Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости
В задачу о колебаниях гравитирующих эллипсоидальных масс однородной несжимаемой жидкости введена вязкость, величина которой прямо пропорциональна давлению. Показано, что такое задание вязкости сохраняет однородный вихревой характер движения. Исследованы малые колебания вращающейся осесимметричной эл...
Saved in:
| Date: | 2008 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | Russian |
| Published: |
Інститут прикладної математики і механіки НАН України
2008
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/10952 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости / С.Н. Судаков // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 563–573. — Бібліогр.: 15 назв. — рос. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859835409041719296 |
|---|---|
| author | Судаков, С.Н. |
| author_facet | Судаков, С.Н. |
| citation_txt | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости / С.Н. Судаков // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 563–573. — Бібліогр.: 15 назв. — рос. |
| collection | DSpace DC |
| description | В задачу о колебаниях гравитирующих эллипсоидальных масс однородной несжимаемой жидкости введена вязкость, величина которой прямо пропорциональна давлению. Показано, что такое задание вязкости сохраняет однородный вихревой характер движения. Исследованы малые колебания вращающейся осесимметричной эллипсоидальной массы жидкости и малые колебания невращающейся сферической массы жидкости.
This paper is devoted to the study of the classical problem of the motion of a gravitating ellipsoidal mass of liquid. The new element is the viscosity of liquid which is determined as a linear homogeneous function of the pressure. It is proved that the so determined viscosity does not destroy the homogeneous rotational flow of liquid.
|
| first_indexed | 2025-12-07T15:34:58Z |
| format | Article |
| fulltext |
Ukrainian Mathematical Bulletin
Volume 5 (2008), № 4, 559 – 568 UMB
The motion of gravitating ellipsoidal masses of
liquid with variable viscosity
Sergey N. Sudakov
(Presented by A. E. Shishkov)
Abstract. This paper is devoted to the study of the classical problem
of the motion of a gravitating ellipsoidal mass of liquid. The new element
is the viscosity of liquid which is determined as a linear homogeneous
function of the pressure. It is proved that the so determined viscosity
does not destroy the homogeneous rotational flow of liquid.
2000 MSC. 76B47, 76D17, 76D27, 76E07, 76U05, 85A30.
Key words and phrases. Gravitating ellipsoidal mass of liquid, vis-
cosity, Dirichlet ellipsoid, Maclaurin ellipsoid.
Introduction
For the first time, the problem on the rotation of a liquid gravitat-
ing ellipsoid was set up and solved by Newton (1686) [8] in order to
investigate the Earth’s shape. Later on, this problem was studied by
Stirling (1735), Maclaurin (1742), Simpson (1743), d’Alembert (1773),
Laplace (1778), Jacobi (1834), Mayer (1842), Liouville (1846), Dirichlet
(1875), Dedekind, Riemann, Poincaré (1885), Cartan, Lyapunov, Roche
(1850), Darwin (1906), Jeans (1916), Chandrasekhar (1969) and many
others. The clear detailed presentation of these results can be found into
monograph of P. Appell [1], article of L. N. Sretenskii [13], monograph
of L. Lichtenstein [5], textbooks of H. Lamb [4] and M. F. Subbotin [14],
and monograph of S. Chandrasekhar [3].
Firstly, the case of a oblate axisymmetric ellipsoid which rotates with
permanent speed around of the symmetry axis (Maclaurin ellipsoid) was
investigated. Jacobi discovered that a liquid figure of equilibrium can be a
triaxial ellipsoid which rotates with permanent speed around of the minor
axis (ellipsoid of Jacobi). In this case, liquid rotates without deformations
Received 4.12.2007
ISSN 1812 – 3309. c© Institute of Mathematics of NAS of Ukraine
560 The motion of gravitating...
as a rigid body. This make it possible to omit any consideration of a
viscosity of liquid. The stability of these figures was investigated by
A. M. Lyapunov [7].
Dirichlet investigated the case of a pulsating rotating ellipsoid (Dirich-
let ellipsoid), by assuming the liquid to be ideal. Riemann [12] investi-
gated the case of a deformation of ideal ellipsoidal liquid. In those studies,
a motion of liquid was homogeneous vortical, and liquid was ideal.
