Developed a system of computer simulation of oscillations of magnetizable microdrops in a wide range
of changing their parameters: surface tension, viscosity, magnetic permeability, density, and radius. Computational experiments of oscillations of magnetizable drops in an alternating magnetic field and the influence of various forces of nature (inertial, viscous, surface and magnetic) on the nature of the oscillations
were carried out. Adequacy of the model, used as the basis for the developed system of computer simulation
was shown on the basis of computational and experimental data

Drozdova, V.I.

Romanenko, M.G.

Shagrova, G.V.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 7 No 4, 04020(4pp) (2015) Том 7 № 4, 04020(4cc) (2015) 
 
 
2077-6772/2015/7(4)04020(4) 04020-1  2015 Sumy State University 
Modeling Oscillations of Magnetizable Microdrops 
 
G.V. Shagrova, V.I. Drozdova, M.G. Romanenko 
 
North-Caucasian Federal University, 2, Kulakova Prosp., 355009 Stavropol, Russia 
 
(Received 15 September 2015; published online 10 December 2015) 
 
Developed a system of computer simulation of oscillations of magnetizable microdrops in a wide range 
of changing their parameters: surface tension, viscosity, magnetic permeability, density, and radius.  Com-
putational experiments of oscillations of magnetizable drops in an alternating magnetic field and the influ-
ence of various forces of nature (inertial, viscous, surface and magnetic) on the nature of the oscillations 
were carried out. Adequacy of the model, used as the basis for the developed system of computer simula-
tion was shown on the basis of computational and experimental data. 
 
Keywords: Magnetized drops, Magnetic fluids, Oscillations of drops, Modeling, Computational experiment. 
 
 PACS numbers: 75.50.Mm, 02.60.Cb 
 
 
1. INTRODUCTION 
 
Development of systems for computer modeling of 
oscillations of magnetizable drops is a topical problem 
related to the study of the sensitivity of their free sur-
face to magnetic fields, for contactless control of mag-
neto-fluidic nano disperse systems, which are currently 
widely used in control and measuring systems, tech-
nical devices and processes. Mathematical modeling of 
the processes occurring at the interfaces between fluids 
with different magnetic and electrical properties is a 
complex nonlinear problem of magnetic hydrodynamics 
and is not only practical but also scientific interest 
related to the development of the theory of dispersed 
phase equilibrium nanosystems. There are several 
approaches to solving this problem, one of them is 
based on the fact that the shape of the interphase 
boundary is unknown in advance and must be obtained 
in the process of solving the problem [1, 2]. This ap-
proach is associated with large computational complex-
ity and is not acceptable when solving practical prob-
lems of managing fluid dispersed nanosystems in real-
time. In another approach, the drop shape is considered 
to be known beforehand [3-6]. In the works of various 
authors [7-9] as a result of the larger number of exper-
iments show that the shape of the magnetizable drops 
in a weak magnetic field corresponds to an ellipsoid of 
rotation. Using this fact allows to simplify the simula-
tion and obtain results that can be used in engineering 
calculations. 
 
2. DESCRIPTION OF THE OBJECT AND 
METHODS OF INVESTIGATION 
 
To build a system of computer simulation of oscilla-
tions of magnetizable drops in a wide range of changes 
in the parameters of the liquid and the external field 
used theoretical description of the change in the shape 
of a magnetizable drops based on the energy method 
obtained in [10] based on the assumptions that the na-
ture of the magnetization is linear and corresponds to 
the shape of a drop is elongated along the field ellipsoid 
of rotation. System of equations describing oscillations 
of magnetizable drops recorded in the form convenient 
for system development computer modeling of the form:  
 
 
   
    
  
 
  
 
 
    
 
 
 
   
      
2 2/3
2
2 2
22
1
23 2
22 2 2
3 2
1
2
2 2 2
0
2 8/3 2/2
1
22 60
3 2 1 2 1
2
1445 2
arcsin 1
2 1 ( 1) 14
2
45 1
2 2
2
i e
dV t t t
V t V t
dt t t tr
tt t
t tt t tr
H t t
r

 
 
  
 
    
   

   
 
     
     
 
  
                     
 
 

 
  
 
 
               
              
 
 
   
3
3/2
2 2 2 2 2 1
2
3/2
2 2 2 2
ln 2 2 1 1 2 1 2 1 1 2
;
/ 1 ln 2 2 1 1 2 1 2 1
;
i e
t t t t t t t
t t t t t t
d t
V t
dt
H t Н t T

      
       


 













 



   
            
   
                   




