Sensors and Actuators A 93 (2001) 173±181
Analytical solutions for the dynamic analysis of a valveless
micropump Ð a ¯uid±membrane coupling study
L.S. Pan, T.Y. Ng*, G.R. Liu, K.Y. Lam, T.Y. Jiang
Institute of High Performance Computing, 89C Science Park Drive, #02-11/12, The Rutherford,
Singapore Science Park 1, Singapore 118261, Singapore
Accepted 24 March 2001
Abstract
A non-linear vibration model for ¯uid±membrane coupling is developed for simulating the behaviour of valveless micropumps. The
analytical solution for ¯uid±membrane coupling vibration has been approximated using the Galerkin and small parameter perturbation
methods. Speci®cally, the vibration response of the membrane and the mean ¯ux through a valveless membrane micropump are investigated
in detail. The effects of the pressure loss coef®cients of the diffuser and the coupling parameter on the mean ¯ux are examined. An optimal
working frequency range for the valveless membrane micropump is obtained. Numerical examples are presented to demonstrate the
effectiveness of the present model for the designing of valveless micropumps. # 2001 Elsevier Science B.V. All rights reserved.
Keywords: Fluid±structure coupling; Micropump; Non-linear vibration; Mean ¯ux; Galerkin method; Small parameter method
1. Introduction
In the past decade, microdevices or micro-electromechanical systems (MEMS) have emerged as a very popular area of research [1,2]. Many forms of microdevices for
various applications have been developed, ranging from
single components such as microsensors and microvalves,
to complex ¯uidic handling systems consisting of pumps,
valves, ¯ow sensors, separation capillaries, chemical detectors, etc. As new areas of application are still being investigated, and integrated systems are becoming more
complicated, there is an increasing requirement for both
experimental data and theoretical exposition on fundamental
physical phenomena at microlevel.
The micropump is one of the various types of important
microdevices, and initial research and development on
micropumps in the 1980s based on microvalves can be
attributed to Smits [3]. Based on these ideas, micromembrane pumps were further developed by other researchers
[4±9]. There are basically two types of the micromembrane
pumps: one with the input and output check valves and the
other without these check valves (or valveless).
These valveless membrane micropumps possess very
wide application potential due to the absence of interior
*
Corresponding author. Tel.: ‡65-7709940; fax: ‡65-7709902.
E-mail address: ngty@ihpc.nus.edu.sg (T.Y. Ng).
moving mechanical parts. Research into their working characteristics and properties, thus, becomes very important.
Some researchers have investigated its characteristics from
the electro-mechanical coupling aspect to obtain reasonable
input electrical signals. Others approach it from the ¯uid
mechanics viewpoint to explore the ¯ux properties of micropump. Volker et al. [10] studied the mechanical properties of
thin ®lms under load deformation. Gerlach et al. [11]
showed experimentally that the working parameters of the
valveless membrane micropump are highly dependent on the
geometric dimensions of the pump and the types of ¯uid
used. Ullmann [12] analysed the performances of single and
double chamber micropumps and discussed the dependence
of the ¯ux on pressure difference between the inlet and
outlet. Galerch [13] discussed the application of the microdiffuser in micropumps as a dynamic passive valve. Olsson
et al. [14] conducted numerical and experimental studies on
¯at-walled diffuser elements for valveless micropumps.
In the valveless membrane micropump, ¯uid ¯ow is
driven by the vibrating membrane, at the same time the
¯uid plays a key role in resistance to this vibration. The
membrane vibration and the ¯uid ¯ow are thus, always
coupled. If the action of the ¯uid is negligible, the membrane
will vibrate at the same frequency as the piezoelectric force
(termed excitation hereafter) for small amplitude vibrations.
However, the effect of ¯uid ¯ow on membrane vibration
is not negligible in actual applications. The effect of
0924-4247/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 6 3 8 - 0
174
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
¯uid±membrane coupling is signi®cant to the overall vibration characteristics, and is thus, one of the primary concerns
when simulating micropumps. Non-linear effects tend to
predominate in the vibration response of the membrane even
when deformations are small.
In this paper, a non-linear ¯uid±membrane coupling
vibration model is established to investigate micropump
characteristics. Under the condition of equal pressures at
the inlet and outlet, the vibration response of membrane due
to a harmonic excitation force is ®rst investigated and the
effects of coupling parameter are then discussed. Finally, the
property on mean ¯ux through the pump is studied and an
optimal working frequency of the micropump is obtained.
