PHYSICS OF FLUIDS
VOLUME 16, NUMBER 2
FEBRUARY 2004
LETTERS
The purpose of this Letters section is to provide rapid dissemination of important new results in the fields regularly covered by
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On the inverse Magnus effect in free molecular flow
Patrick D. Weidman
Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309-0427
Andrzej Herczynski
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467-3811
共Received 4 September 2003; accepted 24 October 2003; published online 15 December 2003兲
A Newton-inspired particle interaction model is introduced to compute the sideways force on
spinning projectiles translating through a rarefied gas. The simple model reproduces the inverse
Magnus force on a sphere reported by Borg, Söderholm and Essén 关Phys. Fluids 15, 736 共2003兲兴
using probability theory. Further analyses given for cylinders and parallelepipeds of rectangular and
regular polygon section point to a universal law for this class of geometric shapes: when the inverse
Magnus force is steady, it is proportional to one-half the mass M of gas displaced by the body.
© 2004 American Institute of Physics. 关DOI: 10.1063/1.1633265兴
The recent article by Borg et al.1 presents a calculation
of the transverse force on a spinning sphere translating in a
rarefied gas, showing that its direction is opposite to that of
the classic Robins2 共studies for spheres兲 or Magnus3 共studies
for cylinders兲 force in continuum flow, which we shall
henceforth refer to as the Magnus force. Their calculation
also includes the effect of heat transferred to the rotating
sphere from the high Knudsen number flow. In this Letter we
show how this ‘‘inverse Magnus force’’ may be calculated
using a particle dynamics model for the rarefied gas flow
over a spinning sphere and other spinning objects.
Newton’s model4 for fluid resistance ‘‘consists of equal
particles freely disposed at equal distances from each other’’
impinging on a body such that their normal components of
momenta are transferred to the body while their tangential
components are preserved. This leads to the famous Newtonian sine-squared law for the pressure coefficient over the
surface of a body.5 While this result is found not to be applicable to subsonic flow, it is fortuitously useful at hypersonic speeds.5
Newton6 is also credited with being the first to document, and offer an explanation for, the curved trajectory of
an obliquely struck tennis ball in his first scientific publication: ‘‘New theory about light and colors.’’ It is this curved
line of flight of a spinning body that we investigate here, not
for a medium in which molecule–molecule interaction is the
dominant effect, but in a rarefied gas where molecular interactions with boundaries are dominant.7 Our model, different
from Newton’s, assumes: 共i兲 the particle mass m is orders of
magnitude smaller than the projectile mass, 共ii兲 collisions
with the body are perfectly elastic, and 共iii兲 the fraction of
tangential momentum acquired by the particle from the rotating body is measured by ␣ ␶ , the Maxwellian accommo1070-6631/2004/16(2)/9/4/$22.00
dation coefficient used by Borg et al.1 We consider in turn a
sphere, a cylinder, and right parallelepipeds of various sections, each in uniform translation and rotating about a primary axis of symmetry normal to the line of flight. Rather
than dealing with the unsteady motion of a body moving
through a cloud of particles, we consider in each case a body
fixed in space, but rotating about an axis of symmetry and
exposed to an oncoming stream of uniformly dispersed molecules.
Figures 1 and 2 show the Cartesian coordinates (x,y,z)
which have unit vectors 共i,j,k兲. Particles impact the body at
velocity v⫽⫺ v j while it rotates at angular velocity ␻⫽␻k
about an axis of symmetry. The number of particle collisions
per unit time on element surface area dS is N
⫽n 0 v 兩 j"n兩 dS, where n 0 is the number of particles per unit
volume of the gas and n is the unit normal to the body
surface. Streamwise and transverse momentum proportional
to v 2 imparted to the body contribute to drag and lift forces,
respectively; imparted streamwise and transverse momentum
proportional to ␻ v contribute to drag and inverse Magnus
forces, respectively.
