2017 International Conference on Innovations in information Embedded and Communication Systems (ICIIECS)
Mathematical Model with Analytical Solution for
Magneto Hydrodynamic flow between
Two parallel plates
R. Delhi Babu, Research Scholar,
S. Ganesh, Research Supervisor
Dept. of Mathematics,
Sathyabama University,
Chennai, India.
delhimsc@gmail.com
Dept. of Mathematics,
Sathyabama University,
Chennai, India.
sganesh19@yahoo.com
Abstract—In this Paper we examine the correct arrangement
of MHD stream between parallel plates both very still with
uniform suction. Outer uniform axial and transverse attractive
field and uniform suction and infusion are connected
transversely to the plates while the smooth movement is
subjected to exponential pivotal and transverse speed and
weight slope. The arrangement of the issue is found with the
assistance of numerical methods. Explanatory expression is
given for the speed field and the impacts of the different
parameters are utilized as a part of this issue.
Keywords— Fluid flow, Parallel porous Plates, MHD flow.
I.INTRODUCTION
Hannes Alfven [1] concentrated the issue of convective
magneto hydrodynamic channel stream between two parallel
plates subjected all the while to a pivotal temperature
inclination and a weight slope numerically. In the conclusion
we have found that a connected transverse attractive field may
lessen the passageway length of the speed impressively, yet has
little impact on the temperature advancement. Hartmann and
Lazarus[2] considered the insecure hydro attractive stream of a
gooey incompressible electrically directing liquid in a pivoting
channel affected by an intermittent weight angle and uniform
attractive field, which is slanted in the hub of turn. An
explanatory arrangement of relentless and temperamental
hydro attractive stream of gooey incompressible electrically
directing liquid affected by consistent and intermittent weight
inclination in nearness of slanted attractive field has been
gotten precisely by Ghosh [3] to concentrate the impact of
gradually pivoting frameworks with low recurrence of swaying
when the conductivity of the liquid is low and the connected
attractive field is feeble. The MHD stream between two
parallel plates is called Hartmann stream. It has numerous
Applications in MHD control generators, MHD pumps,
streamlined warming. At that point a considerable measure of
research work concerning the Hartmann stream has been
acquired under various physical impacts [4-12]. The Steady of
stream has been completed by a few creators. Numerous
scientists have announced that the stream is electrically leading
liquid [13-16]. The electromagnetic force(Lorentz drive)
follows up on the stream and this constrain contradicts the
movement of stream and there by stream is blocked, so that the
outer attractive field can be utilized as a part of the treatment of
a few sorts of ailments. MHD is the liquid mechanics of
electrically leading liquids, some of these liquids incorporate
fluid metals, (for example, mercury, liquid iron) and ironized
gasses referred to as physicists as Plasma, one illustration being
the sun based environment. The subject of MHD is to a great
extent seen to have been started by beyond any doubt dish
electrical architect Yang and Yu [17]. On the off chance that an
electrically leading liquid is put in a consistent attractive field,
the movement of the liquid instigates current which make
drives on the liquid. The creation of these streams has
prompted to the plan of among different gadgets the MHD
produces for electrically generation. The administering
conditions that have been fathomed either systematically or
numerically.
In this paper insecure stream of a leading liquid between
parallel permeable plates is examined. The liquid is misbehave
and down on the time advancement of both speeds. The impact
of suction speed on both bearing of the liquid are examined.
The outcome differential condition is unraveled by an
expository strategy and the arrangement additionally
communicated.
II.MODEL FORMULATION
Consider the transverse magnetic field applied
perpendicular to the walls y = 0 and y = h, in the flow of an
incompressible viscous fluid between the parallel plates. Let
the velocity in the flow field at a given time t be
The equation of continuity is ∂ u + ∂ v = 0
∂x ∂y
Equations of momentum are
ρ
ρ
uiˆ + vˆj .
