (S) and Prandtl number (Pr) on the axial velocity, temperature
distribution and normal velocity are researched and analyzed with the assistance of their graphical
representations.
2. Mathematical analysis of the flow
Consider the unsteady two-dimensional flow and heat transfer of an incompressible Casson fluid over an
exponentially extending/contracting sheet at 𝑦 = 0, with the flow being restricted in 𝑦 > 0. The fluid is
electrically conducting in the presence of a uniform magnetic field applied normal to the sheet, and the
induced magnetic field is ignored under the approximation of small Reynolds number.
The rheological equation of state for an isotropic and incompressible flow of a Casson fluid as follows:
cij
c
yB
cijy
B
ij
ep
ep
22
,22
,
where ij is the (i, j)-th component of the stress tensor, = eij eij and eij are the (i, j)-th component of the
deformation rate, is the product of the component of deformation rate with itself, c is a critical value of this
product based on the non-Newtonian model, B is plastic dynamic viscosity of the non-Newtonian fluid , and
py is the yield stress of the fluid.
So, if a shear stress less than the yield stress is applied to the fluid, it behaves like a solid, whereas if a shear
stress greater than yield stress is applied, it starts to move.
Considering u and v as velocity components in the directions of x and y respectively (axial and normal
respectively) at time t in the flow field, we may write the two dimensional boundary layer equations in
presence of transverse magnetic field as
0
y
v
x
u (1)
)(1
11
0
20
2
2
TTguB
y
u
x
p
t
u
(2)
)( 02
2
TTC
Q
y
T
C
K
t
T
pp
(3)
where 𝜐 is the kinematic fluid viscosity, 𝜌 is the fluid density, = 𝜇𝐵√2𝜋𝑐/𝑝𝑦 is the Casson parameter, 𝜍 is the
electrical conductivity of the fluid, and 𝐻0 is the strength of magnetic field applied in the 𝑦 direction. and
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is the density and viscosity of the blood while p∗ stands for pressure. K‟ is thermal conductivity; CP is the
specific heat at constant pressure. Q is the quantity of heat, T is the temperature and β is the volumetric
expansion parameter while θ is the temperature distribution.
The boundary conditions are taken as:
te2 , teu
2 at 1y
0 , 0u at 1y
Let us introduce the non-dimensional variables,
h
xx * ,
h
yy * ,
hm
uu
2/
* , hm
vv
2/
* /2
*
h
tt ,
32
*
2/),(
hm
dxdptxp
,
32
*
2/ hm
(4)
Substituting equation (4) into equations (1) – (3), we get
0
y
v
x
u (5)
guHay
up
t
u
2
2
211 (6)
rr P
S
yPt
2
21 (7)
where the heat source parameter, TK
QhS
2
, Prandtl number,T
p
rK
CP
3. Analytical Solution of the Problem
With the above discussions in the previous section, let us choose the solutions of the equations (5) – (7)
respectively as
teyFtyu2
)(),( (8)
teyGtyv2
)(),( (9)
teyHty2
)(),( (10)
The boundary conditions are transformed to
,1,1 FH at 1y
,0,0 FH at 1y (11)
By virtue of (8) – (10), we obtain the equations (5) – (7) respectively as
yHgpyFHayF
1)(
1
22 (12)
CG (13)
02 yHPSyH r (14)
Solution of equation (14) using the boundary condition (11) is as follows
yyyH
sinsin2
1cos
cos2
1 (15)
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where rPS 2
From (10) and (15) the temperature distribution is given by
teyyty2
sinsin2
1cos
cos2
1),(
(16)
Using the equation (16) into equation (12) we get
yygpyFHayF
sin
sin2
1cos
cos2
1
1)(
1
22 (17)
From equation (8) and (17) the velocity of the flow of the fluid parallel to the direction of the channel is
obtained as,
teycycycycctyu2
sincossincos),( 54321 (18)
where te
pp
2
*
1
, 22
1Ha
,
2
11
pc ,
cos2 222
g
c ,
sin2 223
g
c
cos2
cos221 214
ccc
,
sin2
sin21 35
cc
From equations (9) and (13), the velocity of the fluid flow perpendicular to the direction of the channel is
given by tCetyv
2
),( (19)
where C is an arbitrary constant.
Equations (16), (18) and (19) show the temperature distribution, the axial velocity and normal velocity
respectively.
3. Results and Discussions
In this section, we discuss the different physical parameters, such as heat source parameter (S), Hartmann
number (M), Prandtl number (Pr) and decay parameter () on temperature distribution, axial velocity and
normal velocity. The obtained computational results are presented graphically and the variations in velocity
and temperature are discussed.
3.1. Effects of different physical parameters on temperature fields
Figure 1 shows the performance of temperature distributions versus y at = 0.5, Pr = 1, = 0.5, and t = 1for
different values of heat source parameter (S = 1, 1.75, 2.5, 3.25, 4). We observe that the temperature field
decreases with increasing the values of S, for y 0.5, and temperature field increases for y 0.5. The
maximum effect of heat source is at y = -1.
