ISSN 1063-7842, Technical Physics, 2024, Vol. 69, No. 10, pp. 2596–2602. © Pleiades Publishing, Ltd., 2024.
Numerical Study of the Influence of the Dispersed Phase Density
on the Interphase Interaction in an Electrically Charged Gas
Suspension for the Mass and Surface Electric Charge Density
D. A. Tukmakova,* and N. A. Tukmakovab,**
a Institute of Mechanics and Engineering, Federal State Research Center “Kazan Scientific Center,”
b
Russian Academy of Sciences, Kazan, 420111 Russia
Kazan National Research Technical University, Kazan, 420111 Russia
*e-mail: tukmakovda@imm.knc.ru
**e-mail: nadejdatukmakova@yandex.ru
Received January 23, 2023; revised January 23, 2023; accepted January 23, 2023
Abstract—The work is devoted to mathematical modeling of the dynamics of inhomogeneous electrically
charged media. The dynamics of gas suspensions (solid particles suspended in a gas) is investigated. The
mathematical model has implemented a continual approach to modeling the dynamics of inhomogeneous
media; the model takes into account the interphase momentum exchange as well as the interphase heat transfer. The carrier medium is described as a viscous, compressible, heat-conducting gas. The equations of the
mathematical model are supplemented with initial and boundary conditions. The influence of the physical
density of the dispersed phase material on the intensity of interaction in electrically neutral and electrically
charged gas suspensions is investigated. Both surface and mass densities of electric charge are considered.
Keywords: numerical simulation, gas suspensions, multiphase media, Coulomb force, electric charge density
models, interfacial interaction
DOI: 10.1134/S1063784224700932
INTRODUCTION
The dynamics of inhomogeneous media is one of
developing fields of the mechanics of liquid, gas, and
plasma [1–18]. In some cases, it is necessary to model
flows of inhomogeneous media in an electric field [4–
16, 18]. In [4], the results of physical experiments and
numerical analysis of the dynamics of an electrically
charged gas suspension are compared without taking
into account the reciprocal effect of mixture phases.
In [5], the formation of the ion–acoustic waves in a
dusty plasma was investigated and mathematical modeling techniques were worked out. In [6], the action of
dusty structures on an electrically charged glow discharge column was investigated and examples of calculations for different formulations of the experimental
conditions were reported. Publication [7] is devoted to
analysis of the dynamics of dispersed particles of different materials used in power stations; a model was
constructed for describing the dust formation, and the
difference in the dynamics of particles with different
densities was demonstrated. In [8], the distribution of
the kinetic energy of particles in an inhomogeneous
dust–plasma structure was investigated. Publication [9]
is devoted to analysis of dynamic properties of a dusty
plasma containing identical negatively charged parti-
cles. In [10], an analytic solution to the Korteweg–
de Vries equation with damping was obtained and the
effect of the plasma parameters on the parameters of
the ion–acoustic waves in a dusty plasma was investigated. Publication [11] was devoted to perfection of
the technology of electrical filters for cleaning of gas
blowouts of industrial enterprises from dispersed
impurities. In [12], using the Brownian dynamics
method, the conditions for the capture and confinement of charged dust particles at the atmospheric air
pressure was investigated for a wide range of parameters characterizing dynamic traps as well as confined
particles. It is shown that the viscosity of the gaseous
medium substantially affects the trapping and confinement of dust particles.
Publication [13] was devoted to analysis of the
effect of an electric charge of dispersed particles on the
gas-dispersed flow filtration efficiency. The effect of
the electric charge on the removal of dispersed inclusions from the flow of a homogeneous medium using
electric and electrically neutral filters for purification
of gas-dispersed media was analyzed. Theoretical calculations showed that the existence of an electric
charge on aerosol particles elevates the filtration efficiency. In [14], the effect of the polydisperse composition of the dispersed phase of an inhomogeneous
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NUMERICAL STUDY OF THE INFLUENCE OF THE DISPERSED PHASE DENSITY
medium on the results of measurements using the
instruments for analyzing dispersed flows was investigated based on the integration of the convection, diffusion, and electric charge transfer equations for the
laminar, steady-state, and incompressible flows of the
carrier medium. Publication [15] was devoted to theoretical aspects in the technologies of jet spraying of an
electrically charged gas–droplet medium aimed at the
precipitation and filtration of solid dispersed particles
suspended in a gas. In [16], a theoretical model was
developed for determining the efficiency of precipitation of aerosol particles by droplets due to the combined action of mass and surface mechanical forces as
well as the forces of the electrical origin, which were
acting silultaneously on the dynamics of an aerosol
particle. Publication [17] was devoted to analysis of the
propagation of electromagnetic waves in a fully ionized dusty plasma. The characteristics of propagation
of a strongly ionized plasma with and without dust in
the Fokker–Planck–Landau and the Bhatnagar–
Gross–Kruk models were compared with the characteristics of a weakly ionized plasma. The effect of the
change in the dust parameters on the propagation of
electromagnetic waves in a strongly ionized dusty
plasma was analyzed using the Fokker–Planck–Landau model. The results of modeling have demonstrated that the densities and average radii of dust particles substantially affect the parameters of electromagnetic wave propagation. In [18], the influence of
various parameters of dispersed particles as well as the
parameters of the carrier medium on the electric filtration efficiency for gas–dispersed media was analyzed. In [19], the results of theoretical investigation of
shock-wave acoustic perturbation in a magnetized
dusty plasma saturated with ions were investigated.
