International Journal of Pure and Applied Mathematics
Volume 75 No. 4 2012, 455-483
ISSN: 1311-8080 (printed version)
url: http://www.ijpam.eu
AP
ijpam.eu
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES
FLOWS DRIVEN BY A TRANSLATING/ROTATING
PROLATE SPHEROID AT ARBITRARY ANGLES
Dali Kong1 , Zhen Cui2 , Yongxin Pan3 , Keke Zhang4 §
1,2,4 Department
of Mathematical Sciences
University of Exeter
Exeter, UK
3 Institute of Geology and Geophysics
Chinese Academy of Sciences
Beijing, P.R. CHINA
Abstract: We investigate, via the Papkovich-Neuber formulation using prolate spheroidal coordinates, a fully three-dimensional Stokes flow in the exterior
of a prolate spheroid driven by its translation or rotation. The Stokes flow is
primarily characterized by four parameters: the eccentricity E of the spheroid,
the angle of attack γ in the case of translation and two rotating angles α and β
in the case of rotation. Our mathematical analysis comprises the three parts:
(i) derive an analytical three-dimensional solution for the Stokes flow driven
by a translating spheroid at an arbitrary angle γ;
(ii) derive an analytical three-dimensional solution for the Stokes flow driven
by a rotating spheroid with arbitrary angles α and β; and
(iii) derive two analytical formulas for the corresponding drag and torque
as a function of E, α, β and γ.
Received:
October 31, 2011
§ Correspondence
author
c 2012 Academic Publications, Ltd.
url: www.acadpubl.eu
456
D. Kong, Z. Cui, Y. Pan, K. Zhang
AMS Subject Classification: 76D07, 35Q35
Key Words: Stokes flow, prolate spheroid, and spheroidal harmonics
1. Introduction
Coccobacillus, a bacterium having a nearly elongated spheroidal shape, can
swim slowly in liquid, such as water, under the influence of viscous drag forces.
The motion of swimming microorganisms is marked by a very small Reynolds
number Re (see [11]), a dimensionless number defined as
Re =
U0 aρ
,
µ
where U0 is the typical velocity of the bacterium, ρ is the liquid density, a
denotes the semi-major axis of the spheroid and µ is the dynamic viscosity
of the liquid. The size of Re provides a measure of the ratio of inertial to
viscous forces. Since the swimming speed U0 is very low and its characteristic
dimension a is extremely small, the Stokes’ approximation (for example, see [6],
[1], [12]), which neglects the inertial term in the Navier-Stokes equation in the
limit Re → 0, is usually adopted for describing the motion of microorganisms
([5]).
Understanding the dynamics of swimming microorganisms having an elongated spheroidal shape requires mathematical solutions of the Stokes’ flow in
connection with the movement of a spheroid with arbitrary eccentricity in an
infinite expanse of viscous and incompressible fluid. The shape of the spheroid,
as depicted in Figure 1, may be written in the form
y2
z2
x2
+
+
= 1,
a2 (1 − E 2 ) a2 (1 − E 2 ) a2
(1)
where E is its eccentricity with 0 < E < 1. Two different types of movement
need to be considered. First, the spheroid makes slow translation in a viscous
fluid and, hence, drives a small-Reynolds-number flow in its exterior, which is
illustrated in Figure 1(a). In general, there exists an angle, γ, between the
direction of translation and the symmetry axis z, which is usually referred to
as angle of attack. Second, a flow can be also driven by a spheroid that is
slowly rotating with an angular velocity Ω, which is depicted in Figure 1(b).
Note that the angle between the symmetry axis z and the angular velocity Ω,
denoted by α, is generally non-zero. Mathematically, the problem of the slow
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
(a)
457
(b)
Figure 1: Sketch of geometry for the Stokes flow in the exterior of a
prolate spheroid: (a) driven by a spheroid moving with a speed U0 =
|U0 | at an arbitrary angle of attack γ, and (b) driven by a rotating
spheroid with an angular velocity Ω marked by two rotating angles
α and β. The bounding surface of the spheroid is described by (1)
in Cartesian coordinates or by ξ = ξ0 = 1/E in prolate spheroidal
coordinates which is discussed in Section 2.
flow is governed by the Stokes equation and the equation of continuity,
(
µ∇2 u = ∇p,
∇ · u = 0,
(2)
where u is the velocity of the flow and p is the pressure, subject to the conditions
that the velocity u coincides with the bounding surface of the spheroid at each
of its points and u → 0 far away from the spheroid.
A classical, well-known Stokes flow is concerned with a sphere i.e., E = 0
in (1), that is immersed in an infinite expanse of very viscous, incompressible
fluid and moving slowly (see, for example, [1]). Spherical geometry removes the
angle of attack γ as a dependent parameter and allows the introduction of a
458
D. Kong, Z. Cui, Y. Pan, K. Zhang
two-dimensional stream function ψ satisfying
2
2
∂
sin θ ∂
1 ∂
+
ψ = 0,
∂r 2
r 2 ∂θ sin θ ∂θ
(3)
where (r, θ, φ), with the corresponding unit vectors (r̂, θ̂, φ̂), are spherical polar
coordinates with θ = 0 at the direction of its movement and r = 0 at the center
of the sphere. Upon writing the stream function ψ as
ψ(r, θ) = f (r) sin2 θ,
the fourth-order partial differential equation (3) for ψ can be reduced a fourthorder ordinary differential equation for f (r) that can be readily solved. The
similar idea was also employed to solve the Stokes problem in triaxial ellipsoidal
geometry ([8]), but the earlier work focused on the symmetric problem in that
the symmetry axis of the object and the uniform flow at distant points (or the
rotation axis in the case of rotational flow) is parallel, i.e., γ = 0 or α = 0
. Detailed discussion and bibliography on the earlier work about spherical or
symmetric ellipsoidal Stokes flows can be found in the Lamb’s book ([6]).
Payne and Pell ([10]) considered the Stokes problem in which the configuration of various obstacles has an axis of symmetry and the uniform flow at
distant points is parallel to the symmetry axis (see also, for example, [14]). In
other words, the Stokes flows are assumed to be axisymmetric with the attack
angle γ = 0 as illustrated in Figure 1(a). By expanding the stream function ψ in
terms of the products combining Gegenbauer functions of various degrees, Dassios and Vafeas ([3]) recently studied the Stokes flow passing a spheroid under
the assumption that it moves parallel to the symmetry axis (γ = 0) and, hence,
the flow is axisymmetric. For a non-zero angle of attack, 0 < γ < 900 , the
spheroidal Stokes flow becomes fully three-dimensional and, consequently, the
approach of employing a scalar stream function ψ seems to be difficult. Chwang
and Wu ([2]) successfully employed the singularity method to construct exact
solutions to the Stokes-flow problem for a spheroid translating or rotating in a
viscous fluid, but the form of the solution with cartesian coordinates is inconvenient for the study of our bacteria swimming problem.
Upon recognizing similarity between the governing equations for an elastic
material and a Stokes flow, Tran-Cong and Blake ([15]) applied the PapkovichNeuber formulation ([9], [7]) to the problem of Stokes flows. They showed that
the general solution of the Papkovich-Neuber type for the Stokes problem (2)
can be written in the form
(
u = ∇(r · Ψ + χ) − 2Ψ,
(4)
p = 2µ(∇ · Ψ),
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
459
where r is the position vector, Ψ, a vector harmonic function, satisfies ∇2 Ψ =
0 and χ, a scalar harmonic function, is a solution to ∇2 χ = 0. They also
provided a mathematical proof of the existence and completeness for the general
solution (4). The powerful Papkovich-Neuber formulation (4) has been then
successfully applied to the three-dimensional Stokes flow within two infinite
cones with coincident apices ([4]) and to the three-dimensional Stokes flow
between concentric spheres ([13]). However, the Papkovich-Neuber formulation
– which would produce the analytical solution in terms of prolate spheroidal
coordinates appropriate for describing the swimming motion of bacteria – has
not been applied to deriving a three-dimensional solution for the spheroidal
Stokes flow with arbitrary angles of α, β and γ. The mathematical complication
and difficulty of deriving such a solution with the Papkovich-Neuber formulation
stem from both spheroidal geometry/coordinates and three-dimensionality that
make the relevant analysis lengthy and cumbersome.
It is thus desirable to apply the Papkovich-Neuber formulation to deriving
the three-dimensional solution that describes the Stokes flow driven by the
translation of a prolate spheroid with arbitrary eccentricity E at an arbitrary
angle γ shown in Figure 1(a) or driven by a rotating spheroid with arbitrary
angles α and β illustrated in Figure 1(b). The primary objective of the present
study is to obtain, via the Papkovich-Neuber formulation (4), an analytical
three-dimensional solution for the Stokes flow driven by either a translating
prolate spheroid at an arbitrary angle of attack γ or a rotating spheroid with
arbitrary angles α and β in an infinite expanse of viscous and incompressible
fluid. We shall also derive an expression for the corresponding drag and torque
on the spheroid as a function of α, β, γ and E. In what follows we shall begin in
§2 by presenting briefly prolate spheroidal coordinates used in our analysis. This
is followed by deriving three-dimensional solutions for the spheroidal Stokes flow
and by obtaining an expression for the drag and torque in §3. The paper closes
in §4 with a summary and some remarks.
2. Prolate Spheroidal Coordinates
It would be helpful to provide a brief introduction to prolate spheroid coordinates that are used in the mathematical analysis of this paper. Our prolate
spheroid coordinates are defined by three sets of orthogonal level surfaces: the
radial coordinate ξ ∈ [ξ0 , ∞) characterizes spheroidal surfaces
x2 + y 2
z2
= 1,
+
c2 ξ 2 c2 (ξ 2 − 1)
460
D. Kong, Z. Cui, Y. Pan, K. Zhang
the angular coordinate η ∈ [−1, 1] determines hyperboloids
x2 + y 2
z2
−
= 1,
c2 η 2 c2 (1 − η 2 )
and, finally, the third coordinate is azimuthal angle φ which is the same as that
in spherical polar coordinates. Here c is the common focal length for all the
spheroids and hyperboloids, the bounding surface of the spheroid is described
by ξ = ξ0 = 1/E in Figure 1 and the domain of the Stokes flow in the exterior
of the spheroid is defined by {ξ0 ≤ ξ < ∞, −1 ≤ η ≤ 1, 0 ≤ φ ≤ 2π}. In the
spherical limit, we have E → 0, c → 0 but cξ0 → a along with cξ → r, η → cos θ
and φ → φ.
The transformation between prolate spheroid coordinates (ξ, η, φ) and the
corresponding Cartesian coordinates (x, y, z) is given by
p

