Applied Mathematics and Computation 206 (2008) 876–884

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

A two-chamber model of valveless pumpingusing the immersed boundary method

Sunmi Lee, Eunok Jung *

Konkuk University, Department of Mathematics, 1 Hwayang-dong, Gwangjin-gu, Seoul 143-701, Republic of Korea

a r t i c l e i n f o

Keywords:ValvelessPumpImmersed boundary methodFluid–stucture interaction

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.09.047

* Corresponding author.E-mail addresses: sunlee@konkuk.ac.kr (S. Lee), j

a b s t r a c t

We present a new mathematical model of valveless pumping for a tube with two elasticchambers, which is motivated by the Liebau’s two-tank model. The tube consists of par-tially soft and partially (almost) rigid and the periodic pumping is applied at an asymmetriclocation of the soft tube. The immersed boundary method is used to investigate the impor-tant characteristics of valveless pumping as the previous experiments and mathematicalmodels have been discovered. We have observed the existence of a unidirectional meanflow and the dependence of mean flows on the frequency and the compression durationof the periodic pumping. We are able to explain the occurrence of local maximum or min-imum mean flows due to the resonances of the system.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

Most computational and experimental models of valveless pumping have been represented by either a closed loop systemor an open system. A unidirectional mean flow is driven by the periodic pumping in these valveless pumps. The mechanismof valveless pumping can be found in many biological systems such as the human embryo before the formation of heartvalves, simple organisms with valveless circulatory systems, and blood circulations during the cardiopulmonary resuscita-tion (CPR) in which the heart valves do not function normally [4,6]. Recently, a great deal of interest in the development ofmicro- and nanosystems has been increased [20]. Due to this importance of valveless pumping, intensive research has beenperformed in physical experiments, theoretical analysis, and numerical simulations. Liebau proposed fluid-mechanical mod-els of valveless pumping to demonstrate the existence of a unidirectional mean flow caused by periodic compressions [12–14]. He suggested two models, a closed loop system and an open system composed of an elastic tube connected with twotanks (referred to as the Liebaus’s two-tank model). He performed experiments using different radii and elastic propertiesof tubes to observe effective functions of valveless pumping. After the Liebau’s work, various models of valveless pumpinghave been proposed.

Thomann used a one-dimensional linear model in a closed loop [24]. Moser et al. introduced a lumped model in a closedloop which consists of two different diameters and two distensible reservoirs [17]. Both models are too simple to give com-plete answers to valveless pumping phenomenon. Jung and Peskin studied the two-dimensional simulations in a closed elas-tic tube consisting of two different elastic materials. They showed that the magnitude and the direction of mean flowsdepend on the frequency and amplitude of the periodic pumping [9,10]. Jung also observed important characteristics ofvalveless flows using a lumped model by dealing with time-dependent compliances, resistances, and inertia [11]. Recently,Jung et al. developed two- and three dimensional valveless pumping models [31]. They showed that the direction and mag-nitude of a net flow can be explained by the sign and amount of power, which is work done on the fluid by the fluid pressure

. All rights reserved.

unge@konkuk.ac.kr (E. Jung).

Fig. 1. The Liebau’s two-tank model: two open tanks are connected by an elastic tube. A unidirectional net flow is observed by periodic pumping applied toan asymmetry part of the elastic tube.

S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884 877

and the elastic wall over one period. Ottesen developed a one-dimensional model and analyzed both analytically and numer-ically [18]. He showed that the frequency, amplitude, compression duration and location of the periodic pumping affectmean flows and confirmed these results with his physical experiments. He also showed the same phenomena of valvelesspumping in micro- and nano-scales using molecular dynamics technique [19]. Hickerson et al presented experiments ofvalveless flows in a fluid filled flexible tube with different impedance [7,8]. They observed a mean flow induced under a vari-ety of conditions such as materials, the frequency and amplitude of the periodic pumping. They observed the role of wavedynamics of the tube and resonances in determining the characteristics of valveless pumping. Manopoulos developed a one-dimensional nonlinear model in a closed loop consisting of a soft and a rigid tube [24]. They took account the effects of thehydraulic losses due to the stenosis. They explained the existence of mean flows by phase differences between the pressurewaves at two edges. Recently, Bringley et al. developed a simple model that can be described by ordinary differential equa-tions and investigated fluid dynamics in a closed loop system consisting of two materials [30]. They showed that the timeseries of flow velocities by the model are consistent with those in the experiment.

