Full text of DOI:10.2514/2.6738

JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER
Vol. 17, No. 1, January–March 2003
Application of Magnetic Fluid Membrane for Flow Control
H. Yamaguchi,¤ Y. Suzuki,† and S. Shuchi‡
Doshisha University, Kyoto 610-0321, Japan
To study the application of a magnetic  uid membrane for  ow rate control, an experimental investigation was
conducted to obtain the basic characteristics and to verify the feasibility of practical use. The working principle of
a magnetically controlled magnetic  uid membrane is based on the magnetic attraction body force exerted on the
magnetic  uid, which is sustained in the cross-sectional area of a pipe. In the experiments, rupture pressure was
–
obtained at various conditions for an oil magnetic  uid interface. Also, from the results obtained for the  ow rate
ontrol characteristics, it was found that the magnetically controlled membrane has a function similar to that of an
ordinary butter y valve, and it was shown that the membrane would be feasible for controlling lower  ow rates at
lower driving pressure gradients.
Nomenclature
posed to date, such as magnetic seals, dampers, bearings, etc.1¡3
Among the many promisingtechnologicalapplicationsof magnetic
 uids are those devices that use surface deformationwith magnetic
attraction force.
A unique application for an aperture control, which is derived
from the interfacial formation of magnetic  uid, was proposed in
our previous research work.4 The basic results were presented for
a sustained magnetic  uid membrane in a pipe section under an
appliedmagneticŽ eld. In viewof possibleengineeringapplications,
detailed characteristics were reported in Ref. 4 for the formation
process and the aperture variation in terms of imposed magnetic
Ž eld intensity.
D
D0
d
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
circular valve diameter of butter y valve, m
pipe diameter of buttery valve, m
pipe diameter of test section, m
magnetic Ž eld intensity, A/m
magnetic Ž eld vector
intensity of magnetic Ž eld in direction, A/m
H
H
Hr
Hz
r
z
intensity of magnetic Ž eld in direction, A/m
H
H
.
I
l
/
maximum of z , A/m
z
max
applied dc current, A
pipe length, m
effective length, m
le
In the present study, which proceeds from our previous study,4 a
newmethodof  ow ratecontrolthatusessustainedmagnetic uid in
a pipe under an appliedmagneticŽ eld is proposed.The basicideaof
using the sustainedmagnetic uid in a sectionofa pipeis to form the
membrane and block the passage of  ow by controlling magnetic
Ž eldintensity.Inthepresentinvestigation,therupturepressureofthe
membrane(whichisanalogoustotheholdingpressureofanordinary
mechanical ow controlvalve)is investigatedexperimentally,using
a water-basedmagnetic uid (W-40)with a working uid of mineral
oil. In the experiment, the rupture pressure of the magnetic  uid
membrane for varying magnetic Ž eld intensity is recorded for a
given volume of magnetic  uid sustainedin the pipe section.While
controllingthe volume  ow rate of the working liquid (oil), the  ow
characteristics (the pressure loss coefŽ cient across the membrane)
are also recorded and examined to simulate a  ow control valve.
lm
M
M
P
length covered by magnetic  uid, m
magnetization,Wb/m2
magnetizationvector, Wb/m2
¢
pressure loss, Pa s
rupture pressure, Pa s
loss
P
¢
r
Re
Reynolds number
r
z
, µ,
cyclindricalcoordinates(see Fig. 6)
volume of sustained magnetic  uid, m3
area mean velocity, m/s2
measured pressure difference, Pa
valve opening angle of butter y valve, rad
pressure loss coefŽ cient
Vm
v
P
1
±
"
¢
´
¸
½
test  uid viscosity, Pa s
friction loss coefŽ cient (¸ 64=
D
Re
)
working  uid density, kg/m3
II. Experimental Apparatus and Procedure
I. Introduction
Figure 1 is a schematic diagram of the experimental apparatus.