2
2

 x = cp(ξ − 1)(1 − η ) cos φ,
(5)
y = c (ξ 2 − 1)(1 − η 2 ) sin φ,


z = cξη.
We shall use (ξ̂, η̂, φ̂) to denote the unit vectors for prolate spheroid coordinates
while (x̂, ŷ, ẑ) for the Cartesian unit vectors. With the above transformation,
we can derive, for example, the following differential operators needed in our
analysis:
s
s
1 1 − η 2 ∂V
1 ξ 2 − 1 ∂V
+ η̂
∇V = ξ̂
c ξ 2 − η 2 ∂ξ
c ξ 2 − η 2 ∂η
∂V
1
1
,
+ φ̂ p
c (ξ 2 − 1)(1 − η 2 ) ∂φ
1
∂
∂V
∂
2
2 ∂V
∇2 V = 2 2
(ξ
−
1)
+
(1
−
η
)
c (ξ − η 2 ) ∂ξ
∂ξ
∂η
∂η
2
1
∂ V
+ 2 2
,
2
c (ξ − 1)(1 − η ) ∂φ2
h
i
p
1
∂
2 − 1)(ξ 2 − η 2 )
(ξ
ξ̂
·
F
∇·F =
c(ξ 2 − η 2 ) ∂ξ
i
p
∂ h
2
2
2
η̂ · F (1 − η )(ξ − η )
+
∂η
∂
1
φ̂ · F,
+ p
2
2
∂φ
c (ξ − 1)(1 − η )
where V is a scalar function while F denotes a vector function.
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
461
3. 3D Spheroidal Stokes Flow, Drag and Torque
3.1. Scalar and Vector Prolate Spheroidal Harmonics
The Papkovich-Neuber formulation (4) requires a solution to the scaler harmonic equation
∇2 χ = 0.
(6)
This part of the analysis is relatively straightforward as, by standard separation
of variables,
χ(ξ, η, φ) = G(ξ)H(η)Φ(φ),
one can readily show that the harmonic equation (6) gives rise to the three
equations
m2
d
2 dG
G + l(l + 1)G = 0, ξ > ξ0 ;
(1 − ξ )
−
dξ
dξ
1 − ξ2
m2
d
2 dH
H + l(l + 1)H = 0, − 1 ≤ η ≤ 1;
(1 − η )
−
dη
dη
1 − η2
d2 Φ
+ m2 Φ = 0, 0 ≤ φ ≤ 2π,
dφ2
whose solutions are given by the associated Legendre functions of the first (Plm )
and second (Qm
l ) kind, where l and m are integers. Applying the conditions at
ξ → ∞ and η = ±1, it can be shown that the general solution for (6) in terms
of prolate spheroidal harmonics is
χ(ξ, η, φ) = c
l
∞ X
X
m
′
sin mφ)Qm
(Dlm cos mφ + Dlm
l (ξ)Pl (η),
l=0 m=0
ξ ≥ ξ0 ,
(7)
′
where c is a scaling factor and Dlm and Dlm
are unknown coefficients to be
determined by the non-slip condition at the bounding surface ξ = ξ0 .
An essential but much more difficult task is to solve the vector harmonic
equation
∇2 Ψ = 0
(8)
in prolate spheroidal coordinates. The vector equation (8) can be written in
component form:
0 = ∇2 (ξ̂ · Ψ) −
h 2ξ 2 (ξ 2 − 1) + (1 − η 2 )
1
ξ̂ · Ψ
c2 (ξ 2 − η 2 )
(ξ 2 − 1)(ξ 2 − η 2 )
462
D. Kong, Z. Cui, Y. Pan, K. Zhang
+
+
+
0 =
−
−
−
0 =
−
2η
p
(ξ 2 − 1)(1 − η 2 ) ∂ η̂ · Ψ
ξ 2 − η2
∂ξ
s
(ξ 2 − 1)(1 − η 2 ) ∂ η̂ · Ψ
2ξη
ξ2 − 1
−
η̂ · Ψ
ξ 2 − η2
∂η
ξ 2 − η2 1 − η2
s
2ξ
ξ 2 − η 2 ∂ φ̂ · Ψ i
,
ξ 2 − 1 1 − η 2 ∂φ
h (ξ 2 − 1) + 2η 2 (1 − η 2 )
1
∇2 (η̂ · Ψ) − 2 2
η̂ · Ψ
c (ξ − η 2 )
(1 − η 2 )(ξ 2 − η 2 )
p
(ξ 2 − 1)(1 − η 2 ) ∂ ξ̂ · Ψ
2η
ξ 2 − η2
∂ξ
s
p
(ξ 2 − 1)(1 − η 2 ) ∂ ξ̂ · Ψ
2ξη
1 − η2
2ξ
−
ξ̂ · Ψ
ξ 2 − η2
∂η
ξ 2 − η2 ξ 2 − 1
s
2η
ξ 2 − η 2 ∂ φ̂ · Ψ i
,
1 − η2
ξ 2 − 1 ∂φ
h
ξ 2 − η2
1
φ̂ · Ψ
∇2 (φ̂ · Ψ) − 2 2
c (ξ − η 2 ) (ξ 2 − 1)(1 − η 2 )
s
s
2ξ
2η
ξ 2 − η 2 ∂ ξ̂ · Ψ
ξ 2 − η 2 ∂ η̂ · Ψ i
.
+
ξ 2 − 1 1 − η 2 ∂φ
1 − η2
ξ 2 − 1 ∂φ
2ξ
p
(9)
(10)
(11)
Evidently, finding an analytical solution to the three scalar equations(9)–(11)
in prolate spheroidal coordinates is not straightforward. It is significant to note
that, however, because of spheroidal geometry and the non-slip condition, all
the quantities in the Papkovich-Neuber formulation (4) must be expressed in
prolate spheroidal coordinates.
After making several attempts in various ways, it is unveiled that a mathematically convenient way of tackling (8) is to adopt a Cartesian system first and,
then, transform it to the prolate spheroidal system by using the transformation
s
s