 
 (1) 
 G.V. SHAGROVA, V.I. DROZDOVA, M.G. ROMANENKO J. NANO- ELECTRON. PHYS. 7, 04020 (2015) 
 
 
04020-2 
Magnetic fluid microdrop has the following parame-
ters: viscosity; surface tension ; density 1; radius r; 
the magnetic permeability i, and the magnetic fluid, in 
which is placed a drop, has a density of 2 and the per-
meability e.  
The state of the magnetizable drops in the magnetic 
field strength  Н t , characterized by a parameter 
 t  that represents the ratio of the semiaxes of the 
ellipsoid of rotation  a t  and  b t  with equal 
     /t a t b t   speed    t V t   and acceleration of 
the     /t V t t    . 
The strength of the external magnetic field
   H t Н t T   acting on the drop can vary as sinus-
oidal    0 sin 2аH t H H f t       , and provide a 
sawtooth change    0 аH t H Н k t     for 0 t T  . 
 
3. COMPUTER SIMULATION SYSTEM  
 
Computer simulation system (Fig. 1) is implemented 
in Simulink matrix laboratory Matlab. The Simulink 
allows you to use modern technology visually-oriented 
programming for simulation block modeling of various 
systems and devices. Basic computer simulation system 
contains a subsystem "Subsystem".  The subsystem 
"Subsystem" (Fig. 2) contains four modules to determine 
the contribution of forces of different nature on the na-
ture of the oscillations. 
The subsystem "1" represents a module, to calculate 
the first term of the right part of the first equation of the 
system of equations (1), the result of the calculation 
shown in the figure as E(). 
The second and the third module allow calculating 
respectively the second and third summands of the right 
part of the first equation of the system of equations (1). 
The result of the calculation of the second module is 
indicated in figure 2, as E()/E(). 
It describes the action of viscous and inertial forces 
on the drop and is output to the evaluation unit Scope. 
  
 
Fig. 1 – Diagram of computer simulation Fig. 2 – Diagram "Subsystem" 
 
 
 MODELING OSCILLATIONS OF MAGNETIZABLE MICRODROPS J. NANO- ELECTRON. PHYS. 7, 04020 (2015) 
 
 
04020-3 
 
 
 
 
  
2/3
2
22
1
60
2 1
2
E t
E tr
 
 

 
    
 
 (2) 
 
The result of the calculation of the third module is "3" 
(Fig. 2) E()/E(), characterizes the effect of surface and 
inertial forces on the drop and also displayed on the 
device Scope.  
 
 
 
 
    
 
 
 
   
3
2 2
3 2
1
22
22
45
2 1 ( 1)4
2
14 2
arcsin 1
1
E t
E t tr
tt
t tt
 
  

 

  
     
 
  
      
     
 (3) 
 
 
 
 
 
      
 
               
              
2
2 2 2
0
2 8/3 2/32
1
3/2
2 2 2 2 2 1
2
3/2
2 2 2 2
45 1
2 2
2
ln 2 2 1 1 2 1 2 1 1 2
/ 1 ln 2 2 1 1 2 1 2 1
i e
i e
H t tE
E
r
t t t t t t t
t t t t t t
   

  
      
       
 

 

 
 
 
  
 
   
           
   
   
          
   
 (4) 
 
The result of the calculation of the fourth module 
E()/E() (Fig. 2) is registered by the instrument Scope 
and characterizes the action of the magnetic and inertial 
forces on the drop. Signal generated by the device Gen-
erator. 
 
4. COMPUTER SIMULATION 
 
The developed system allows studying the nature of 
the oscillations of magnetizable microdroplets in a wide 
range of changes in interfacial tension  from 10 – 7 N/m 
to 10 – 3 N/m and the strength of magnetic field acting on 
the drop, like a sine and a sawtooth. If the magnetic field 
acting on the microdrop, larger than the critical field 
jumpwise elongation Н1, the computer simulation sys-
tem allows obtaining the hysteretic nature of forced 
oscillations. When Н(t)  Н1 the result of numerical 
simulation correspond to nonhysteretic nature of forced 
oscillations. 
Numerical simulation adequately describes obtained 
[11] when Н0  Н1, f  0.02 Hz and f  0.03 Hz experi-
mental results.  Time dependence (t) is periodic, micro-
droplets have time to change shape during the half-
period of oscillations of the external field, the period of 
time when (t)  1 is reduced with increasing frequency. 
Figures 3 -5 present the simulation results of forced 
oscillations microdroplets with parameters: 
1  1900 kg/m3; 2  1060 kg/m3; е   1; i  50; 
  3,7·10 – 7 N/m,   0,068 Pas. Curves 1 correspond to 
the values (t); curves 2 correspond to the shape of an 
external magnetic field; curves 3, 4 и 5  present respec-
tively the dependency relations E()/E(), E()/E() and 
E()/E() from time.  
Fig. 3 presents the simulation results obtained in the 
sawtooth field at H0  0; На  160 A/m, r  5 mkm.  
Fig. 4 presents the simulation results obtained in si-
nusoidal field at H0  0; Ha  240 A/m, r  5,6 mkm. 
Fig. 5 shows the simulation results obtained in the 
alternating magnetic field with magnetic bias when 
H0  80 A/m; Ha  240 A/m; f  0,02 Hz, r  5,6 mkm. 
 
