Behavior of passive admixture in a vortical hydrodynamic field
The motion of passive admixture of spherical particles in the stationary hydrodynamic field of a swirling flow is studied. A spherical particle of a given mass in the hydrodynamic field of a swirling flow is located on a certain circular orbit, where the centrifugal force is compensated by the rad...
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Інститут фізики конденсованих систем НАН України
2006
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Behavior of passive admixture in a vortical hydrodynamic field / R.O. Bobrov, A.V. Kyrylyuk, A.V. Zatovsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 103–108. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1212902017-06-14T03:07:36Z Behavior of passive admixture in a vortical hydrodynamic field Bobrov, R.O. Kyrylyuk, A.V. Zatovsky, A.V. The motion of passive admixture of spherical particles in the stationary hydrodynamic field of a swirling flow is studied. A spherical particle of a given mass in the hydrodynamic field of a swirling flow is located on a certain circular orbit, where the centrifugal force is compensated by the radial drag force due to the sink. This leads to the separation of the host fluid and admixture. A theory of Brownian motion of admixture in dilute solutions with a non-uniform flow is constructed. 2006 Article Behavior of passive admixture in a vortical hydrodynamic field / R.O. Bobrov, A.V. Kyrylyuk, A.V. Zatovsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 103–108. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 47.32.-y, 82.40.Ck, 02.60.Cb DOI:10.5488/CMP.9.1.103 http://dspace.nbuv.gov.ua/handle/123456789/121290 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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The motion of passive admixture of spherical particles in the stationary hydrodynamic field of a swirling flow is
studied. A spherical particle of a given mass in the hydrodynamic field of a swirling flow is located on a certain
circular orbit, where the centrifugal force is compensated by the radial drag force due to the sink. This leads
to the separation of the host fluid and admixture. A theory of Brownian motion of admixture in dilute solutions
with a non-uniform flow is constructed. |
| format |
Article |
| author |
Bobrov, R.O. Kyrylyuk, A.V. Zatovsky, A.V. |
| spellingShingle |
Bobrov, R.O. Kyrylyuk, A.V. Zatovsky, A.V. Behavior of passive admixture in a vortical hydrodynamic field Condensed Matter Physics |
| author_facet |
Bobrov, R.O. Kyrylyuk, A.V. Zatovsky, A.V. |
| author_sort |
Bobrov, R.O. |
| title |
Behavior of passive admixture in a vortical hydrodynamic field |
| title_short |
Behavior of passive admixture in a vortical hydrodynamic field |
| title_full |
Behavior of passive admixture in a vortical hydrodynamic field |
| title_fullStr |
Behavior of passive admixture in a vortical hydrodynamic field |
| title_full_unstemmed |
Behavior of passive admixture in a vortical hydrodynamic field |
| title_sort |
behavior of passive admixture in a vortical hydrodynamic field |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121290 |
| citation_txt |
Behavior of passive admixture in a vortical hydrodynamic field / R.O. Bobrov, A.V. Kyrylyuk, A.V. Zatovsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 103–108. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bobrovro behaviorofpassiveadmixtureinavorticalhydrodynamicfield AT kyrylyukav behaviorofpassiveadmixtureinavorticalhydrodynamicfield AT zatovskyav behaviorofpassiveadmixtureinavorticalhydrodynamicfield |
| first_indexed |
2025-07-08T19:32:31Z |
| last_indexed |
2025-07-08T19:32:31Z |
| _version_ |
1837108457194389504 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 103–108
Behavior of passive admixture in a vortical
hydrodynamic field
R.O.Bobrov1, A.V.Kyrylyuk2, A.V.Zatovsky1
1 Department of Theoretical Physics,
Odessa National Mechnikov University,
2, Dvoryanskaya Str., 65026 Odessa, Ukraine
2 Eindhoven University of Technology,
P.O.Box 513, 5600 MB Eindhoven, The Netherlands
Received July 18, 2005
The motion of passive admixture of spherical particles in the stationary hydrodynamic field of a swirling flow is
studied. A spherical particle of a given mass in the hydrodynamic field of a swirling flow is located on a certain
circular orbit, where the centrifugal force is compensated by the radial drag force due to the sink. This leads
to the separation of the host fluid and admixture. A theory of Brownian motion of admixture in dilute solutions
with a non-uniform flow is constructed.