The investigation of a motion of several gravitating liquid masses is
a difficult problem of celestial mechanics. The problem of the motion
of two gravitating masses of liquid was posed by E.V. Pitkevich [10, 11].
The case where a motion of liquid is homogeneous vortical is a significant
special case of this problem.
At the end of the XIXth – the beginning of the XXth century, the
homogeneous vortical (rotational) motion of the ideal liquid was involves
in the studies of the motion of the Earth’s liquid core. The aim of these
investigations was an adequate description of the motion of the Earth’s
pole. The results can be found in the monograph of H. Moritz and I.I.
Mueller [8].
According to the article of V. V. Brazhkin [2], the viscosity of an iron
melt increases very much with the pressure, and the Earth’s liquid core
consists of iron. Hence, the viscosity of the core must increase toward
the Earth’s center. If we assume that a similar effect is correct for el-
lipsoidal liquid celestial bodies, we must take into account this effect in
mathematical models for the motion of a liquid celestial body. In the au-
thor’s work [15], the problem of the motion of an ellipsoidal gravitating
mass of a viscous liquid was studied. The viscosity of liquid was take
into account as a function of coordinates and the lengths of the ellipsoid
semiaxes. The function setting the viscosity of liquid was chosen so that
the motion of ellipsoidal mass of liquid is homogeneous vortical.
In the present article, a similar problem is solved in the case where
the viscosity is a linear function of the pressure.
1. The equations of motion
Let Oξ1ξ2ξ3 be the immovable Cartesian coordinate system, and let
Ox1x2x3 be the moving Cartesian coordinate system which can rotate
with respect to its origin O. The boundary of the liquid ellipsoid is given
in the coordinate system Ox1x2x3 by the equation
x2
1/c2
1 + x2
2/c2
2 + x2
3/c2
3 = 1, (1.1)
S. N. Sudakov 561
where c1, c2, c3 are continuous differentiable functions of t. We assume
that the condition
c1c2c3 = R3 = const (1.2)
holds. The ellipsoid contain a viscous gravitating incompressible liquid
without voids. The kinematic liquid viscosity ν is given by the formula
ν = kp, (1.3)
where k is a constant, and p is the pressure. The equations of motion of
the liquid with respect to the moving axes Ox1x2x3 have the form [6]
∂v
∂t
+ (v · ∇)v = −
1
ρ
∇p + ν(p)∆v + 2σ∇ν(p)
− ω̇ × x − ω × (ω × x) − 2ω × v − ∇Φ, (1.4)
div v = 0, (1.5)
where v is the liquid velocity vector with respect to the moving axes
Ox1x2x3; σ is the strain rate tensor of the liquid with components
σij =
1
2
(
∂vi
∂xj
+
∂vj
∂xi
)
, i, j = 1, 2, 3; (1.6)
ω is the absolute angular velocity vector in the coordinate system
Ox1x2x3; Φ is the potential of gravitation forces [4],
Φ = πργ(α1x
2
1 + α2x
2
2 + α3x
2
3 − χ0), (1.7)
αi = c1c2c3
∞
∫
0
dλ
(c2
i + λ)D
, i = 1, 2, 3, χ0 = c1c2c3
∞
∫
0
d λ
D
,
D = [(c2
1 + λ)(c2
2 + λ)(c2
3 + λ)]
1
2 ;
and γ is the gravitational constant.
We assume that the pressure on the liquid boundary is equal to zero.
Then the liquid viscosity (1.3) must be equal to zero on the boundary as
well. Therefore, the boundary condition for Eqs. (1.4) and (1.5) must
have the form
(v − u) · n |S= 0, (1.8)
562 The motion of gravitating...
where S is the liquid boundary (1.1), n is the unit vector of a normal
to the boundary S, and u is the velocity of the liquid boundary in the
moving coordinate system Ox1x2x3.