Analysis was based on the integration of the Korteweg–de Vries equation. In publication [20], it has
been revealed using numerical simulation of a gas–
dispersed flow that electric fields can be formed by
aerodynamic processes in two-fraction suspensions on
two-fractional suspensions of oppositely charged particles. It has been revealed that the mechanism of generating electric fields is self-regulating and is based on
turbulence. In [21], the influence of electric fields on
dust particles suspended in air was investigated for an
electric field varying in the vertical direction. The particle distribution function was defined with account
for the vertical coordinate. The effect of gravitational,
aerodynamic, and electrostatic forces acting on the
particle dynamics and the electric field distribution
was investigated. In [22], the dynamics of natural rotation (about the center of mass) of dust particles in a
magnetic field was studied. The angular velocity of the
natural rotation in the dust trap in a high-frequency discharge was measured. In publication [23], the ponderomotive force exerted by a high-intensity rapidly oscillating ion–acoustic wave on a grain with a varying charge
in a dusty plasma was investigated. The account for the
oscillations of the grain charge in the field of the ion–
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acoustic wave has made it possible to separate new components of this force, which are proportional to the
wave vector and to the third power of the field amplitude. The resulting components are responsible for the
directional transport of the dust fraction of the plasma.
It follows from analysis of the aforementioned publications that in various investigations of the dynamics
of electrically charged inhomogeneous media, the
combination of electrophysical and hydrodynamic
processes has been studied. For perfecting technologies and setups operating with electrically charged gas
suspensions, it is necessary to establish regularities in
the dynamics of such media in electric and aerodynamic fields. In this study, the effect of the density of
the particle material on the intensity of the high-speed
slip of the mixture phases is considered for the mass
and surface charge densities of the dispersed phase.
The mathematical model takes into account the action
of the mixture phases in the course of propagation of a
low-intensity shock wave on the electrically charge gas
suspension. It is assumed that the electric field is
formed by charged dispersed particles.
1. METHODS OF INVESTIGATION
For describing the dynamics of an electrically
charged gas suspension, the mathematical model implementing the continual approach to simulation is used,
which takes into account the influence phase-heterogenic mixture component on one another [1, 2, 24–28].
The flow of a carrier medium is described by the system
of the Navier–Stokes equations [29] for a viscous compressible heat-conducting gas with account for the force
interaction and heat exchange between phase [24–28]:
∂ρ
+ ∇ (ρV ) = 0,
∂t
∂ρV k
+ ∇i (ρV kV i + δik p − τi k ) = −Fk + α∇ k p, (1)
∂t
∂ (e )
+ ∇i (V i ( e + p − τii ) − V k τki − λ∇iT )
∂t
= −Q − Fk (V k − V1k ) + α∇ k ( pV k ).
The viscous stress tensors are written in the form
(
)


τ11 = μ 2 ∂u − 2 D , τ22 = μ  2 ∂v − 2 D  ,
∂x 3
∂
y
3


 ∂u ∂v 
u
∂
∂
v
τ12 = μ  +  , D =
+ .
∂x ∂y
 ∂y ∂x 
The dispersed phase dynamics is described by the
conservation equation for the average density, the conservation equations for the momentum components, and
the energy conservation equation written with account
for the interaction between phases of the mixture:
∂ρ1
+ ∇ (ρ1V1 ) = 0,
∂t
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TUKMAKOV, TUKMAKOVA
y
p2
0
x
p1
p2 > p1
Fig. 1. Schematic of the process being modeled.
