Gen. Math. Notes, Vol. 31, No. 2, December 2015, pp.29-33
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
The Derivation of a Goldstein Formula
L.J. Crane1 and A.G. McVeigh2
1,2
Institute for Numerical Computation and Analysis
Suite 6, 5 Clarinda Park North, Dún Laoghaire
Dublin, Ireland
2
E-mail: gotham1961@gmail.com
(Received: 19-10-15 / Accepted: 28-11-15)
Abstract
This technical note presents the derivation of an integral function credited
to Goldstein [2] in 1932 and recently employed in the authors’ previous work [1]
in Archive of Applied Mechanics. The particular form of this improper integral
is developed using techniques involving contour integration and the calculus of
residues.
Keywords: Bingham Number, Slip Flow, Inversion Theorem, Laplace
Transform.
1
Introduction
The problem of axially-symmetric slip flow generated by an infinite cylinder
undergoing impulsive motion was recently investigated by Crane and McVeigh
[1]. In accounting for momentum slip close to the cylinder wall, they obtained
the non-dimensional shear stress analytically in terms of the Bingham number,
Bn, in the cases where the cylinder moved under both uniform velocity and
acceleration. In denoting the non-dimensional variables of axial velocity, cylinder radius and time by U , R and T , respectively, they presented the unsteady
Navier Stokes momentum equation as follows:
∂U
1 ∂
=
∂T
R ∂R
∂U
R
∂R
(1)
30
L.J. Crane et al.
subject to, for T > 0
UR=1+
λ
=1+
2
∂U
∂R
, U → 0 as R → ∞
(2)
R=1+
and, for T > 0:
U = 0 for R > 1
(3)
where λ is an empirically-derived slip-length parameter. In this work, the
Laplace transform of f (T ) is the function f¯(p); taken to be:
Z ∞
exp(−pT )f (T )dT = f¯(p)
L {f (T )} =
0
Now, investigating radiating heat flow from an infinite region of constant initial temperature and bounded internally by a circular cylinder, Goldstein [2],
derived the transform:
√
K0 ( p)
1
Ψ̄(p) =
1+ √ 0 √
(4)
√
p
µ̂ pK0 ( p) − K0 ( p)
where K0 denotes the modified Bessel function of the second kind of order 0,
and in the work herein, Crane and McVeigh [1] specify µ̂ = 2λ. The associated
inverse is thus:
Z ∞
exp(−b2 T )
1
4
Ψ(T ) =
db
(5)
µ̂π 2 0
b
(bJ1 + J0 /µ̂)2 + (bY1 + Y0 /µ̂)2
where J0 and J1 are cylindrical Bessel functions of the first kind of order 0 and
1, respectively and where Y0 and Y1 denote the cylindrical Bessel functions of
the first kind having order 0 and 1. Accordingly, Crane and McVeigh [1], give:
Bn =
2
Ψ(T )
λ
and
2
Bn =
Tλ
2
Z
(uniform velocity)
(6)
(uniform acceleration)
(7)
T
Ψ(T )dT
0
Derivation
From (4), the complex inversion integral is:
#
√ Z γ+i∞ "
K0 p
1
1
Ψ(T ) =
1+ √ 0 √ √ exp(pt)dp,
2πi γ−i∞ p
µ̂ pK0 p − K0 p
t > 0 (8)
31
The Derivation of a Goldstein Formula
The integration in (8) is to be performed along a line, p = γ, in the complex
plane where p is a point having coordinates (x + iy). The real number, γ, is
to be so large that all singularities of the integrand lie to the left of the line
(γ-i∞, γ+i∞). Since p = 0 is a branch point of the integrand, the adjoining
Bromwich contour is chosen as the integration path (Fig. 1). This comprises
iy
γ+i∞
D B R ^ ε
● p=γ
H ^ L ^ E K J N x A γ-i∞
1.pdf
Figure 1: The modified Bromwich contour
the line AB (p = γ + iy), the arcs BDE and LN A of a circle of radius R and
centre at (0, 0), and the arc HJK of a circle of radius, , with centre at (0, 0).