2

1
−
η
ξ2 − 1


x̂
=
ξ
cos
φ
ξ̂
−
η
cos φη̂ − sin φφ̂,



ξ 2 − η2
ξ 2 − η2



s
s


1 − η2
ξ2 − 1
(12)
sin
φ
ξ̂
−
η
sin φη̂ + cos φφ̂,
ŷ = ξ

ξ 2 − η2
ξ 2 − η2



s
s



2−1

ξ
1 − η2


ξ̂ + ξ
η̂.
 ẑ = η
2
2
ξ −η
ξ 2 − η2
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
463
We find that, after a considerable effort and lengthy analysis, the general solution to (9)–(11) in prolate spheroidal coordinates satisfying the conditions at
ξ → ∞ and η = ±1, is
ξ̂ · Ψ =
∞ X
l h
X
l=0 m=0
+ ξ
s
+ η
s
ξ
s
1 − η2
cos φ(Alm cos mφ + A′lm sin mφ)
ξ 2 − η2
1 − η2
′
sin φ(Blm cos mφ + Blm
sin mφ)
ξ 2 − η2
i
ξ2 − 1
′
sin
mφ)
(C
cos
mφ
+
C
lm
lm
ξ 2 − η2
m
(13)
× Qm
l (ξ)Pl (η),
s
∞
l
XXh
ξ2 − 1
cos φ(Alm cos mφ + A′lm sin mφ)
−η
η̂ · Ψ =
ξ 2 − η2
l=0 m=0
s
ξ2 − 1
′
sin φ(Blm cos mφ + Blm
sin mφ)
− η
ξ 2 − η2
s
i
1 − η2
′
sin
mφ)
+ ξ
(C
cos
mφ
+
C
lm
lm
ξ 2 − η2
m
× Qm
l (ξ)Pl (η),
∞ X
l h
X
− sin φ(Alm cos mφ + A′lm sin mφ)
φ̂ · Ψ =
(14)
l=0 m=0
i
′
m
sin mφ) Qm
+ cos φ(Blm cos mφ + Blm
l (ξ)Pl (η).
(15)
By using the vector harmonic function Ψ given by (13)–(15), we can now derive
r · Ψ and ∇ · Ψ which are needed in the Papkovich-Neuber formulation:
r·Ψ = c
∞ X
l hp
X
l=0 m=0
(ξ 2 − 1)(1 − η 2 ) cos φ
× (Alm cos mφ + A′lm sin mφ)
p
′
(ξ 2 − 1)(1 − η 2 ) sin φ(Blm cos mφ + Blm
sin mφ)
+
i
′
m
+ ξη(Clm cos mφ + Clm
sin mφ) Qm
l (ξ)Pl (η)
(16)
464
D. Kong, Z. Cui, Y. Pan, K. Zhang
and
∇·Ψ =
×
×
+
+
+
×
−
×
+
p
∞
l
1 X X n (ξ 2 − 1)(1 − η 2 )
c
ξ 2 − η2
l=0 m=0
dPlm (η) m
dQm
l (ξ) m
Pl (η) − η
Ql (ξ)
ξ
dξ
dη
cos φ(Alm cos mφ + A′lm sin mφ)
′
sin mφ
sin φ Blm cos mφ + Blm
dQm
1
2
l (ξ) m
Pl (η)
η(ξ
−
1)
2
2
ξ −η
dξ
m
2 dPl (η) m
ξ(1 − η )
Ql (ξ)
dη
′
sin mφ
Clm cos mφ + Clm
m
m
p
Qm
l (ξ)Pl (η)
2
2
(ξ − 1)(1 − η )
h
sin φ(−Alm sin mφ + A′lm cos mφ)
io
′
cos mφ) .
cos φ(−Blm sin mφ + Blm
(17)
All coefficients in the above expressions, such as Alm and A′lm , need to be
determined as a function of three characteristic angles α, β and γ of the problem.
Our remaining task is, according to whether a prolate spheroid is in translation
or rotation, to derive a three-dimensional solution for the corresponding Stokes
flow by determining all the unknown coefficients.
3.2. Flow Driven by Translation at an Arbitrary Angle γ
Consider first a prolate spheroid of eccentricity E moving slowly with the speed
U0 = |U0 | at an angle 0 ≤ γ ≤ 90o . Geometry of the problem, together with
the coordinate system, is illustrated in Figure 1(a). Without loss of generality,
we shall assume the translation velocity U0 has only x and z components, i.e.,
the spheroid is always moving within the xoz plane.
The no-slip boundary condition at the bounding surface of the spheroid
ξ = ξ0 imposes that
u(η, ξ = ξ0 , φ) = [∇(r · Ψ + χ) − 2Ψ] (η, ξ = ξ0 , φ)
s
s
!
ξ02 − 1
1 − η2
η + sin γ
ξ0 cos φ ξ̂
= U0 cos γ
ξ02 − η 2
ξ02 − η 2
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
+ U0 cos γ
s
1 − η2
ξ0 − sin γ
ξ02 − η 2
s
465
!
ξ02 − 1
η cos φ η̂
ξ02 − η 2
− U0 (sin γ sin φ) φ̂,
(18)
where ∇χ(η, ξ = ξ0 , φ) can be obtained from (7) while ∇r · Ψ(η, ξ = ξ0 , φ)
and Ψ(η, ξ = ξ0 , φ) can be
pderived from (13)–(16). By noticing that P0 (η) =
1,P1 (η) = η and P11 (η) = 1 − η 2 , we can deduce from (18) that
′
′
= 0, if l ≥ 2.
= Dlm = Dlm
Alm = A′lm = Blm = Blm
In consequence, the expressions (7) and (13)–(16) can be simplified as
h
χ(ξ, η, φ) = c D00 Q0 (ξ) + D10 Q1 (ξ)η
i
p
′
+ (D11 cos φ + D11
sin φ)Q11 (ξ) 1 − η 2 ,
ξ̂ · Ψ =
+
+
+
+
+
+
×
η̂ · Ψ =
s
1 − η2
1 − η2
cos
φA
+
ξ
sin φB00
ξ
00
ξ 2 − η2
ξ 2 − η2
s
!
ξ2 − 1
η
C00 Q0 (ξ)
ξ 2 − η2
s
s
1 − η2
1 − η2
ξ
cos
φA
+
ξ
sin φB10
10
ξ 2 − η2
ξ 2 − η2
s
!
ξ2 − 1
η
C10 Q1 (ξ)η
ξ 2 − η2
" s
1 − η2
ξ
cos φ(A11 cos φ + A′11 sin φ)
ξ 2 − η2
s
1 − η2
′
sin φ(B11 cos φ + B11
sin φ)
ξ
ξ 2 − η2
s
#
ξ2 − 1
′
η
(C11 cos φ + C11
sin φ)
ξ 2 − η2
p
Q11 (ξ) 1 − η 2 ,
s
s
ξ2 − 1
ξ2 − 1
cos
φA
−
η
sin φB00