2. Fluid±membrane coupling model
Fig. 1 shows the schematic of the cross-section of a
valveless membrane micropump being considered here.
Under the action of a periodic excitation (piezoelectric)
force fe, the membrane deforms periodically, thus, pumping
the ¯uid inside the chamber. The vibration of the membrane
can be described through the change of its vertical displacement (de¯ection) W in space and with time. For most
membrane micropumps, the de¯ection of the membrane
is reasonably small compared to the shortest characteristic
length of the membrane. Therefore, the bending theory of
thin plates is applicable, leading to the following governing
equation for the de¯ection W
Eh3
@2W
4
r
W
‡
hr
ˆ fe
m
@t2
12 1 m2 †
P
(1)
where E, rm, h and m are the elastic modulus, density,
thickness and Poisson's ratio of the membrane, respectively,
t the time variable, r4 is the two-dimensional double
Laplacian operator @ 2 =@x2 ‡ @ 2 =@y2 †2, and P the dynamic
pressure exerted on the membrane by the fluid. It can be seen
from Fig. 1 that besides Pin and Pout, the pressure P under the
membrane depends also on the instantaneous velocity of
fluid. As the flow of the fluid results from the vibration of the
membrane, the pressure P changes with the position and
velocity of the membrane. Eq. (1) is thus, a non-linear partial
differential equation. It should also be noted that the pressure
Fig. 1. Schematic of the cross-section of a micropump.
used here denotes the actual absolute pressure minus the
pressure acting on the upper surface of the membrane.
From the solid mechanics viewpoint, the pressure P must
be known a priori before solving Eq. (1). From the ¯uid
mechanics viewpoint, the pressure P can be obtained only
after having the solution of Navier±Stokes equations governing the ¯uid ¯ow. However, it is necessary when solving the
Navier±Stokes equations to know the velocity of the membrane or the pressure on it. Hence it is impossible to
determine the pressure P before obtaining the solutions of
both Eq. (1) and the Navier±Stokes equations. In other words,
the pressure P represents the coupling between membrane
vibration and ¯uid ¯ow. In order to investigate the vibration
response of membrane due to the excitation force, the
strictest way is thus, to solve both Eq. (1) and the Navier±
Stokes equations simultaneously. However, this is not only be
very complicated, but would also require extensive computational effort. An alternative would be to obtain a relationship
between the pressure and the vibration velocity of membrane
through setting up an approximate physical model. This can
be done through the retaining of the primary features of the
¯ow ®eld and neglecting certain factors. For this purpose, it is
assumed that inside the micropump:
the local pressure loss of flow through diffuser and nozzle
predominates in the total loss; and
the pressure inside the chamber is uniform.
These assumptions are reasonable if the sectional area of
the chamber in the micropump is much larger than that of the
diffuser or nozzle.
Based on these two assumptions, the pressure in Eq. (1)
can be expressed by the volumetric ¯uxes through the
diffuser and the nozzle, as well as the pressures at the inlet
and the outlet. Further, by applying the relationship between
the transverse motions of the membrane and the volumetric
¯uxes, the pressure can be expressed as a function of the
velocity of the membrane. From Fig. 1, if Pin > Pout , the
¯uid will ¯ow automatically from the inlet to the outlet
under the action of pressure difference (Pin Pout ) even if
the membrane is at rest. The case to be discussed here is that
of Pin Pout . For simplicity, the geometrical shapes of the
diffuser and the nozzle are the same.
2.1. For the case of P Pout
In this case, the ¯uid inside the micropump will ¯ow out
through both the diffuser and nozzle. According to the above
assumptions and the relationship between pressure loss and
volumetric ¯ux, we have for the diffuser
r
P ˆ Pout ‡ 2 zdo Qd †2
(2)
2A
and for the nozzle
r
P ˆ Pin ‡ 2 zno Qn †2
(3)
2A
where r is the fluid density, A the typical section area of the
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
diffuser or the nozzle, Qd and Qn the volumetric fluxes
through the diffuser and the nozzle from the inside to the
outside, respectively and zdo and zno the total loss coefficients
of the diffuser and nozzle, respectively. It should be noted
that irrespective of diffuser or nozzle, the loss coefficient
consists mainly of two parts, one from the entrance of
diffuser/nozzle and the other from the exit. For the convenience of subsequent discussion, the values of Qd and Qn are
defined as positive when flowing out of the chamber through
the diffuser and nozzle.