The solution of Eqs. (1.4) and (1.5) is sought in the form
v1 = c1
(
ω∗
2
x3
c3
− ω∗
3
x2
c2
)
+
ċ1
c1
x1 (123), (1.9)
p = −p0(t)
(
x2
1
c2
1
+
x2
2
c2
2
+
x2
2
c2
2
− 1
)
, (1.10)
where v1, v2, v3 are the components of the velocity vector, p is the pres-
sure, ω∗
1
, ω∗
2
, ω∗
3
, and p0(t) are the unknown functions of t. The symbol
(123) means that the other expressions can be found by a cyclic permu-
tation of indices. If we substitute expressions (1.9) and (1.10) in Eqs.
(1.4), we have three equalities ki1x1 + ki2x2 + ki3x3 = 0, i = 1, 2, 3,
which must be fulfilled for all x1, x2, x3. It is possible only if the equali-
ties kij = 0, i, j = 1, 2, 3 are correct. These equalities can be represented
in the form
c̈1 = c1(ω
∗
2
2 + ω∗
3
2 + ω2
2 + ω2
3) + 2c3ω
∗
2ω2 + 2c2ω
∗
3ω3
− 2πργα1c1 +
2p0
c1
(
1
ρ
+ 2k
ċ1
c1
)
(123), (1.11)
ω̇∗
1
c2
c3
+ ω̇1 = −2ω∗
1
ċ2
c3
+ ω∗
2ω
∗
3
c2
c3
+ 2kp0
ω∗
1
c2
3
(
c3
c2
−
c2
c3
)
+ ω2ω3 + 2ω∗
2ω3
c2
c3
− 2ω1
ċ3
c3
(123), (1.12)
ω̇∗
1
c3
c2
+ ω̇1 = −2ω∗
1
ċ3
c2
+ ω∗
2ω
∗
3
c3
c2
+ 2kp0
ω∗
1
c2
2
(
c3
c2
−
c2
c3
)
− ω2ω3 − 2ω∗
3ω2
c1
c2
− 2ω1
ċ2
c2
(123), (1.13)
where ω1, ω2, ω3 are the components of the angular velocity ω in the
coordinate system Ox1x2x3.
The algebraic equation (1.2) and the system of ordinary differential
equations (1.11)–(1.13) are the system of ten equations for ten unknowns
p0, ci, ω
∗
i , ωi, i = 1, 2, 3.
S. N. Sudakov 563
2. Small oscillations of the axisymmetric ellipsoid
Let c1 = c2, ω1 = ω2 = ω∗
1
= ω∗
2
= ω∗
3
= 0. Then, if we exclude c3
with the help of Eq. (1.2), the motion equations take the form
c̈1 = c1ω
2
3 − 2πργα1c1 +
2p0
c1
(
1
ρ
− 2k
ċ1
c1
)
,
−2R3(c−3
1
c̈1 − 3c−4
1
ċ2
1) = −2πργα3
R3
c2
1
+
2p0c
2
1
R3
(
1
ρ
+ 4k
ċ1
c1
)
, (2.1)
ω̇3 = −2ω3
ċ1
c1
.
The last equation yields the equality
c2
1ω3 = l, l = const (2.2)
which represents the conservation law of angular momentum. The vari-
able ω3 can be excluded from the first equation (2.1) by equality (2.2).
Then we can exclude p0 from two first equations (2.1) and introduce new
variables
ζ = c1/R, τ = T−1t, η =
dζ
dτ
,
where T is the characteristic time. Then the motion equations take the
form
dζ
dτ
= η,
dη
dτ
=
{
L2ζ3 − 2πργT 2ζ(α1ζ
6 − α3) + 6
η2
ζ
−
4kρ
T
η
[
3
η2
ζ2
− L2ζ2 + πργT 2(2α1ζ
6 + α3)
]}
×
[
ζ6 + 2 +
4kρ
T
η
ζ
(ζ6 − 1)
]−1
, (2.3)
where L = lT/R2. For the oblate axisymmetric ellipsoid, the quantities
α1 and α3 can be expressed by the formulas [4]
α1 = α2 = (ξ2 + 1)ξ arcctg ξ − ξ2,
α3 = 2(ξ2 + 1)(1 − ξ arcctg ξ),
ξ = (ζ6 − 1)−
1
2 .