−η
00
ξ 2 − η2
ξ 2 − η2
(19)
s
(20)
466
D. Kong, Z. Cui, Y. Pan, K. Zhang
+
+
+
+
−
+
×
s
!
1 − η2
C00 Q0 (ξ)
ξ
ξ 2 − η2
s
s
ξ2 − 1
ξ2 − 1
−η
cos
φA
−
η
sin φB10
10
ξ 2 − η2
ξ 2 − η2
s
!
1 − η2
ξ
C10 Q1 (ξ)η
ξ 2 − η2
" s
ξ2 − 1
−η
cos φ(A11 cos φ + A′11 sin φ)
ξ 2 − η2
s
ξ2 − 1
′
η
sin φ(B11 cos φ + B11
sin φ)
ξ 2 − η2
s
#
1 − η2
′
(C11 cos φ + C11
sin φ)
ξ
ξ 2 − η2
p
Q11 (ξ) 1 − η 2 ,
(21)
φ̂ · Ψ = (− sin φA00 + cos φB00 )Q0 (ξ)
+ (− sin φA10 + cos φB10 )Q1 (ξ)η
+ − sin φ(A11 cos φ + A′11 sin φ) + cos φ(B11 cos φ
p
′
+ B11
sin φ) Q11 (ξ) 1 − η 2 ,
np
r·Ψ = c
(ξ 2 − 1)(1 − η 2 ) (cos φA00 + sin φB00
(22)
+ξηC00 ) Q0 (ξ)
p
(ξ 2 − 1)(1 − η 2 ) (cos φA10 + sin φB10
+
+ξηC10 ) Q1 (ξ)η
hp
(ξ 2 − 1)(1 − η 2 ) cos φ(A11 cos φ + A′11 sin φ)
+
p
′
sin φ)
+ (ξ 2 − 1)(1 − η 2 ) sin φ(B11 cos φ + B11
i
o
p
′
+ξη(C11 cos φ + C11
sin φ) Q11 (ξ) 1 − η 2 ,
(23)
which now contain only 16 unknown coefficients.
To determine the 16 unknown coefficients in (19)–(23), we first look at the
azimuthal component, the simplest of the three components, of (18):
1
∂ r·Ψ+χ
φ̂ · u(ξ = ξ0 ) = p
− 2φ̂ · Ψ
c
(ξ 2 − 1)(1 − η 2 ) ∂φ
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
467
(a)
(b)
(c)
(d)
Figure 2: Flow structure in the xz−plane for a spheroid of eccentricity
E = 0.8, computed from (31)–(33), is plotted for four different angles
of attack: (a) γ = 0; (b) γ = 30o ; (c) γ = 60o ; and (d) γ = 90o . The
symmetry axis of the spheroid is vertical with the dashed line indicating
the direction of translation.
= [− sin 2φA11 + cos 2φA′11 + cos 2φB11
p
′
+ sin 2φB11
]Q11 (ξ) 1 − η 2
ξη
′
+p
(−C11 sin φ + C11
cos φ)Q11 (ξ)
ξ2 − 1
1
′
(−D11 sin φ + D11
cos φ)Q11 (ξ)
+p
2
ξ −1
468
D. Kong, Z. Cui, Y. Pan, K. Zhang
+(A00 sin φ − B00 cos φ)Q0 (ξ)
+(A10 sin φ − B10 cos φ)Q1 (ξ)η
+2[sin φ(A11 cos φ + A′11 sin φ)
p
′
− cos φ(B11 cos φ + B11
sin φ)]Q11 (ξ) 1 − η 2
= U0 (sin γ sin φ) ,
which immediately leads to
′
′
′
A10 = A11 = A′11 = B00 = B10 = B11 = B11
= C11 = C11
= D11
=0
and
Q1 (ξ0 )
= −U0 sin γ.
A00 Q0 (ξ0 ) − D11 p 12
ξ0 − 1
(24)
(25)
In other words, of the 16 unknown coefficients in (19)–(23) the ten of them are
zero. The 6 remaining coefficients can be determined by examining the ξ̂ and
η̂ components of (18), which are
s
ξ2 − 1 ∂ r · Ψ + χ
ξ̂ · u(ξ = ξ0 ) =
− 2ξ̂ · Ψ
ξ 2 − η 2 ∂ξ
c
s
!
( s
2
ξ −1
1 − η2
cos φA00 + ηC00 Q0 (ξ)
ξ
=
ξ 2 − η2
ξ 2 − η2
p
(ξ 2 − 1)(1 − η 2 ) cos φA00 + ξηC00 + D00
+
×
dQ0 (ξ)
dξ
dQ1 (ξ)
+ C10 Q1 (ξ)η 2 + (ξηC10 + D10 )η
dξ
1
p
dQ (ξ)
1 − η 2 cos φ
+D11 1
dξ
s
s
!
1 − η2
ξ2 − 1
−2 ξ
cos φA00 + η
C00 Q0 (ξ)
ξ 2 − η2
ξ 2 − η2
s
ξ2 − 1
−2η
C10 Q1 (ξ)η
ξ 2 − η2
s
s
!
ξ02 − 1
1 − η2
= U0 cos γ
η + sin γ
ξ0 cos φ
(26)
ξ02 − η 2
ξ02 − η 2
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
469
and
η̂ · u(ξ = ξ0 ) =
s
1 − η2 ∂
ξ 2 − η 2 ∂η
r·Ψ+χ
c
− 2η̂ · Ψ
s
1 − η2
ξ 2 − η2
η cos φ
p
+ A00 ξ 2 − 1Q0 (ξ) − D11 Q11 (ξ) p
ξ 2 − η2
s
1 − η2
= U0 cos γ
ξ0
ξ02 − η 2
s
!
ξ02 − 1
− sin γ
η cos φ .
ξ02 − η 2
= (−C00 ξQ0 (ξ) + D10 Q1 (ξ))
(27)
From (26)–(27) we can deduce that
C10 = D00 = 0
and
dQ1 (ξ0 )
dQ0 (ξ0 )
− Q0 (ξ0 ) + D10
= U0 cos γ,
C00 ξ0
dξ
dξ
dQ1 (ξ0 )
dQ0 (ξ0 )
− Q0 (ξ0 ) + D10
= U0 cos γ,
C00 ξ0
dξ
dξ
p
2
ξ0 − 1 dQ0 (ξ0 )
ξ02 − 1 dQ11 (ξ0 )
A00
− Q0 (ξ0 ) + D11
ξ0
dξ
ξ0
dξ
= U0 sin γ.
(28)
(29)
(30)
Solving the linear system of equations (25) and (28)–(30) gives the 4 non-zero
coefficients:
2
−1
ξ0 − 3 ξ0 + 1 ξ0
A00 = U0 sin γ
ln
−
,
4
ξ0 − 1
2
−1
2
ξ0 + 1 ξ0 + 1
ln
+ ξ0
,
C00 = U0 cos γ −
2
ξ0 − 1
2
−1
ξ + 1 ξ0 + 1
1
D10 = U0 cos γ 0 2 ln
,
−
ξ0 − 1 ξ0
2ξ0
−1
2
ξ0 + 1
ξ0
ξ0 − 3
ln
−
.
D11 = U0 sin γ
2(ξ02 − 1) ξ0 − 1 ξ02 − 1
470
D. Kong, Z. Cui, Y. Pan, K. Zhang
It follows, from the Papkovich-Neuber formulation (4), that the three-dimensional
solution describing a Stokes flow driven by the translation of a prolate spheroid
with arbitrary eccentricity E at an arbitrary angle γ is