To obtain a dependence of the pressure P on the vibration
velocity of the membrane, a volume changing with time is
introduced as follows:
Z b=2 Z a=2
Vˆ
W dx dy
(4)
b=2
a=2
Based on the geometrical relationship and continuity equation of fluid flow, the volume defined in Eq. (4) and the
fluxes in Eqs. (2) and (3) satisfy
V_ ˆ Qd ‡ Qn
(5)
where V_ denotes the rate of change of the volume. From the
present pressure condition, each term in Eq. (5) satisfies the
inequalities V_ 0, Qn 0, Qd 0. Solving Eqs. (2), (3)
and (5) simultaneously, we obtain
q
2
zno V_
zno zdo V_ ‡ 2 zno zdo †A2 DP=r
(6)
Qd ˆ
zno zdo
q
2
zdo V_ ‡ zno zdo V_ ‡ 2 zno zdo †A2 DP=r
(7)
Qn ˆ
zno zdo
r zno zdo V_
2A2 zno zdo †2
zn zo
o
q
2
zno ‡ zdo †V_ 2 zno zdo V_ ‡ 2 zno zdo †A2 DP=r (8)
P ˆ Pin ‡
zno
DP ‡
d
where DP ˆ Pout Pin . For the special case of DP ˆ 0, the
Eqs. (6) to (8) are respectively reduced to
pn
z
Qd ˆ p oq V_
(9)
n
zo ‡ zdo
q
zdo
Qn ˆ p q V_
(10)
zno ‡ zdo
P ˆ Pin ‡
r
zno zdo
2
q V_

p
2
2A
n
d 2
zo ‡ zo †
(11)
2.2. For the case of P Pin
In this case, the ¯uid ¯ows into the chamber through both
the diffuser and the nozzle. By use of the same principle, the
175
pressure inside the chamber can be expressed as
P ˆ Pout
P ˆ Pin
r d
z Q d †2
2A2 i
r n
z Q n †2
2A2 i
(12)
(13)
where zni and zdi are the loss coefficients of the nozzle and the
diffuser, respectively when fluid flows into the chamber. For
the present case, Eq. (5) is still valid, but each term in it is
negative. Solving Eqs. (5), (12) and (13) simultaneously, we
obtain
q
2
zni V_ ‡ zdi zni V_ ‡ 2 zdi zni †A2 DP=r
Qd ˆ
(14)
zdi zni
q
2
d_
zi V ‡ zdi zni V_ ‡ 2 zdi zni †A2 DP=r
Qn ˆ
(15)
zdi zni
r zdi zni V_
2A2 zdi zni †2
q
d
n _
d n_2
d
n
2
zi ‡zi †V ‡2 zi zi V ‡2 zi zi †A DP=r
(16)
P ˆ Pin
zdi
zni
zni
DP
For the special case of DP ˆ 0, Eqs. (14) to (16), respectively reduce to
pn
z
(17)
Qd ˆ q ip V_
zdi ‡ zni
q
zdi
Qn ˆ q p V_
(18)
zdi ‡ zni
P ˆ Pout
r
zni zdi
_2

q
pn 2 V
2A2
d
zi ‡ zi †
(19)
2.3. For the case of Pin P Pout
In this case, the ¯uid outside the outlet will ¯ow into the
chamber through the diffuser and at the same time the ¯uid
in the chamber ¯ows out through the nozzle. Hence, the
equations governing the pressure loss are
r d
P ˆ Pout
z Q d †2
(20)
2A2 i
r
(21)
P ˆ Pin ‡ 2 zno Qn †2
2A
The continuity equation is still as in Eq. (5), but for the
present case, Qn 0 and Qd 0. From Eqs. (5), (20) and
(21), we have
q
2
n_
2 zdi ‡ zno †A2 DP=r zdi zno V_
zo V
(22)
Qd ˆ
zdi ‡ zno
176
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
Qn ˆ
zdi V_ ‡
q
2
2 zdi ‡ zno †A2 DP=r zdi zno V_
zdi ‡ zno
P ˆ Pin ‡
2
zno
zdi ‡ zno
4 zdi
DP ‡
(23)
r zdi zno V_
2A2 zdi ‡ zno †2
v 3
u d
u 2 z ‡ zno †A2 DP
n _
5
zo † V ‡ 2 t i
2
r zdi zno V_
(24)
_ In
the equations
above, V_ must satisfy jVj

q
p


2A2 DP= rzdi † or
2A2 DP= rzno † .