(2.4)
564 The motion of gravitating...
The parameters of the stationary solutions of system (2.3) are connected
by the relation
L2 = 2πργT 2(α1ζ
4 − α3ζ
−2), (2.5)
where 1 ≤ ζ ≤ ∞. We denote, by L0 and ζ0, the value of L and ζ
which satisfy relation (2.5). Then every solution of Eqs. (2.3) can be
represented in the form
ζ = ζ0 + δ, (2.6)
where δ is the unknown function of the dimensionless time τ . If we
substitute (2.6) into Eqs. (2.3) and make the linearization by η and δ
assuming them small, the motion equations take the form
dδ
dτ
= η,
(2.7)
dη
dτ
= aδ + bη,
where
a = −2πργT 2(4ζ6
0α10 + ζ7
0α11 + 2α30 − ζ0α31)(ζ
6
0 + 2)−1,
b = −4kπρ2γT (2α10ζ
6
0 + α30)(ζ
6
0 + 2)−1,
α10 = ζ6
0 (ζ6
0 − 1)−3/2arcctg(ζ6
0 − 1)−1/2 − (ζ6
0 − 1)−1,
α11 = −3ζ5
0 (ζ6
0 − 1)−2
[
(ζ6
0 + 2)(ζ6
0 − 1)−1/2arcctg(ζ6
0 − 1)−1/2 − 3
]
,
α30 = 2ζ6
0 (ζ6
0 − 1)−1
[
1 − (ζ6
0 − 1)−1/2arcctg(ζ6
0 − 1)−1/2
]
,
α31 = 6ζ5
0
[
(ζ6
0 + 2)(ζ6
0 − 1)−5/2arcctg(ζ6
0 − 1)−1/2 − 3(ζ6
0 − 1)−2
]
.
The characteristic equation of system (2.7)
∣
∣
∣
∣
−λ 1
a b − λ
∣
∣
∣
∣
= 0
can be written as
λ2 − bλ − a = 0.
This equation has the solutions
λ1,2 =
1
2
(b ±
√
b2 + 4a).
The coefficient a is independent of k. If ζ0 > 1, then a < 0. The
coefficient b < 0, and |b| is directly proportional to k. Then if k = 0 (the
case of the ideal liquid), we have b = 0, and roots of the characteristic
S. N. Sudakov 565
equation are purely imaginary λ = ±
√
|a|. In this case, the nonlin-
ear equations of motion describe the undamped periodic oscillations of a
Dirichlet ellipsoid [4].
If k is enough small and b2+4a < 0, then the roots of the characteristic
equation are complex conjugate, and their real parts are negative. In this
case, the Dirichlet ellipsoid approaches the Maclaurin ellipsoid by means
of decaying oscillations.
If k is enough large and b2 + 4a ≥ 0, then the roots of the character-
istic equation are real and negative. In this case, the Dirichlet ellipsoid
asymptotically approaches the Maclaurin ellipsoid.
3. Oscillations of a nonrotating spherical
mass of liquid
If we assume that
ω∗
1 = ω∗
2 = ω∗
3 = ω1 = ω2 = ω3 = 0,
then Eqs. (1.12) and (1.13) are satisfied identically, and Eqs. (1.11) takes
the form
c̈1 = −2πργα1c1 +
2p0
c1
(
1
ρ
− 2k
ċ1
c1
)
(123). (3.1)
In Eqs. (3.1), we exclude c3 by using Eqs. (1.2). Then we exclude p0
and introduce the new variables ζi = ci/R, i = 1, 2, τ = T−1t. Thus,
we obtain the system of equations
dζi
dτ
= ηi, ai1
dηi
dτ
+ ai2
dη2
dτ
= fi, i = 1, 2, (3.2)
where
a11 = ζ1 + 2kρT−1(η1 + ζ1ζ
−1
2
η2) + ζ−3
1
ζ−2
2
(1 − 2kρT−1ζ−1
1
η1),
a12 = ζ−2
1
ζ−3
2
(1 − 2kρT−1ζ−1
1
η1),
a21 = ζ−3
1
ζ−2
2
(1 − 2kρT−1ζ−1
2
η2),
a22 = ζ2 + 2kρT−1(ζ−1
1
ζ2η1 + η2) + ζ−2
1
ζ−3
2
(1 − 2kρT−1ζ−1
2
η2),
f1 = −2πργT 2α1ζ
2
1 [1 + 2kρT−1(ζ−1
1
η1 + ζ−1
2
η2)]
+ [2ζ−2
1
ζ−2
2
(ζ−2
1
η2
1 + ζ−1
1
ζ−1
2
η1η2 + ζ−2
2
η2
2)