s
ξ+1
ξ
ξ+1
ξ
1
1
2
ln
+
ln
−
ξ̂ · u
ξ − 1  2 ξ−1 ξ 2 −1
2
ξ−1
ξ 2 −1 
=
+ ξ 2 +1
η cos γ
ξ
+1
1
U0
ξ 2 − η 2 ξ02 +1 ln ξ0 +1 − ξ
0
0
ln
−
0
2
2
ξ0 −1
ξ
−1
ξ
2ξ0
0
0


p
ξ+1
ξ 2 −2
ξ
ξ+1
ξ
2
ln
−
ln
−
1
−
1−η 
2
ξ−1
2
ξ−1
ξ 2 −1

+ ξ 2 −3
+p
2 −3
ξ
2
2
ξ
+1
ξ
ξ0
ξ
+1
0
0
0
0
0
ξ −η
ln ξ0 −1 − ξ 2 −1
4 ln ξ0 −1 − 2
2(ξ 2 −1)
0
× cos φ sin γ,

p
ξ
ξ+1
2
1−η 
η̂ · u
2 ln ξ−1
=p
+
2
U0
ξ 2 − η 2 ξ0 +1 ln ξ0 +1 − ξ0
2
ξ0 −1
 √
√
+p
ξ 2 −1
ln ξ+1

ξ−1
 2 2
−

ξ
−3
2
2
ξ
+1
ξ0
0
0
ξ −η
ln
−
4
ξ0 −1
2
1
× η cos φ sin γ,

ξ+1
1
φ̂ · u 
2 ln ξ−1
= ξ 2 −3
ξ0 +1
U0
0
4 ln ξ0 −1 −
ξ0
2
−
× sin φ sin γ,
s
2h ξ
p
1 − η2
=−
2
2
µU0
c ξ −η
ξ2 − 1
+
ξ 2 −1
2
0
(31)
ξ
ξ+1
2 ln ξ−1 − 1
ξ02 +1
1
ln ξξ00 +1
−1 − ξ0
2ξ02
ln
ξ02 −3
2(ξ02 −1)
ξ+1
ξ−1
−

 cos γ

√ξ 
ξ 2 −1
+1
−
ln ξξ00 −1
ξ0
ξ02 −1


(32)

ξ+1
ξ
1
2 ln ξ−1 − ξ 2 −1

ξ02 −3
ξ0
ξ0 +1
−
ln
ξ0 −1
2(ξ02 −1)
ξ02 −1
sin γ
ξ02 −3
ξ0 +1
4 ln ξ0 −1 −
i
η
cos γ
.
ξ 2 − η 2 − ξ02 +1 ln ξ0 +1 + ξ
0
2
ξ0 −1
ξ0
2
(33)
cos φ
(34)
Computed from the expressions (31)–(33), Figure 2 depicts the structure of the
Stokes flow in the xz−plane for a spheroid of eccentricity E = 0.8 at four different angles, γ = 0, 30o , 60o and 90o . It can be seen that, while the axisymmetry
of the flow (independent of φ) is clearly displayed when γ = 0, the Stokes flow
becomes fully three-dimensional when γ 6= 0.
It may be worth mentioning that the classical solution for spherical geometry (see, for example, Lamb, 1932) can be recovered by taking the limits in
(31)–(34),
E → 0, γ → 0, cξ0 → a = r0 , η → cos θ, cξ → r,
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
471
which yield