In the present investigation,oil, whose physical properties are pre-
sentedin Table1, isusedas a testworking uid.Todrivetheworking
 uid into the test section, the working  uid, which is contained in
the cylinder, is forced to  ow by inward pushing of the piston with
driving units that consist of an electric motor, speed controller,and
other mechanical parts, as shown in Fig. 1. In this investigation,
the pressure and volumetric  ow rate are kept constant throughout
the experiments.The volumetric  ow rate of the working  uid was
obtained by knowing the speed and area of the piston. Differen-
tial pressure across the test section, which is used to obtain to the
pressure loss coefŽcient, was measured by the differential pressure
gauge, as indicatedin Fig. 1. The inlet length to the test section was
taken sufŽ ciently long so that the  ow coming into the test section
was a fully developed steady laminar pipe  ow. The test section is
made of acrylic pipe with an inner diameter of 12 mm and total
length of 800 mm, where a given volume of magnetic  uid is sus-
tained by a magnetic Ž eld forming a membrane. A cylindrical coil
electromagnet is placed around the test section, and it is cooled by
circulating water from a cooler unit. The electromagnetis powered
by a dc power supply unit, capable of producing a magnetic Ž eld
MAGNETIC  uid is a colloidal suspension of 10-nm ferro-
magneticparticlesthatare suspendedwith a surfactantcoating
A
in a carrier liquid such as hydrocarbon,ester, or water.1¡3 Magnetic
 uid behaves as an electrically nonconducting homogeneous me-
dia in which single-domain magnetic particles are stabilized even
in strong magnetic Ž elds. Because magnetic  uids have magnetic
properties along with  uidity, many applications have been pro-
Received 12 April 2002; revision received 22 July 2002; accepted for
°c
publication 24 July 2002. Copyright
2002 by the American Institute of
Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper
may be made for personal or internal use, on condition that the copier pay
the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose-
wood Drive, Danvers, MA 01923; include the code 0887-8722/03 $10.00 in
correspondence with the CCC.
¤Professor, Department of Mechanical Engineering; hyamaguc@mail.
doshisha.ac.jp.
†Research Student, Department of Mechanical Engineering; eta1305@
mail4.doshisha.ac.jp.
‡Research Student, Department of Mechanical Engineering; dta0379@
mail4.doshisha.ac.jp.
89

90
YAMAGUCHI, SUZUKI, AND SHUCHI
Fig. 1 Experimental apparatus: 1, electric motor; 2, speed controller; 3, ball screw; 4, piston rod; 5, coupling plate; 6, cylinder; 7, Ž ller tank; 8,  ow
valve; 9, test section; 10, tank; 11, electromagnet; 12, cooler; 13, dc power supply; and 14, differential pressure gauge.
Fig. 2 Details of test section.
5
»
£
strength of 0 0.50 10 A/m in the test section. A visual obser-
vation was also conducted during each  ow test to obtain visual
information regarding the surface of the magnetic  uid membrane
under the magnetic Ž eld. The visual observationwas done by view-
ing the test section from the side of the reservoir tank as shown in
Fig. 1.
Fig. 2. Beforethe ow experiments,withoutthe presenceofworking
 uid and magnetic  uid, detailed data were recorded for magnetic
Ž eld distribution in the test section by supplying dc electric power
to the magnet. In Figs. 3 and 4, representative distributions of the
magnetic Ž eld intensity in the test section are plotted for the axial
H
H
and for the radial component r , respectively. In
component
z
H
H
/
max
The arrangement of the test section is basically the same as in
the previous investigation. In Fig. 2, details of the test section are
Fig. 5, the normalized axial magnetic Ž eld intensity z =.
is
z
4
z
plottedfor the axialcoordinate in the test sectionwhen a typicaldc
I D
shown. Before each  ow experiment,exactexperimentalconditions
were set Ž rst by pouring the working  uid into the cylinderthrough
the Ž ller tank (as shown in Fig. 1). Then a given volume of mag-
netic  uid was inserted into the test section by a syringe through
the stainless Ž lling pipe as shown in Fig. 2, while an appropriate
magnetic Ž eld to sustain the magnetic  uid in the test section was
applied. Experiments were carried out in a temperature controlled
room so that the physical properties (Table 1) of the working  uid
currentof
100 mA was applied to the magnet. As seen in Fig. 5,
H¤
there is a smooth and symmetric magnetic Ž eld for z , which is
z D
the normalized value with the peak value occurring at
0 (which
0). Note
r
H
r
z D r D
varies for the direction and zero value of
at
0,
from Figs. 3 and 4 that the magnetic Ž elds are symmetric because
the magneticŽ eld was generatedby the solenoidcoilelectromagnet.
