Stokes Equation - Solution in Spherical Polar Coordinates

d ⎡ 2 dΦ
⎤
( 1 η ) n( n 1)
0
dη
⎢ − + + Φ=
dη
⎥
(5)
⎣ ⎦
Many important properties of the Legendre Polynomials can be found in
MacRobert (1967) and in Abramowitz and Stegun (1965).
Some useful relationships involving the Gegenbauer Polynomials are given
below.
( η) − P ( η)
P
2n
−1
η
−1/2
C
n ( η ) = − ∫ P
1 ( )
1
n−
s ds
(7)
−
−1/2
n−2
n
C
n ( η ) =
(6)
The first few Gegenbauer and Legendre Polynomials are
C
C
C
P
P
P
−
( ) 1 C ( )
η = η = η
−1/2 1/2
0 1
1 1
2 2
1 1
8 5 1
−
( η) = ( 1− η ) C ( η) = η ( 1−η
)
−1/2 2 1/2 2
2 3
( η) = ( −η )( η − )
−1/2 2 2
4
( ) 1
P ( )
0 1
2 2
( η) = ( 3η − 1) P ( η) = η( 5η
−3)
2 3
4
η = η = η
1 1
2 2
1 35
8 30 3
4 2
( η) = ( η − η + )
The Gegenbauer Polynomials satisfy the orthogonality property
for values of mn≥ , 2.
−1/2 −1/2
+ 1
m ( η) n ( η)
2
∫ C C dη
=
δ
1
2
mn
(8)
−
1−η
n( n−1)( 2n−1)
2