Set
Z
Z
Ψ(T ) =
Z
+
AB
Z
+
+
BDE
EH
Z
+
HJK
Z
+
KL
(9)
LN A
and since the only singularity, p = 0, of the integrand is not inside the contour, the integral on the left is zero by Cauchy’s theorem. Further, it is readily
shown that, as R tends to infinity, the integrals along BDE and LN A vanish
in the limit. Along the inner circle, HJK, where p = exp(iθ), then, on taking
the limit as becomes vanishingly small:
Z
Ψ(T ) =
Z
=i
HJK
π
−π
K0 (0)
1−
dθ = 0
K0 (0)
(10)
32
L.J. Crane et al.
and so,
Z
Z
=−
AB
Z
−
EH
(11)
KL
Along the path, EH, where p = xexp(iπ) = −x:
√
Z
EH
1
=
iπ
Z
√
R
K0 (ib)
exp(−b2 t)
db
1+
b
iµ̂bK00 (ib) − K0 (ib)
(12)
Introducing the identities:
1
1
ib = bexp( π) and K00 (ib) = π [J1 (b) + iY1 (b)]
2
2
so that, along EH, as R → ∞ and → 0:
Z
Z
1 0 exp(−b2 t)
µ̂b (−J1 + iY1 )
=
db
iπ ∞
b
−µ̂bJ1 − J0 + i (Y0 + µ̂bY1 )
EH
(13)
and, on taking the complex conjugate, then:
Z
=
EH
Z
1 0 exp(−b2 t) µ̂2 b2 (J12 + Y12 ) + µ̂b (J0 J1 + Y0 Y1 ) + iµ̂b (J1 Y0 − J0 Y1 )
db
iπ ∞
b
2µ̂b (Y0 Y1 + J0 J1 ) + µ̂2 b2 (J12 + Y12 ) + J02 + Y02
(14)
Similarly, for the path KL, where p = xexp(−iπ) = −x.
Z
=
KL
Z
1 ∞ exp(−b2 t) µ̂2 b2 (J12 + Y12 ) + µ̂b (J0 J1 + Y0 Y1 ) + iµ̂b (J0 Y1 − J1 Y0 )
db
iπ 0
b
2µ̂b (Y0 Y1 + J0 J1 ) + µ̂2 b2 (J12 + Y12 ) + J02 + Y02
(15)
Denoting the real and imaginary parts of the integrand in (14) by Re(A) and
Im(A), respectively; likewise, for KL in (15) respectively by Re(B) and Im(B),
so that (11) can be written:
Z
Z
Ψ(T ) = −
−
EH
1
=
iπ
Z
0
∞
KL
1
exp(−b2 t)
[Re(A) + Im(A)] db −
b
iπ
Z
0
∞
exp(−b2 t)
[Re(B) + Im(B)] db
b
(16)
The Derivation of a Goldstein Formula
and so, from (14) and (15), Re(A)=Re(B) and Im(A)=-Im(B); hence:
Z
2 ∞ exp(−b2 t)
Ψ(T ) =
Im(A)db
iπ 0
b
where
Im(A) =
iµ̂b (J1 Y0 − J0 Y1 )
2µ̂b (Y0 Y1 + J0 J1 ) + µ̂2 b2 (J12 + Y12 ) + J02 + Y02
33
(17)
(18)
Introducing the identities:
Y00 = −Y1 and J00 = −J1
and, using the Wronskian relation:
J0 Y00 − Y0 J00 = 2/πb
returns (17) as the real-valued function, that is:
4µ̂
Ψ(T ) = 2
π
Z
∞
0
exp(−b2 t)
db
b
2µ̂b (Y0 Y1 + J0 J1 ) + µ̂2 b2 (J12 + Y12 ) + J02 + Y02
(19)
and finally, following some algebra, Goldstein’s result (5) is recovered; namely:
4
Ψ(T ) =
µ̂π 2
Z
0
∞
exp(−b2 t)
1
db
b
(bJ1 + J0 /µ̂)2 + (bY1 + Y0 /µ̂)2
(20)
References
[1] L.J. Crane and A.G. McVeigh, Slip flow along an impulsively started
cylinder, Arch. Appl. Mech., 85(2015), 831-836.
[2] S. Goldstein, Some two-dimensional diffusion problems with circular symmetry, Proc. Lond. Math. Soc., 2(34) (1932), 51-88.