y - International Journal of Applied Information Systems

International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868
Foundation of Computer Science FCS, New York, USA
Volume 4– No.7, December 2012 – www.ijais.org
*
The x -axis is taken along the plate in the vertically upward
*
direction and the y -axis is taken normal to the plate.
Initially the plate and the fluid are at the same temperature
*
*
T in the stationary condition with concentration level
at all points. At time,
accelerated with a velocity
C
*
t 0 , the plate is exponentially
**
* at
o
u U e
in its own plane
and the temperature of the plate is raised at a uniform rate and
the level of concentration near the plate is raised linearly with
time. All the physical properties of the fluid are considered to
be constant. The unsteady flow is governed by the following
equations:
* 2 *
u u
g T T g C C
* *2
t y
* 2 *
C C
D K C C
* *2
t y
Tk T Q
t y
* * * * *
* * *
*
* T T
* 2 *
*
C *2
0
C
The initial and boundary conditions relevant to the fluid flow
are:
* * * * * *
t* 0 : u 0, T T , C C For all y
*
**
* at T
q * * *
t* 0 : u U e , , C C at y 0
o *
w
y
k
* * * * * *
u 0, T T , C C as y
(4)
*
*
*
Where u is the axial velocity, t is time, T the fluid
*
temperature, is coefficient of volume, C is the fluid
concentration,
(1)
(2)
(3)
*
C w is wall concentration, D the mass
*
is the
diffusivity, K is chemical reaction parameter,
coefficient of concentration expansion.
3. ANALYTICAL SOLUTION
To solve the governing equations in dimensionless form, the
following non-dimensional quantities were introduced:
* * * 2
y U0 u t U0 Q
0
y , U , t ,, F 2
U0 cU
0
* * * * *
T T C C a
, c , a,
* * 2
q k Cw CU0
*
(5)
K c
K , Sc ,Pr ,
2
U0 D k
2
* * *
gq g CwC Gr , Gc 4 3
kU 0 U0
The governing equations on using (5) into (1) to (4) reduce to
the following
2
u u
Nc
2
t y
tScy 2
c 1 c
Kc
2
2
1
F
2
t Pr y
Subject to the boundary conditions
For t 0: u c 0,
t
for 0 y 1
at ue , 1,
y
u 0, 0, c 0
For 0:
c 1 at y 0
, as y
To obtain the solutions, we apply the Laplace transform to (6)
– (8) and the boundary conditions to get
2
d
Pr 2 PF 0
(9)
dy
2
dC
Sc 2 P K C
dy
(6)
(7)
(8)
0
(10)
2
dU
PU NC
2
dy
(11)
The boundary conditions are
t 0 : U C 0, y 1
1 d
1 1
t 0 : U , , C , at y 0
P a dy P P
U 0, 0, C 0, as y
The solutions to (9) to (11) subject to (12) give
(
y, p)
e
C( y, p)
Pr
e
k1
y Z
Z F
k2Z
Z K
Z
(12)
(13)
(14)
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