ABSTRACT
L = ∂ 2
∂x2 + ∂
∂y2 (4.3)
The equation of continuity is
∂u ∂x
+ ∂υ ∂y
= 0 (4.4)
If we introduce the stream function ψ(x, y, t) defined by
u = ∂ψ ∂x
, υ = −∂ψ ∂y
(4.5)
then the equation (4.4) is identically satisfied. Eliminating p between (4.1) and (4.2) and then using (4.5) we get the governing equation of motion in terms of ψ(x, y, t) as
∂
∂t (Lψ) + ∂ψ
∂y · ∂ ∂x
(Lψ) − ∂ψ ∂x
· ∂ ∂y
(Lψ) = νL2ψ (4.6)
which we write for our purpose in the form
∂4ψ
∂x4 + ∂
4ψ
∂y4 = ν−1 ∂
∂t (Lψ) − 2 ∂
4ψ
∂x2∂y2 + ν−1 Nψ (4.7)
where the nonlinear term Nψ is defined by
Nψ = ∂ψ ∂y
· ∂ ∂x
(Lψ) − ∂ψ ∂x
· ∂ ∂y
(Lψ) = ∂(Lψ, ψ) ∂(x, y)
(4.8)
Equation (4.7) cannot be solved analytically except in some special cases because of the nonlinear character, but it can be solved numerically by means of traditional numerical techniques. The modern powerful method known as Adomian’s decomposition method can provide analytical approximations to this nonlinear equation and is applied here in order to get the solution that demands to be parallel to any modern supercomputer.