Bredow constructed a system with a elastic tube connecting two reservoirs [3]. He observed the direction of a mean flowis dependent on the frequency and location of the periodic pumping. Rath and Teipel developed a one-dimensional nonlinearsystem of the Bredow’s experiments [22]. They showed the elastic wall and the nonlinear terms are the crucial factors of theexistence of a unidirectional mean flow. Takagi, Saijo and Takahashi presented experiments in a valveless piston pump ofrigid pipes with a T-junction connected to two tanks. They observed the relationship between resonances of the systemand the direction of mean flows [28,29]. Propst and Borzi proposed a one-dimensional model of elastic tubes or rigid T-pipesconnecting reservoirs [2,21]. They discussed various factors and conditions where unidirectional mean flows can occur.

The main motivation of the present work comes from a detailed and better understanding of the complex behavior invalveless pumping by constructing a two-dimensional computational model, which is an analogue of the Liebau’s two-tankmodel. The original Liebau’s two-tank model is an open system consisting of two open tanks connected by a tube. Fig. 1shows a schematic description of his model. In this paper this model is modified to a two-chamber model, which containsa tube connecting two flexible chambers (see Fig. 2). Although two open tanks are replaced with two chambers, we couldobserve similar phenomena of valveless pumping as the previous work of the Liebau’s models reported, as well as the detailsof the fluid-structure dynamics. The purpose of this paper is to present numerical studies in the two-chamber model. In thismodel we solve the two-dimensional Navier–Stokes equations coupled with elastic boundaries using the immersed bound-ary method. Results of parameter studies such as the frequency and the compression duration of the periodic pumping arediscussed. Further special cases are compared qualitatively in mean flows, resonance behaviors, and wave motions along thetube.

0 5 10 15 20 25 300

2

4

6

7.5

X (cm)

Y (

cm)

Fig. 2. The two-chamber model: a long thin tube consisting of partially soft (thin) and partially rigid (thick) and two elastic chambers are illustrated.Periodic pumping is applied to the left second sixth of the tube (arrows).

878 S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884

The rest of the paper is organized as follows: a mathematical model and the respective numerical method are presentedin Section 2. In Section 3, mean flows as functions of the frequency and the compression duration of the periodic pumpingare discussed and three special cases are compared qualitatively. Conclusions are given in Section 4.

2. Two-chamber model

In this section, we give a mathematical formulation of the two-chamber model. As we mentioned in the previous section,we modify the Liebau’s two-tank model to our two-chamber model. The main reason of this modification is that our mod-eling is based on the immersed boundary (IB) method. In the IB method, the boundaries should be immersed in a fluid, there-fore two open tanks are replaced with two chambers. We consider a fluid filled rectangular box as our computational domainX. A long and thin tube (partially soft and partially rigid) and two elastic chambers are immersed in X. The immersed bound-aries for a tube and two chambers form a collection C. Fig. 2 illustrates the initial configuration of the two-dimensional two-chamber model. The thick and thin parts of the tube represent (almost) rigid and soft, respectively. Two chambers consist ofmore flexible materials than the soft parts of the tube. The fluid markers are placed only inside the model since we are inter-ested in the motions of fluid inside. Periodic pumping (an external force or a driving oscillation) is applied to the left secondsixth of the tube to derive fluid motions.

Numerical simulations have been presented using the IB method. The IB method is both a mathematical formulation anda numerical scheme. The IB method is originally developed by Peskin to simulate blood flow in a heart. The IB method is veryuseful to solve fluid-structure interaction problems and has been applied to many biophysics and biomedical applications,including two-dimensional and three-dimensional simulations of blood flow in the heart [25,26], the design of prostheticcardiac valves [27], wave propagation in the cochlea [1], valveless pumping in a closed loop [9,10], simulations of whirlinginstability [15], a higher order IB method [5], a single-cell modeling the dynamics of tumor [23].

