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Two-dimensional flow of magnetic fluid between two parallel plates
327
Journal of Magnetism and Magnetic Materials 65 (1987) 327-329 North-Holland, Amsterdam
TWO-DIMENSIONAL T. SAWADA, Faculty
FLOW OF MAGNETIC FLUID BETWEEN TWO PARALLEL PLATES
T. TANAHASHI
Science and
and T. AND0
Keio University,
Two-dimensional flows magnetic fluid experimentally. The rate of field and Reynolds
Kohoku-ky
Yokohama,
two parallel in the coefficient is
1. Introduction many interests arisen in tion of fluids (MF) engineering deand a of practical have been out, e.g. device, levitation, netic ink abrasive. As flow phenomena MF, many have been But, most them are with respect pipe flows applied magnetic [l-5]. SpasojeviC al. [6] a channel of a suspensions experimentally. treated a viscous carrier mainly clarified characteristics of suspensions. In present paper, deal with sional flow MF between parallel plates in some types of netic fields. effect of magnetic field flow resistance clarified in laminar re-
fields are magnetic
fields (a), magnetic magnets
and (c); (a) is transverse produced by permanent X 40 10 mm3). magnetic flux B along the center line is B = 40 mT. Type (b) is a longitudinal magnetic field produced by some solenoidal coils whose lengths are 50 mm. The magnetic flux densities due to the excited currents I = 1, 2, 3, 4 A are B = 7, 14, 21, 29 mT, respectively. Changing the number of permanent magnets or solenoidal coils in the case of (a) and (b), one can alter the length of the applied magnetic field. Type (c) is a longitudinally nonuniform magnetic field produced by a solenoidal coil whose turn changes along the axial direction, and distributions of magnetic flux density, which is measured by a gauss meter, are shown in fig. 1. When this solenoidal coil is set inversely, two different magnetic fields can be obtained, that is, the direction of magnetic gradient is opposite.
50
2. Experiment The test channel is made of acrylic resin and its length is 998 mm. The cross section of a test channel is rectangular whose width and height are 40 mm and 2 mm, respectively. Test liquid is MF with 24% weight concentration of fine magnetite particles in a water carrier. Viscosity and density of the MF are q = 4.83 mPas and p = 1.22 X lo3 kg/m3 at 15OC, respectively. We use the following three kinds of magnetic
and longitudinal and discussed respect to
60 co
I 3oc 20 10
I
-?oo t
Fig. 1. Distributions of magnetic flux density for solenoidal 3 coil.
0304~8853/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
328
T. Saw&
et al. / Two-dimensional flow
The pressure difference between the upstream and downstream taps is measured by the digital manometer. The pressure difference includes the static pressure change due to a magnetic force and the pressure drop due to a flow resistance. However, if the pressure taps are connected to a manometer, the indicator gives directly the pressure drop since the effect of magnetic force is cancelled out by the magnetic force acting on the MF in both comecting tubes [3]. In type (a), the taps are situated just in the middle of the magnetic field region at intervals of 100 mm. The difference pressure ‘is measured outside of the solenoidal coil apart from both sides at intervals SQ mm in type (b) and (c)-
3. Results and discussion The resistance coefficient X and number Re are respectively defined AP _j@!f 4h ’ I
Reynolds
Re=apvh rl
’
where Ap is the pressure drop through the channel length, h the width of the two parallel plates, u the mean velocity, p the fluid density, and 71the fluid viscosity. For the laminar flow of Newtonian fluid, X is represented by X = 96/Re. Fig. 2 shows the influence of a transverse mag-
of a magnetic fluid
netic field length on increasing rate of resistance coefficient AA/A. When the magnetic field length L is small, the increasing rate of A makes a large difference with the one of Re. But Ax/x becomes a constant for L > 300 mm. This figure indicates that the increasing rate AX/X depends on the magnetic field length L and the Reynolds number ReforL~3OOmm.ForL~3OOmmthelineof AX/X - L is nearly flat with a little slope. Since the magnetic field is almost constant except for the 10 mm region from the end of the magnet, that is, the differential pressure is measured in the uniform magnetic field, large changes of Ax/x with L is caused by the hydraulic disturbance due to an abrupt change of a magnetic field at the entrance. In the present experiment, influence of a magnetic field on the development of an inlet flow cannot be clarified because the length of a permanent magnet is relatively long, i.e. 40 mm. The relation between a longitudinal magnetic field length and AX/h is shown in fig. 3. The influence of longitudinal magnetic field length is not evident. Fig. 3 shows a tendency that AX/A increases gradually with L. Since the differential pressure is measured outside of the solenoidal coil, this figure indicates global AX/X due to a solenoidal coil. The magnetic field is almost uniform for L > 100 mm except in the vicinity of the end of a solenoidal coil. Then Ax/x does not change largely with L. Magnetic forces caused by the non-uniform magnetic field near both ends of a solenoidal coil z~fecancelled. Figs. 4 and 5 S~QF #e relation between Ah/A and & under the inverse magnetic fields, respectively. The line Ah/X-Re is based on the pressure
1
15 8=29
5-
mT
118
;
_V
0
I
I
I
200
I
L mm
I
LOO
’ 0 4 100
I
Fig. 2. Increasing rate of resistance coefficient versus verse magnetic field length.