Key words: vortical hydrodynamic field, Brownian motion, passive admixture
PACS: 47.32.-y, 82.40.Ck, 02.60.Cb
1. Introduction
At present, there are many studies that try to explain the behavior of admixture or microorgan-
isms in various hydrodynamic fields (see for example [1–4]). Much attention has been devoted to
stochastic fields [1,2] and external noise [6,7]. It is known that if passive admixture is subjected to a
stochastic hydrodynamic field, then the foreign particles form clusters. In 1970–1971 in Argentina,
500 balloons of the same density were released [5]. They spread over the entire southern hemisphere
on the height about 12 km. This modeling showed that the balloons were concentrated in groups
corresponding to a clasterization process. The mathematical model demonstrated that the balloons
moved in regular and stochastic hydrodynamic fields. Interaction between particles typically leads
to a bound state in a system of particles. Apart from this, a spatially non-uniform distribution
of both interacting and non-interacting particles in the velocity field of a hydrodynamic flow is
possible. The problems of this kind attract much attention (see for example [6] and references
therein). In the present study, we consider the motion of non-interacting foreign particles in the
hydrodynamic field of a swirling flow. A separate particle in the field of a swirling flow moves in a
planar circular orbit [8]. In this orbit, the angular and radial velocities of the stream are inversely
proportional to the distance from the axis of the drain, so that the centrifugal force is compensated
by the radial drag force due to the sink. Such an orbit appears to be steady only for the field of a
swirling flow, for which the angular and radial velocities of the stream are inversely proportional to
the distance from the particle to the center of the drain. However, this cannot be the case for other
hydrodynamic fields. In the presence of random sources passive admixture can be considered as a
set of Brownian particles. In this case, the most probable position of the particles is an equilibrium
orbit.
c© R.O.Bobrov, A.V.Kyrylyuk, A.V.Zatovsky 103
R.O.Bobrov, A.V.Kyrylyuk, A.V.Zatovsky
2. The motion of particles in a hydrodynamic field
We consider the motion of non-interacting particles of identical mass in the vortical hydrody-
namic field of a rotating cylindrical chamber of radius R:
Ur = −V
r
R, Uϕ =
ωR2
r
, r > R. (1)
This field is the solution of the hydrodynamical problem of steady rotation of the infinite porous
cylinder in a motionless fluid with the angular velocity ω and the speed of the suction V on its
border. We suppose that each particle in the stream experiences only Stokes’ force and that the
particles do not affect the stream. The planar motion of any particle in the stream is described by
the equation
~̇V = −λ(~V − ~U). (2)
Here, ~V is the velocity of the particle, λ = 6πηd/M , η is the kinematic viscosity, d is the radius
of the particle, and M is the mass of the particle. In the polar reference frame the equations of
motion have the following form:
r̈ − rϕ̇2 = −λ
(
ṙ +
a
r
)
, rϕ̈ + 2ṙϕ̇ = −λ
(
rϕ̇ − b
r
)
, (3)
where a = V R and b = ωR2. After multiplying the second equation in equations (3) by r, intro-
ducing a new variable ψ = r2ϕ̇, and making some mathematical transformations we obtain
r̈ − ψ2
r3
= −λ
(
ṙ +
a
r
)
, ψ̇ = −λ (ψ − b) , (4)
where
ψ = b+ C exp (−λt) .
Here, C is the constant of integration, which is determined by the initial conditions. Using the
solution of equations (4) one can reduce equations (3) to a single equation
r̈ + λṙ − f(r) = F (t)/r3 (5)
with
f(r) = −λa
r
+
b2
r3
, F (t) = 2bC exp(−λt) + C2 exp(−2λt).
Figure 1. The distribution of non-interacting spherical particles (N=200) in the hydrodynamic
field of a swirling flow at different times. The circle of radius r0 represents the equilibrium orbit.
In the following we suppose that the friction in the system is large, λt � 1. In other words, we
consider the motion of the particles at times which are bigger than characteristic hydrodynamic
times, t� 1/λ. The term in equations (4) which contains r̈ has the magnitude of the order of Ur/t
and λṙ ≈ λUr. The ratio of the first term to the second one in equations (5) is λt and therefore
the first term, which is an inertial term, can be omitted. Since the function F (t) ∼ exp(−λt), the
104
Behavior of passive admixture in a vortical hydrodynamic field
term in the right part of equations (5) decreases more rapidly than the inertial term. As a result,
in this approximation the equation of motion turns into
λṙ − f(r) = 0. (6)
Let us consider the stationary motion, when the radial velocity of a particle stays constant in
time and is equal to zero. Then equations (6) read as follows f(r) = 0 or −λa/r+ b2/r3 = 0. From
this equation it follows that a particle moves along an equilibrium orbit of radius
r0 =
b√
λa
, (7)
which is entirely determined by the vortical field and Stokes’ friction. The obtained numerical solu-
tion of equations (3) evidently shows a trend for the motion of the particles toward the equilibrium
orbit, figure 1.