The above discussion on the three cases shows that the
dynamic pressure exerted on the membrane by the ¯uid
depends on the geometrical shapes of the diffuser and nozzle
(evident in the loss coef®cients), the pressures at the inlet
and the outlet and the instantaneous velocity of the membrane itself. Obviously, dependence of this sort can be
numerically very complicated. However, as the diffuser
and nozzle used here are similar in the shape and size, their
loss coef®cients satisfy zdo ˆ zni and zdi ˆ zno . The purpose of
using four coef®cients rather than two coef®cients in the
discussion above is to demonstrate its validity even when the
shapes of diffuser and nozzle are different from each other. If
zdi and zdo are replaced by zL and zS, respectively, the
vibration equation of membrane can be expressed as
2
Eh3
4
_ ‡ hrm @ W
r
W
‡
P
V†
1
@t2
12 1 m2 †
V_
zL
ˆ fe Pin
DP
_ zL zS
jVj
p
_ >
when jVj
2A2 DP= rzL † , otherwise
2
Eh3
_ ‡ hrm @ W
r4 W ‡ P2 V†
2
@t2
12 1 m †
V_
DP
ˆ fe Pin
_
2jVj
(25a)
(25b)
where P1 and P2 are respectively
_
_ ˆ r zL zS V
P1 V†
2
2A zL zS †2
q _ 2 zL zS V_ 2 ‡ 2 zL zS †A2 DP=r
zL ‡zS †jVj
s
_
r
4zL A2 DP
V
_ ˆ
P2 V†
_ 2
A2 4
r
zL V†
(26)
(27)
After obtaining the solution of Eq. (25), the mean flux
through the micropump can be calculated through the
following formula
Z
1
Qˆ
Qd dt
(28)
T T
where T is the changing periodicity of the excitation force.
For the special case of Pin ˆ 0 and DP ˆ 0, Eqs. (25) and
(28) reduce to
_
Eh3
rzL zS jVj
 ˆ fe (29)
r4 W ‡
p p 2 V_ ‡hrm W
2
2
12 1 m †
2A
zL ‡ zS †
Z p
p
1
_ V_ dt
_ ‡ zS H V††
zL H V†
Q ˆ p p
T zL ‡ zS † T
(30)
_ is the step function, i.e. it is unity when V_ > 0
where H(V)
and zero when V_ 0.
3. Approximate solution
Although many simpli®cations have been made for the
description of the ¯uid ¯ow inside the membrane micropump, the resulting membrane vibration equation, Eq. (25),
as well Eqs. (26) and (27), are quite complicated. Even for
the case of DP ˆ 0, Eq. (29) is still a non-linear integral_ V.
_ It is thus,
partial differential equation due to the term jVj
necessary to utilise approximate methods for obtaining
analytical solutions to Eq. (29). For simplicity, it is assumed
that the excitation force fe is harmonic (o1, the ®rst natural
frequency of the membrane), and the membrane is square
(a ˆ b) and ®xed on its four sides.
