Viscous Linear Instability of an Incompressible Round Jet T.M. ...

2.4 Final form of the equations used in the program
It is now more convenient to write the set (2.13) - (2.16) in the form of a set
of first-order equations. This is done simply by defining two new variables –
u ′ and w ′
where
⎡
⎢
A = ⎢
⎣
Y ′ = A(r)Y (2.18)
⎡
⎢
Y = ⎢
⎣
u
v
w
p
u ′
w ′
⎤
⎥
⎦
0 0 0 0 1 0
(iα) ( 1)
(in)
0 0 0
r r
(β 2 − 2α 2 + n2
0 0 0 0 0 1
0 (2α 2 − β 2 + n2
r 2 ) ( in
r 2) 0 (iα) ( in
r )
r2 ) ( Ū ′ R) 0 (iαR) (−1) 0 r
0 ( 2in
r 2 ) (β 2 + n2 +1
r2 ) ( inR
r
) 0 −(1
r )
where β = [α2 + iαR( Ū − c)]12.
Because these equations are singular at
r = 0, in order to develop a numerical technique to solve these equations it
is necessary to find the form of the solution close to the axis of symmetry.
Also, because the numerical integration cannot be carried out to infinity,
boundary conditions are sought in the nearly undisturbed fluid surrounding
the jet. These asymptotic solutions will be developed in the next subsection.
9
⎤
⎥
⎦