5
trans-
A I
200
I
300 L mm
600 600 1500 I
LOO
Fig. 3. Increasing rate of resistance coefficient versus longitudinal magnetic field length.
T. Sawada et al. / Two-dimensionalflow
329
of a magneticfluid
of the magnetic force iU VH acting on MF, and the fluid flow are parallel. When I = 5 A in fig. 4, Ax/x increases with Re. This phenomenon is peculiar in comparison with that in fig. 5. But we could not clarify this phenomenon in the present experiment. l
3A 2A
4
2
1 A
a
10-21 102
10s
4. Concluding remarks
Re Fig. 4. Increasing rate of resistance coefficient versus Reynolds number in longitudinal magnetic field.
,0-Zlo2
Re
103
Fig. 5. Increasing rate of resistance coefficient versus Reynolds number in longitudinal magnetic field.
drop obtained by a least squares method. The magnetic field in fig. 4 acts on MF restraining the fluid flow, while that in fig. 5 yields a inverse effect. Kamiyama et al. [5] carried out the similar experiment for a pipe flow under the transverse magnetic field whose strength is linearly changed along the axis. In their experiment, the relation between AA/A and Re hardly depends on the gradient of the magnetic field. In our experiment, however, when the strength of the magnetic field increases with the flow, the decreasing rate of Ah/X with Re is smaller than that in the other magnetic field. This phenomena in two inverse magnetic fields are owing to the longitudinally nonuniform magnetic field, that is, both directions
The effect of some different magnetic fields on the resistance coefficient in the two-dimensional flow has been examined experimentally. It is found that there is a relation between the length and strength of the magnetic field, and the increasing rate of resistance coefficient. The detailed experiment about an entrance region due to a magnetic field should be necessary in order to clarify this relation. The influence of inverse magnetic fields on Ah/h is quantitatively clarified.
Acknowledgements This work was supported by the Foundation Hattori-Hokokai, Grant-in-Aid for Scientific Research of the Japanese Ministry of Education and the Iwatani Naoji Foundation’s Research. We thank Mr. M. Oshie for his help in experiments.
References [l] S. Kamiyama, K. Koike and N. Iizuka, Bull. JSME 22 (1979) 1205. [2] M.M. Miorov, E.Ya. Blums and A.E. Malmains, Magnetrohydrodynamics 11 (1976) 529. (31 S. Kamiyama, K. Koike, and T. Oyama, J. Magn. Magn. ,Mat. 39 (1983) 23. [4] K. Sudo, Y. Tomita, R. Yamane, Y. Ishibashi and H. Otowa, Bull. JSME 26 (1983) 2120. [S] S. Kamiyama, T. Oyama and K. Mokuya, Jpn. Sot. Mech. Eng. 47 B (1981) 2299. [6] D. Spasojevic, T.F. Irvine and N. Afgan, Intern. J. Multiphase Flow 1 (1974) 607.