The IB method involves elastic boundaries and fluid as Lagrangian and Eulerian variables, respectively. These are linkedby the interaction equations in which the Dirac delta function plays an important role. The Eluerian variables are defined in afixed Cartesian coordinates, and the Lagrangian variables are defined a curvilinear coordinates that moves freely through theCartesian coordinates. By transformation of two coordinate systems, the immersed boundary is treated as a part of a fluid,which means a boundary exerts forces to the fluid and the boundary moves with the local fluid velocity. The governing equa-tions for our two-dimensional IB model (a fluid-elastic boundary system) are given as follows:

q@uðx; tÞ@t

þ ðuðx; tÞ � rÞuðx; tÞ� �

þrpðx; tÞ ¼ lr2uðx; tÞ þ fðx; tÞ; ð1Þ

r � uðx; tÞ ¼ 0; ð2Þ@Xðs; tÞ@t

¼ Uðs; tÞ ¼Z

Xuðx; tÞdðx� Xðs; tÞÞdx; ð3Þ

fðx; tÞ ¼Z

CFðs; tÞdðx� Xðs; tÞÞds; ð4Þ

Fðs; tÞ ¼ �jtðXðs; tÞ � Zðs; tÞÞ þ js@2Xðs; tÞ@s2

!: ð5Þ

Eqs. (1) and (2) are the Navier–Stokes equations for the fluid in Cartesian coordinates with x = (x,y). The fluid velocityu(x, t), the fluid pressure p(x, t), and the external force density f(x, t) are unknown functions of x and t. The constant fluidviscosity l and constant fluid density q are considered.

Eqs. (3) and (4) represent the interactions between the fluid and the immersed boundaries. X(s, t) and U(s, t) are the im-mersed boundaries and the boundary velocity, respectively, which are treated in the Lagrangian form, where s is theLagrangian parameter s and marks a material point of the immersed boundary, 0 6s 6 L, L is the unstressed length of theboundary. As mentioned above, the immersed boundary moves at the local fluid velocity which is the no-slip conditionfor a viscous fluid. This is done by the Dirac delta function over the fluid domain, X, in Eq. (3). In Eq. (4), the boundary forcedensity F(s, t) defined on the boundaries are distributed to the fluid force density f(x, t) by a Dirac delta function where theintegral is taken over the boundary points. The interaction equations are in integral form with kernel of two one-dimensionalDirac delta functions: d2(x) = d(x)d(y). The force density, f(x, t) is singular like a one-dimensional delta function, the singular-ity being supported only on the boundary. Although the force density, f(x, t) is infinite on the immersed boundary, its integralover any finite domain is finite, the total force applied to the fluid by the part of the immersed boundary contained within thedomain.

Eq. (5) defines the boundary force density, F(s, t), consisting of two terms. For the first term, we introduce the target func-tion, Z(s, t), which provides a restoring force that keeps the boundary points near their target positions. This target functionhas two purposes: the one is to maintain the shape of the boundary and the other one is to apply the periodic forcing. Thesecond term in (5) models an elastic membrane under tension. Together, the two terms model the boundaries as tetheredelastic membranes. Note that the second term actually is the response to stretching of the membrane. The parameter jt

is a stiffness constant between the physical boundary and the target position and js is a stiffness constant of a linear springbetween two adjacent boundary points. The target function is defined only in the tube. On two chambers, there is only the

S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884 879

second term so that they are stretched or compressed freely. The target function in the pumping location (the left sixth of thetube as shown in Fig. 2) are modeled to give periodic pumping in vertical direction as if the fingers compress the pumpinglocation. This target function is the time-dependent function and described below. The target function in the rest of the tubeis time-independent and defined as the same as the initial tube. The time-dependent target function is given as a sinusoidalin time and space.

The driving oscillatory function with one period in space and time is defined as follows:

Aðs; tÞ ¼A0 sin p ZxðsÞ�Xtarget

16L

� �sin pt

d�T� �

if 0 6 mod ðt; TÞ 6 d � T;

0 if d � T 6 mod ðt; TÞ 6 T;

(

where T is the period, A0 is the amplitude, d is the compression duration of periodic oscillations. The compression duration isdefined as a ratio of the compression time to the period of the periodic driving oscillations. Xtarget is the starting point of theleft second sixth of the tube, and L is the length of the tube. Having defined A(s, t), the y component Zy(s, t) of the target func-tion Z(s, t) is defined as follows: Zy(s, t) = 0.5Yscale + r � A(s, t) for the upper tube and Zy(s, t) = 0.5Yscale � r + A(s, t) for the lowertube, where Yscale is the length of the y-directional computational domain. Note that the x component Zx(s) of the target func-tion Z(s, t) is the same as the initial position of the tube.

