Stokes Equation - Solution in Spherical Polar Coordinates

Notes on the Solution of Stokes’s Equation for Axisymmetric
Flow in Spherical Polar Coordinates
R. Shankar Subramanian
Department of Chemical and Biomolecular Engineering
Clarkson University
The details of the solution of Stokes’s Equation for the streamfunction
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E ψ = 0 in spherical polar coordinates ( r, θφ , ) by the method of separation
of variables are given in the book “Low Reynolds Number Hydrodynamics”
by J. Happel and H. Brenner. The solution is obtained as the sum
1
, 2
( 1)
ψ r θ + ψ r,
θ where ψ satisfies
( ) ( )
( 2)
( ) ( )
and ψ is the particular solution of
( 1)
2
E ψ = 0
(1)
2
E ψ
( 2) ( 1)
= ψ
(2)
The solution, in the form of infinite series, is specialized to typical problems
in the spherical geometry in a domain that includes at least a portion of the
symmetry axis by requiring that the velocity field be bounded along that
axis, which corresponds to θ = 0, π . The result is given below.
∞
∑ n n n n
C
n
(3)
n=
2
−1/2
C
n
η are the Gegenbauer Polynomials of order n
n − n+ 1 n+ 2 − n+ 3 −1/2
( r,
) = ( Ar + Br + Cr + Dr ) ( )
ψ θ η
Here, η cosθ
and degree − 1/2. Their properties are discussed in Sampson (1891), and in
Abramowitz and Stegun (1965). The Gegenbauer Polynomials are solutions
of the second order ordinary differential equation
= , and ( )
2
2 d Φ
( 1− η ) + n
2
( n−1)
Φ= 0
(4)
dη
where n is an integer. The second linearly independent solution of this
equation is a function that becomes unbounded along the symmetry axis.
The Gegenbauer Polynomials are closely related to the Legendre
Polynomials ( ) n
P η , which satisfy Legendre’s equation.
1

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

The function Qn
( )
−1/2
Gegenbauer Polynomials through Q ( η) = −C ( η)
.
η used by Leal in the textbook is related to the
n
n+
1
References
M.A. Abramowitz and I.A. Stegun, “Handbook of Mathematical Functions,”
Dover Publications (1965).
J. Happel, and H. Brenner, “Low Reynolds Number Hydrodynamics,”
Prentice-Hall (1965); republished by Noordhoff International (1973).
L.G. Leal, “Laminar Flow and Convective Transport Processes,”
Butterworth-Heinemann (1992).
T.M. MacRobert, “Spherical Harmonics,” Pergamon Press (1967).
R.A. Sampson, “On Stokes’s Current Function,” Phil. Trans. Royal Soc. A
182, 449-518 (1891).
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