r03
3r0


−
U0 cos θ,
r̂
·
u
=


2r
2r 3





3r0
r03

θ̂ · u =
+ 3 U0 sin θ,
(35)
4r
4r




φ̂ · u =0,





3

p = 2 µr0 U0 cos θ,
2r
in spherical polar coordinates, where r0 = a denotes the radius of the sphere.
3.3. Flow Driven by Rotation at Arbitrary Angles
Consider now the three-dimensional Stokes flow driven by a prolate spheroid
that is slowly rotating with the angular velocity Ω. In prolate spheroidal coordinates, Ω can be written in the form
s
Ω
1 − η2
=
sin α cos βξ
cos φ
Ω0
ξ 2 − η2
s
s
1 − η2
ξ2 − 1 sin
φ
+
cos
αη
ξ̂
+ sin α sin βξ
ξ 2 − η2
ξ 2 − η2
s
ξ2 − 1
cos φ
+
− sin α cos βη
ξ 2 − η2
s
s
2
ξ −1
1 − η2 sin φ + cos αξ
η̂
− sin α sin βη
ξ 2 − η2
ξ 2 − η2
+ (− sin α cos β sin φ + sin α sin β cos φ) φ̂,
(36)
where α ∈ [0, π] and β ∈ [0, 2π]. The geometry of the problem, as well as the
definition of α and β, is shown in Figure 1(b).
In this case, the no-slip boundary condition imposes the following condition
at the bounding surface, ξ = ξ0 , of the spheroid:
u(η, ξ = ξ0 , φ) = [∇(r · Ψ + χ) − 2Ψ] (η, ξ = ξ0 , φ)
n
= cΩ0 (− sin α cos β sin φ + sin α sin β cos φ)
s
1 − η2
ξ̂
×η
ξ02 − η 2
472
D. Kong, Z. Cui, Y. Pan, K. Zhang
(a)
(b)
(c)
(d)
Figure 3: Flow structure in the plane perpendicular to, and viewed
from, the axis of rotation with a spheroid of eccentricity E = 0.8 is
plotted at β = 0 for four different angles of α: (a) α = 0, (b) α = 30,
(c) α = 60 and (d) α = 90.
+ (sin α cos β sin φ − sin α sin β cos φ)
s
ξ02 − 1
×ξ0
η̂
ξ02 − η 2
+ − (sin α cos β cos φ + sin α sin β sin φ)ξ0 η
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
q
o
+ cos α (ξ02 − 1)(1 − η 2 ) φ̂ .
473
(37)
Here ∇χ(η, ξ = ξ0 , φ) can be obtained from (7), while ∇r · Ψ(η, ξ = ξ0 , φ) and
Ψ(η, ξ = ξ0 , φ) can be derived from (13)–(16). Similar to the case of translation,
we can also deduce from (37) that
′
′
Alm = A′lm = Blm = Blm
= Dlm = Dlm
= 0, if l ≥ 2.
It follows there also exist 16 unknown coefficients to be determined by the nonslip condition (37) at ξ = ξ0 . First, the azimuthal component of (37) gives the
equation
′
− sin 2φA11 + cos 2φA′11 + cos 2φB11 + sin 2φB11
p
× Q11 ξ 1 − η 2
ξη
′
(−C11 sin φ + C11
cos φ)Q11 (ξ)
+ p
ξ2 − 1
1
′
+ p
(−D11 sin φ + D11
cos φ)Q11 (ξ)
2
ξ −1
+ (sin φA00 − cos φB00 )Q0 (ξ) + (sin φA10 − cos φB10 )Q1 (ξ)η
′
+ sin 2φA11 + 2 sin2 φA′11 − 2 cos2 φB11 − sin 2φB11
p
× Q11 ξ 1 − η 2
= cΩ0 − sin α cos β cos φξ0 η − sin α sin β sin φξ0 η
q
+ cos α (ξ02 − 1)(1 − η 2 ) ,
(38)
from which we can obtain that
′
A00 = B00 = D11 = D11
=0
and
−B10
A10
1
Q1 (ξ0 )
′ Q1 (ξ0 )
p
= − sin α cos βΩ0 c,
+ C11
ξ0
ξ02 − 1
Q1 (ξ0 )
Q1 (ξ0 )
= − sin α sin βΩ0 c,
− C11 p 12
ξ0
ξ0 − 1
Q1 (ξ0 )
Q1 (ξ0 )
− B11 p 12
= cos αΩ0 c.
A′11 p 12
ξ0 − 1
ξ0 − 1
From the ξ−component of (37), we obtain that
p
ξ 2 − 1Q1 (ξ)
p
(A10 cos φ + B10 sin φ)
ξ 2 − η2
(39)
(40)
(41)
474
D. Kong, Z. Cui, Y. Pan, K. Zhang
ξQ1 (ξ)
′
p 1
(C11 cos φ + C11
sin φ)
ξ 2 − η2
p
p
C00 1 − η 2 ξQ0 (ξ) D10 1 − η 2 Q1 (ξ)
p
p
−
+
ξ 2 − η2
ξ 2 − η2
s
ξ02 − 1
,
= (sin α cos β sin φ − sin α sin β cos φ)ξ0
ξ02 − η 2
−
which, after making use of (39) and (40), leads to
C00 = D10 = 0.
Finally, the ξ−component of (37) yields the equation:
n
h
p
p
1
p
− ηQ1 (ξ) C10 η ξ 2 − 1 + A10 ξ 1 − η 2 cos φ
ξ 2 − η2
i
p
+ B10 ξ 1 − η 2 sin φ
h
p
p
−Q11 (ξ) C11 η ξ 2 − 1 1 − η 2 cos φ + A11 ξ(1 − η 2 ) cos2 φ
′
+ B11
ξ(1 − η 2 ) sin2 φ
+(A′11 + B11 )ξ sin φ cos φ(1 − η 2 )
i
p
p
′
+C11
η 1 − η 2 ξ 2 − 1 sin φ
p
dQ0 (ξ)
dQ1 (ξ)
+ C10 η 2 ξ ξ 2 − 1
dξ
dξ
p
dQ
(ξ)
1
+A10 η 1 − η 2 (ξ 2 − 1) cos φ
dξ
p
dQ1 (ξ)
+B10 η 1 − η 2 (ξ 2 − 1) sin φ
dξ
p
p
dQ1 (ξ)
+C11 η 1 − η 2 ξ ξ 2 − 1 cos φ 1
dξ
p
p
dQ1 (ξ)
′
+C11
η 1 − η 2 ξ ξ 2 − 1 sin φ 1
dξ
1
dQ (ξ)
+A11 (ξ 2 − 1)(1 − η 2 ) cos2 φ 1
dξ
dQ1 (ξ)
+(A′11 + B11 )(ξ 2 − 1)(1 − η 2 ) sin φ cos φ 1
dξ
o
1
dQ (ξ)
′
+B11
(ξ 2 − 1)(1 − η 2 ) sin2 φ 1
dξ
+D00
p
ξ2 − 1
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
n
= cΩ0 (− sin α cos β sin φ + sin α sin β cos φ)η
s
475
1 − η2
,
ξ02 − η 2
which gives
′
A11 = B11
= C10 = D00 = 0,
A′11 + B11 = 0,
(42)
and
dQ1 (ξ0 )
A10
− 1)
− ξ0 Q1 (ξ0 )
dξ
q
dQ11 (ξ0 )
1
2
− Q1 (ξ0 )
+C11 ξ0 − 1 ξ0
dξ
= sin α sin βΩ0 c,
dQ1 (ξ0 )
2
B10 (ξ0 − 1)
− ξ0 Q1 (ξ0 )
dξ
1
q
dQ1 (ξ0 )
′
1
2
+C11 ξ0 − 1
− Q1 (ξ0 )
dξ
= − sin α cos βΩ0 c.
(ξ02
(43)
(44)
The six remaining non-zero coefficients are then determined by solving the linear
system of equations(39)–(43):
−1
ξ02 + 1 ξ0 + 1
1
−
,
ln
A10 = sin α sin βΩ0 c
2ξ0
ξ0 − 1
4ξ02
2ξ0
ξ0 + 1 −1
A′11 = cos αΩ0 c − 2
,
+ ln
ξ0 − 1
ξ0 − 1
−1
1
ξ02 + 1 ξ0 + 1
ln
B10 = sin α cos βΩ0 c −
,
+
2ξ0
ξ0 − 1
4ξ02
ξ0 + 1 −1
2ξ0
,
− ln
B11 = cos αΩ0 c
ξ0 − 1
ξ02 − 1
−1
ξ0
1 ξ02 + 1 ξ0 + 1
C11 = sin α sin βΩ0 c
,
−
ln
ξ02 − 1 2 ξ02 − 1 ξ0 − 1
−1
1 ξ02 + 1 ξ0 + 1
ξ0
′
.
+
ln
C11 = sin α cos βΩ0 c − 2
ξ0 − 1 2 ξ02 − 1 ξ0 − 1
Substitution of χ, Ψ, r · Ψ and ∇ · Ψ with the six non-zero coefficients into
(4) yields the three-dimensional solution describing the Stokes flow driven by
476
D. Kong, Z. Cui, Y. Pan, K. Zhang
the rotation of a prolate spheroid with arbitrary eccentricity E and arbitrary
rotating angles α and β:

s
ξ+1
1
1 − η2 
ξ̂ · u
2 ln ξ−1
= −η
Ω0 c
ξ 2 − η 2 ξ02 +1 ln ξ0 +1 − 1
ξ0 −1
2ξ0
4ξ02

ξ
+ 21 ln ξ+1
ξ−1
ξ 2 −1
 sin α sin (φ − β),
(45)
− ξ 2 +1
ξ0
ξ0 +1
1 0
−
ln
2
2
2 ξ0 −1
ξ0 −1
ξ0 −1

p
ξ+1
1
ξ 2 − 1  2 ξ ln ξ−1 − 1
η̂ · u
=p
2
+1
Ω0 c
ξ 2 − η 2 ξ0 +1
− 2ξ10
ln ξξ00 −1
4ξ02

ξ+1
ξ2
1
ξ
ln
−
2
ξ−1
ξ 2 −1
 sin α sin (φ − β),
(46)
− ξ 2 +1
ξ0
ξ0 +1
1 0
−
ln
2
2 ξ02 −1
ξ0 −1
ξ0 −1


