ResearchGate
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/257856382
Thermoconvective flow velocity in a high-speed magnetofluid seal after it has 
stopped
Article in Technical Physics • September 2012
DOI: 10.1134/S1063784212090150 *12
CITATION
1
READS
58
2 authors, including:
M. S. Krakov
Belarusian National Technical University
70 PUBLICATIONS 361 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Project Thermal and concentration convection in cylindrical enclosure. View project
A ll content fo llow ing this page was uploaded by M. S. Krakov on 14 February 2016.
The user has requested enhancement of the downloaded file.
IS S N  1063-7842, Technical Physics, 2012, Vol. 57, No. 9, pp. 1308—1311. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © M.S. Krakov, I.V. Nikiforov, 2012, published in Zhurnal Tekhnicheskoi Fiziki, 2012, Vol. 82, No. 9, pp. 126—129.
SHORT
COMMUNICATIONS
Thermoconvective Flow Velocity 
in a High-Speed Magnetofluid Seal After it Has Stopped1
M. S. Krakov*" and I. V. Nikiforov*
a Belarusian National Technical University, pr Nezavisimosti 65, Minsk, 220013 Belarus 
b Belarusian State University, pr Nezavisimosti 4, Minsk, 220050 Belarus 
*e-mail: mskrakov@gmail.com 
Received November 7, 2011
Abstract— Convective flow is investigated in the high-speed (linear velocity of the shaft seal is more than 
1 m/s) magnetofluid shaft seal after it has been stopped. Magnetic fluid is preliminarily heated due to viscous 
friction in the moving seal. After the sea^has been stopped, nonuniform heated fluid remains under the action 
of a high-gradient magnetic field. Numerical analysis has revealed that in this situation, intense thermomag­
netic convection is initiated. The velocity of magnetic fluid depends on its viscosity. For the fluid with viscos­
ity of 2 X 10-4 m2/s the maximum flow velocity within the volume of magnetic fluid with a characteristic size 
of 1 mm can attain a value of 10 m/s.
DOI: 10.U34/S1063784212090150
IN TRODU CTIO N
During the 11th International Conference on 
Magnetic Fluids, Dr. R.E. Rosensweig paid attention 
to the fact that after the complete stoppage of a high­
speed magnetofluid shaft seal (M FSS), radial flow is 
seen at a free surface, but the cause of it is not clear [1]. 
It is well known that magnetic fluid in the MFSS is 
strongly heated due to viscous friction. The tem pera­
ture distribution in the fluid volume is nonuniform. 
The magnetic field in the MFSS is also nonuniform 
and the strength gradient is very high (~109 A /m 2). 
U nder these conditions, in the magnetic fluid volume, 
there must appear very intense thermomagnetic con­
vection that can be the cause of magnetic fluid flow at 
the free surface in spite of a motionless shaft. O f fun­
damental importance is the question: what is the value 
of the convective flow velocity and can it be registered 
when observing the free surface of the magnetic fluid 
in the MFSS.
GOVERNING EQUATIONS
Let R  be the shaft radius and a be the width of the 
gap between the shaft and the pole of the MFSS. As for 
the MFSS, where usually R > a, an axisymmetrical 
flow in the plane r—z  could be considered as plane. It 
is possible to use a stream function у  as vr =  —d ^/d z ,
vz =  d y /d r  and a vortex ш =  curlV . Then, the system 
of dimensionless equations for axisymmetric flow and
temperature in the meridional plane in Boussinesq 
approximation can be written as:
ш = - A y , (1)
-  (3\|/(d(o 
dz dr dr  dz
л 2 л 2
= ^  ^  -  G r m [ V x ( MTVH ) ],
dz. d r
P r f -  ddydrTx = AT  
X dr dz dz drl
(2)
(3)
The article was translated by the authors.