This model is implemented by the IB method using the finite difference schemes and discrete delta functions for discret-izing the governing Eq. (1) and periodic boundary conditions are imposed in the x and y directions. The brief outline of thenumerical scheme is given as follows: first, compute the boundary force density Fn on the boundary from the given boundaryconfiguration Xn. The boundary force density is distributed to the fluid force density fn using the Dirac delta function. Second,the fluid velocity un+1 and the fluid pressure pn+1 are updated from the given un and fn by solving the Navier–Stokes Eqs. (1)and (2). Third, the boundary velocity Un+1 is interpolated by the fluid velocity, then the boundary Xn+1 is updated by the up-dated boundary velocity. The details of this numerical scheme are found in [9,27].

3. Numerical results and discussions

In this section, we present numerical results of valveless pumping in the two-chamber model. There are many parametersthat affect flows of valveless pumping such as the frequency, the compression duration, and the location of periodic pump-ing, and material or geometrical properties reported in the previous research [8,9,11,16,18]. In this study, two most impor-tant parameters that affect a net flow are examined, the frequency and the compression duration of periodic pumping.Influence on the amount and the direction of a mean flow by these two factors and three special cases are further discussedin the following two subsections.

Physical and numerical parameters used in our simulations are given as follow. The computational domain is taken to be30 cm � 7.5 cm. The elastic tube is 11 cm in length with a diameter 0.75 cm. Each chamber has a radius 2 cm. We assume afluid density of 1 g/cm3 and a viscosity of 0.01 g/cm s. With regard to the material parameters, stiffness constants, jt, be-tween target and boundary points, 26,000 g/s2 cm for the (almost) rigid part and 900 g/s2 cm for the flexible part of the tubeare used. The spring constants, js, between boundary points 120 g cm/s2 for the whole tube and 30 g cm/s2 for two chambersare used. For numerical parameters, 512 � 128 grid and mesh width h = 30 cm/512 = 0.0586 cm are used. The resting lengthof all adjacent boundary points is equal to Ds = h/4 = 0.0146 cm. For the time step, Dt = h2 = 0.0034 s is chosen. Since theamount of the fluid inside two chambers are finite, we must limit the simulation duration. It takes about 15 s for the mostactive case where no more fluid left in one chamber. In our simulation, we choose the simulation duration 15 s.

3.1. Mean flow and resonance

In this subsection, we investigate mean flows as functions of the frequency and the compression duration of the periodicpumping. A mean flow is defined as a space- and time-averaged value of the normal velocities along a cross-section of thetube. Mean flows are computed at two locations of the tube, x = 10.5 cm and x = 19.5 cm. The frequency is chosen between0.05 and 10 Hz and the compression duration is in the range 0.1–1 with the amplitude, A0 = 0.4r, with the radius of the tube,r.

Fig. 3 displays mean flows as a function of the frequency at the compression duration d = 0.6 of the periodic pumping.Each data point is computed as a separated numerical simulation. A positive mean flow corresponds to a flow from the leftto the right and a negative mean flow corresponds to an opposite directional flow. As seen on this figure, there exists a strongand complicated influence of the pumping frequency on the direction and the magnitude of a mean flow. In our simulation,positive mean flows are observed at most frequencies, except frequencies between 4.3 and 6.4 Hz, which negative meanflows occur. Almost zero mean flows are observed at several frequencies, 4.2, 4.3, 6.4, and 6.5 Hz. At these frequenciesthe curve crosses the horizontal axis (dotted) and changes the direction of a mean flow.

There are several peaks (local maxima and minima) between f = 0.5 Hz and f = 5 Hz. For instance, the positive and nega-tive maximum mean flows occur at f = 1.5 Hz and f = 4.95 Hz, respectively. At these peak frequencies, mean flows are localmaxima or minima due to the resonances of the system. To find a natural frequency of the system, we take the free vibrationtest at f = 1 Hz. Fig. 4 displays the free vibration at the point, x = 12.5 cm, on the upper tube and its Fourier transformation(FT). Displacements of y-direction of the point are plotted as a function of time on the top frame and its Fourier transform

0 2 4 6 8 10−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency (Hz)

Mea

n flo

w (

cm2 /

sec)

Right cross−sectionLeft cross−section

Fig. 3. The space- and time-averaged flows as a function of the frequency at the compression duration d = 0.6 are displayed. Mean flows are compared attwo cross-sections of the tube, x = 10.5 cm and x = 19.5 cm. Positive values represent mean flows from left to right and negative values represent oppositedirectional mean flows. This result shows that the frequency affects on the direction and magnitude of mean flows in a great deal.