+ 2πργT 2α3ζ
−2
1
ζ−2
2
](1 − 2kρT−1ζ−1
1
η1),
566 The motion of gravitating...
f2 = −2πργT 2α2ζ
2
2 [1 + 2kρT−1(ζ−1
1
η1 + ζ−1
2
η2)]
+ [2ζ−2
1
ζ−2
2
(ζ−2
1
η2
1 + ζ−1
1
ζ−1
2
η1η2 + ζ−2
2
η2
2)
+ 2πργT 2α3ζ
−2
1
ζ−2
2
](1 − 2kρT−1ζ−1
2
η2).
System (3.2) can be written in the form
dζ1
dτ
= η1,
dη1
dτ
= (f1a22 − f2a12)(a11a22 − a12a21)
−1,
dζ2
dτ
= η2,
dη2
dτ
= (f2a11 − f1a21)(a11a22 − a12a21)
−1.
(3.3)
System (3.3) has a stationary solution ζ1 = ζ2 = 1, η1 = η2 = 0.
This solution describes the equilibrium of the spherical mass of liquid.
Then, every another solution of system (3.3) can be represented in the
form
ζ1 = 1 + δ1, ζ2 = 1 + δ2, η1, η2, (3.4)
where δ1 and δ2 are new unknown functions of τ . We substitute (3.4) in
system (3.3) and assume that δ1, δ2, η1, η2 are small. The linearization in
δ1, δ2, η1, η2 gives the system
dδi
dτ
= ηi,
dηi
dτ
= −
8
3
πργT 2
(
2
5
δi + kρT−1ηi
)
,
i = 1, 2. (3.5)
Thus, the systems of linearized equations for the variables ζ1, η1 and
ζ2, η2 are independent of each other, and both can be solved separately.
Taking into account that the systems differ from each other only by the
unknown variables and the initial conditions, they can be solved similarly.
The solution of system (3.5) is sought as
(δi, ηi)
T = (b1, b2)
T exp(λτ),
where b1, b2, λ are the unknown constants. The characteristic equation
has the form
∣
∣
∣
∣
−λ 1
−16
15
πργT 2 −8
3
kπρ2γT − λ
∣
∣
∣
∣
= 0.
This equation can be written as the equation for λ,
λ2 +
8
3
kπρ2γTλ +
16
15
πργT 2 = 0.
The solution of this equation is
S. N. Sudakov 567
λ1,2 =
4
3
(
−kπρ2γT ±
√
(kπρ2γT )2 −
3
5
πργT 2
)
.
For k = 0, the system becomes conservative and will perform the
undamped oscillations in the vicinity of the stable equilibrium state, i.e.
the sphere. In this case, the roots of the characteristic equation are purely
imaginary λ1,2 = ±4
3
√
3
5
πργT 2 i.
In the case of 0 < k <
√
3/(5πρ3γ), the roots of the characteristic
equation are complex conjugate with a negative real part. Into this case,
the ellipsoidal mass of liquid will approach a sphere by performing the
damped oscillations.
In the case of k >
√
3/(5πρ3γ), the ellipsoidal mass of liquid, the
kinetic energy of which is sufficiently small and the shape is enough close
to a spherical one, will asymptotically approach the sphere.
The author is grateful to the reviewer for the careful review and the
useful remarks that make possible to find new approaches to the problem.
References
[1] P. E. Appell, Figures d’Equilibres d’Une Masse Liquid Homogene en Rotation,
Gauthier-Villars, Paris, 1932.
[2] V. V. Brazhkin, A. G. Lyapin, Universal viscosity growth in melted metals at
megabar pressures: the vitreous state of the Earth’s inner core // Uspekhi Fiz.
Nauk 170 (2000), 535–551.