 
Fig. 3 – Time functions in a sawtooth magnetic field 
 
5. CONCLUSION 
 
The calculation results which are shown in figures 
3-5 correspond to the conception of the competition 
between capillary and magnetic stresses. Minimum 
instantaneous values (t) correspond to a maximum 
contribution of surface forces. 
 
 
 
 
 
 
 
 
 
 
 
 G.V. SHAGROVA, V.I. DROZDOVA, M.G. ROMANENKO J. NANO- ELECTRON. PHYS. 7, 04020 (2015) 
 
 
04020-4 
 
 
 
Fig. 4 – Time functions in a sinusoidal magnetic field 
 
 
Fig. 5 – Time functions in a alternating magnetic field with 
magnetic bias 
 
The use of a sinusoidal field with a magnetic bias 
allows us to manage the form of droplets so as, during 
halftime of the external sinusoidal field on microdrop to 
the sum of fixed and variable components, and in an-
other of half their difference. 
The increase of instantaneous values (t) corre-
sponds to an increase of the contribution of magnetic 
forces to the sum of the forces acting on microdrops. 
The maximum value of the ratio of viscous and inertial 
forces corresponds to a transition of microdrops to its 
undeformed state in which the influence of inertial 
forces is minimal. 
 
 
REFERENCES 
 
1. I.E. Tarapov, Izvest. AS USSR. Mech. Fluid Gas. No 5, 141 
(1974). 
2. S.V. Smetanin, G.R Shrager, V.A. Yakutenok, Fluid Dy-
namic. 35 No 6, 813 (2000). 
3. J.C. Bacri, D. Salin, J. Physique Lett. 44, 415 (1983). 
4. А.О. Ceber, 21 Magn. Gidrodin. No 1, 25 (1985). 
5. А.N. Golubyatnikov, G.I. Subhankulov, Magn. Gidrodin. 
22 No 1, 73 (1986). 
6. А.О. Kuzubov, Kand. Dissertation. Phys.-Mat. Sci. (Mos-
cow: 1988). 
7. V.I. Arhipenko, Yu.D. Barkov, V.G. Bashtovoy, Magn. 
Gidrodin. 14 No 3, 131 (1978). 
8. V.V. Chekanov, М.D. Halupovsky, I.Yu. Chuenkova,  
Magn. Gidrodin. 24 No 3, 124 (1988). 
9. N.V. Candaurova, I.Yu. Chuenkova, Magn. Gidrodin. 27 
No 1, 114 (1991). 
10. M.G. Romanenko, Kand. Dissertation. Phys.-Mat. Sci. 
(Stavropol: 2011). 
11. V.I. Drozdova, G.V. Shagrova, M.G. Romanenko, J. Nano- 
Electron. Phys. 6 No 3, 03013 (2014). 
 