0 0.5 1 1.5 2 2.5 3 3.5 43.5

4

4.5

Time (sec)

Y (

cm)

0 5 10 15 20 25 30 35 40 45 50−20

−10

0

10

20

Frequency (Hz)

Am

plitu

de

Fig. 4. Displacements of y-direction at x = 12.5 cm on the upper tube are plotted as a function of time on the top frame. Its Fourier transform shows anatural frequency at 0.75 Hz and harmonics at 1.5, 3 Hz on the bottom frame. These frequencies approximately coincide peak frequencies in Fig. 3, whichare f = 0.65, 1.5 and 2.6 Hz.

880 S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884

(FT) is displayed on the bottom frame. The FT shows a natural frequency at f = 0.75 Hz and harmonics at f = 1.5 Hz, andf = 3 Hz. This confirms a natural frequency of the system approximately coincides peak frequencies of mean flows, whichare f = 0.65 Hz, f = 1.5 Hz, and f = 2.6 Hz in Fig. 3.

Next we present the effect of the compression duration of the periodic pumping on mean flows. Fig. 5 displays mean flowsas functions of the frequency and the compression duration of the periodic pumping. The mean flow at the right cross-sec-tion of the tube is chosen since they (at the left and the right section) are almost same. One hundred curves at frequency from0.1 to 10 Hz with 10 different compression durations are plotted. Positive mean flows are dominant in this result. For thecompression duration either too small (0.1 and 0.2) or close to the full ratio (0.9 and 1.0), the amount of a mean flow is rel-atively small. As the compression duration taking the middle values, the magnitude of a mean flow becomes larger. Note thatthe positive maximum mean flow appear at d = 0.5 and f = 1.5 Hz and the negative maximum mean flow happens at d = 0.6and f = 4.95 Hz.

Flo

w a

nd m

ean

flow

(cm

2 /se

c) 0 1 2 3 4 5 6 7 8 9 10

−2

0

2

4

0 1 2 3 4 5 6 7 8 9 10

−1

0

1

0 1 2 3 4 5 6 7 8 9 10

−0.5

0

0.5

1

1.5

Time (sec)

f = 1.5 Hz

f = 4.3 Hz

f = 4.95 Hz

Fig. 6. Flowmeters are displayed as a function of time for the maximum positive, an almost zero, and the maximum negative cases in the top, middle, andbottom frames, respectively. Thick and thin line present the space- and time-averaged flows and the space-averaged flows, respectively.

Mea

n flo

w (

cm2 /

sec)

012345678910

0

0.5

1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequecny (Hz)

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Duration

Fig. 5. The space- and time-averaged flows as functions of the frequency and the compression duration are illustrated. This results confirm that thedirection and magnitude of mean flows are heavily dependent on the frequency and the compression duration.

S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884 881

3.2. Case study

In this subsection, we concern three special cases chosen from the result in Fig. 3: the maximum positive mean flow, analmost zero mean flow, and the maximum negative mean flow. These cases have been investigated and compared qualita-tively in flowmeters, wall wave motions of the upper tube, and fluid motions. Three mean flows and frequencies are given asfollows:

� f = 1.5 Hz (mean flow: 0.76 cm2/s, the maximum positive net flow)� f = 4.3 Hz (mean flow: 0.006 cm2/s, an almost zero net flow)� f = 4.95 Hz (mean flow: �0.16 cm2/s, the maximum negative net flow)

12 13 14 15 16 17 18

3.9

4

4.1

4.2

4.3

f=1.5 Hz

12 13 14 15 16 17 18

3.9

4

4.1

4.2

4.3Y

(cm

)

f=4.3 Hz

12 13 14 15 16 17 18

3.9

4

4.1

4.2

4.3

X (cm)

f=4.95 Hz

Fig. 7. Wave motions along the top tube for the maximum positive, an almost zero, and the maximum negative cases are displayed in the top, middle, andbottom frames, respectively. Sixteen equal-time snapshots of the wave motions over one cycle after the almost periodic steady-state are plotted.