[3] S. Chandrasekhar, Ellpsoidal Figures of Equilibrium, Yale Univ. Press, New Haven,
1969.
[4] H. Lamb, Hydrodynamics, Dover, New York, 1945.
[5] L. Lichtenstein, Gleichgewichtsfiguren der Rotierenden Flüssigkeiten, Springer,
Berlin, 1933.
[6] L. G. Loitsyanskii, Mechanics of Liquid and Gas, Nauka, Moscow, 1973 (in Rus-
sian).
[7] A. M. Lyapunov, About stability of the ellipsoidal forms of equilibrium of the rotat-
ing liquid, Collected Works, Izd. AN SSSR, Moscow, 1959, pp. 5–113 (in Russian).
[8] H. Moritz, I. I. Mueller, Earth Rotation: Theory and Observation, Ungar, New
York, 1987.
[9] I. Newton, Mathematical Foundations of Natural Philosophy, Nauka, Moscow, 1989
(in Russian).
[10] E. V. Petkevich, The problem of liquid bodies // Pis’ma Astron. Zh., 3 (1977),
N 9, 424–428.
[11] E. V. Petkevich, The equations of the external problem of two liquid bodies //
Pis’ma Astron. Zh., 3 (1977), N 11, 522–525.
[12] G. F. B. Riemann, About the motion of a liquid homogeneous ellipsoid, Selected
Works, OGIZ, Moscow-Leningrad, 1948, pp. 339–366 (in Russian).
[13] L. N. Sretenskii, Theory of equilibrium figures of a liquid rotating mass // Uspekhi
Mat. Nauk, (1938), N 5, 187–230.
568 The motion of gravitating...
[14] M. F. Subbotin, Course of Celestial Mechanics, Gostekhizdat, Moscow, 1949 (in
Russian).
[15] S. N. Sudakov, About oscillations of rotating gravitating liquid ellipsoids with
variable viscosity // Mekh. Tverd. Tela (2002), Iss. 32, 217–226.
Contact information
Sergey Nikitovich
Sudakov
Institute of Applied Mathematics
and Mechanics of NASU,
R. Luxemburg str. 74,
Donetsk 83114,
Ukraine
E-Mail: techmech@iamm.ac.donetsk.ua
|
| id | nasplib_isofts_kiev_ua-123456789-10952 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | Russian |
| last_indexed | 2025-12-07T15:34:58Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Судаков, С.Н. 2010-08-10T10:24:29Z 2010-08-10T10:24:29Z 2008 Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости / С.Н. Судаков // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 563–573. — Бібліогр.: 15 назв. — рос. 1810-3200 https://nasplib.isofts.kiev.ua/handle/123456789/10952 В задачу о колебаниях гравитирующих эллипсоидальных масс однородной несжимаемой жидкости введена вязкость, величина которой прямо пропорциональна давлению. Показано, что такое задание вязкости сохраняет однородный вихревой характер движения. Исследованы малые колебания вращающейся осесимметричной эллипсоидальной массы жидкости и малые колебания невращающейся сферической массы жидкости. This paper is devoted to the study of the classical problem of the motion of a gravitating ellipsoidal mass of liquid. The new element is the viscosity of liquid which is determined as a linear homogeneous function of the pressure. It is proved that the so determined viscosity does not destroy the homogeneous rotational flow of liquid. ru Інститут прикладної математики і механіки НАН України Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости The motion of gravitating ellipsoidal masses of liquid with variable viscosity Article published earlier |
| spellingShingle | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости Судаков, С.Н. |
| title | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости |
| title_alt | The motion of gravitating ellipsoidal masses of liquid with variable viscosity |
| title_full | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости |
| title_fullStr | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости |
| title_full_unstemmed | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости |
| title_short | Движение гравитирующих эллипсоидальных масс жидкости переменной вязкости |
| title_sort | движение гравитирующих эллипсоидальных масс жидкости переменной вязкости |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/10952 |
| work_keys_str_mv | AT sudakovsn dviženiegravitiruûŝihéllipsoidalʹnyhmassžidkostiperemennoivâzkosti AT sudakovsn themotionofgravitatingellipsoidalmassesofliquidwithvariableviscosity |