In the present investigation, of interest is the coupled
¯uid±membrane vibration response due to the excitation
force and the mean ¯ux through the micropump. The
transient stage caused by initial displacement will thus,
not be discussed. It is inevitable that the non-linear equation,
Eq. (29), will have many response frequencies even when
under the action of a single excitation frequency. Hence the
displacement W of the membrane may be a complex function
of the variables x, y, and t. It is well known that for the given
geometric structure of the membrane, the normal mode
shape functions under free vibration are orthogonal to each
other. It is thus, possible to expand the displacement W into
X
Wˆ
Fj x; y†Zj t†
(31)
j
where Zj(t) is a generalised coordinate and Fj x; y† is the
normal mode shape function corresponding to the jth natural
frequency oj, i.e. Fj x; y† satisfies
E
12rm 1
m2 †
r4 Fj x; y† ˆ o2j Fj x; y†
(32)
Generally, the first few terms in Eq. (31) should be dominant. In the following investigation, the first four terms are
taken into account and Eq. (29) can thus, be re-written
approximately as
4
X
jˆ1
o2j Fj Zj ‡
ˆ
4
X
_
rzL zS jVj
V_ ‡
Fj 
Zj
p



p
2
zL ‡ zS †
2rm hA2
jˆ1
fe
sin ot†
hrm
(33)
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
According to the distribution characteristics of the normal
mode shape functions of the thin square plate, it is easy to
verify that
Z a=2 Z a=2
Fj †jˆ2;3;4 dx dy ˆ 0
(34a)
a=2
a=2
Thus, from Eq. (4), we have
Z a=2
_V ˆ Z_ 1
~
F1 dx dy ˆ Z_ 1 V
(34b)
a=2
Substituting the above equation into Eq. (33) and utilising
the orthogonality of Fj(x, y), Eq. (33) is reduced into

Z1 ‡ bjZ_ 1 jZ_ 1 ‡ o21 Z1 ˆ o21 F sin ot†
(35)
and

Zj ‡ o2j Zj ˆ 0;
j ˆ 2; 3; 4
(36)
where
bˆ
~ L zS
a2 Vz
r
p

p 2
2
r
2A h
zL ‡ zS † m
(37)
~ 21 †:
and F ˆ fe a2 = hrm Vo
As emphasised earlier, the present objective is to examine
the steady-state vibration response of the membrane due to
the excitation force. The solutions of Eq. (36) need not to be
considered. In Eq. (37), b is a dimensionless parameter
representing the effects of ¯uid±membrane coupling and
the diffuser (or nozzle) structure. It is termed the coupling
parameter hereafter. For an aluminium membrane with
a ˆ 1000 mm and h ˆ 50 mm, and if A is taken as a2/100,
~ ˆ ha2 =2, zL ˆ 1:1, and zS ˆ 0:1, b can be
rm =r ˆ 2:3, V
calculated to be around 185.5. Generally, this parameter is
larger than one. For the convenience of subsequent discussion, we introduce a new variable t ˆ ot and omitting the
subscript, Eq. (35) can be written as
o2
d2 Z
dZ dZ
‡ o21 Z ˆ o21 F sin t
‡ bo2
dt2
dt dt
(38)
The contribution of each term on the left side of Eq. (38)
to the solution depends on the magnitude of the coef®cients,
2
177
For the case of o o1 or b being sufficiently large, Eq. (38)
becomes a highly non-linear equation. In this case, the
second term on the left side of Eq. (38) is much larger than
the other two terms. The small parameter perturbation
method is still applicable and the derivative of the function
Z can be approximately expressed as
s
8
>
o1
F p
>
>
sin t
0t<p
>
<
b
o
dZ
s


(40)
ˆ
dt >
>
o1
F p
>
>
sin t p t < 2p
:
b
o
The resultant solution, is thus
s (R p
R p=2 p
t
o1
F
sin u du
sin u du 0 t < p
0
0 p
Zˆ
R p=2 p
Rt
b
o
sin
u
du
sin u du p t < 2p
0
p
(41)
In general, Eq. (38) can be strongly non-linear, with the
small parameter perturbation method no longer being
applicable as the non-linear (second) term may be almost
of the same order of magnitude as the other terms. For this
case, the solution is assumed to be
Z ˆ Z0 sin t
y†
(42)
where the amplitude Z0 and the phase difference y lagging
the excitation force are functions of the natural and excitation frequencies (o1 and o), as well as the loading amplitude
F and the coupling parameter b. Substituting Eq. (42) into
Eq. (38), we obtain
o21
ˆ
o2 †Z0 sin t
y† ‡ Z20 bo2 jcos t
Z ˆ F sin t
o2 b 2
F jcos tj cos t
o21
(39)
y†
(43)
By the application of the Galerkin method, Z0 and y can be
determined uniquely as
Z0 ˆ
‰0:5 1
R2o †2 ‡
F
q
2
64F 2 b R4o =9p2 ‡0:25 1 R2o †4 Š1=2
(44)
3
F
6
y ˆ arc sin4 q
2
F 2 ‡ 3p=16† 1 R2o † =bR2o †2 ‡ 3p=16† 1
or more strictly speaking, on the ratio of these coef®cients.
As the ratios of coef®cients change with the excited frequency o and the coupling parameter b, thus, also will the
vibration response change with o and b for a given loading
amplitude F. For the case of o2/b/o1 < 0:01, Eq. (38) is
weakly non-linear. By means of small parameter perturbation method, the approximate solution can be expressed as
y†j cos t
o21 F sin t
R2o †2 =bR2o
7
5
(45)
where Ro is the ratio of frequencies (o/o1).