§ ∂ 2u ∂ 2u ·
∂u
∂p
= − + μ ¨¨ 2 + 2 ¸¸
∂t
∂x
∂y ¹
© ∂x
§ ∂ 2v ∂ 2v ·
∂v
∂p
= − + μ ¨¨ 2 + 2 ¸¸ − σ e B02 sin 2 β v
∂t
∂y
∂y ¹
© ∂x
‹,(((
(1)
(2)
(3)
2017 International Conference on Innovations in information Embedded and Communication Systems (ICIIECS)
Assumption
(i)
(ii)
(iii)
(iv)
(v)
(vi)
The parallel plates are porous.
The liquid going ahead here and there.
Magneto Hydrodynamic flow is considered.
Flow between non conducting two parallel plates.
Viscosity of the fluid is considered to constant.
u and v are velocity components in x and y
directions respectively.
∂2 p
∂2
∂2ψ
∂2ψ
= −μ 2 ∇2ψ 9 + ρiω 2 −σe B02 sin2 β 2
∂x∂y
∂x
∂x
∂x
( )
From equations (9) and (10),
ª 2 § iωρ ·º 2
¸¸»∇ ψ = 0
«∇ − ¨¨
© μ ¹¼
¬
ρ
h
μ
ψ
η
σ
- Density of the fluid
- Height of the channel
- Coefficient of viscosity
- Stream function
- Dimensionless distance
- Electrical conductivity of the fluid
B0 - Electromagnetic induction
Equation of continuity can be satisfied by a stream function is
§ u0 v2 x ·
−
¸ f (η )
h ¹
©a
where η
III.SOLUTION OF MATHEMATICAL MODEL
½
¾
v ( x, h ) = v 2 ¿
v( x,0) = v1
∂ψ
∂ψ
& v ( x, y ) = −
∂y
∂x
(4)
− ρiω
(
)
∂ψ
∂p
∂ 2
∂ψ
=−
+μ
∇ ψ − σ e B02 sin 2 β
∂x
∂y
∂x
∂x
(
)
(5)
f (0) = 1 − a, f (1) = 1
f ′(0) = 0,
f ′(1) = 0
(
)
½
¾
¿
(15)
f (η ) =
1
2αh sinh αh + 4(1 − cosh αh)
ªαh sinh αh(1 − a + aη ) + a (cosh αh(η − 1) − cosh αhη )º
׫
»
¬+ (a − 2) cosh αh + (2 − a )
¼
(6)
(7)
(8)
Substituting the value of
(9)
f (η ) in the stream function
v x·
§u
ψ (x,η ) = h¨ 0 − 2 ¸ f (η )
h ¹
©a
Hence
u = u( x, y)e iωt
=
Differentiating equations (7) & (8) w.r.to ‘y’ & ‘x’ Partially,
∂2 p
∂2
∂ 2ψ
= μ 2 ∇ 2ψ − ρiω 2
∂x∂y
∂y
∂y
(14)
Hence the solution of equation (14) with the boundary
condition (15) is
Equations (2), (3) and (6), we get
∂ 2
∂ψ
∂p
=− +μ
∇ψ
∂y
∂x
∂y
μ
(D4 − α 2 h 2 D2 ) f (η) = 0
Define the stream functions are
u ( x, y ) =
iρω
with the boundary conditions
With the following boundary conditions
u ( x, h ) = 0
α2 =
(13)
Equation (13) reduces to
Consider the solutions of the equations (1) to (3) as follows
u ( x,0) = 0,
v
y
, a = 1 − 1 , 0 ≤ v1 ≤ v2 and u 0 is the
v2
h
f iv (η ) − α 2 h 2 f ′′(η ) = 0 ,
where
u = u ( x , y ) e i ωt ½
°
v = v ( x , y ) e i ωt ¾
p = p ( x, y )e iωt °¿
=
(12)
average velocity.