ξ+1
ξ2
ξ+1
1
1
ξ
ln
−
ξ
ln
−
1
η̂ · u
2
ξ−1
ξ 2 −1
2
ξ−1

= η  ξ 2 +1
− ξ 2 +1
ξ
+1
ξ
ξ
1 0
1
Ω0 c
0
0 +1
0
0
ln
ln
−
−
2 ξ02 −1
ξ0 −1
ξ02 −1
4ξ02
ξ0 −1
2ξ0
× sin α cos (φ − β)
+
ξ+1
ξ
1
2 ln ξ−1 − ξ 2 −1 p 2
(ξ
ξ0 +1
ξ0
1
2 ln ξ0 −1 − ξ02 −1
p
−8η
= 2
µΩ0
ξ − η2
s
− 1)(1 − η 2 )Ωc cos α,
1 − η 2 sin α sin (φ − β)
.
ξ 2 − 1 (ξ02 + 1) ln ξ0 +1 − 2ξ0
(47)
(48)
ξ0 −1
Figure 3 shows the flow structure, computed from (45)–(47) in the plane passing
z = 0 and perpendicular to the axis of rotation Ω, with four different angles of
α at β = 0 for a spheroid of eccentricity E = 0.8. It can be seen in Figure 3 that,
while the flow at α = 0 is axisymmetric (i.e., independent of φ), it becomes
fully three-dimensional when α 6= 0.
As discussed in the previous section for the case of translation, the classical solution for a rotating sphere (see, for example, Lamb, 1932) can be also
recovered by taking the limits E → 0, γ → 0 and cξ0 → a = r0 in (45)–(48):
r̂ · u = 0,
r02
Ω0 r0 sin α sin (φ − β),
r2
i
r0
Ω0 r03 h
sin
θ
cos
α
,
−
cos
θ
sin
α
cos
(φ
−
β)
+
r2
r
θ̂ · u = −
φ̂ · u =
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
477
p = 0.
Since the equations (2) for the Stokes problem are linear, the solution of the
general Stokes flow, driven simultaneously by both the translation and rotation
of a spheroid, can be written as a linear combination of (31)–(34) and (45)–(48).
4. Drag and Torque
On the basis of two solutions for the translation and rotation of a prolate
spheroid, (31)–(34) and (45)–(48), we can compute two important quantities,
the corresponding drag force D and the torque T on the spheroid. For practical
applications, such as to the dynamics of swimming microorganisms, it is usually
helpful to express D and T in the Cartesian coordinate system.
Consider first the drag force D on a translating spheroid at an angle of
attack γ, which may be expressed as
Z
f dS,
D=
S
R
where S denotes the surface integration over the bounding surface S of the
spheroid, f in tensor notation is
fi = (−pδij + 2µσij )nj ,
with nj being unit normal at the bounding surface S and
∂uj
1 ∂ui
+
.
σij =
2 ∂xj
∂xi
By performing an analysis for the tensor σij in prolate spheroidal coordinates,
we can put f at the surface S in the form
s
(
#)
"
p
2µ ξ 2 − 1 ∂ ξ̂ · u
(η̂ · u)η 1 − η 2
p
f =
−p +
ξ̂
−
c
ξ 2 − η2
∂ξ
(ξ 2 − η 2 ) ξ 2 − 1
( s
"
#
p
µ 1 − η 2 ∂ ξ̂ · u
(η̂ · u)ξ ξ 2 − 1
p
+
−
c ξ 2 − η2
∂η
(ξ 2 − η 2 ) 1 − η 2
s
"
#)
p
(ξ̂ · u)η 1 − η 2
µ ξ 2 − 1 ∂ η̂ · u
p
+ uξ
η̂
+
c ξ 2 − η2
∂ξ
(ξ 2 − η 2 ) ξ 2 − 1
478
D. Kong, Z. Cui, Y. Pan, K. Zhang
+
+
(
"
∂ ξ̂ · u
p
− (φ̂ · u)ξ
∂φ
c (ξ 2 − 1)(1 − η 2 )
s
)
µ ξ 2 − 1 ∂ φ̂ · u
φ̂,
c ξ 2 − η 2 ∂ξ
µ
s
1 − η2
ξ 2 − η2
#
which can be then transformed into the corresponding Cartesian coordinates
s
s
(
)
1 − η2
ξ2 − 1
f = ξ̂ · f
ξ cos φ − η̂ · f
η cos φ − φ̂ · f sin φ x̂
ξ 2 − η2
ξ 2 − η2
s
s
)
(
1 − η2
ξ2 − 1
ξ sin φ − η̂ · f
η sin φ + φ̂ · f cos φ ŷ
+ ξ̂ · f
ξ 2 − η2
ξ 2 − η2
s
s
(
)
ξ2 − 1
1 − η2
+ ξ̂ · f η
+ (η̂ · f )ξ
ẑ.
(49)
ξ 2 − η2
ξ 2 − η2
By virtue of the expressions for u given by (31)–(33) and the expression for p
given by (34) and, then, by evaluating them at the bounding surface ξ = ξ0 , we
can obtain f , according to (49), as a function of η and φ in the direction of the
Cartesian coordinates. After a further integration over the spheroidal surface,
we are able to derive an analytical formula for the drag force D on a translating
spheroid:
2 − 1) −2 + ξ ln ξ0 +1
h
8
+
4(ξ
0
0
ξ0 −1
D
= −
ξ
+1
2
0
2πµU0
2ξ0 − (ξ0 − 3) ln
+
−
+
2ξ02
−
ξ0
2
ξ0 (ξ02
−
− 1) ln
ξ02 −3
4
ξ0 −1
ξ0 +1 i
ξ0 −1
+1
ln ξξ00 −1
h 4ξ02 (ξ02 − 1) 2 −
ξ02 −1
ξ0
c sin γ x̂
ln ξξ00 +1
−1
+1
2ξ0 − 2ξ03 + (ξ04 − 1) ln ξξ00 −1
i
(ξ02 − 1) 2 − ξ0 ln ξξ00 +1
−1
c cos γẑ,
ξ02 +1
ξ0 +1
ξ0
−
ln
4
4
ξ0 −1
(50)
which is valid for a prolate spheroid having arbitrary eccentricity E and moving
at an arbitrary angle γ. It can be demonstrated that the general formula (50)
in the spherical limit E → 0 becomes
D = −6πµr0 U0 (sin γ x̂ + cos γẑ),
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
E
0.9000000
0.8526842
0.8053684
0.7580526
0.7107368
0.6634211
0.6161053
0.5687895
0.5214737
0.4741579
0.4268421
0.3795263
0.3322105
0.2848947
0.2375789
0.1902632
0.1429474
0.0956316
0.0483158
0.0010000
x̂ · D/(Uo µ)
-6.0965387
-6.6360387
-7.0631336
-7.4178095
-7.7201962
-7.9820636
-8.2109703
-8.4120920
-8.5891394
-8.7448645
-8.8813600
-9.0002464
-9.1027937
-9.1900036
-9.2626660
-9.3213992
-9.3666784
-9.3988555
-9.4181734
-9.4247751
ẑ · D/(Uo µ)
-8.9913082
-10.1181591
-11.0311030
-11.8015304
-12.4662984
-13.0473907
-13.5591593
-14.0115690
-14.4118497
-14.7654189
-15.0764333
-15.3481365
-15.5830878
-15.7833174
-15.9504341
-16.0857028
-16.1900982
-16.2643445
-16.3089426
-16.3241877
479
|D|/(Uo µ)
10.8633055
12.1001716
13.0985911
13.9391542
14.6632202
15.2953503
15.8515246
16.3428075
16.7772086
17.1607182
17.4979255
17.7924065
18.0469798
18.2638789
18.4448727
18.5913506
18.7043830
18.7847647
18.8330454
18.8495488
Table 1: The values of D/(Uo µ) and |D|/(Uo µ), computed from (50),
for a = 1 and γ = 30o as a function of eccentricity E.
which is consistent with the classical result for a sphere (see, for example,
Batchelor, 1967).
Table 1 shows various values of D/(Uo µ) with a = 1, computed from formula
(50) at the attack angle γ = 30o , for different values of eccentricity E. Of
particular interest is that the drag |D| on a spheroid with E = 0.9 is nearly half
of that for a sphere having the same radius a. In the spherical limit E → 0, the
formula (50) for γ = 30o gives
D/(Uo µ) = −9.4247780x̂ − 16.3241943ẑ, |D|/(Uo µ) = 18.8495559.
Our results suggest that the drag |D| on prolate spheroids with different E but
the same radius a attains its maximum in the spherical limit E → 0 at any
angle of attack γ.
Finally, consider the torque T on a rotating spheroid which may be written
480
D. Kong, Z. Cui, Y. Pan, K. Zhang
as
T=
Z
S
r × f dS,
where f is derived by using the expression (49) but with u given by (45)–(47)
evaluated at the spheroidal surface ξ = ξ0 . After an analysis analogous to that
for the drag D, the torque T on a rotating spheroid is found to be
T =
−8π
+1
−2ξ0 + (ξ02 + 1) ln ξξ00 −1
"
#
2 − 3ξ (ξ 2 − 1) ln ξ0 +1
−4
+
8ξ
0
1
0
0
ξ
−1
0
× 2ξ0 (ξ02 − 1) tanh−1
+
ξ0
3
× Ω0 µc3 (sin α cos β x̂ + sin α sin β ŷ)
#
"
1
32πc3 Ω0 µ(ξ02 − 1)