In each case the transverse (x-) and streamwise (y-)
components of pre-impact particle momenta are p x ⫽0 and
p y ⫽⫺m v , and the respective post-impact momenta are denoted p x⬘ and p ⬘y . Changes in body momenta ⌬ P x and ⌬ P y
are then given by conservation of linear momentum: ⌬ P x
⫽⫺ 关 p x⬘ ⫺ p x 兴 and ⌬ P y ⫽⫺ 关 p ⬘y ⫺ p y 兴 . The component forces
acting on the surface area element are then f x ⫽N⌬ P x and
f y ⫽N⌬ P y . The body experiences impacts only on its upwind side; hence integration of f x and f y over this exposed
surface gives the total component forces F x and F y .
Figure 1共a兲 shows particle m impacting a sphere of radius R rotating about the z-axis. Spherical coordinates
L9
© 2004 American Institute of Physics
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L10
Phys. Fluids, Vol. 16, No. 2, February 2004
P. D. Weidman and A. Herczynski
F y ⫽⫺mn 0
冕 冕
2␲
0
␲ /2
0
关 2 v 2 R 2 sin ␪ cos3 ␪
⫹ ␣ ␶ R 3 v ␻ sin2 ␪ cos ␪ sin ␾ 兴 d ␪ d ␾ ⫽⫺ ␲ mn 0 R 2 v 2 .
共3兲
Denoting M ⫽mn 0 4 ␲ R /3 the mass of particles displaced by
the sphere and A⫽ ␲ R 2 its frontal area, the total force on the
sphere is
3
F⫽ 12 M ␣ ␶ ␻ v i⫺Amn 0 v 2 j.
FIG. 1. Schematic showing particles of mass m impacting 共a兲 spherical and
共b兲 cylindrical bodies rotating about their z-axes of symmetry, with incident
and reflected angles ␺ and ␺⬘ as given in the text.
(r, ␪ , ␾ ) are defined, for consistency with later calculations,
such that ␪ is the polar angle measured from the y-axis and ␾
is the latitudinal angle measured from z-axis. The particle
striking the sphere at angular position 共␪,␾兲 has incident
angle ␺⫽␪ and is deflected through angle ␺ ⬘
⫽tan⫺1冑(tan ␪⫺␥ sin ␾/cos ␪)2⫹␥2 cos2 ␾,
where
␥
⫽ ␣ ␶ ␻ R/ v . The number of particles per unit time colliding
on
surface
element
area
R 2 sin ␪d␪d␾
is
N
2
⫽n 0 R v sin ␪ cos ␪d␪d␾ and the momentum changes 共per
collision兲 are
⌬ P x ⫽⫺m v sin 2 ␪ sin ␾ ⫹ ␣ ␶ mR ␻ cos ␪ ,
共1a兲
⌬ P y ⫽⫺m v共 1⫹cos 2 ␪ 兲 ⫺ ␣ ␶ mR ␻ sin ␪ sin ␾ .
共1b兲
From symmetry considerations, the integrated momentum
change ⌬ P z produces no force on the sphere. Multiplying
共1a兲 and 共1b兲 by N and integrating over the hemispherical
surface exposed to the oncoming particles gives
F x ⫽mn 0
冕 冕
2␲
0
␲ /2
0
关 ⫺ v 2 R 2 sin 2 ␪ sin ␪ cos ␪ sin ␾
⫹ ␣ ␶ R 3 v ␻ sin ␪ cos2 ␪ 兴 d ␪ d ␾ ⫽
2␲
mn 0 ␣ ␶ R 3 v ␻ ,
3
Note that 共2兲 is precisely the Magnus force calculated using a
Maxwellian distribution function reported as Eq. 共18兲 in
Borg et al.1 The result 共3兲 is the steady drag force on the
sphere and there is no lift for this symmetric configuration.
We now analyze the Magnus force in a rarefied gas for
some geometries not heretofore considered. Referring to Fig.