4. Membrane vibration characteristics
Although structural and geometrical non-linearities have
been neglected, the vibration response of membrane to the
excitation force is still non-linear due to ¯uid±membrane
coupling. As observed in the solution process of Eq. (38),
the vibration characteristics are determined by three factors:
178
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
the inertial motion of the membrane (®rst term), the ¯uid±
membrane coupling (second term), and the elastic deformation (third term). With the change of the excitation frequency
o and the coupling parameter b, the effect of each factor on
the membrane vibration is different and the corresponding
vibration characteristics, are thus, also different.
When the frequency o is very low relative to the ®rst
natural frequency
(o ! o1 ), in other words, when
p
o < 0:1o1 = b, both the reaction of ¯uid±membrane coupling and the inertial force of membrane are very small
relative to the elastic deformation force of membrane. Hence
in this case, the excitation force is mainly balanced against
the elastic deformation, and the linear characteristics of the
vibration response, thus, dominate. This can be clearly seen
from the approximate solution of Eq. (39). Although, the
amplitude of displacement contains the square of the excitation amplitude F and the coupling parameter b, their effect
does not exceed 1% of the linear component. Thus, when the
excited frequency is very low, the vibration response of
membrane to the excitation force is linear and the effect of
the coupling parameter b is minimal.
When the frequency o is close to the natural frequency
(o o1 ) or if the coupling parameter b is suf®ciently large,
the inertial and the elastic deformation forces of membrane
are very small compared with the reaction to the ¯uid±
membrane coupling and are thus, negligible. The corresponding approximate solution is given in Eqs. (40) and
(41). Although, the non-linearity dominates in this case, the
vibration response of membrane to the excited amplitude
and frequency, as well as the coupling parameter, is of
simple form. In fact, the amplitude response is proportional
to the square root of the excitation amplitude F and inversely
proportional to o and the square root of b. The phase
difference is p/2, regardless of the values of F, o and b.
Usually, in general forced vibration, the amplitude of the
velocity depends on the excitation frequency o. However,
Eq. (40) indicates that the resultant amplitude of the velocity
is independent of o, but proportional to the square root of the
ratio of F to b. This is a characteristic property of the
velocity response in the ¯uid±membrane coupled valveless
micropump. On the other hand, the larger is the coupling
parameter b, the smaller will be the amplitude. Thus, when
the excitation frequency is close to the ®rst natural frequency, the coupling parameter b plays an important role in
determining the vibration response.
Fig. 2 shows the comparison of vibration responses for the
two cases discussed above, with F ˆ 1 and b ˆ 185:5. It can
be observed that for the case of o ˆ 0:002o1 , the vibration
response function is almost similar to the excitation force. In
other words, the vibration of the membrane is almost
synchronous with the excitation force. However, in the case
of o ˆ o1 , the phase response falls behind the excitation
force by a quarter of a period. This implies that the nonlinearity associated with ¯uid±membrane coupling plays a
similar role as the damping term in linear vibration. The
amplitude response is much smaller compared to that in the
Fig. 2. Comparison of vibration responses for two excitation frequency
cases.
case of o ˆ 0:002o1 . This is contrary to that in forced linear
vibration, in which resonance or maximum amplitude will
appear. Therefore, the ¯uid±membrane coupling plays an
active role in preventing the membrane from resonating.
The discussion above is based on two special cases. For a
general case, the ¯uid±membrane coupling, the deformation
and the inertia are all not negligible relative to the each other
even though the contribution of each component changes
with the excitation frequency and coupling parameter. By
means of the Galerkin method, an approximate solution is
found, see Eqs. (42), (44) and (45). When the excitation
frequency o is very low, Eq. (42) reduces to Eq. (39),
whereas when o ˆ o1 or if b is suf®ciently large, Eq.
(42) becomes
s
o1
3pF
cos t
(46)
Zˆ
8b
o
Comparison with Eq. (41) shows that there is a difference
between the solutions obtained using different approximate
methods. The numerical comparison among the solutions is
given in Fig. 3. It is obvious that for b ˆ 185:5, the two
curves based on Eqs. (39) and (42) are almost coincident
when o ˆ 0:002o1 , and the two curves at o ˆ o1 from Eqs.