For simplicity, all the particles are uniformly distributed in the field equations (1) with the
initial coordinates and velocities equal in module to those of the stream at these positions:
r(t)|t=0
= r(0), ϕ(t)|t=0
= ϕ(0), ṙ(t)|t=0
= − V R
r(0)
, ϕ̇(t)|t=0
=
ωR2
r2(0)
.
The time needed to reach an equilibrium orbit is entirely determined by the initial positions of
the particles. The dependence of the radial coordinate on the number of time steps is represented
in figure 2. This dependence is shown for five particles with different initial coordinates. Since the
angular component of the particle velocities is determined purely by the angular velocity of the
stream and the radial coordinate, the centripetal acceleration and the field of the swirling flow make
a bigger effect on the motion of a particle at smaller distances to the drain. With the increase in
the distance this effect decreases inversely proportionally to the coordinate. Consequently, the time
needed to reach an equilibrium orbit is determined purely by the initial position. If a particle is
closer to the drain, then it comes to an equilibrium orbit faster, which is demonstrated in figure 2.
1
2
3
10 2 10 3 10
5 5 5. .
C
o
o
rd
in
a
te
x
Time (in step) τ
particle-5
particle-4
particle-3
particle-2
particle-1
Figure 2. The dependence of the dimensionless radial coordinate x = r/r0 on the number of
time steps τ = tλ (N=5), β = ω/λ = 10−2, γ = V/ (λR) = 5 · 10−5.
The hydrodynamic field studied can model a hurricane in the air. Based on the results of the
present study one can conclude that all the objects sucked by the hurricane will not leave the funnel
of a hurricane under fixed conditions. From equations (7) it follows that the objects of different
mass will rotate in different orbits. The radius of an equilibrium orbit is proportional to the square
root of the mass of the object. This means that heavy particles rotate in equilibrium orbits of a
bigger radius. Similarly, our results can be applied to the analysis of the fish schooling in such a
hydrodynamic field. The motion of large predator fish can cause the appearance of a swirling flow
similar to the one studied in the present work.
The condition for a particle not to be sucked within the rotating cylinder (see figure 2) is given
by the inequality
105
R.O.Bobrov, A.V.Kyrylyuk, A.V.Zatovsky
r0 > R, ω >
√
6πηaV
MR
. (8)
After introducing the dimensionless constants
β =
ω
λ
, γ =
V
λR
, (9)
equations (8) can be rewritten in a dimensionless form with the following constraint on these
constants:
β√
γ
> 1. (10)
From the physical point of view the function f(r)in equations (5) and equations (6) can be
interpreted as an additional force which acts on the particle. This function contains information
about the centripetal acceleration and the field of the swirling flow equations (1). Since in case
of λt � 1 the decay of the function F (t) is much faster than the decay of the inertial terms,
equations (5) can be rewritten as
r̈ + λṙ − f(r) = 0. (11)
This equation can be interpreted as the equation of motion of the particle in the effective potential
field Φ(r):
f(r) = − ∂
∂r
Φ(r),
Φ(r) = λa ln (r) +
b2
r2
1
2
+ const.
Since the constant in the expression for the effective potential energy is arbitrary, we have chosen
Φ(r) = λa ln
(
r
r0
)
+
b2
r2
1
2
. (12)
1 1.5 2
0.2
0.4
0.6
0.8
1
Coordinate x
T
h
e
p
o
te
n
ti
a
l
e
n
e
rg
y
Φ
(r
)/
λ
a
Figure 3. The dependence of the potential energy on the dimensionless coordinate x = r/r0.
Here β = 10−2 and γ = 5 · 10−5.
In figure 3 the potential energy as a function of the dimensionless coordinate x = r/r0 is
drawn. The minimum of the energy is at x = 1, which corresponds to the fact that the most
probable location of the particle is in the equilibrium orbit. The detailed analysis shows that in
the vicinity of the minimum, the equilibrium orbit represents a steady state. The potential energy
has an asymptote Φ(x)|
x→∞
→ ∞. This suggests that independently of the energy each particle
eventually reaches the equilibrium orbit.