1共b兲 the particle impacts a solid cylinder of radius R and
length L rotating about its z-axis of symmetry, where (r, ␾ ,z)
are cylindrical coordinates. The particle striking the cylinder
at incidence angle ␺⫽␲/2⫺␾ deflects through angle ␺ ⬘
⫽tan⫺1(cot ␾⫺␥/sin ␾) with ␥ as previously defined. The
number of particles per unit time impacting surface area element Rd ␾ dz is N⫽n 0 R v sin ␾d␾dz and the momentum
changes for the element area are
⌬ P x ⫽⫺m v sin 2 ␾ ⫹ ␣ ␶ mR ␻ sin ␾ ,
共5a兲
⌬ P y ⫽⫺m v共 1⫺cos 2 ␾ 兲 ⫺ ␣ ␶ mR ␻ cos ␾ .
共5b兲
Multiplying 共5a兲 and 共5b兲 by N and integrating over the cylinder surface exposed to the oncoming particles gives
F x ⫽mn 0
冕冕
L
0
␲
0
关 ⫺ v 2 R sin 2 ␾ sin ␾
⫹ ␣ ␶ R 2 ␻ v sin2 ␾ 兴 d ␾ dz⫽
F y ⫽⫺mn 0
共2兲
共4兲
冕冕
L
0
␲
0
␲
mn 0 ␣ ␶ R 2 L ␻ v ,
2
共6兲
关 2 v 2 R sin3 ␾
8
⫹ ␣ ␶ R ␻ v sin ␾ cos ␾ 兴 d ␾ dz⫽⫺ mn 0 RL v 2 .
3
共7兲
Denoting M ⫽mn 0 ␲ R 2 L the mass of gas particles displaced
by the cylinder and A⫽2RL its frontal area, the total force
on the cylinder is
F⫽ 12 M ␣ ␶ ␻ v i⫺ 43 Amn 0 v 2 j.
FIG. 2. Schematic showing particles of mass m impacting right parallelepipeds whose sections are 共a兲 rectangular and 共b兲 regular n-sided polygons
共shown here for n⫽6), with incident and reflected angles ␺ and ␺⬘ as given
in the text. Each body rotates about its z-axis of symmetry, the angles ␤
locating radii r 1 and r are measured positive from ␣, and the polygon has
side length a and fixed ‘‘radius’’ R.
共8兲
Like the sphere, the cylinder experiences a steady inverse
Magnus force, a steady drag force and zero lift.
Bodies of noncircular section will now be considered.
Depicted in Fig. 2共a兲 is a right parallelepiped of length L and
rectangular section a⫻b undergoing uniform rotation ␻
about the z-axis placed at the centroid of the section. The
counter-clockwise positive angle ␣ ⫽ ␻ t is measured from
the x-axis. With an eye on forthcoming generalizations, we
donote the right and left exposed faces by subscripts 1 and 2,
respectively. Then coordinates r 1 and ␤, measured positive
from ␣, locate points on the right face whose lower and
upper corners are at ␤ ⫺ ⫽⫺tan⫺1(a/b) and ␤ ⫹ ⫽tan⫺1(a/b),
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Phys. Fluids, Vol. 16, No. 2, February 2004
Inverse Magnus effect in free molecular flow
respectively. From Fig. 2共a兲, expressions for radial r 1 and
horizontal x 1 positions on this exposed surface, and the factors ␬ 1 and ␬ ⬘1 for respective projections of momentum imparted to the body along the x- and y-axes, are given by
r 1⫽
b
,
2 cos ␤
x 1 ⫽r 1 cos共 ␣ ⫹ ␤ 兲 ,
␬ 1 ⫽sin共 ␣ ⫹ ␤ 兲 ,
␬ ⬘1 ⫽⫺cos共 ␣ ⫹ ␤ 兲 .