Journal of Magnetism and Magnetic Materials 65 (1987) 327-329 North-Holland, Amsterdam
TWO-DIMENSIONAL T. SAWADA, Faculty
FLOW OF MAGNETIC FLUID BETWEEN TWO PARALLEL PLATES
T. TANAHASHI
Science and
and T. AND0
Keio University,
Two-dimensional flows magnetic fluid experimentally. The rate of field and Reynolds
Kohoku-ky
Yokohama,
two parallel in the coefficient is
1. Introduction many interests arisen in tion of fluids (MF) engineering deand a of practical have been out, e.g. device, levitation, netic ink abrasive. As flow phenomena MF, many have been But, most them are with respect pipe flows applied magnetic [l-5]. SpasojeviC al. [6] a channel of a suspensions experimentally. treated a viscous carrier mainly clarified characteristics of suspensions. In present paper, deal with sional flow MF between parallel plates in some types of netic fields. effect of magnetic field flow resistance clarified in laminar re-
fields are magnetic
fields (a), magnetic magnets
and (c); (a) is transverse produced by permanent X 40 10 mm3). magnetic flux B along the center line is B = 40 mT. Type (b) is a longitudinal magnetic field produced by some solenoidal coils whose lengths are 50 mm. The magnetic flux densities due to the excited currents I = 1, 2, 3, 4 A are B = 7, 14, 21, 29 mT, respectively. Changing the number of permanent magnets or solenoidal coils in the case of (a) and (b), one can alter the length of the applied magnetic field. Type (c) is a longitudinally nonuniform magnetic field produced by a solenoidal coil whose turn changes along the axial direction, and distributions of magnetic flux density, which is measured by a gauss meter, are shown in fig. 1. When this solenoidal coil is set inversely, two different magnetic fields can be obtained, that is, the direction of magnetic gradient is opposite.
50
2. Experiment The test channel is made of acrylic resin and its length is 998 mm. The cross section of a test channel is rectangular whose width and height are 40 mm and 2 mm, respectively. Test liquid is MF with 24% weight concentration of fine magnetite particles in a water carrier. Viscosity and density of the MF are q = 4.83 mPas and p = 1.22 X lo3 kg/m3 at 15OC, respectively. We use the following three kinds of magnetic
and longitudinal and discussed respect to
60 co
I 3oc 20 10
I
-?oo t
Fig. 1. Distributions of magnetic flux density for solenoidal 3 coil.
0304~8853/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
328
T. Saw&
et al. / Two-dimensional flow
The pressure difference between the upstream and downstream taps is measured by the digital manometer. The pressure difference includes the static pressure change due to a magnetic force and the pressure drop due to a flow resistance. However, if the pressure taps are connected to a manometer, the indicator gives directly the pressure drop since the effect of magnetic force is cancelled out by the magnetic force acting on the MF in both comecting tubes [3]. In type (a), the taps are situated just in the middle of the magnetic field region at intervals of 100 mm. The difference pressure ‘is measured outside of the solenoidal coil apart from both sides at intervals SQ mm in type (b) and (c)-
3. Results and discussion The resistance coefficient X and number Re are respectively defined AP _j@!f 4h ’ I
Reynolds
Re=apvh rl
’
where Ap is the pressure drop through the channel length, h the width of the two parallel plates, u the mean velocity, p the fluid density, and 71the fluid viscosity. For the laminar flow of Newtonian fluid, X is represented by X = 96/Re. Fig. 2 shows the influence of a transverse mag-
of a magnetic fluid
netic field length on increasing rate of resistance coefficient AA/A. When the magnetic field length L is small, the increasing rate of A makes a large difference with the one of Re. But Ax/x becomes a constant for L > 300 mm. This figure indicates that the increasing rate AX/X depends on the magnetic field length L and the Reynolds number ReforL~3OOmm.ForL~3OOmmthelineof AX/X - L is nearly flat with a little slope. Since the magnetic field is almost constant except for the 10 mm region from the end of the magnet, that is, the differential pressure is measured in the uniform magnetic field, large changes of Ax/x with L is caused by the hydraulic disturbance due to an abrupt change of a magnetic field at the entrance. In the present experiment, influence of a magnetic field on the development of an inlet flow cannot be clarified because the length of a permanent magnet is relatively long, i.e. 40 mm. The relation between a longitudinal magnetic field length and AX/h is shown in fig. 3. The influence of longitudinal magnetic field length is not evident. Fig. 3 shows a tendency that AX/A increases gradually with L. Since the differential pressure is measured outside of the solenoidal coil, this figure indicates global AX/X due to a solenoidal coil. The magnetic field is almost uniform for L > 100 mm except in the vicinity of the end of a solenoidal coil. Then Ax/x does not change largely with L. Magnetic forces caused by the non-uniform magnetic field near both ends of a solenoidal coil z~fecancelled. Figs. 4 and 5 S~QF #e relation between Ah/A and & under the inverse magnetic fields, respectively. The line Ah/X-Re is based on the pressure
1
15 8=29
5-
mT
118
;
_V
0
I
I
I
200
I
L mm
I
LOO
’ 0 4 100
I
Fig. 2. Increasing rate of resistance coefficient versus verse magnetic field length.
