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 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 Pout 2 zdo Qd 2 (2) 2A and for the nozzle r P 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 V 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_ 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 zno zdo q 2 zdo V_ zno zdo V_ 2 zno zdo A2 DP=r (7) Qn 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 Pin zno DP d where DP Pout Pin . For the special case of DP 0, the Eqs. (6) to (8) are respectively reduced to pn z Qd p oq V_ (9) n zo zdo q zdo Qn p q V_ (10) zno zdo P 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 Pout P 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 (14) zdi zni q 2 d_ zi V zdi zni V_ 2 zdi zni A2 DP=r Qn (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 Pin zdi zni zni DP For the special case of DP 0, Eqs. (14) to (16), respectively reduce to pn z (17) Qd q ip V_ zdi zni q zdi Qn q p V_ (18) zdi zni P Pout r zni zdi _2 q pn 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 Pout z Q d 2 (20) 2A2 i r (21) P 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 zdi zno 176 L.S. Pan et al. / Sensors and Actuators A 93 (2001) 173±181 Qn zdi V_ q 2 2 zdi zno A2 DP=r zdi zno V_ zdi zno P 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 zni and zdi 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 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 fe Pin _ 2jVj (25a) (25b) where P1 and P2 are respectively _ _ 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 _ 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 Q Qd dt (28) T T where T is the changing periodicity of the excitation force. For the special case of Pin 0 and DP 0, Eqs. (25) and (28) reduce to _ Eh3 rzL zS jVj 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 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 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 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 W Fj x; yZj 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 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 j1 o2j Fj Zj 4 X _ rzL zS jVj V_ Fj Zj p p 2 zL zS 2rm hA2 j1 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 j2;3;4 dx dy 0 (34a) a=2 a=2 Thus, from Eq. (4), we have Z a=2 _V Z_ 1 ~ F1 dx dy 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 o21 F sin ot (35) and Zj o2j Zj 0; j 2; 3; 4 (36) where b ~ L zS a2 Vz r p p 2 2 r 2A h zL zS m (37) ~ 21 : and F 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 1000 mm and h 50 mm, and if A is taken as a2/100, ~ ha2 =2, zL 1:1, and zS 0:1, b can be rm =r 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 ot and omitting the subscript, Eq. (35) can be written as o2 d2 Z dZ dZ o21 Z 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) 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 Z 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 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 o2 Z0 sin t y Z20 bo2 jcos t Z 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 0:5 1 R2o 2 F q 2 64F 2 b R4o =9p2 0:25 1 R2o 4 1=2 (44) 3 F 6 y 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 yj 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 1 and b 185:5. It can be observed that for the case of o 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 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 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 o1 or if b is suf®ciently large, Eq. (42) becomes s o1 3pF cos t (46) Z 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 185:5, the two curves based on Eqs. (39) and (42) are almost coincident when o 0:002o1 , and the two curves at o 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 1 and b 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 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 1 and o 0:002o1 . When b 102 , the vibration function Z is almost the same function as the excitation force (Z 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 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 VZ ing Eq. (42), the mean ¯ux through the valveless membrane micropump can be written as p p ~ zL z V pS oZ0 Q 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/ (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 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 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 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 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 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 o1 . If o(a) is de®ned as the frequency p corresponding to Q aQmax, or oZ0 =o1 a 3pF=8b, then v s ! u u o a t 4Fb 3p (50) g2 g g 1 o1 3p 2Fb where g 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 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. 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