With this notation the particle m in Fig. 2共a兲 strikes the right
(i⫽1) face with incidence angle ␺⫽␲/2⫺␣ and deflects
through angle ␺ ⬘ ⫽tan⫺1关(cos ␣⫺␥ cos ␤)/(sin ␣⫺␥ sin ␤)兴,
where ␥ ⫽ ␣ ␶ ␻ r 1 / v . The number of collisions per unit time
impacting area element dx 1 dz is N⫽n 0 v dx 1 dz and the momentum changes imparted to this surface are
⌬ P x ⫽⫺m v sin 2 ␣ ⫹ ␣ ␶ mr 1 ␻ ␬ 1 ,
共9a兲
⌬ P y ⫽⫺m v共 1⫺cos 2 ␣ 兲 ⫹ ␣ ␶ mr 1 ␻ ␬ ⬘1 .
共9b兲
Multiplying 共9a兲 and 共9b兲 by N and integrating over the exposed i⫽1 共right兲 surface yields
mn 0 b
sin ␣
共 F 1 兲x⫽
2
⫺ ␣ ␶␻ v b
冕 冕冋
␤⫺
␤⫹
L
0
sin共 ␣ ⫹ ␤ 兲
2 cos3 ␤
v sin 2 ␣
2
册
cos2 ␤
d ␤ dz
⫽⫺mn 0 aL v 2 sin 2 ␣ sin ␣
⫹
共 F1兲y⫽
abL
mn 0 ␣ ␶ ␻ v sin2 ␣ ,
2
mn 0 b
sin ␣
2
⫹ ␣ ␶␻ v b
冕 冕冋
␤⫺
␤⫹
L
v2
0
cos共 ␣ ⫹ ␤ 兲
2 cos3 ␤
册
共10兲
共 1⫺cos 2 ␣ 兲
cos ␤
I⫽
n⫹1
,
2
I⫽
n⫺1
,
2
冉
冉
2k
冊
␲
␲
;
⬍ ␣ ⬍ 共 2k⫹1 兲
n
n
冊
␲
␲
,
⭐ ␣ ⭐2 共 k⫹1 兲
n
n
共 2k⫹1 兲
where k⫽0,...,n⫺1. Referring to the geometry in Fig. 2共b兲
we have
x i ⫽r cos共 ␣ ⫹ ␤ ⫹ ␦ i 兲 ,
␬ i ⫽sin共 ␣ ⫹ ␤ ⫹ ␦ i 兲 ,
共14a兲
␬ i⬘ ⫽⫺cos共 ␣ ⫹ ␤ ⫹ ␦ i 兲 ,
where r and ␦ i are
冉冊
␲
1
a
,
r⫽ cot
2
n cos ␤
␦ i ⫽ 共 i⫺1 兲
2␲
n
共14b兲
and ␤, measured positive from ␣, lies in the range
关 ⫺ ␲ /n, ␲ /n 兴 . Again in Fig. 2共b兲 the particle m strikes the
i⫽1 face with incidence angle ␺⫽␲/2⫺␣ and deflects
through angle ␺ ⬘ ⫽tan⫺1关(兩cos (␣⫹␦i)兩⫺␥ cos ␤)/(sin (␣⫹␦i)
⫺␥ sin ␤)兴, where ␥ ⫽ ␣ ␶ ␻ r/ v . The number of particles per
unit time impacting the ith face is N⫽n 0 v dx i dz. Following
previous methodology, the transverse force acting over an
element of the ith face is ( f i ) x ⫽N( ␣ ␶ m ␻ r ␬ i ) and integration over this surface yields
冉冊
a 2L
␲
sin2 共 ␣ ⫹ ␦ i 兲 .
cot
2
n
共15兲
The total inverse Magnus force on the parallelepiped is then
d ␤ dz
I
F x⫽
⫽⫺2mn 0 aL v 2 sin3 ␣
abL
⫺
mn 0 ␣ ␶ ␻ v sin ␣ cos ␣ .