Substituting (12) in (11), we have
H 0 - Transverse magnetic field
ρiω
(11)
ψ (x,η ) = h¨
NOTATIONS
(10)
∂ψ iωt
e
∂y
§ 2 sin αh − 2 sinα ( y − h) − 2 cosαy ·º
v x ·ª
§u
¸¸»
= ¨ 0 − 2 ¸«aαheiωt ¨¨
h ¹¬
©a
© 2αh sinhαh + 4(1 − coshαh) ¹¼
2017 International Conference on Innovations in information Embedded and Communication Systems (ICIIECS)
20
v = v( x, y )e iωt
∂ψ iωt
=−
e
∂x
16
iωt
v2 e
2αh sinh αh + 4(1 − cosh αh)
14
wt=0
ª(1 − a )[4 + 2αh sin αh − 2 cos αh − 2a cos αh + 2a]º
׫
»
¬+ a[2αy sin αh + 2 cos α ( y − h) + 2 cos αy
¼
The Pressure drop is given up
η
v x·
μv
§u
p(x,η ) − p(0,0) = K ¨ 0 − 2 ¸ x + 22 f ′(η ) − iρωv 2 ³ f (η )dη
h
© a 2h ¹
0
Where K =
μ
h2
wt=180
12
y axis
=
18
wt=45
10
6
wt=90
4
2
0
-5
f ′′′(0)
wt=135
8
-4
-3
-2
-1
0
u axis
1
2
3
4
5
-4
x 10
Fig. 3 u0=1.5, v2=-1, x=1, y=0:0.5:20, h=20, a=1, Į=0.5
IV.GRAHPICAL REPRESENTATION
20
20
18
18
16
16
14
wt=0
14
wt=180
12
y ax is
y axis
12
10
wt=135
8
8
6
6
wt=90
4
2
wt=45
-0.15
wt=90
wt=180
wt=0
4
0
-0.2
wt=45
10
-0.1
2
wt=135
-0.05
0
u axis
0.05
0.1
0
-8
0.15
Fig. 1 u0=1.5, v2= -1, x=1, y=0:0.5:20, h=20, a=1, Į=0.25
-6
-4
-2
0
v axis
2
4
6
8
-5
x 10
Fig. 4 u0=1.5, v2=-1, x=1, y=0:0.5:20, h=20, a=1, Į=0.5
20
20
18
18
16
16
14
14
wt=0
wt=180
12
10
y axis
y axis
12
8
wt=45
10
wt=135
8
6
wt=90
6
wt=90
4
2
0
-0.02
wt=180
wt=0
4
wt=45
wt=135
2
-0.015
-0.01
-0.005
0
v axis
0.005
0.01
0.015
0.02
Fig. 2 u0=1.5, v2=-1, x=1, y=0:0.5:20, h=20, a=1, Į=0.25
0
-1
-0.8
-0.6
-0.4
-0.2
0
u axis
0.2
0.4
0.6
0.8
1
-8
x 10
Fig. 5 u0=1.5, v2= -1, x=1, y=0:0.5:20, h=20, a=1, Į=1
2017 International Conference on Innovations in information Embedded and Communication Systems (ICIIECS)
20
18
16
14
wt=0
wt=180
y axis
12
wt=45
10
wt=135
8
6
wt=90
4
2
0
-5
-4
-3
-2
-1
0
v axis
1
2
3
4
5
-9
x 10
Fig. 6 u0=1.5, v2=-1, x=1, y=0:0.5:20, h=20, a=1, Į=1
V.SENSITIVE ANALYSIS AND CONCLUSION
In this Paper we examine the transient Hartman
stream of a directing liquid affected by attractive field,
considering the two plates are in parallel impacts within the
sight of uniform suction and infusion. An explanatory answer
for the conditions of movement has been produced while the
differential condition has been unraveled diagnostically. The
impact of attractive field, suction and infusion on the speed
and weight dissemination has been examined. It is found that
the impact of the suction and infusion speed u and v relies on
the attractive field. For vast estimation of attractive field
expanding MHD stream builds u. For little estimations of
attractive field expanding the MHD stream marginally
diminishes u. A correct arrangement of insecure transverse
MHD stream between parallel permeable plates both very still
with uniform suction. Outside uniform hub and transverse
attractive field and uniform suction and infusion are connected
oppositely to the plates while the smooth movement is
subjected to exponential pivotal and transverse speed and
weight slope. The arrangement of the issue is acquired with
the guide of numerical method. Expository expression is given
for the speed field and the impact of the different parameters
going into the issue is talked about with the assistance of
diagram.
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