∂ρ1V1k
(2)
+ ∇i (ρ1V1iV1k ) = Fk + FCk − α∇ k p,
∂t
∂ ( e1 )
k
k
+ ∇ (eV
1 1 ) = Q.
∂t
In analysis of the dynamics of the carrier medium,
it is assumed that ρ = (1 – α)ρg [1, 2]. Here, p, ρg, u,
and v are the pressure, density, and the Cartesian
components of the carrier medium velocity in the
direction of the х and у axes, respectively; Т and е are
the temperature and total energy of the gas; ρ1, Т1, е1, u1,
and v1 are the average velocity, temperature, internal
energy, and the Cartesian components of the dispersed
phase velocity in the direction of the х and у axes, respectively. The carrier medium temperature can be determined from equation T = (γ – 1)(e/ρ – 0.5(u2 + v 2 )/R,
where R is the gas constant carrier phase, μ is the viscosity of the gas, λ is its thermal conductivity, and γ is the
adiabatic exponent. The internal energy of the dispersed
phase suspended in a gas is defined as e1 = ρ1CpT1, where
Ср is the heat capacity of the unit mass of the dispersed
phase material; the average density of the dispersed
phase is calculated using expression ρ1 = αρ10, where
α is the volume concentration of the dispersed phase,
which is a function of the temporal and spatial variables; ρ10 is the physical density of the dispersed phase
material in the mixture, which remains a constant
quantity; Fk are the spatial components of the aerodynamic drag force; FСk are the spatial components of
the Coulomb force acting on the particles, and Q is the
heat flux between the carrier and dispersed phases of
the mixture. The electric field potential in the computational domain is determined from the solution of the
Poisson equation. The right-hand side of the Poisson
equation contains the (mass or surface) charge density
of the gas suspension, which is divided by the absolute
permittivity of the carrier medium [30]:
divE =
ρE
,
εε0
ρ
Δϕ=− E,
εε0
2
E = −∇ϕ,
−9
ε0 = 10 F/m;
36π
ρE = SqS = α qS ,
3r
∂ϕ
∂ϕ
FCx = −ρ1 , FCy = −ρ1 .
∂x
∂y
ρE = αρ10qm,
(3)
Here, ρE is the charge density; qi is the specific
charge of the unit mass (m) or density (s) of the solid
fraction; ϕ is the electric field potential; ε = 1 is the
relative permittivity of air, and ε0 is the absolute permittivity of air. System of equations (1), (2) is integrated
busing the explicit MacCormack finite difference
method of the second order of accuracy [31]. For suppressing numerical oscillations, the algorithm of nonlinear correction of the grid function is used [32, 33].
The system of equations is supplemented with the
corresponding initial and boundary conditions. In
analysis of the flows of a two-phase mixture, the Dirichlet homogeneous boundary conditions are specified on allmsurfaces for the velocity components of
the carrier medium and the dispersed phase. For
remaining dynamic functions of the mixture, the Neuman homogeneous boundary conditions are specified
on the lateral surfaces in accordance with the method
of finite-difference simulation of the dynamics of a
compressible heat-conducting gas [31] and the technique for modeling the dynamics of the dispersed
phase with varying “average density” and energy,
which is suspended in a compressible heat-conducting
gas [2, 3]. Poisson equation (3) [30, 34] describing
electric field potential is solved using the finite difference method with the help of the iteration algorithm of
the stabilization method [34] on the computational grid
generated for gasdynamic calculations to take into
account the effect of the Coulomb force in the solution
of the dynamic equations for a two-phase medium as
well as the average density distribution for the dispersed
phase at the points of discretization of a physical
domain in the solution of the Poisson equation.
2. RESULTS OF CALCULATIONS
Figure 1 shows the general diagram of a gas suspension flow in the channel being modeled. A homogeneous gas compressed to a high pressure is in the lefthand part of the channel; the pressure in the right-hand
part of the channel is lower (x < L/2, p = p2, x ≥ L/2,
p = p1, p1 < p2). The gas in the right-hand part of the
channel also contains dispersed particles, i.e., is a gas
suspension (x < L/2, α = 0, x > L/2, α = 0.001). The
particle diameter is d = 10 μm; the volume concentration of the dispersed phase is α = 0.001. The surface
charge density of the dispersed phase was qs =
‒0.0001195 C/m2; the mass charge density of the dispersed phase was qm = –0.0001 C/kg.