2
the Magnus force will be computed. The number of surfaces
exposed to the particle stream depends on whether n is even
or odd. For n even, there are I⫽n/2 exposed surfaces except
for those discrete times when (n⫺2)/2 sides are instantaneously exposed. For n odd, the number of exposed sides is
given by
共 F i 兲 x ⫽mn 0 ␣ ␶ ␻ v
2
共11兲
A similar analysis made for the i⫽2 共left兲 face yields the
force components (F 2 ) x and (F 2 ) y and summation of results
gives the total force
兺
i⫽1
冉 冊兺
a 2L
␲
cot
共 F i 兲 x ⫽mn 0 ␣ ␶ ␻ v
2
n
I
i⫽1
sin2 共 ␣ ⫹ ␦ i 兲 .
J
兺
j⫽0
sin2 共 ␣ ⫹ ␦ j 兲 ⫽S 1 sin2 ␣ ⫹S 2 cos2 ␣ ⫹S 3 sin ␣ cos ␣ ,
共17a兲
共12兲
where M ⫽mn 0 abL is the mass of the gas displaced by the
parallelepiped. Note that setting L⫽b⫽a in 共12兲 gives the
result for a perfect cube. The limit b→0 for a flat plate
Fplate⫽⫺2mn 0 aL v 2 sin2 ␣ 共 cos ␣ i⫹sin ␣ j兲
共16兲
It is convenient to introduce j⫽i⫺1 in which case, for the
resulting summation limit J, one must evaluate
F⫽ 关 mn 0 v 2 L 共 b cos ␣ ⫺a sin ␣ 兲 sin 2 ␣ ⫹ 21 M ␣ ␶ ␻ v兴 i
⫺2mn 0 v 2 L 共 a sin3 ␣ ⫹b cos3 ␣ 兲 j,
L11
共13兲
gives no Magnus force, though there are unsteady lift and
drag components.
Finally, we take the general case of a right parallelepiped
rotating about the centroid of its n-sided regular polygon
section of side a, as depicted in Fig. 2共b兲. For brevity, only
where
J
S 1⫽
兺
j⫽0
J
S 3⫽
兺
j⫽0
冉 冊
冉 冊
2␲ j
cos
,
n
2
4␲ j
sin
.
n
J
S 2⫽
兺
j⫽0
sin2
冉 冊
2␲ j
,
n
共17b兲
Explicit formulas for the summations in 共17b兲 found in Ref.
8 allow us to present the following results.
Case 1: n even. For n⭓4, the upper limits in 共17b兲 are
J⫽(n⫺2)/2 and evaluation of the sums gives S 1 ⫽S 2 ⫽n/4
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L12
Phys. Fluids, Vol. 16, No. 2, February 2004
P. D. Weidman and A. Herczynski
and S 3 ⫽0. Adding contributions according to 共17a兲, inserting that result into 共16兲, and simplifying yields the steady
inverse Magnus force
冉冊
M
a 2L
␲
⫽ ␣ ␶␻ v ,
n cot
F x ⫽mn 0 ␣ ␶ ␻ v
8
n
2
共18兲
where M ⫽mn 0 a 2 Ln cot(␲/n)/4 is the mass of gas displaced
by the body. Note that the result for n⫽4 in 共18兲 agrees with
the steady Magnus force component of Eq. 共12兲 with b⫽a
for a parallelepiped of square section.
For n⭓3 and odd, the force is calculated in two phases,
one for the maximum and the other for the minimum number
of exposed faces.
Case 2a: n odd; maximum number of exposed faces.
Here the summation limit in 共17b兲 is J⫽(n⫺1)/2 and the
evaluated sums are S 1 ⫽(n⫹2)/4, S 2 ⫽n/4 and S 3
⫽⫺1/2 tan(␲/n). Adding components according to 共17a兲 and
inserting the result into 共16兲 gives
F x ⫽mn 0 ␣ ␶ ␻ v
冉 冊冋
a 2L
␲
cot
8
n
n⫹2 sin2 ␣ ⫺tan
冉冊 册
␲
sin 2 ␣ .
n
共19兲
Case 2b: n odd; minimum number of exposed faces.