The distributionof the magnetic Ž eld intensity was measured by
§
a Hall probe, whose instrumentationaccuracyis within 1% for the
±
±
§
and magnetic  uid were kept constant at 25 C( 1 C). For a given
volume of magnetic  uid, the magnetic Ž eld was varied where the
externallyapplied magnetic Ž eld was imposed on the membrane by
the coil electromagnet,which is held in a copperholderas shown in
rangeof magneticŽ eld intensity.However, the largestmeasurement
error comesfrom positioningthe Hall probe in the measuringspace,
which was achieved by constructing a procession stage. The mar-
gin of error associated with the measurement of the magnetic Ž eld

YAMAGUCHI, SUZUKI, AND SHUCHI
91
Table 1 Properties of test  uids at 25±C
Test  uid
Density, kg/m3
Viscosity, Pa ¢ s
Mineral oil SM-2
Magnetic  uid W-40
0:832 £ 103
1:86 £ 10¡3
1:407 £ 103
30 £ 10¡3
Fig. 5 Normalized axial magnetic  uid intensity.
(
)
max
Fig. 3 Distribution of magentic Ž eld Hz/ Hz
.
Fig. 6 Cylindrical coordinate system for applied magnetic Ž eld H.
(
)
max
Fig. 4 Distribution of magentic Ž eld Hr/ Hr
.
§
intensityis approximatelya maximum of 4.0%. The procedureof
the measurement (positioningthe Hall probe) was repeated several
times to minimize the error. The data shown in Figs. 3–5 are the
average values from repeated measurements.
In Fig. 6, a schematic diagram describingthe sustainedmagnetic
 uid membrane with applied magnetic Ž eld H is shown, with the
o
cylindricalcoordinatesystem also presented.The origin of the co-
ordinate system, as shown in Fig. 6, is taken at the center of electro-
magnet on the axis of the pipe in the test section. Note that pressure
measurements were made by differential pressure transducersboth
during the static experiment, where the rupture pressure across the
membrane was recorded,and duringthe  ow experiment,where the
pressure difference across the membrane was recorded.
The magnetic  uid used in the present investigationwas a water-
based magnetic  uid (W-40) as supplied by industry. Its magneti-
zation characteristic is displayed in Fig. 7, where the curve is well
expressedby theLangevinfunction(seeRef. 5).The working uid is
mineral oil SM-2 as supplied by industry with the propertiesshown
in Table 1. The mineral oil provides excellent separation from the
magnetic  uid.
Fig. 7 Magnetization characteristics of magnetic  uid W-40.
§
intensity would be approximately maximum 4.0% due to certain
uncertainty in determining the exact space position. The sustained
volume of the magnetic  uid was measured by a Ž ne syringe scale
§
with a maximum possible error of 3.0%. The determination of
volume  ow rate of the working  uid had some uncertaintyinvolv-
ing measuring piston speed, particularlyat lower piston speed, due
to friction of the ball screw at lower torque. The volume  ow rate
measurement error by repeated measurements would be approxi-
§ »
mately
5
7% maximum.
The measurement error associated with the pressure difference
and the rupture pressures purely comes from the instrumentation
III. Results and Discussion
§
accuracy, which has a maximum of approximately 1.5%. The
A. Rupture Pressure
measurement of the recorded magnetic Ž eld distributionand the in-
tensity involvedcertain human measuring errors, although the Ž eld
was measuredby a Hole probewitha precisioncoordinatetable.The
error associated with measuring the magnetic Ž eld distributionand
Before  ow rate controlling characteristics were examined, the
rupture pressure was obtained for various magnetic Ž eld intensi-
ties. The rupture pressure is the critical pressure difference when

92
YAMAGUCHI, SUZUKI, AND SHUCHI
Fig. 9 Transient behavior of ¢P across magnetic  uid membrane,
m = 3.00 ml.
V
–
(
)
Fig. 8 Rupture pressure oil magnetic  uid .