5
trans-
A I
200
I
300 L mm
600 600 1500 I
LOO
Fig. 3. Increasing rate of resistance coefficient versus longitudinal magnetic field length.
T. Sawada et al. / Two-dimensionalflow
329
of a magneticfluid
of the magnetic force iU VH acting on MF, and the fluid flow are parallel. When I = 5 A in fig. 4, Ax/x increases with Re. This phenomenon is peculiar in comparison with that in fig. 5. But we could not clarify this phenomenon in the present experiment. l
3A 2A
4
2
1 A
a
10-21 102
10s
4. Concluding remarks
Re Fig. 4. Increasing rate of resistance coefficient versus Reynolds number in longitudinal magnetic field.
,0-Zlo2
Re
103
Fig. 5. Increasing rate of resistance coefficient versus Reynolds number in longitudinal magnetic field.
drop obtained by a least squares method. The magnetic field in fig. 4 acts on MF restraining the fluid flow, while that in fig. 5 yields a inverse effect. Kamiyama et al. [5] carried out the similar experiment for a pipe flow under the transverse magnetic field whose strength is linearly changed along the axis. In their experiment, the relation between AA/A and Re hardly depends on the gradient of the magnetic field. In our experiment, however, when the strength of the magnetic field increases with the flow, the decreasing rate of Ah/X with Re is smaller than that in the other magnetic field. This phenomena in two inverse magnetic fields are owing to the longitudinally nonuniform magnetic field, that is, both directions
The effect of some different magnetic fields on the resistance coefficient in the two-dimensional flow has been examined experimentally. It is found that there is a relation between the length and strength of the magnetic field, and the increasing rate of resistance coefficient. The detailed experiment about an entrance region due to a magnetic field should be necessary in order to clarify this relation. The influence of inverse magnetic fields on Ah/h is quantitatively clarified.
Acknowledgements This work was supported by the Foundation Hattori-Hokokai, Grant-in-Aid for Scientific Research of the Japanese Ministry of Education and the Iwatani Naoji Foundation’s Research. We thank Mr. M. Oshie for his help in experiments.
References [l] S. Kamiyama, K. Koike and N. Iizuka, Bull. JSME 22 (1979) 1205. [2] M.M. Miorov, E.Ya. Blums and A.E. Malmains, Magnetrohydrodynamics 11 (1976) 529. (31 S. Kamiyama, K. Koike, and T. Oyama, J. Magn. Magn. ,Mat. 39 (1983) 23. [4] K. Sudo, Y. Tomita, R. Yamane, Y. Ishibashi and H. Otowa, Bull. JSME 26 (1983) 2120. [S] S. Kamiyama, T. Oyama and K. Mokuya, Jpn. Sot. Mech. Eng. 47 B (1981) 2299. [6] D. Spasojevic, T.F. Irvine and N. Afgan, Intern. J. Multiphase Flow 1 (1974) 607.