(41) and (42) are very close to each other except for a very
small region near the peak. This indicates that although Eq.
(42), as well as Eqs. (44) and (45), is an approximate
solution of Eq. (38), it possesses high accuracy.
Hence Eqs. (44) and (45) are able to provide the amplitude
and phase difference in general cases of F, o and b. The
variation of the vibration response to the excitation frequency o is plotted in Fig. 4 for the case, where F ˆ 1 and
b ˆ 185:5. It can be observed that with the increase of the
excitation frequency, the amplitude response decreases and
the phase lag increases. This shows that the vibration
response of the membrane is very sensitive to the excitation
frequency. They arrive to their limits (curve 6), when
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
179
Fig. 3. Comparison of solutions obtained by different methods.
Fig. 5. Variation of vibration response for different coupling parameter.
o ˆ o1 , and the maximum phase difference is p/2. These
vibration characteristics are obviously different from that of
a linear, forced vibration system with damping, in which
case the phase difference is always p/2 and the response
amplitude increases as the excitation frequency approaches
the natural frequencies.
Fig. 5 shows the variation of the vibration response with
the coupling parameter b for the case where F ˆ 1 and
o ˆ 0:002o1 . When b ˆ 102 , the vibration function Z is
almost the same function as the excitation force
(Z ˆ Fsin t†). With the increase of b, the amplitude of Z
decreases and the phase lag increases gradually. The limit
value of the phase difference is p/2, as shown in Fig. 4. The
amplitude response to the coupling parameter b is different
from that to the excitation frequency o. For a given excitation frequency, the amplitude response approaches zero
when b is suf®ciently large, while for a given b, the response
reaches its minimum value when o ˆ o1 . This behaviour is
reasonable because the larger is the b, the larger is the
resistance of ¯uid±membrane coupling to the vibration. So
long as b is suf®ciently large, the membrane will not vibrate.
5. Mean flux
The following discussion is based on Eq. (42) as it
represents the complete vibratory characteristics of the
~ t† into Eq. (30) and applymembrane. Substituting V ˆ VZ
ing Eq. (42), the mean ¯ux through the valveless membrane
micropump can be written as
p p
~
zL
z V
pS oZ0
Q ˆ p
(47)
z L ‡ zS p
From Eq. (44), the amplitude Z0 is a function of the
coupling parameter b which varies with the loss coef®cients
zL and zS, and the dependence of the mean ¯ux Q on zL and
zS is thus, quite complicated. However, when the excited
frequency o is low enough, the action of b is negligible and
the Q can, thus, be expressed as
p
p p
1
zS =zL
zL
zS
p
p

p

Q/
ˆ
(48)
zS
zL ‡
1‡
zS =zL
Clearly, the smaller is the ratio of two loss coefficients, zS/
zL, the larger will be the mean flux Q. In the case of o ˆ o1 ,
and according to Eq. (46), we have
p p
zL
z
p S
Q/
(49)
zL zS
Fig. 4. Variation of vibration response to excitations of different
frequencies.
It indicates that the mean flux Q increases with the increase
of zL and the decrease of zS, that is, Q increases with the
decrease of zS/zL. This conclusion is in agreement with that
in the very low excitation frequency range. In fact, it is easy
to verify from Eq. (47) that the increase of mean flux Q with
180
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
the decrease of zS/zL is also true for any excitation frequency
in the range of o o1 .
From the ¯uidics viewpoint, this conclusion is reasonable
because for the given frequency and amplitude, the smaller
is the loss coef®cient ratio zS/zL, the higher will be the
amount of ¯uid ¯owing out, as less ¯uid ¯ows in through the
diffuser, all within a single period. Thus, in the process of
selecting the diffuser or nozzle design in a valveless micropump, it is necessary to keep the zS/zL ratio as low as
possible if high ¯ow rates are a requirement.
The coupling parameter b is another parameter affecting
the mean ¯ux Q of the pump. In fact, Eq. (47) shows that the
Q is proportional to the response amplitude Z0, which
decreases with the increase of b. This is clearly observed
in Fig. 5. Hence to attain a large ¯ux Q, the value of coupling
parameter b should be as small as possible. From Eq. (37),
one way to deduce b is to increase the ratio of the section
area A of diffuser to the membrane area a2. This is in line
with the requirement for decreasing the energy loss in this
¯uid±membrane coupled vibration model.