Modeling Oscillations of Magnetizable Microdrops

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 7 No 4, 04020(4pp) (2015) Том 7 № 4, 04020(4cc) (2015)   2077-6772/2015/7(4)04020(4) 04020-1  2015 Sumy State University Modeling Oscillations of Magnetizable Microdrops  G.V. Shagrova, V.I. Drozdova, M.G. Romanenko  North-Caucasian Federal University, 2, Kulakova Prosp., 355009 Stavropol, Russia  (Received 15 September 2015; published online 10 December 2015)  Developed a system of computer simulation of oscillations of magnetizable microdrops in a wide range of changing their parameters: surface tension, viscosity, magnetic permeability, density, and radius.  Com-putational experiments of oscillations of magnetizable drops in an alternating magnetic field and the influ-ence of various forces of nature (inertial, viscous, surface and magnetic) on the nature of the oscillations were carried out. Adequacy of the model, used as the basis for the developed system of computer simula-tion was shown on the basis of computational and experimental data.  Keywords: Magnetized drops, Magnetic fluids, Oscillations of drops, Modeling, Computational experiment.   PACS numbers: 75.50.Mm, 02.60.Cb   1. INTRODUCTION  Development of systems for computer modeling of oscillations of magnetizable drops is a topical problem related to the study of the sensitivity of their free sur-face to magnetic fields, for contactless control of mag-neto-fluidic nano disperse systems, which are currently widely used in control and measuring systems, tech-nical devices and processes. Mathematical modeling of the processes occurring at the interfaces between fluids with different magnetic and electrical properties is a complex nonlinear problem of magnetic hydrodynamics and is not only practical but also scientific interest related to the development of the theory of dispersed phase equilibrium nanosystems. There are several approaches to solving this problem, one of them is based on the fact that the shape of the interphase boundary is unknown in advance and must be obtained in the process of solving the problem [1, 2]. This ap-proach is associated with large computational complex-ity and is not acceptable when solving practical prob-lems of managing fluid dispersed nanosystems in real-time. In another approach, the drop shape is considered to be known beforehand [3-6]. In the works of various authors [7-9] as a result of the larger number of exper-iments show that the shape of the magnetizable drops in a weak magnetic field corresponds to an ellipsoid of rotation. Using this fact allows to simplify the simula-tion and obtain results that can be used in engineering calculations.  2. DESCRIPTION OF THE OBJECT AND METHODS OF INVESTIGATION  To build a system of computer simulation of oscilla-tions of magnetizable drops in a wide range of changes in the parameters of the liquid and the external field used theoretical description of the change in the shape of a magnetizable drops based on the energy method obtained in [10] based on the assumptions that the na-ture of the magnetization is linear and corresponds to the shape of a drop is elongated along the field ellipsoid of rotation. System of equations describing oscillations of magnetizable drops recorded in the form convenient for system development computer modeling of the form:                                  2 2/322 222123 222 2 23 2122 2 202 8/3 2/2122 603 2 1 2 121445 2arcsin 12 1 ( 1) 14245 12 22i edV t t tV t V tdt t t trtt tt tt t trH t tr                                                                                           33/22 2 2 2 2 123/22 2 2 2ln 2 2 1 1 2 1 2 1 1 2;/ 1 ln 2 2 1 1 2 1 2 1;i et t t t t t tt t t t t td tV tdtH t Н t T                                                      (1)  G.V. SHAGROVA, V.I. DROZDOVA, M.G. ROMANENKO J. NANO- ELECTRON. PHYS. 7, 04020 (2015)   04020-2 Magnetic fluid microdrop has the following parame-ters: viscosity; surface tension ; density 1; radius r; the magnetic permeability i, and the magnetic fluid, in which is placed a drop, has a density of 2 and the per-meability e.  The state of the magnetizable drops in the magnetic field strength  Н t , characterized by a parameter  t  that represents the ratio of the semiaxes of the ellipsoid of rotation  a t  and  b t  with equal      /t a t b t   speed    t V t   and acceleration of the     /t V t t    . The strength of the external magnetic field   H t Н t T   acting on the drop can vary as sinus-oidal    0 sin 2аH t H H f t       , and provide a sawtooth change    0 аH t H Н k t     for 0 t T  .  3. COMPUTER SIMULATION SYSTEM   Computer simulation system (Fig. 1) is implemented in Simulink matrix laboratory Matlab. The Simulink allows you to use modern technology visually-oriented programming for simulation block modeling of various systems and devices. Basic computer simulation system contains a subsystem "Subsystem".  