Figure 2 emphasizes that the temperature field distribution for different values of Prandtl number (Pr = 1, 3, 5,
7, 9) at S = 1, = 0.5, = 0.2, and t = 1. The effect of Prandtl number on temperature steadily decreases with
increasing the values of Prandtl number.
It is clear from Figure 3 that temperature field distribution decreases with increasing the decay parameter up
to y 0.16 and it increases with increasing the decay parameter for y 0.16, at S = 1, Pr = 1, = 0.2, and
t = 1 for different values of decay parameter ( = 0.5, 0.75, 1, 1.25, 1.5). The maximum effect of decay
parameter on the temperature field is between -0.8 y -0.4.
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Figure 1. Temperature Field for different values of values of Heat Source Parameter (S)
Figure 2. Temperature Field for different values of values of Prandtl Numer (Pr)
Figure 3. Temperature Field for different values of values of Prandtl Numer (Pr)
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3.2. Effects of different physical parameters on velocity fields
Figure 4 exhibits the axial velocity profiles for several values of heat source parameter (S = 1, 1.75, 2.5, 3.25,
4) at = 1.5, Pr = 1, = 0.2, t = 1, = 0.2, g = 9.8, = 0.5, p = 0.5 and Ha = 1. It is observed that axial
velocity increases with increasing the heart source parameter S for y -0.3 and the effect reverses for
-0.3 y 1.
Figure 5 indicates the effect of magnetic field on the axial velocity for different values of Hartmann number
(Ha = 1, 1.25, 1.5, 1.75, 2) at S = 1, = 0.5, Pr = 1, = 0.2, t = 1, = 0.2, g = 9.8, = 0.5 and p = 0.5. It is
observed that the axial velocity decreases with increasing the magnetic field up to y -0.3 and for y -0.3
axial velocity increases with increasing the magnetic field.
The effect of Prandtl number on the distribution of the axial velocity at S = 1, = 0.5, Pr = 1, = 0.2, t = 1,
= 1, g = 9.8, = 0.5, p = 0.5 and Ha = 1is shown in Figure 6. It is observed that axial velocity decreases
with increasing the Prandtl number.
Figure 7 defines the effect of decay parameter on the axial velocity for different values of decay parameter
( = 0.5, 0.75, 1, 1.25, 1.5) at S = 1, = 0.5, Pr = 1, = 0.2, t = 1, = 1, g = 9.8, = 0.5, Ha = 2 and p = 0.5.
It is observed that the axial velocity increases with increasing the decay parameter up to y -0.3 and for
y -0.3 axial velocity decreases with increasing the decay parameter.
Effects of Casson parameter on velocity profiles for unsteady motion are clearly exhibited in Figure 8 the
behavior of velocity with increasing is noted at S = 1, = 0.5, Pr = 1, = 0.2, t = 1, = 0.2, g = 9.8, = 0.5
and p = 0.5. It is observed that the axial velocity decreases with increasing the magnetic field up to y -0.2
and for y -0.2 axial velocity increases with increasing the magnetic field.
Figure 4. Axial velocity for different values of values of Heat Source Parameter (S)
Figure 5. Axial velocity for different values of values of Magnetic Field Parameter (Ha)
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Figure 6. Axial velocity for different values of values of Prandtl Numer (Pr)
Figure 7. Axial velocity for different values of Decay Parameter (S).
Figure 8. Axial velocity for different values of Casson Parameter ().
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Figure 9. Normal velocity for different values of Decay Parameter ().
Normal velocity for different values of decay parameter is shown in Figure 9. It is depicted that normal
velocity is decreasing with increasing values of and also for increasing values of t. The normal velocity is
decreasing slowly at low values of the decay parameter ( = 1) while it decreases very fast and tends to zero
at high values of decay parameter ( = 3).
4. Conclusion
The present study provides the solution for the unsteady Casson fluid flow through the parallel plate channel
with heat source and external transverse magnetic field is presented. The present work is the effect of
magnetic field, heat source and Casson parameter seems to be significant.
The present mathematical model gives a simple form of axial velocity, temperature distribution and normal
velocity of the flow. Analytical expressions are obtained by choosing the axial velocity; temperature
distribution and the normal velocity of the flow depend on y and t only.
The temperature field decreases with increasing the heat source parameter (S), Prandtl number(Pr) and the
decay parameter(). And temperature field increases with increasing the heat source parameter (S) and decay
parameter () for y 0.5 and y 0.16 respectively. For y -0.3, the axial velocity increases with increasing
the heart source parameter (S) and decay parameter () and the effect reverses in -0.3 y 1. The axial
velocity decreases with increasing the magnetic field (Ha), Prandtl number (Pr) and Casson parameter ().
And axial velocity increases with increasing the magnetic field (Ha) and Casson parameter () for y -0.5 and
y -0.2 respectively. The effect of increasing values of the Casson parameter is to suppress the velocity field.
Prandtl number can be used to increase the rate of cooling in conducting flows. The normal velocity decreases
with increasing the decay parameter and tending to zero very fast for higher values of the decay parameter.
The results may be helpful for possible technological applications in liquid-based systems involving
stretchable materials.
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