where v /a , T0 =  1K, and a are used as scales for veloc­
ity, temperature, and distance, Pr =  v/ k  is the Prandtl 
number, Grm =  p QppToHQMSa'2/ pQV2 is the magnetic 
Grashof number, р0 is the magnetic permeability of 
vacuum, p p is the coefficient of fluid thermal expan­
sion, V, po, and k  are viscosity, density, and thermal 
diffusivity of the magnetic fluid, H0 is the maximum 
strength of a magnetic field under the pole of the 
MFSS, M  =  M(H, T) is the magnetic fluid magnetiza­
tion, the state equation is assumed to be M (H , T) =  
M*(H)[1 -  pp( T  -  ^ ) ] ,  M*(H) =  M SH /(H T + H), the 
equilibrium values are marked by symbol “*” , M S is 
the magnetic fluid magnetization saturation, and H T is 
the experimental value of the magnetic field strength 
at which fluid magnetization is equal to half of the 
magnetization saturation (for magnetic fluids used in 
MFSS H T-  50 -  100 kA/m).
The problem was studied numerically in the geom­
etry as presented in [2, 3] for the shape of the pole of 
the MFSS described by a hyperbola with an angle 
between asymptotes 2p . For an adequate description
1308
THERMOCONVECTIVE FLOW VELOCITY 1309
T, overheating, K
Fig. 1. Calculation doman. Fig. 2. Temperature variation along the shaft of the MFSS.
of the problem one used the system of n , ^ coordinates 
of the elliptic cylinder so that the coordinate line n = 
e coincided with the pole surface, the line n coincided 
with the shaft surface, and the coordinate lines ^ were 
normal to them  (Fig. 1). A finite-difference scheme 
based on the control volume method is used for the 
solution of Eqs. (1)—(3). In the frames of this m ethod 
the linear interpolation function was used for the 
stream function and exponential Patankar function [4] 
for the vortex, which gives one the possibility to take 
into account both the direction and intensity of flow in 
the control volume.
The analysis used the typical values of magnetic 
fluid properties and MFSS parameters: a =  2 x 10-4 m, 
angle between the pole surface and plane symmetry 
e =  45°, thermal diffusivity к  =  'k/p0Cp =  0.2/(1.2 x 
103 x 1.7 x 103) =  10-7 m 2/s, coefficient of thermal 
expansion вр =  10-3 K-1, maximum magnetic field 
strength in the MFSS gap H0 =  2 x 106 A/m , magnetic 
fluid magnetization saturation M* =  4 x 104 A/m , and 
density of magnetic fluid p0 =  1.2 x 103 kg/m 3.
The viscosity o f m agnetic fluids used in the 
M FSS is in the range from 3 x 10-5 to 1.5 x 
10“3 m 2/s . C alculations were m ade on the grid with 
251 x 151 nodes only for viscosities greater than  or 
equal to 2 x 10-4 m 2/s , for which steady temperature 
fields in the seal with the moving shaft, are found in 
[3]. These temperature fields were used as boundary 
conditions when solving Eq. (3). The examples of the 
temperature distribution for one of the versions are
illustrated in Figs. 2 and 3. Here, the temperature T  is 
the difference o f the absolute tem perature and the 
cooling system tem perature. So, the tem perature on 
the solid boundaries o f the magnetic fluid volume 
was assumed to be defined and was taken from the 
calculations o f high-speed M FSS with the rotating 
shaft [3]. The heat flux through the magnetic fluid free
T, overheating, K
Fig. 3. Temperature variation along the pole of the MFSS.
TECHNICAL PHYSICS Vol. 57 No. 9 2012
1310 KRAKOV, NIKIFOROV
Fig. 4. Convective flow streamlines. Pr = 2040, Gr,„ = 
0.0838, V = 2 X 10-4 m2/s, Tmax = 47 K. ymax = 0.0369, 
¥min = —0.00225. The same flow structure exists for all vis­
cosities.
Fig. 5. Temperature profiles in the magnetic fluid volume.
Pr = 2040, Gr„ = 0.0838, v = 2 x 10-4 m2/s, Tmax = 
42.7 K.
surface and plane symmetry was assumed to be equal 
to zero.