cos α ẑ,
+
3 −2ξ0 + (ξ02 − 1) ln ξ0 +1
(51)
ξ0 −1
which is valid for a prolate spheroid of arbitrary eccentricity E rotating with
any angles of α and β. It can be also demonstrated that (51) in the spherical
limit E → 0 (cξ0 → r0 ) gives rise to
T = −8πµr03 Ω0 (sin α cos β x̂ + sin α sin β ŷ + cos αẑ),
which is again consistent with the classical result for spherical geometry (see,
for example, Batchelor, 1967).
Table 2 shows the various values of T/(Ω0 µ) and |T|/(Ω0 µ) for a = 1,
computed from formula (51), with rotating angles α = 30o and β = 0 for
different values of eccentricity E. In the spherical limit E → 0, the expression
(51) with α = 30o and β = 0 gives
T/(Ω0 µ) = −12.5663706x̂ − 21.7655924ẑ, |T|/(Ω0 µ) = 25.1327412.
It is of interest to notice that the torque |T| on a spheroid with E = 0.9 and
α = 30o is only about 20% of that on a sphere with the same radius a. Our
results indicate that the torque |T| on spheroids of different E at any rotating
angles α and β attains its maximum in the spherical limit E → 0.
5. Summary and Some Remarks
This work is primarily motivated by the desire to understand the dynamics
of slowly swimming microorganism that has the shape of an elongated prolate
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
E
0.9000000
0.8526842
0.8053684
0.7580526
0.7107368
0.6634211
0.6161053
0.5687895
0.5214737
0.4741579
0.4268421
0.3795263
0.3322105
0.2848947
0.2375789
0.1902632
0.1429474
0.0956316
0.0483158
0.0010000
x̂ · D/(Ω0 µ)
-3.5254670
-4.5556942
-5.4994100
-6.3674815
-7.1661988
-7.8999458
-8.5721112
-9.1854721
-9.7423803
-10.2448633
-10.6946841
-11.0933790
-11.4422820
-11.7425432
-11.9951406
-12.2008898
-12.3604505
-12.4743319
-12.5428961
-12.5663605
ẑ · D/(Ω0 µ)
-3.2402146
-4.8409623
-6.4329125
-7.9956578
-9.5137719
-10.9749500
-12.3690549
-13.6875615
-14.9232064
-16.0697530
-17.1218265
-18.0747942
-18.9246770
-19.6680817
-20.3021503
-20.8245194
-21.2332907
-21.5270076
-21.7046388
-21.7655664
481
|D|/(Ω0 µ)
4.7883095
6.6475007
8.4632071
10.2213191
11.9107624
13.5225246
15.0490734
16.4839995
17.8217862
19.0576543
20.1874518
21.2075751
22.1149094
22.9067842
23.5809394
24.1354991
24.5689513
24.8801329
25.0682186
25.1327137
Table 2: The various values of T/(Ω0 µ) and |T|/(Ω0 µ), computed from
(51), for a = 1 with rotating angles γ = 30o and β = 0 as a function of
eccentricity E.
spheroid. It represents the first application of the Papkovich-Neuber formulation (4) to the three-dimensional spheroidal Stokes flow. We have successfully
obtained, for the first time, the Papkovich-Neuber-type solutions in prolate
spheroidal coordinates, (31)–(34) and (45)–(48), for the Stokes flow driven by a
translating spheroid at an arbitrary angle γ or by a rotating spheroid with arbitrary angles α and β. We have also derived, based on the two three-dimensional
solutions (31)–(34) and (45)–(48), two useful vector formulas for the corresponding drag and torque as a function of E, α, β and γ.
With two formulas for the drag vector D given by (50) and the toque vector
T given by (51), we are now in a position to write down the governing equations
for the swimming motion of a magnetotactic or non-magnetotactic bacterium
that has the shape of an elongated prolate spheroid. Since the main geometric
482
D. Kong, Z. Cui, Y. Pan, K. Zhang
and physical parameters of the swimming bacteria are known (for example, Pan
et al., 2009), we would be capable of comparing the trajectories of swimming
microorganisms observed in laboratories to those computed from a dynamic
model using (50) and (51) and, hence, offering helpful insight into the complex
dynamics of swimming microorganisms, which will be discussed in a future
paper. Finally, an interesting result of this study – a more elongated spheroidal
shape with the same radius a can enjoy a much smaller drag and torque which
is revealed for the first time – may indicate, from a fluid dynamical point of
view, why there exist very few swimming bodies in nature that have spherical
shape.
Acknowledgments
DK and CZ are supported by the University of Exeter Studentship. KZ is
supported by UK NERC, Leverhulme and STFC grants and YP is supported
by the China CAS grant KZCX2-YW-T10.
References
[1] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University
Press, Cambridge, (1967).
[2] A.T. Chwang, T.Y. Wu, Hydromechanics of low-Reynolds-number flow.
Part 2. Singularity method for Stokes flows, J. Fluid Mech., 67 (1975),
787-815.
[3] G. Dassios, P. Vafeas, On the Spheroidal Semiseparation for Stokes flow,
Research Letters in Physics (2008), 135289.
[4] O. Hall, C.P. Hills, A.D Gilbert, Nonaxisymmetric stokes flow between
concentric cones, Q. J. Mech. Appl. Math., 62 (2009), 131-148.
[5] J. Koiller, K. Ehlers, R. Montgomery, (1996) Problems and Progress in
Microswimming, J. Nonlinear Sci., 6 (1996), 507-541.
[6] H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, UK
(1932).
[7] H. Neuber, Ein neuer Absatz zuröjsung räumlicher Probleme der
Elastiziätstheorie, Z. Anger. Math. Mech., 14 (1934), 203-212.
ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES...
483
[8] H.
Oberbeck,
über
stationäre
Flüssigkeitsbewegungen
mit
Berücksichtigung der innere Reibung, J. Reine Angew. Math., 81
(1876), 62-80.
[9] P.F. Papkovich, The representation of the general integral of the fundamental equations of elasticity theory in terms of harmonic functions, Izr.
Akad. Nauk. SSSR Ser. Mat., 10 (1932), 1425-1435, In Russian.
[10] L.E. Payne, W.H. Pell, The Stokes flow problem for a class of axially
symmetric bodies, J. Fluid Mech., 7 (1960), 529-549.
[11] E.M. Purcell, Life at low Reynolds number, AM J. Phys., 45 (1977), 3-11.
[12] P.N. Shankar, Slow Viscous Flows – Qualitative Features and Quantitative
Analysis Using the Method of Complex Eigenfunction Expansions, Imperial
College Press (2007).
[13] P.N. Shankar, Exact solutions for Stokes flow in and around a sphere and
between concentric spheres, J. Fluid Mech., 631 (2009), 363-373.
[14] H. Taseli, M. Demiralp, A new approach to the classical Stokes flow problem: Part I Methodology and first-order analytical results, Journal of Computational and Applied Mathematics, 78 (1997), 213-232.
[15] T. Tran-Cong, J. Blake, General Solution of the Stokes’ Flow Equations,
Journal of Mathematical Analysis and Applications, 90 (1982), 72-84.
484