1180
Ch. Baby Rani, N. Vedavathi, G. Dharmaiah & K. S. Balamurugan
(
v′ = −V0 1 + ε A expiω ′t ′
)
(8)
Where, A is a real positive constant, and ε is small such that ε<<1, εA<<1, and
V0 is a non-zero positive
constant, the negative sign indicates that the suction is towards the plate. In order to write the governing equations and the boundary conditions in dimensionless form, the following nondimensional quantities are introduced.
u=
V y′ t ′V 2 u′ v′ u′ k1′k 4ω ′ν , v = , y = 0 , u = , t = 0 ,ω = ,Q = , 2 ∗ V0 V0 V0 V0 4ν 4σ T∞′3ν 2 ν
θ=
νρ c p T ′ − T∞′ C ′ − C∞′ K ′V 2 V L′ K ′ν ,φ = , K = 2 0 , Pr = , h = 0 , Kr = r2 , Tw′ − T∞′ Cw′ − C∞′ k V0 ν ν ν
Sc =
,M =
D
νβ g (Tw′ − T∞′ ) νβ ∗ g ( Cw′ − C∞′ ) σ B02ν , Gr = , Gc = , 2 3 V0 V03 ρV0
(9)
In view of Equations (6), (7) (8) and (9), Equations (2), (3) and (4) can be reduced to the following dimensionless form.
(
)
2 1 ∂u 1 iω t ∂u ∂ u − 1+ε Ae = 2 + Gr1θ + Gc1φ − M 1 + ∂y ∂y 4 ∂t K
(
u (10)
)
2 1 ∂θ 4 ∂ θ iωt ∂θ 1 − 1+ε Ae = 1+ ∂y Pr 3 Q ∂y 2 4 ∂t
(11)
(
)
2 1 ∂φ iω t ∂φ 1 ∂ φ − 1+ε Ae = − Krφ 4 ∂t ∂y Sc ∂y 2
(12)
Gr1 = Gr cos α , Gc1 = Gc cos α , M 1 = Where
m M 1 + m2
The corresponding dimensionless boundary conditions are
∂u , ∂y u → 0, u=h
θ = 1 + ε e iωt , φ = 1 + ε eiω t
at
θ → 0,
as
φ →0
y=0 y→∞
(13)
SOLUTION OF PROBLEM The equations (10) to (12) are coupled, non-linear partial differential equations and these cannot be solved in closed form. However, these equations can be reduced to a set of ordinary differential equations, which can be solved analytically. So this can be done, when the amplitude of oscillations flow velocity
u,
temperature field
Impact Factor (JCC): 7.6197
θ
(ε ≪ 1) is very small, we can assume the solutions of
and concentration C in the neighborhood of the plate as:
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