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(a)
2.0
1.8
2599
p, kPa
108
1.6
0
4
M, kV
1.2
8
0.6
12
0.8
1.4
103
1.0
m
x,
98
(a)
0.4
0.2
0.1
y, 0
m
0.5
x, 1.0
m
U1, kg/m
3
(b)
1.0
0.5
0.1
y, 0
m
0.2
0.4
0.6
1.0
0.8 m
x,
1.2
1.4
1.6
1.8
1.5
2.0
2.0 0
p, kPa
108
0.1
y, m
103
(b)
Fig. 2. Spatial distribution of (a) the “average density” of
the dispersed phase and (b) the electric field potential.
Figure 2a shows the spatial distribution of the electric field potential and of the average density of particles in the dispersed phase at the initial instant. Zero
potential corresponds to the part of the channel, in
which there are no electrically charged particles, while
a negative potential is observed in the part of the channel filled with electrically charged particles (Fig. 2b).
As a result of the initially nonuniform distribution
of the gas pressure in the channel (Fig. 3a), a lowintensity shock wave propagating from the region with
an elevated pressure to the low-pressure region is
formed (Fig. 3b) [29].
On account of the viscosity of the carrier medium,
a parabolic viscous profile of the velocity modulus of
the carrier liquid is formed, V = u2 + v 2 (Fig. 4) [29].
The velocity modulus in a homogeneous gas
exceeds the velocity modulus of the carrier medium in
the gas suspension; an increase in the gas suspension
density leads to a decrease in the velocity of the carrier
medium (Fig. 5a). Numerical calculations demonstrate that with increasing physical density of the particle material in an electrically neutral gas suspension,
the velocity of the phase slip in the mixture becomes
higher, V = (u − u1 ) + ( v − v1 ) (Fig. 5b).
The spatial distribution of the longitudinal component of the Coulomb force along the х axis demonstrates that the Coulomb force attains the highest
value at the interface between the homogeneous gas
2
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98
0.5
1.0
x, m
1.5
2.0 0 0.1
y, m
Fig. 3. Spatial distribution of the gas pressure (a) at the initial instant and (b) at instant t = 2 ms.
0
0.2
0.4
_V _, m/s
7
0.6
0.8
5
1.4
3
1
y, 0 0.1
m
1.0
1.2 , m
x
1.6
1.8
Fig. 4. Spatial distribution of the carrier medium velocity
modulus.
2600
TUKMAKOV, TUKMAKOVA
(a)
(b)
Homogeneous gas
U10 = 1000 kg/m3
U10
= 1200 kg/m3
8
U10
= 1400 kg/m3
_V _, m/s
10
U10 = 1600 kg/m3
6
4
U10 = 1000 kg/m3
2.0
U10 = 1200 kg/m3
U10 = 1400 kg/m3
1.5
_VV1_, m/s
12
U10 = 1600 kg/m3
1.0
0.5
2
0
0.5
1.0
x, m
1.5
2.0
0
1.0
1.2
1.4
1.6
x, m
1.8
2.0
Fig. 5. Distributions of physical parameters of the dynamics of mixture phases along the х coordinate: (a) gas velocity modulus
in a homogeneous medium and in electrically neutral gas suspensions with different densities of the dispersed phase; (b) modulus
of the difference in the velocities of mixture phases for different densities of the dispersed phase material.
concentration of the dispersed phase. The mathematical model implements the continual approach and
takes into account the momentum transfer between
the phases. We have considered the mathematical
models of an electrically charged gas suspension with
a surface and mass charge density of the dispersed
phase. Upon an increase in the physical density of the
material for electrically neutral gas suspensions, the
intensity of the high-speed slip between the phases
becomes higher. In electrically charged gas suspensions, the intensity of the high-speed slip for the mass
as well as the surface electric charge density is higher
due to the action of the Coulomb force on the dispersed phase. The largest difference in the high-speed
slip intensities of the carrier and dispersed phases of
the mixture for neutral and electrically charged gas
suspensions is observed in the part of the channel, in
which the Coulomb force has the highest value. Upon
an increase in the density of the particle material, the
increase in the high-speed slip intensity due to an
0
Fx, N/m3
and the gas suspension (Fig. 6). Figures 7a–7d show
the distribution the high-speed slip modulus of the
carrier and dispersed phases of a gas suspension during
the propagation of a low-intensity shock wave in electrically charged and neutral gas suspensions. It can be
seen that in the left-hand part of the channel, in the
region unperturbed by the flowing gas, particles move
due to the action of the Coulomb force [24]. The
results of calculations of the mass and surface electric
charge densities of the dispersed phase, which have
been performed for physical density ρ10 = 1000 kg/m3
of the particle material, coincide. Upon an increase in
the particle material density, the difference between
the results of calculations of the high-speed slip of
electrically charged gas suspensions with the surface
and mass charge densities increases. A high intensity
of the high-speed slip is observed for gas suspensions
with a mass density of the electric charge.
