On the Analytical Determination of the Contour of Well - Styeamlined
Bodies
S. O. GLADKOV, N. S. NAGIBIN
Moscow Aviation Institute (National Research University),
125993, Moscow, Volokolamskoye sh., 4,
RUSSIA
Abstract. - The problem is solved with the help of a modified Prandtl equation applied to the case under study.
This is a two-dimensional problem of flowing around a flat body when the essential factor is to take into account
the limitation of its dimensions in the longitudinal and transverse directions. Thanks to the above Prandtl equation
it was possible to reduce the problem to a self-similar equation. An analytical solution has been found. Thanks
to this solution the shape of the body is analytically determined when the resistance is at its lowest. An analysis
of the solution of the problem for different Reynolds numbers is carried out. The resulting equation is solved
numerically for different values of its included parameters. With the help of a graphic illustration the different
shapes of such contours are shown.
Key-Words: - a continuity equation, the Navier–Stokes equation, a non-linear equation, a self-similar solution,
Prandtl equation, Reynolds number, numerical equation.
Received: January 12, 2023. Revised: November 15, 2023. Accepted: December 14, 2023. Published: December 31, 2023.
1 Introduction
This article is devoted to the issue related to the
analytical determination of the shape of bodies with
the least resistance force.
As it is known, [1], [2], [3], [4], [5], [6], [7], [8],
[9], [10], [11], [12], [13], [14], [15], [16], [17], [18],
[19], [20], [21], [22], [23], [24], [25], the shape of a
raindrop also belongs to such bodies. This is quite
understandable because due to the lack of turbulence
in its tail section, the resistance force is greatly
reduced. At the same time, we are not aware of any
work where the drop shape would be described
analytically.
In this regard, we have set ourselves another task
namely to find a body shape for which the resistance
force will be significantly less compared to other
bodies. As follows from this statement this is about
the application of methods of the variational
calculus. At the same time, the question comes to the
fore: what should be chosen as the functional
extremum we must findThe answer is obvious since
the role of the desired function must be assigned to
the resistance force. Let us start with its calculation.
2 The Function of the Resistance Force
We will represent the shape of the body in the form
of a flat two-dimensional figure of finite thickness
h
which corresponds to the longitudinal section of
the spatial body similar to the task of N.E.
Zhukovskiy on calculating the lift on the wing.
To find the full resistance force value that it
experiences the resistance force not only on the end
part of the body must be taken into account but also
on both side surfaces. Considering the body
symmetrical concerning the direction of the
streamline flow let us use a general expression for
the resistance force which we will write in the form,
[1]:
k
i ik
S
F ds
,
where
S
is the body's total surface,
ik
is
the viscous stress tensor.
Having expanded the integral we have
0 1 0 1
22
kn
i ik iy in
S S S S
F ds dxdz ds
.
Projecting this force onto the axis
x
along
which the flow is directed we will get
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01
1
0
2
2
4
n
x xy xn
SS
x
h
xy xn
y
x
F dxdz ds
x dx h dL




,
where

is the flow contour. Or
1
0
1
0
2
2
4
2 1 .
x
h
c xy y
x
x
xn
x
F x dx
h x dx




, (1)
where the components of the viscous resistance
tensor are
v
vv
y
xx
xy y x y




(2)
vv
xn
xn
n
xx







(3)
where
is the dynamic viscosity.
Note that a quite obvious condition was taken
into account [1] in formula (2)
v v ,
yx a
The second inequality means that the typical
variation range of the velocity argument along the
axis
y
should be of the order
and along the axis
z
it is the order of the average linear size of the
body
ab
.
The normal derivative can be calculated as
follows
v v v
sin cos
x x x
n
x x z


(4)
Therefore we get the following from the
expression (3)
vv
vv
v
2 sin cos
v cos v sin
xn
xn
n
xx
z
xz
zx
xx
x z x











(4)
Considering that
22
33
22
22
1
cos ,sin ,
11
sin , cos ,
11










(5)
the functional (1) can be rewritten as follows:
1
0
1
0
2
v
4
v
vv
2 2 v
x
x
ch
xy
x
x
zz
x
xz
F x dx
y
h dx
x z x











(6)
To calculate the velocity distributions appearing
in (6) we can use the modified Prandtl equation.
Indeed, near the flow surface, the Navier–Stokes
equation taking into account the finite dimensions of
the body can be brought to the following equation
(for a detailed analysis, [1])
22
22
v v v v
1
vv
x x x x
xz P
x z x x z



