In this paper, the problem of optimum cross-section profile in axisymmetric Stokes flow has been discussed under the restriction of specific drag. We take up a class of bodies to be of given maximum cross section with fore and aft symmetry about this section. The possible shape under the stationary value drag has been obtained by making use of method of extremals (Fox, C., 1950, An Introduction to the Calculus of Variations, Oxford University Press, Oxford; Elsgolc, L. E., 1962, Calculus of Variations, Pergamon, New York; Sagan, H., 1969, Introduction to the Calculus of Variations, McGraw-Hill, New York). It has been found that body profile possesses conical front and rear ends. The value of the cross-sectional area has also been calculated for the profile and compared with some known values.

References

References
1.
Fox
,
C.
, 1950,
An Introduction to the Calculus of Variations,
Oxford University Press
,
Oxford
.
2.
Elsgolc
,
E. C.
, 1962,
Differential Equations and the Calculus of Variations,
Pergamon
,
New York
.
3.
Sagan
,
H.
, 1969,
Introduction to the Calculus of Variations,
McGraw-Hill
,
New York
.
4.
Watson
,
S. R.
, 1971, “
Towards the Minimum Drag on a Body of Given Volume in Slow Viscous Flow
,”
IMA J. Appl. Math.
,
7
(
3
), pp.
367
376
.
5.
Pironneau
,
O.
, 1973, “
On Optimum Profiles in Stokes Flow
,”
J. Fluid Mech.
,
59
(
1
), pp.
117
128
.
6.
Pironneau
,
O.
, 1974, “
On Optimum Design in Fluid Mechanics
,”
J. Fluid Mech.
,
64
(
1
), pp.
97
110
.
7.
Bourot
,
J. M.
, 1974, “
On the Numerical Computation of the Optimum Profile in Stokes Flow
,”
J. Fluid Mech.
,
65
(
3
), pp.
513
516
.
8.
Glowinski
,
R.
, and
Pironneau
,
O.
, 1975, “
On the Numerical Computation of the Minimum Drag Profile in Laminar Flow
,”
J. Fluid Mech.
,
72
(
2
), pp.
385
389
.
9.
Mironov
,
A. A.
, 1978, “
Optimization of Body Shape at Small Reynolds Numbers
,”
J. Appl. Mech. Tech. Phys.
,
19
(
1
), pp.
70
74
(Translated from Russian into English).
10.
Bessho
,
M.
, and
Himeno
,
Y.
, 1983, “
On Optimization Profiles in Two-Dimensional Stokes Flow
,”
The Japan Society of Naval Architects and Ocean Engineers Kansai Division
,
193
, pp.
115
125
.
11.
Bessho
,
M.
, and
Himeno
,
Y.
, 1985, “
A Method for Drag Minimization in Stokes Flow
,”
Second International Symposium on Ship Viscous Research AAPA-Goteborg
, No. 21.
12.
Taseli
,
H.
,
Demiralp
,
M.
, and
Kafali
,
K.
, 1989, “
Drag Minimization in Stokes Flow
,”
Int. J. Eng. Sci.
,
27
(
6
), pp.
633
640
.
13.
Ganesh
,
R. K.
, 1994, “
The Minimum Drag Profile in Laminar Flow
,”
ASME J. Fluids Eng.
,
116
(
3
), pp.
456
462
.
14.
Richardson
,
S.
, 1995, “
Optimum Profiles in Two-Dimensional Stokes Flow
,”
Proc. R. Soc. London, Ser. A
,
450
(
1940
), pp.
603
622
.
15.
Kim
,
D. W.
, and
Kim
,
M. U.
, 1995, “
Minimum Drag Shape in Two Dimensional Flow
,”
Int. J. Numer. Methods Fluids
,
21
(
2
), pp.
93
111
.
16.
Gunzberger
,
M. D.
, 2000, “
On a Shape Control Problem for the Stationary Navier-Stokes Equations
,”
Math. Modell. Numer. Anal.
,
34
(
6
), pp.
1233
1258
.
17.
Mohammadi
,
B.
, and
Pironneau
,
O.
, 2001,
Applied Optimal Shape Design
,
Oxford University Press
,
Oxford
.
18.
Mohammadi
,
B.
, and
Pironneau
,
O.
, 2004, “
Shape Optimization in Fluid Mechanics
,”
Annu. Rev. Fluid Mech.
,
36
, pp.
255
279
.
19.
Datta
,
S.
, and
Srivastava
,
D. K.
, 2002, “
Optimum Drag Profile in Axi-symmetric Stokes Flow
,”
Indian Journal of Pure and Applied Mathematics
,
33
(
3
), pp.
409
426
.
20.
Borrvell
,
T.
, and
Petersson
,
J.
, 2002, “
Topology Optimization of Fluids in Stokes Flow
,”
Int. J. Numer. Methods Fluids
,
41
(
1
), pp.
77
107
.
21.
Yagi
,
H.