The discussion above is based on a ®xed excitation force.
However, the variation of mean ¯ux Q with the excitation
amplitude and frequency can be very large. Although the
amplitude response Z0 decreases with increase of the excitation frequency (see Fig. 4), oZ0 to which the mean ¯ux Q is
proportional increases unidirectionally for the given F and
with b ®xed. Fig. 6 shows the change of oZ0 with the
excitation frequency in the case where b ˆ 185:5. It can
be observed that for any excitation amplitude F, the curves
are all almost horizontal, when o=o1 0:15. This implies
that the increment of mean ¯ux Q is very small even with
large increments of o when o/o1 0:15. It can be veri®ed
that the mean power provided by the excited force is
proportional to o3. Thus, if the micropump works at the
frequency o > 0:15o1, the energy loss will be very large.
For instance, when the excited frequency is increased
from 0.1o1 to 0.2o1, the mean ¯ux sees no any apparent
Fig. 7. Mean flux response to the excitation force at b ˆ 500.
increment for any F 1. However, the input energy is
required to increase up to as much as eight times. In the
region o/o1 < 0:08, the curves are all very sharp. This
implies that with the decrease of excitation frequency, the
decrease of mean ¯ux is also very rapid, and thus, the
working frequency of the pump should not go below
0.08o1. Therefore, at a ®xed coupling parameter b, there
is an optimal region for the working frequency for a valveless micropump, and this is in the vicinity of o 0:1o1 .
When the coupling parameter b is varied, the optimal
working frequency region may shift along the frequency
axis. Fig. 7 shows the change of the mean ¯ux with the
excitation frequency at b ˆ 500. Comparing with Fig. 6, the
optimal working frequency region has shifted towards the
lower frequencies. This indicates that the higher the coupling parameter, the lower will be the optimal working
frequency range. However, the shift of the optimal working
frequency region remains quite small and it is observed that
the region generally remains near o ˆ 0:1o1.
The optimal work frequency region can also be determined analytically. From Eq. (44),
for a given
F and b, (oZ0/
p

o1) reaches its maximum of
3pF=8b at o ˆ o1 . If o(a)
is de®ned as the
frequency
p
 corresponding to Q ˆ aQmax, or
oZ0 =o1 † ˆa 3pF=8b, then
v
s
!
u
u
o a† t
4Fb
3p
(50)
g2 ‡
g g
ˆ
1
o1
3p
2Fb
where g ˆ a 2 a2 . The optimal working frequency can be
obtained by Eq. (50) after defining the value of a.
6. Concluding remarks
Fig. 6. Mean flux response to the excitation force at b ˆ 185:5.
A non-linear vibration model for ¯uid±membrane coupling has been developed based on pressure loss formulae in
¯uid mechanics. The analytical solution for ¯uid±membrane
L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181
coupling vibration was approximated using the Galerkin and
small parameter perturbation methods, with no pressure
difference between the inlet and the outlet. Also, structural
and geometric non-linearities have been neglected. The
solution shows the following vibration properties. For the
case of a ®xed coupling parameter, if the excitation frequency is suf®ciently small, the vibration response is linear
and synchronised with the excitation force. With the
increase of the excitation frequency, the response amplitude
decreases, the phase difference increases and the non-linear
component in the response becomes more signi®cant. When
the excitation frequency is increased to the ®rst natural
frequency of membrane, not only does resonance fail to
occur, the response amplitude is also at a minimum with the
phase difference at its maximum value (p/2). This phenomenon is the converse of that observed in forced linear damped
vibration.
The vibration response also changes with the coupling
parameter. The larger is the coupling parameter, the smaller
will be the amplitude of membrane, and the larger will be the
phase difference lagging the excitation force. When the
coupling parameter is suf®ciently large, the amplitude
response can be almost zero, as though no vibration occurs.
This is reasonable because the coupling parameter effectively represents the resistance to the vibration. Some properties on the mean ¯ux are investigated based on the solution
obtained. It was found that the ratio of the pressure loss
coef®cients of diffuser and the coupling parameter play
important roles. Generally, the lower this ratio is, the higher
will be the mean ¯ux. Also, for a valveless membrane
micropump, there exists an optimal working frequency
range. Usually, this optimal working frequency region is
around 0.1 times the ®rst natural frequency of the membrane.
181
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