(7)
where
is the kinematic viscosity,
P
is
the pressure,
is the fluid density.
Equation (7) is valid if
vv
xz
.
It should be emphasized that equation (7)
describes the flow around the end part of the body
the thickness of which is
h
. The longitudinal
dimensions of the body are finite and that is why
both second-order partial derivatives had to be taken
into account in the Laplace operator in contrast to
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the equation solved by Blasius, [1], who considered
the body semi-infinite.
Further, it will be seen that it is the account of
the two-dimensionality of the Laplace operator that
allows us to find a self-similar solution that differs
significantly from the solution found by Blasius.
If we use the Bernoulli's principle
22
v
22
xU
P

(8)
and consider the velocity
U
to be constant,
equation (7) will be simplified, and using the
continuity equation we come to the following
22
22
v v v
v,
vv0.
x x x
z
xz
z x z
xz






(9)
We will look for the solution of the continuity
equation in the following form
v,
v.
x
z
uz
x


(10)
where the function
,xz

is to be found.
Both velocity components in (10) must satisfy
two conditions
vxu
, (11)
v0
z
. (12)
If we now substitute formula (10) into the upper
equation of system (9) we get
2 2 2
2 2 2 0
x z z x z



(13)
With the introduction of a dimensionless
function
u ab
(14)
equation (13) is brought to the form:
2 2 2
2 2 2
R0
x z z x z



(15)
where
Ru ab
is the Reynolds number.
It is easy to see that equation (15) admits a self-
similar solution. Indeed, if a new argument is
entered
z
x
(16)
then equation (15) is brought to the following
equation
21 4 2 R 0

(17)
According to (10) we have as a result
v 1 ,
v.
x
z
ab
ux
u ab
x





(18)
If a notation that reduces the order of the
equation is entered
G
(19)
we will get
21 4 2 R 0G G G GG

(20)
Or
21 R 0G GG



.
The first integral is
22
1
R
12
GGС


(21)
where
1
C
is the constant of integration.
Assuming
10C
we come to Bernoulli's
equation the solution of which can be represented as
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22
2
1
R
11
4
GC arctg


(22)
where
2
С
is another dimensionless constant
of integration.
Thus the velocity distribution for arbitrary
values of the Reynolds number according to (10) and
(22) can be represented as follows
22
2
22
2
v1
1
1,
R
11
4
v
1,
R
11
4
.
x
z
ab
uG
x
ab
uxC arctg
u ab G
x
u ab
xC arctg
z
x













(23)
Taking into account solutions (23) functional (6)
eventually becomes as follows
1
0
1
0
2
2
23
v
4
2
2
2
1
x
x
ch
xy
x
x
zx
F x dx
y
xx
hu ab G G
xx
G x x
GG
x x x
Gx
ax



















(24)
By introducing an abbreviated notation for the
second integral
2c
F
we obtain the following for it as
a result of simple transformations
1
0
22
x
c
x
F hu ab Hdx
, (25)
where the sub-integral function is
2
22
2
121
21
zx
H G G G
x x x
xa
GG
x a x











(26)
And thus for functional (25) the Euler-Poisson
equation (see [21]) has the form:
2
20
dd
H H H
dx dx
(27)
leads us to the equation
2
2
3 1 2
21
0
z
GG
x x G x x x G










(28)
The solution of equation (28) allows us to find
out the shape of the contour
x
we are interested
in.
3 The Solution of Equation (28)
With the accordance decision (22) at
R1
we
have
2
2
1
1
GC
It means that equation (28) is somewhat
simplified and brought to the following
22
2 2 2
2 2 2
2 2 2
33
1 3 2
10
x
x x x
x
xx






(29)
where it is taken into account that
22
22
3G x x
Gx

 


Due to substitution
fx
(30)
equation (29) is transformed into the
following
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2 2 2 5 2 1
1 3 1 0
ff
f x f f x f f

.
(31)
Its stationary solution (
0ff

) leads to
four "fixed points":
25 17
4
f
, (32)
The numerical solution of equation (31)
according to (30) is shown in Fig. 1-3.
Fig. 1: The contour shape for the case
2
5 17
0,01 4
f
Fig. 2: The contour shape for the case
2
5 17
0,01 4
f
but on a larger scale
Fig. 3: The flow contour shape in the case when
1
5 17
0, 01 4
f
As can be seen in these figures the shape of a fish is
one of the possible well-streamlined bodies.
4 Arbitrary Reynolds numbers
In this case, solution (22) and equation (28) can be
written as a system
22
2
2
22
1,
R
11
4
3 1 2
2 1 0,
xx
GC arctg
GG
x x G x x x G














(33)
Where
2
2
,
dG d G
GG
dd


.
And new argument
x
. After differentiation
of the function
G
we find the following expression
2
1 2 2
12
2
R, q q q
G
pG q q


, (34)
where the functions are
22
12
2 1 2
R
1 1 ,
4
R
2 1 .
2
q C arctg
q q C arctg