, 2003, “
Shape Optimization of a Body Located in Incompressible Viscous Flow
,” Annual Report, Kawahara Lab.
22.
Ketamine
,
E.
and
Azegami
,
H.
, 1994, “
Solution to Viscous Flow Field Domain Optimization Problems
,”
Jpn. Soc. Mech. Eng.
,
60
, pp.
1479
1486
.
23.
Ketamine
,
E.
,
Azegami
,
H.
,
Tsubata
,
T.
, and
Itoh
,
S.
, 2005, “
Solution to Shape Optimization Problems of Viscous Flow Fields
,”
Int. J. Comput. Fluid Dyn.
,
19
(
1
), pp.
45
51
.
24.
Nojima
,
K.
, and
Kawahara
,
M.
, 2006, “
Three-Dimensional Shape Identification of Body Located in Viscous Fluid Flow
,”
The 7th International Conference on Hydroscience and Engineering (ICHE-2006)
, Sep. 10-13,
Philadelphia, USA
.
25.
Guest
,
J. K.
, and
Prevost
,
J. H.
, 2006, “
Topology Optimization of Creeping Fluid Flows Using a Darcy-Stokes Finite Element
,”
Int. J. Numer. Methods Eng.
,
66
, pp.
461
484
.
26.
Srivastava
,
D. K.
, 2007, “
Optimum Volume Profile in Axisymmetric Stokes Flow
,”
Meccanica
,
42
(
3
), pp.
239
245
.
27.
Yagi
,
H.
, and
Kawahara
,
M.
, 2007, “
Optimal Shape Determination of a Body Located in Incompressible Viscous Fluid Flow
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
49–52
), pp.
5084
5091
.
28.
Shinohara
,
K.
,
Okuda
,
H.
,
Ito
,
S.
,
Nakajima
,
N.
, and
Ida
,
M.
, 2006, “
Shape Optimization Using an Adjoint Variable Method in ITBL Grid Environment
,”
14th International Conference on Nuclear Engineering
,
Miami, USA
, p.
89568
.
29.
Shinohara
,
K.
,
Okuda
,
H.
,
Ito
,
S.
,
Nakajima
,
N.
, and
Ida
,
M.
, 2006, “
Shape Optimization Using an Adjoint Variable Method for Reducing Drag in Stokes Flow
,”
Int. J. Numer. Methods Fluids
,
58
, pp.
119
159
.
30.
Ishiyama
,
H.
, and
Kawahara
,
M.
, 2008, “
Shape Optimization of Body Located in Incompressible Viscous Flow
,”
Int. J. Comput. Math.
,
25
(
10
), pp.
1515
1530
.
31.
Yoshida
,
H.
, and
Kawahara
,
M.
, 2008, “
Shape Optimization of an Oscillating Body in Fluid Flow by Adjoint Equation and ALE Finite Element Methods
,”
Int. J. Comput. Math.
,
22
(
4
), pp.
229
239
.
32.
Roper
,
M.
,
Pepper
,
R. E.
,
Brenner
,
M. P.
, and
Pringle
,
A.
, 2008, “
Explosively Launched Spores of Ascomycete Fungi Have Drag Minimizing Shapes
,”
Proc. Natl. Acad. Sci. U.S.A.
,
105
(
52
), pp.
20583
20588
.
33.
Roper
,
M.
, and
Michael
,
P. A.
, 2008, “
Nonperturbative Approximation for the Moderate Reynolds Number Navier-Stokes Equations
,” Proc. Natl. Acad. Sci. U.S.A.,
106
(9), pp.
2977
2982.
34.
Roper
,
M.
,
Squires
,
T. M.
, and
Brenner
,
M. P.
, 2008, “
Symmetry Unbreaking in the Shapes of Perfect Projectiles
,”
Phys. Fluids
,
20
, p.
093606
.
35.
Abdelwahed
,
M.
, and
Masmoudi
,
M.
, 2009, “
Optimal Shape Design for Fluid Flow Using Topological Perturbation Technique
,”
J. Math. Anal. Appl.
,
356
(
2
) pp.
548
563
.
36.
Abdelwahed
,
M.
, and
Hassine
,
M.
, 2009, “
Topological Optimization Method for a Geometric Control Problems in Stokes Flow
,”
Appl. Numer. Math.
,
59
(
8
), pp.
1823
1838
.
37.
Shinohara
,
K.
, 2009, “
Adjoint Variable Method for Drag Reduction Under Oseen Flow
,”
Int. J. Numer. Methods Appl.
,
2
(
1
), pp.
146
.
38.
Datta
,
S.
, and
,
and Srivastava D. K.
, 1999, “
Stokes Drag on Axially Symmetric Bodies: A New Approach
,”
Proc. Indian Acad. Sci., Math. Sci.
,
109
(
4
), pp.
1
12
.
39.
Forsyth
,
A. R.
, 1927,
Calculus of Variations
,
Cambridge University Press
,
Cambridge
.
You do not currently have access to this content.