P
Figure 9 shows transient behavior of pressure difference 1
the magnetic  uid membrane is broken by applying a pressure dif-
ference across the interfacial surface of the membrane. Figure 8
for some representative Reynolds numbers. In Fig. 9, for the
sake of comparison, the sustained volume was kept constant at
3:00 ml, and the Reynolds number was varied for each given
magnetic Ž eld intensity z . The diagram shows the readings of the
differentialpressureoutput1 as time elapsesafter the  uid is sud-
denly driven to  ow at a constant speed by the pushing piston. It is
seen in Fig. 9 that as the magnetic Ž eld becomes higher, the magni-
tudeof thestartuptransitionfor 1 becomeslarger.Also,forhigher
0.461 10 A/m),
the magnitude of the startup transition for 1 becomes larger, that
is, with a higher magnetic Ž eld the rupture pressure is high (as al-
ready seen in Fig. 8), thus,1 becomeshigh. The startuptransition
iscausedafterthe ruptureofmembranewhen theapertureareastarts
toexpandastheworking uid owsthroughtheopeningareaastime
elapses. When the aperture area becomes constant with increasing
time, the opening of the aperture reaches steady state, resulting in
P
shows the results of rupture pressure
against magnetic Ž eld in-
r
V D
H¤
m
tensity
for given sustainedvolume of magnetic  uid. As can be
z
H¤
seen from Fig. 8, the rupture pressure increases linearly with ap-
plied magnetic Ž eld intensity. Also, the rupture pressure becomes
higher when the sustained volume is increased. With a higher sus-
V D
P
tained volume, such as
4:00 and 3.75 ml in Fig. 8, the thick-
m
P
ness of membrane (particularly toward the lower wall region) was
spread over the measuring section (over the width of the electro-
5
Re D
H¤ D
£
Reynolds numbers, such as
100 (at
z
P
magnet) when magnetic Ž eld intensity was reduced (for example,
5
Hz¤
5
£
V D
0:35 10 A/m for
4:00 ml), due to the gravity force, so
m
P
that data were not recorded. However, when a lower sustained vol-
V D
ume is used, such as
3:00, 3.25, and 3.50 ml in Fig. 8, and the
m
magnetic Ž eld intensity is reduced, the membrane can be ruptured
(or opened) only by applying the magnetic Ž eld without a pressure
difference across the membrane, shown in Fig. 8 as points on the
P
1
reaching a plateau, the steady state, as seen in Fig. 9. Note that
Hz¤
P D
axis for
0. The formationprocessof the membranewithout
pressure difference is fully discussed in Ref. 4.
r
5
H¤ D
£
for the lower magnetic Ž eld case, such as
0.264 10 A/m in
Fig. 9, thedependencyofthetransientbehavzioron Reynoldsnumber
becomes small. This is mainly because, with small magnetic Ž eld
intensity, the effect of magnetic body force1 becomes small due to
the very small rupturepressure,so that the effectof the volume  ow
rate, that is, Reynolds number and inertia, in opening the aperture
area wider becomes minimal. As seen in Fig. 9, at 600 s after the
startup the movement of piston was suddenly stopped to verify the
B. Pressure Loss CoefŽ cient
With regard to controlling the  ow rate with the membrane by
applying a magnetic Ž eld, the pressure drop across the membrane
was measured when the working  uid started to  ow through the
ruptured (opened) hole. In the present study, the pressure loss co-
efŽ cient across the membrane was obtained for a given sustained
volume of magnetic uid with the effectsof magneticŽ eld intensity
P
decay transitionof 1 . When the  ow throughthe aperture area of
the membrane is ceasing, the aperture area tends to close itself and
then Ž nally reaches a completely closed state, where the pressure
Hz¤
and Reynolds number,
P
difference 1 reaches the value of the rupture pressure. Thus, in
Re ´
d
½v =´
(1)
the present investigation,the decay plateau has implications on the
rupture pressure, as seen in Fig. 8.
for dimensionlessvolume  ow rate of the working  uid. The pres-
sure loss coefŽcient " is deŽ ned as follows:
The control of volume  ow rate ( ow rate control characteris-
tics) by the membrane of the sustainedmagnetic uid is represented
by a relationship between the pressure loss coefŽ cient " [Eq. (2)]
and Reynolds number (volume  ow rate), much like ordinary
mechanical  ow control valves.6 In Figs. 10–14, the  ow rate con-
trol characteristics are presented for different sustained volumes
of magnetic  uid, and, in each case, the magnetic Ž eld intensity
is varied. As seen in Figs. 10–14, the pressure loss coefŽ cient "
¯
1
2
"
.1
/
½v2
(2)
D
P ¡ P
loss
P
where v is given for the working  uid.