0 10 20 30

02468

0.00

sec

0 10 20 30

02468

1.84

sec

0 10 20 30

02468

3.67

sec

0 10 20 30

02468

5.51

sec

0 10 20 30

02468

7.34

sec

0 10 20 30

02468

9.18

sec

0 10 20 30

02468

11.0

2 se

c

0 10 20 30

02468

12.8

5 se

c

Fig. 8. Eight equal-time snapshots of the fluid motions are displayed at the maximum positive case, f=1.5 Hz.

882 S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884

Flowmeters are displayed in Fig. 6 for the maximum positive, an almost zero, the maximum negative cases in the top,middle, and bottom frames, respectively. The space-and time-averaged flows (thick curves) and the space-averaged flows(thin curves) are plotted in time. Mean flows are computed at the right cross-section of the tube, x = 19.5 cm. In all cases,motions of flows are oscillatory and the oscillations are settled down to the almost periodic steady states. It is clear tosee nonzero mean flows in the top and bottom frames validating the existence of unidirectional mean flows. As many otherwork pointed the role of resonances [7,9,16], we also observe resonances at peak frequencies. The period of acceleration andthe period of deceleration are almost the same in the top and bottom frames.

Fig. 7 presents the wall wave motions along the upper part of the flexible tube. The top, middle, and bottom frames arewave motions for the positive, an almost zero, and the negative cases, respectively. Sixteen equal-time snapshots of wallwave motions along the upper tube over the last one cycle are displayed. The wave motions are generated by the sum of

S. Lee, E. Jung / Applied Mathematics and Computation 206 (2008) 876–884 883

the pumping incidence waves and the reflected waves at the rigid parts. The wave motions are related to the pumping fre-quency such as a harmonic oscillation where a string fixed at both ends (standing waves patterns occur in a linear manner offrequencies). Especially the wave motions in the top and middle frames seem to be the standing waves. For example there isone node at x = 15 cm in the top frame. The amplitude of wave motions are bigger for a lower frequency in the top frame thanfor higher frequencies in the middle and bottom frames.

Eight equal-time snapshots of the fluid motions are given in Fig. 8 with the simulation duration, 12.85 s. The frequency1.5 Hz, (the maximum positive flow), is chosen to show the most active fluid motions. In the initial configuration, the firstframe at t = 0.0 s, blue1 (markers inside the left chamber), red (markers inside the tube), and green (markers inside the rightchamber) colors are used to show the details of the fluid motions. At time t = 3.67 s, some red markers start flowing into theright chamber. At the middle frame t = 9.18 s it is clearly observed that most of red markers (fluid inside the tube) move tothe right chamber and at t = 12.85 s most blue markers (fluid inside the left chamber) move to the tube and flow into the rightchamber. Note that this corresponds to the fact the difference of water level in the two-tank model [2,3,22]. We observe thatfluid markers are back to the initial state (same amount each chamber) if no pumping is applied. For our numerical simulations,the first order IB method is used and this results in slight volume leakage due to the fact that the numerical divergence is notcompletely conserved. The maximum change of volume in our simulations is about 6%.

4. Conclusions

We have developed a new mathematical model of valveless pumping in a two-chamber model which consists of two elas-tic chambers connected by a tube. Fluid dynamics is investigated as the periodic pumping is applied to the end of the softtube. This model is motivated by the Liebau’s two-tank model. Although two open tanks are replaced with two chambers, weare capable of observing main phenomena of valveless pumping. The IB method is used to investigate the most importantcharacteristics of valveless pumping as follows: the existence of a unidirectional mean flow is observed by applying periodicpumping to the asymmetric part of the tube. The direction and the magnitude of a mean flow is sensitively depend on thefrequency and the compression duration of the periodic pumping. We are able to explain the occurrence of peak frequenciesdue to the resonances of the system. Taking an advantage of a two-dimensional model, details of flowsmeters, the wall wavemotions of the tube, and fluid motions are discussed for three special cases, the maximum positive, the maximum negativeand an almost zero mean flow. Flowmeters show almost periodic oscillatory behavior in all cases and confirm the existenceof unidirectional mean flows. The standing wave patterns are observed in the wall wave motions of the tube at peak frequen-cies. The snapshots of fluid motions presented for the maximum positive case f = 1.5 Hz. This validates that a clear unidirec-tional flow exists in this new model. For more fluid dynamics, figures and movies of the simulations are provided at thereferenced website: http://math.konkuk.ac.kr/~junge/vp_chamber.html.

Acknowledgement

Jung’s work was supported by Korea Research Foundation Grant (KRF-2004-015-C00063).

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