RESULTS
Numerical solutions to Eqs. (1)—(3) show that in 
the calculation domain two convective cells are 
formed with the flow in the small cell being much 
more intense than in the large one (Fig. 4). The largest 
velocity of convective flow is observed in the small cell 
in the vicinity of the pole tip. The intensity of fluid 
flow is so high that isotherms are essentially distorted 
(Fig. 5), though usually thermal conductivity prevails 
in the volume of the small part of a millimeter size and 
isotherms coincide with volume boundaries. It could 
be seen from the presented pattern that the fluid moves 
counter clockwise in the inner cell and clockwise in 
the outer one; i.e., the external observer has to see the 
fluid movement at the free surface from shaft to pole. 
The flow velocity at the free surface is minimal near 
the shaft and the pole and has a maximum. Figure 6 
plots the maximum fluid flow velocity in the volume 
and the maximum velocity at the fluid surface as a 
function of maximum overheating temperature of the 
shaft surface. As it should be expected, the intensity of 
the convective flow increases with growing overheating 
temperature. Calculations were made for the viscosity 
values of v =  15 x 10-4 m 2/s  (Fig. 6), v =  5 x 10-4 m 2/s  
(Fig. 7), and v =  2 x 10-4 m 2/s (Fig. 8). Of the greatest 
interest are the values of the surface velocity of a mag­
netic fluid drop that can be compared to the experi­
mental data. It is seen that at T  =  50 K the maximum 
velocity at the free surface of the magnetic fluid varies 
with decreasing viscosity from 0.15 m m /s for the vis-
Velocity, 10 3 m/s
Fig. 6. Variation of the maximum fluid flow velocity and 
surface velocity vs. the overheating temperature of the
pole. V = 15 X 10 4 m2/s.
TECHNICAL PHYSICS Vol. 57 No. 9 2012
THERMOCONVECTIVE FLOW VELOCITY 1311
Velocity, 10-3 m/s Velocity, m/s
Fig. 7. Variation of the maximum fluid flow velocity and 
surface velocity vs. the overheating temperature of the 
pole. V = 5 X 10-4 m2/s.
Fig. 8. Variation of the maximum fluid flow velocity and 
the surface velocity with the overheating temperature of 
the pole. V = 2 x 10-4 m2/s.
cosity V = 15 X 10-4 m 2/s to 16 cm /s for the fluid vis­
cosity V = 2 X 10-4 m 2/s.
Thus, natural thermomagnetic convection in the 
volume of the magnetic fluid, due to its heating during 
the operation of high-speed MFSS, can appear to be 
the cause of the m otion of this fluid after the MFSS 
has been stopped.
CONCLUSIONS
The magnetic fluid in the high-speed MFSS is 
strongly heated due to viscous friction. After the seal 
has been stopped, nonuniform heated fluid is under 
the action of a high-gradient magnetic field. Num eri­
cal analysis has revealed that in this situation, intense 
thermomagnetic convection is initiated. The velocity 
of the magnetic fluid depends on its viscosity. For the 
fluid with viscosity of 2 x 10-4 m 2/s the maximum flow 
velocity within the volume of magnetic fluid in the 
MFSS with a characteristic size of 1 mm can attain a
value of 10 m /s, and the free surface velocity attains a 
value about 30—50 cm /s and could be seen visually. 
Thus, natural convection in the volume of the mag­
netic fluid due to its heating during the operation of 
high-speed MFSS can appear to be the cause of the 
m otion of this fluid after the MFSS has been stopped.
REFERENCES
1. R. E. Rosensweig, in Proceedings of 11th International 
Co^^^^e^ce on Magnetic Liquids, Koshitse, Slovakia, 
2007 (private communication).
2. V. K. Polevikov, Izv. Ross. Akad. Nauk, Mekh. Zhidk. 
Gaza, No. 3, 170 (1997).
3. M. S. Krakov and I. V. Nikiforov, Tech. Phys. 56 , 1745 
(2011).
4. S. Patankar, Numerical Heat Transfer and Fluid Flow. 
Hemisphere Series on Computational Methods in 
Mechanics and Thermal Science (Taylor and Fransis, 
London, 1980).
SPELL:OK
TECHNICAL PHYSICS Vol. 57 No. 9 2012
View publication stats