to wall friction, as follows:
is the pressure loss due
(3)
loss
1
2
¸. = / ½v2
P
loss
D
le d ¢
Re
decreases as Reynolds number
is increased, where " can be in-
where ¸ is given for the pipe. It is noted that, for Eqs. (2) and (3),
creased by applying a higher magnetic Ž eld intensity. There are
strong dependencies of the sustained volume of magnetic  uid on
the  ow rate control characteristics, as seen in Figs. 10–14. With
highersustainedvolumesof magnetic uid, suchas3.75and 4.00ml
(Figs. 13 and 14), " increasessubstantially.Thus, with a highersus-
tained volume and the application of a higher magnetic Ž eld inten-
sity, the membrane could control a higher range of applied pressure
with a higher volume  ow rate. This results from a higher mag-
netic body force causedby the highervolume and an applied higher
d D
l
12 mm and is the effective pipe length between positions of
e
l D l ¡ l l D
differentialpressuretransducersdeŽ ned as
(
400mm).
e
m
l
Also note that m , the distance covered by the magnetic  uid in
l
the section of the pressure measurement, was subtracted from
in Eq. (3). This was done in determining " in Eq. (2), though the
l
pressure loss associated with the distance
was effectively less
m
than the maximum 0.08% in the range of Reynolds number used in
the present study.

YAMAGUCHI, SUZUKI, AND SHUCHI
93
–
Fig. 10 Pressure loss coefŽ cient for oil magnetic uid at Vm = 3.00 ml.
–
Fig. 13 Pressure loss coefŽ cient for oil magnetic  uid at Vm = 3.75 ml.
–
Fig. 11 Pressure loss coefŽ cient for oil magnetic uid at Vm = 3.25 ml.
–
Fig. 14 Pressure loss coefŽ cient for oil magnetic  uid at Vm = 4.00 ml.
–
Fig. 12 Pressure loss coefŽ cient for oil magnetic uid at Vm = 3.50 ml.
magnetic Ž eld intensity that holds the membrane’s shape4 against
the applied pressure across the membrane. Note that the disconti-
Fig. 15 Comparision of membrane  ow control characteristic to that
of butter y valve, Vm = 3.75 ml.
5
H¤ D
£
nuity of data points, such as that seen for
0. 461 10 A/m in
H¤ D
£
z H¤ D
£
5
5
Fig. 13 and for
0:41 10 A/m and
0:395 10 A/m
the major cause for the breaking up phenomena in the range of the
magnetic Ž eld intensity in the present study.
z
z
in Fig. 14, indicates the breaking up points of the membrane. At
these breaking up points, for an associated Reynolds number, the
surface around the aperture area (the opening edge) tends to break
up intodropletsand  ow away with the streamof working uid. The
breaking up probably occurs due to a surface instability based on
the Kelvin–Helmholtz instability1 of the contact interfacialsurface.
Thus, this breaking up phenomenon would impose limitations on
the utilization of the membrane for controlling volume  ow rate.
The surfaceinstability1¡3 is anotherimportantsurfacephenomenon
of the magnetic  uid membrane to consider. However, in the range
Also note that the membrane for an air working  uid (the air–
magnetic  uid interface) would not function in controlling the vol-
ume  ow rate. This is due to bubblingfrom the membrane when the
air is passed through the aperture area, a phenomenon that is also
observed in soap bubbles. It is thought that the surfactantused as a
coating substance to disperse the magnetic particles is responsible
7
for the bubbling phenomenon.
Finally, in Fig. 15, the  ow rate controlling characteristic of the
V D
membranefor a representativecase,
3:75 ml, is comparedwith
m
of the magnetic Ž eld intensity used in the present investigation,
a typical butter y valve (see Appendix).8 Although there would not
be any reason for a direct comparisonwith the mechanicalbutter y
valve, there exists a similarity in the controlling of the volume  ow
rate. In the Appendix ± is the opening angle of the buttery valve,
5
5 H¤
5
£
0
0:461 10 A/m, the surface instabilitywas not observed
z
by the visual observation before the breaking up. Thus, the inter-
facial instability due to the Kelvin–Helmholtz instability would be

94
YAMAGUCHI, SUZUKI, AND SHUCHI
and it exhibits a similar effect as that of the applied magnetic Ž eld
intensity, althoughthe controllingrange for the membrane is rather
limited at a lower Reynolds number in this comparison. The chief
advantageas opposedto an ordinarybutter y valve is that the action
of closing and opening valve can be achieved directly by applying
the magneticŽ eldtothe sustainedmagnetic uid, resultingina rapid
response,whereas with the butter y valve, the action of closing and
openingthe valveis usuallydoneby a servomechanism.In addition,
it is thought that the magnetically controlled membrane would be
more appropriateforcontrollinglow owrategaseswith anaccurate
and Ž ne  ow control characteristic.