In this case the summation limit in 共17b兲 is J⫽(n⫺3)/2 and
one finds
S 1⫽
冉冊
␲
n⫺2
⫹sin2
,
4
n
S 3 ⫽tan
冉 冊冋
␲
n
冉冊
␲
n
S 2 ⫽ ⫺sin2
,
4
n
冉 冊册
3
␲
⫺2 sin2
2
n
F x ⫽mn 0 ␣ ␶ ␻ v
冉冊
LR 2
␲
n tan
.
2
n
Since limn→⬁ 关 n tan(␲/n)兴⫽␲ we recover the expected result
共6兲 for the inverse Magnus force on a rotating cylinder. The
same limit to the cylinder is obtained in either 共19兲 or 共20兲
for n odd.
In conclusion, we have derived the result of Borg et al.1
for the inverse Magnus force on a spinning sphere translating
in a rarefied gas using a Newtonian-inspired model. This
approach displays the physics in a transparent manner which
readily can be applied to other simple geometries. We find
that cylinders and parallelepipeds of square and rectangular
section exhibit a steady inverse Magnus force proportional to
one-half the mass M of gas displaced by the projectile, just
like for the sphere. A more general study for parallelepipeds
having regular n-sided polygon sections reveals that for n
even the force is steady and proportional to M /2, while for n
odd it is unsteady, both wobbly and jerky. Finally, our calculation for a parallelepiped of rectangular section provides
an example showing that polygons with n even, stretched
such that any two opposing sides maintain equal length, still
possess a steady inverse Magnus force. We arrive, therefore,
at the following universal hypothesis: A planar body rotating
about the centroid of its convex section, possessing reflectional symmetry about two orthogonal axes, will experience
a steady inverse Magnus force of magnitude (M /2) ␣ ␶ ␻ v .
Obvious extensions to spinning regular polyhedra—Platonic,
Archimedean, or Catalani—can be made.
.
This gives, after appropriate rearrangement,
冉冊
冉 冊
冉
冉 冊
1
a 2L
␲
cot
F x ⫽mn 0 ␣ ␶ ␻ v
8
n
冋
3␲
n
␲
sin 3 ␣
⫻ n⫺
⫹2
cos 2 ␣ ⫺
sin ␣
2␲
n
sin
n
sin
冊
册
. 共20兲
Recall that terms containing ␣ represent unsteady force contributions.
Of interest is the limit n→⬁ of a regular polygon of
fixed ‘‘radius’’ R⫽a n cot(␲/n)/2 关cf. Fig. 2共b兲兴 and diminishing side length a n . In terms of R, the n even result 共18兲
becomes
K. I. Borg, L. H. Söderholm, and H. Essén, ‘‘Force on a spinning sphere
moving in a rarefied gas,’’ Phys. Fluids 15, 736 共2003兲.
2
B. Robins, New Principles of Gunnery, edited by C. Hutton 共F. Wingrave,
London, 1805; originally published in 1742兲.
3
G. Magnus, ‘‘Ueber die Abswichung der Geschosse, und eine auffallende
Erscheinung bei rotierenden Körpern,’’ Poggendorfs Annalen der Physik
und Chemie 88, 1 共1853兲.
4
I. Newton, Mathematical Principles of Natural Philosophy, Book II,
Propositions 34 and 35, translated by A. Motte and F. Cajori, The Regents
of the University of California 共Encyclopedia Britannica, Chicago, 1952兲.
5
J. D. Anderson, Jr., Hypersonic and High Temperature Gas Dynamics
共McGraw-Hill, New York, 1989兲, pp. 46 – 48.
6
I. Newton, ‘‘New theory about light and colors,’’ Philos. Trans. R. Soc.
London 5–6, 3078 共1671–72兲.
7
H. W. Liepmann and A. Roshko, Elements of Gasdynamics 共Wiley, New
York, 1957兲, p. 380.
8
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th edition, edited by A. Jeffrey 共Academic, Boston, 1994兲.
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