When a shock wave propagates in a gas suspension, the carrier medium dynamics is affected by the
density of the dispersed phase material as well as by
the electric charge of the dispersed phase of the mixture
(Figs. 8a, 8b). Upon an increase in the density of the
dispersed phase material, the difference in the values
of the gas pressure in the shock wave, which have been
obtained using the mass and surface models of the
electric charge, becomes very important (see Fig. 8b).
An increase in the density of the dispersed phase
material in electrically neural and electrically charged
gas suspensions leads to a decrease in the flow velocity
of the carrier medium and to an increase in the pressure in the shock wave.
U10 = 1000 kg/m3
2
U10 = 1600 kg/m3
4
6
0
0.5
CONCLUSIONS
1.0
x, m
In this study, the action of a low-intensity shock
wave propagating from a homogeneous gas on electrically neutral and electrically charged gas suspensions
has been simulated numerically for a high volume
Fig. 6. Spatial distribution of the х component of the
Coulomb force for the mass charge density distribution in
the dispersed phase for different densities of the particle
material.
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NUMERICAL STUDY OF THE INFLUENCE OF THE DISPERSED PHASE DENSITY
(a)
1.0
0.5
1.5
x, m
1.5
1.0
0
1.0
2.0
Electrically neutral particles
Urface charge densities of particles
Mass density of particle charge
(c)
1.2
(d)
2.0
1.5
_VV1_, m/s
_VV1_, m/s
Electrically neutral particles
Urface charge densities of particles
Mass density of particle charge
(b)
0.5
0
1.0
2.0
2.0
_VV1_, m/s
_VV1_, m/s
1.5
Electrically neutral particles
Urface charge densities of particles
Mass density of particle charge
2601
1.0
0.5
1.4
1.6
x, m
1.8
2.0
Electrically neutral particles
Urface charge densities of particles
Mass density of particle charge
1.5
1.0
0.5
0
1.0
1.2
1.4
1.6
x, m
1.8
0
1.0
2.0
1.2
1.4
1.6
x, m
1.8
2.0
(a)
1
2
3
4
4
6
7
_V _, m/s
10
8
6
4
0.6
0.8
1.0
x, m
1.2
1.4
р, kPa
Fig. 7. Distribution of the modulus of the velocity difference between the phases of the mixture for electrically neutral gas and
electrically charged gas suspensions with the mass and the surface charge density for different densities of the dispersed phase
material along the x coordinate: (a) ρ10 = 1000; (b) 1200; (c) 1400, and (d) 1600 kg/m3.
1
(b)
2
104800
3
104400
4
104000
4
6
103600
7
103200
102800
102400
102000
101600
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
x, m
Fig. 8. Distribution of (a) gas velocity modulus and (b) gas pressure along the x coordinate; (curve 1) homogeneous gas; (2) electrically neutral gas suspension with dispersed phase density ρ10 = 1000 kg/m3; (3) electrically charged gas suspension with dispersed phase density ρ10=1000 kg/m3 and surface density of the electric charge; (4) electrically charged gas suspension with dispersed phase density ρ10 = 1000 kg/m3 and the mass electric charge density; (5) electrically neutral gas suspension with dispersed
phase density ρ10 = 2500 kg/m3; (6) electrically charged gas suspension with dispersed phase density ρ10 = 2500 kg/m3 and the
surface electric charge density, and (7) electrically charged gas suspension with dispersed phase density ρ10 = 2500 kg/m3 and the
mass electric charge density.
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TUKMAKOV, TUKMAKOVA
increase in the Coulomb force is most significant in
the electrically charged gas suspension with the mass
charge density.
FUNDING
This work was performed under the State assignment of
the Federal Research Center “Kazan Scientific Center of
the Russian Academy of Sciences.”
CONFLICT OF INTEREST
The authors of this work declare that they have no conflict of interests.
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TECHNICAL PHYSICS
Vol. 69
No. 10
2024