106
Behavior of passive admixture in a vortical hydrodynamic field
3. Brownian motion of particles
The particles in the hydrodynamic field of the swirling flow can be considered as Brownian par-
ticles [9]. In this case the stochastic force should be added to the equation of motion equations (5),
so that the corresponding Langevin equation reads
r̈ + λṙ − f(r) = n(t), ṙ = Vr .
Let us suppose that the medium is in equilibrium and that no external force is present. Then, all
directions of the stochastic force are equivalent and therefore its mean value is equal to zero:
〈n(t)〉 = 0.
We also suppose that the characteristic time of the correlations of the Langevin force, τcor, is
much smaller than the hydrodynamic relaxation time, τrel = 1/λ. In the first-order approximation
this characteristic time is set to zero. Since no direction is more preferable than others, different
components of the Langevin force do not correlate with one another (collisions far in time are
statistically independent):
〈n(t)n(t′)〉 = 2Dδ (t− t′) ,
where 2D is the intensity of the Langevin source, which has the meaning of the average intensity
of the random collisions with atoms of the medium. As τcor � τrel, the stochastic process can be
considered as a Gaussian one. Due to the large friction coefficient, the change in coordinates takes
place in two stages. First, the Maxwell distribution of velocities Vr is rapidly established and then
the considerably slower process of establishing the Boltzmann distribution over the coordinates r
occurs. We consider only the second stage of slow development of the equilibrium. Such a process
is described by the reduced Langevin equation
λṙ − f(r) = n(t).
The corresponding Einstein-Fokker-Planck equation for the probability of the particle to leap from
one state to another can be written as follows:
∂W
∂t
= − 1
λ
∂
∂r
f(r)W +D
∂2W
∂r2
. (13)
1 1.5 2
0.3
0.4
0.5
0.6
0.7
Coordinate x
P
ro
b
a
b
ili
ty
d
e
n
s
it
y
W
(
r)
in
f
Figure 4. The dependence of the probability density on the dimensionless coordinate x = r/r0.
Here β = 10−2 and γ = 5 · 10−5.
Within the framework of the assumptions made, it is possible for the system of Brownian
particles in liquid medium to relax to an equilibrium state. In particular, the probability distri-
bution function relaxes to the Boltzmann distribution which is given by the stationary solution of
equations (13):
Winf(r) = C exp
[
1
λD
∫
f(r)dr
]
∼ exp
[
− 1
λD
Φ(r)
]
107
R.O.Bobrov, A.V.Kyrylyuk, A.V.Zatovsky
or
Winf(r) ∼ exp
[
− a
D
(
ln
(
r
r0
)
+
1
2
(r0
r
)2
)]
. (14)
In figure 4 the dependence of the probability density on the dimensionless coordinate x = r/r0
is depicted. From figure 4 one can see that the maximum of this function corresponds to the
equilibrium orbit of radius r0. Therefore, the numerical modeling shows that both representations
of the particles, either Brownian particles or passive admixture, lead to the maximum probability
of finding the particles in the equilibrium orbit.
4. Conclusion
We have investigated the motion of passive admixture of spherical particles in a stationary
hydrodynamic field of a swirling flow. The analysis shows that the most probable position of a
foreign particle is along an equilibrium orbit which is determined by the characteristics of the
vortical field. Considering the particles as Brownian ones, the distribution of the particles in the
swirling flow in a stationary regime is found.
References
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2. Klyatskin V.I., Koshel’ K.V., Physics-Uspekhi, 2000, No. 7, 170, 771.
3. Ivanitsky G.R., Medvinsky A.B., Tsyganov M.A., Physics-Uspekhi, 1994, No. 10, 164, 1041.
4. Ivanitsky G.R., Medvinsky A.B., Tsyganov M.A., Physics-Uspekhi, 1991, No. 4, 161, 13.
5. Mesinger F., Mintz Y., Physics-Uspekhi, Los Angeles: Dep. Meteorology, Univ. of California, Tech. Rep.,
1970, 4, 5.
6. Czirok A., Vicsek T., Physica A, 2000, 281, 17.
7. Horsthemke W., Lefever R. Noise-induced Transitions. Springer-Verlag, Berlin, 1984.
8. Goldshtik M.A. Vortex Flows. Nauka, Novosibirsk, 1981 (in Russian)
9. Reimann P., Phys. Reports, 2002, 361, 57.
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PACS: 47.32.-y, 82.40.Ck, 02.60.Cb
108
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