Fig. A1 Schematic of butter y valve.
In the present investigation, the basic characteristic of control-
ling the volume  ow rate with a magnetically sustained membrane
was presentedfor application as a  ow control device. Because the
response time of forming the membrane when the magnetic Ž eld
intensity is altered is on the order of 10¡6 s (the typical relaxation
time of the magnetic uid),1 the action of controllingthe  ow would
be superior to ordinary mechanically controlled  ow valves. Note
that magnetic rheological (MR)  uid is another interesting smart
 uid for possible utilizationin  ow control such as that proposedin
the presentstudy because the magnetizationis usually much higher
than the magnetic  uid. A detailed study of a magnetically con-
trolled MR  uid membrane is forthcoming. As stated earlier, the
working  uid (mineral oil SM-2; Table 1) provides excellent sepa-
rationof the magnetic uid, and the magnetizationcharacteristicsof
the magnetic uid did not changewith repeatedusage in the present
experiment.However, long-termdegenerationwould occur, and the
problem, includingevaporationof the carrier liquid of the magnetic
 uid in the event of air contact,must be consideredin a practicalin-
dustrialuse. Much attentionand improvementwouldbe requiredfor
more practical uses, but the feasibility of a magnetically sustained
membrane is veriŽ ed in the present study.
The feasibility of utilizing the membrane as a  ow control device
was positively veriŽ ed.
Appendix: Butter y Valve
FigureA1 showsa typicalbutter yvalve,wherethe circularvalve
D
of diameter is situated in the middle of a pipe whose diameter is
Re
D
Re D
0; ± is the valve opening angle. Reynolds number is deŽ ned as
D
½v 0=´, where is v the area mean velocity.
Acknowledgment
This work was partly supported by a Grant-in-Aid for ScientiŽ c
Research (Category C) from the Japan Ministry of Education, Cul-
ture, Sports, Science, and Technology.
References
1Rosensweig, R. E., Ferrohydrodynamics, 1st ed., Cambridge Mono-
graphs on Mechanics and Applied Mathematics, Cambridge Univ. Press,
New York, 1985, pp. 1–8, 61–63, 110–119, and 202–206.
2Berkovsky, B. M., Medvedev, F. V., and Krakov, M. S., Magnetic Fluids
Engineering Applications, Oxford Univ. Press, New York, 1993, pp. 1–24.
3Blums, E., Cebers, A., and Maiorov, M. M., Magnetic Fluids, 1st ed.,
Walter de Gruyter, New York, 1997, pp. 343–375.
IV. Conclusions
4Yamaguchi, H., Suzuki, Y., and Shuchi, S., “Membrane Formation Pro-
cess in Magnetic Fluid and Application for Aperture Control,” Journal of
Thermophysics and Heat Transfer (to be published).
A study to investigate the use of a magnetically sustained mem-
brane of magnetic uid for controllingvolume  ow rate was carried
out experimentally. From the experimental results, the following
conclusionswere drawn:
1) The rupturepressure(whichimpliesthe valveholdingpressure
in an ordinary  ow controldevice)has an almost linear relationship
with applied magneticŽ eld intensity.For a highersustainedvolume
of magnetic  uid, the rupture pressure is higher.
5Chantrell, R. W., Popplewell, J., and Charles, S. W., “Measurements of
Particle Size Distribution Parameters in Ferro uids,” IEEE Transactions on
Magnetics, Vol. 14, No. 5, 1978, pp. 975–977.
6Joseph, A. S., and Allen, E. F., “Geometric Characteristics of Selected
Valves,” Handbook of Fluid Dynamics and Fluid Machinery, Vol. 3, Appli-
cations of Fluid Dynamics, Wiley-Interscience, New York, 1996, pp. 2043–
2054.
2) The pressure loss coefŽ cient " is a decreasing function of
7Jacob, N. I., Intermolecular and Surface Force, 2nd ed., Academic
Press, London, 1992; Asakura Book Corp., Ltd., Tokyo, 1996, pp. 301–325
(Japanese translation).
Re
increasing Reynolds number
in the  ow rate controlling char-
acteristic of the membrane. For a higher sustained volume of mag-
8Murata, S., and Miyake, Y., Hydrodynamics, Rikogakusha,Tokyo, 1982,
pp. 65–67 (in Japanese).
Re
netic  uid, the  ow rate controllingcharacteristic("– relation) is
greater, and it has a trend similar to that an ordinary butter y valve.