(35)
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Therefore the lower equation in system (33)
according to the replacement
x

is brought to
the form
22
R, 1 R, 0x xp p

, (36)
where
R,p
is given by formulas (34), (35).
The asymptotic behaviour of the function
R,p
is the following.
If it is
0x
then it is
0
3
lim
x
p
x

and if it
is
x
then it is
lim
x
pb
x

where
22
2
2
R4
R
C
bC
(37)
The numerical solution of equation (36) is
shown in Fig. 4-7. The boundary conditions were
assumed as follows
0 1, 0 0GG

(38)
Fig. 4: The contour shape for arbitrary Reynolds
numbers and in the case when
1
5 17
0, 01 4
f
(small scale)
Fig. 5: The same as in Fig. 8 that is, with
1
5 17
0, 01 4
f
but in a more distant
coordinates of the abscissa
Fig. 6: The flow contour shape for the case
2
5 17
0,01 4
f
Fig. 7: The count shape for
2
5 17
0,01 4
f
with the abscissa axis prolongation
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5 Conclusion
1. A mathematical model is described that allows
analytically determining the most optimal type of
contour of a well-streamlined body;
2. A numerical solution of the problem for arbitrary
Reynolds numbers is obtained.
3. In the next paper we will considering in more
detail the influence of the Prandtl number.
References:
[1] L. D. Landau, E. M. Lifshitz. Hydrodynamics.
V. 6. Moscow: Science. 1988.
[2] L. Prandtl, O. Titiens. Hydro- and
Aeromechanics In 2 vols. Moscow, GITTL,
1933-1935.
[3] H. Lamb. Hydrodynamics. Moscow, GITTL,
1947.
[4] S.A. Khristianovich, V.G. Galperin, M.D.
Millionshchikov, L.A. Simonov. Applied Gas
Dynamics. Moscow, TsAGI, 1948.
[5] N.E. Zhukovskiy. Collected Works. Volume
2. Hydrodynamics. – Moscow, GITTL, 1949.
[6] G.V. Lipman, A.E. Paket. Introduction to the
Aerodynamics of a Compressible Fluid. –
Moscow, IL, 1949.
[7] N.A. Slezkin. Dynamics of a Viscous
Incompressible Fluid. Moscow, GITTL,
1955.
[8] V.G. Levich. Physicochemical
Hydrodynamics. Moscow, Fizmatgiz, 1959.
[9] G. Birkhoff. Hydrodynamics: a Study in
Logic, Fact, and Similitude. Moscow, GIIL,
1963
[10] J. Serrin. Mathematical Principles of
Classical Fluid Mechanics. Moscow, GIIL,
1963
[11] N.E. Kochin, I.A. Kibel, N.V. Roze.
Theoretical Hydromechanics. In 2 parts. –
Moscow, Fizmatlit, 1963.
[12] L. M. Milne-Thomson. Theoretical
Hydrodynamics. Moscow, Mir, 1964.
[13] A.S. Monin, A.M. Yaglom. Statistical
Hydromechanics. In 2 parts. – Moscow,
Nauka, 1965-1967.
[14] H. Rouse. Mechanics of Fluids. Moscow,
Stroyizdat, 1967.
[15] L.I. Sedov. Continuum Mechanics. In 2 parts.
Moscow, Nauka, 1970.
[16] I.S. Sokolnikov. Tensor Calculus. Theory and
Applications in Geometry and Continuum
Mechanics. Moscow, Nauka, 1971.
[17] A.A. Ilyushin. Continuum Mechanics.
Moscow, MSU, 1971-1990.
[18] O.V. Golubeva. Course of Continuum
Mechanics. Moscow, Vysshaya Shkola, 1972.
[19] G. Batchelor. An Introduction to Fluid
Dynamics. Moscow, Mir, 1973.
[20] M.A. Lavrentyev, B.V. Shabat. Problems of
Hydrodynamics and their Mathematical
Models. Moscow, Nauka, 1973.
[21] L.E. Elsholz. Differential Equations and
Calculus of Variations. Moscow, Science,
1971.
[22] S.O. Gladkov. On One Proof of the
Uniqueness of the Stokes Hydrodynamic
Solution. Russian Physics Journal. V.61, N6,
2018. pp. 103-105.
[23] S.O. Gladkov. On Calculating the Stopping
Time of a Cylindrical Body Rotating in a
Viscous Continuum and the Time of
Entrainment of a Coaxial External Cylinder.
Technical Physics. V.59, N3, 2018, pp. 377-
341.
[24] W. Chester. The Forces on a Body Moving
through a Viscous Fluid. Mathematical and
Physical Science. V.437, N1899, 1992, pp.
185-193.
[25] N.V. Chemetov, S.Necasova. The motion of
the rigid body in the viscous fluid includes
collisions. Global solvability result. Nonlinear
Analisis: Real Word Applications. V.34, N4,
2017, pp. 416-445.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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