The subsystem "Subsystem" (Fig. 2) contains four modules to determine the contribution of forces of different nature on the na-ture of the oscillations. The subsystem "1" represents a module, to calculate the first term of the right part of the first equation of the system of equations (1), the result of the calculation shown in the figure as E(). The second and the third module allow calculating respectively the second and third summands of the right part of the first equation of the system of equations (1). The result of the calculation of the second module is indicated in figure 2, as E()/E(). It describes the action of viscous and inertial forces on the drop and is output to the evaluation unit Scope.    Fig. 1 – Diagram of computer simulation Fig. 2 – Diagram "Subsystem"    MODELING OSCILLATIONS OF MAGNETIZABLE MICRODROPS J. NANO- ELECTRON. PHYS. 7, 04020 (2015)   04020-3       2/32221602 12E tE tr         (2)  The result of the calculation of the third module is "3" (Fig. 2) E()/E(), characterizes the effect of surface and inertial forces on the drop and also displayed on the device Scope.                32 23 212222452 1 ( 1)4214 2arcsin 11E tE t trttt tt                          (3)                                          22 2 202 8/3 2/3213/22 2 2 2 2 123/22 2 2 245 12 22ln 2 2 1 1 2 1 2 1 1 2/ 1 ln 2 2 1 1 2 1 2 1i ei eH t tEErt t t t t t tt t t t t t                                                            (4)  The result of the calculation of the fourth module E()/E() (Fig. 2) is registered by the instrument Scope and characterizes the action of the magnetic and inertial forces on the drop. Signal generated by the device Gen-erator.  4. COMPUTER SIMULATION  The developed system allows studying the nature of the oscillations of magnetizable microdroplets in a wide range of changes in interfacial tension  from 10 – 7 N/m to 10 – 3 N/m and the strength of magnetic field acting on the drop, like a sine and a sawtooth. If the magnetic field acting on the microdrop, larger than the critical field jumpwise elongation Н1, the computer simulation sys-tem allows obtaining the hysteretic nature of forced oscillations. When Н(t)  Н1 the result of numerical simulation correspond to nonhysteretic nature of forced oscillations. Numerical simulation adequately describes obtained [11] when Н0  Н1, f  0.02 Hz and f  0.03 Hz experi-mental results.  Time dependence (t) is periodic, micro-droplets have time to change shape during the half-period of oscillations of the external field, the period of time when (t)  1 is reduced with increasing frequency. Figures 3 -5 present the simulation results of forced oscillations microdroplets with parameters: 1  1900 kg/m3; 2  1060 kg/m3; е   1; i  50;   3,7·10 – 7 N/m,   0,068 Pas. Curves 1 correspond to the values (t); curves 2 correspond to the shape of an external magnetic field; curves 3, 4 и 5  present respec-tively the dependency relations E()/E(), E()/E() and E()/E() from time.  Fig. 3 presents the simulation results obtained in the sawtooth field at H0  0; На  160 A/m, r  5 mkm.  Fig. 4 presents the simulation results obtained in si-nusoidal field at H0  0; Ha  240 A/m, r  5,6 mkm. Fig. 5 shows the simulation results obtained in the alternating magnetic field with magnetic bias when H0  80 A/m; Ha  240 A/m; f  0,02 Hz, r  5,6 mkm.   Fig. 3 – Time functions in a sawtooth magnetic field  5. CONCLUSION  The calculation results which are shown in figures 3-5 correspond to the conception of the competition between capillary and magnetic stresses. Minimum instantaneous values (t) correspond to a maximum contribution of surface forces.             G.V. SHAGROVA, V.I. DROZDOVA, M.G. ROMANENKO J. NANO- ELECTRON. PHYS. 7, 04020 (2015)   04020-4    Fig. 4 – Time functions in a sinusoidal magnetic field   Fig. 5 – Time functions in a alternating magnetic field with magnetic bias  The use of a sinusoidal field with a magnetic bias allows us to manage the form of droplets so as, during halftime of the external sinusoidal field on microdrop to the sum of fixed and variable components, and in an-other of half their difference. The increase of instantaneous values (t) corre-sponds to an increase of the contribution of magnetic forces to the sum of the forces acting on microdrops. The maximum value of the ratio of viscous and inertial forces corresponds to a transition of microdrops to its undeformed state in which the influence of inertial forces is minimal.   REFERENCES  1. I.E. Tarapov, Izvest. AS USSR. Mech. Fluid Gas. No 5, 141 (1974). 2. S.V. Smetanin, G.R Shrager, V.A. Yakutenok, Fluid Dy-namic. 35 No 6, 813 (2000). 3. J.C. Bacri, D. Salin, J. Physique Lett. 44, 415 (1983). 4. А.О. Ceber, 21 Magn. Gidrodin. No 1, 25 (1985). 5. А.N. Golubyatnikov, G.I. Subhankulov, Magn. Gidrodin. 22 No 1, 73 (1986). 6. А.О. Kuzubov, Kand. Dissertation. Phys.-Mat. Sci. (Mos-cow: 1988). 7. V.I. Arhipenko, Yu.D. Barkov, V.G. Bashtovoy, Magn. Gidrodin. 14 No 3, 131 (1978). 8. V.V. Chekanov, М.D. Halupovsky, I.Yu. Chuenkova,  Magn. Gidrodin. 24 No 3, 124 (1988). 9. N.V. Candaurova, I.Yu. Chuenkova, Magn. Gidrodin. 27 No 1, 114 (1991). 10. M.G. Romanenko, Kand. Dissertation. Phys.-Mat. Sci. (Stavropol: 2011). 11. V.I. Drozdova, G.V. Shagrova, M.G. Romanenko, J. Nano- Electron. Phys. 6 No 3, 03013 (2014).  

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Modeling Oscillations of Magnetizable Microdrops

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