Proceedings of ASCHT07
1st Asian Symposium on Computational Heat Transfer and Fluid Flow
October 18-21, 2007, Xi’an, CHINA

Paper No. ASCHT2007-002
NUMERICAL STUDY ON FLUID FLOW CHARACTERISTICS OF PERISTALTIC PUMP
K. Tatsumi, Y. Matsunaga§, Y. Miwa and K. Nakabe*
Department of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan§ Denso corporation, Kariya, Aichi 448-8661, Japan*nakabe@me.kyoto-u.ac.jp

ABSTRACT
Fluid flow characteristics in a peristaltic micro pump,which can be applied to Lab-on-a-Chip or Micro Total Analysis
Systems (μ-TAS), were numerically investigated. For thepresent computational analysis, we assumed a simple sinusoidal
wave pattern to simulate the peristaltic motion of flexible
channel wall and applied the immersed boundary (IB) method
to the treatment of the wall boundary. The effects of theparameters of the peristaltic motion such as the amplitude andvelocity of the sine wave upon the flow characteristics andpump performance were evaluated in the present paper. The
shapes of the wall motion were also changed from thesinusoidal pattern to saw-toothed or trapezoidal patterns to
examine the potentiality of the pumping flow rate increase.
INTRODUCTION
The length scale of micro fluidic devices in the systems of
μ-TAS or power-MEMS is in the range of order 1 ~ 100μm,
accompanied by a large decrease of the flow Reynolds number.
Then, fluid flows in the channels of such devices becomelaminar and hardly mix in comparison to turbulent mixing [1 ~3]. Figure 1 shows a schematic illustration of micro fluidic
device which is composed of micro pumps, mixers and sensors.
The authors’ research group has investigated the enhancementof mixing using ribs, trenches and jet flows through somebranch channels for designing effective micro mixers [4] andalso produced micrometer-scale thermo-sensitive films for thedevelopment of micro temperature sensors [5]. In the present
article, the authors paid attention to micro pumps aiming at thedevelopment to effectively feed fluids to the micro fluid
devices.
Peristaltic pumps are regarded as one of the promisingtypes of micro pumps for fluid-MEMS of μ-TAS. In Ref. [6], aproto-type peristaltic pump was manufactured andexperimentally examined. The active part consisted of
piezoelectric (PZT) elements was attached to a micrometerscale flexible channel wall from the outside, which isschematically illustrated in Fig. 2. Fluids can be transported
through the channel from the left side to the right side, in
response to the peristaltic motion of the channel wall. Thismechanism can simplify the structure of the pump device. As amore convenient feature, this pump makes it available formultiple medical diagnoses to only replace disposable testsection channels but not the pump itself, resulting in cost andtime reductions.

Fig. 1

Schematic illustration of μ-TAS.

Moving wallFig. 2

Flow

Schematic view of peristaltic micro-pump driving.

In order to design this type of the pump with emphasis onan effectively-controlled fluid transportation, the flow
characteristics in the channel are investigated with1

computational method, and the effects of the geometricalparameters such as the amplitude and velocity of the peristaltic
motion of the wall on the pump performance was examined.
The effect of the moving wall patterns are also discussed in thepresent study.

simultaneously move with the sinusoidal amplitude halved ascompared to the other cases. The last two cases labeled as
A1c01s and A1c01t in Table 1 represent, respectively, sawtoothand trapezoidal patterns of the wall motion. These four cases
will be argued with more details in a later section.

NUMERICAL PROCEDURES
The governing equations solved in the computational
domain shown in Fig. 3 were two-dimensional, time-dependentand incompressible continuity and Navier-Stokes (N-S)
equations:
∂U i
=0
∂xi

Flow directionHyh( i = 1, 2 )Δτ

Wave direction

Moving wall applied by using Immersed-Boundary method( i = 1, 2 )Fig. 3where Ui is the velocity of the Cartesian xi direction. ρ and μ
are the fluid density and viscosity, respectively. The working
fluid was water as a Newtonian fluid.
A simple sinusoidal wave pattern was assumed to simulate
the peristaltic motion of the channel wall, which can be
represented using the IB method [7, 8] adopted in the finitevolume computational code. Following the IB method, virtual
body forces fiτ expressed below can be added to the N-S
equation at every computational time step τ and at suitable grid
locations to adjust the fluid velocities to the wall speeds
corresponding to the peristaltic motion:
τ
τ −Δτ
U IB
,i − U i

A1

L

DU i
∂P
∂ ⎛ ∂U i ⎞
ρ
=−
+
⎜μ
⎟
∂xi ∂x j ⎜⎝ ∂x j ⎟⎠Dt

f iτ = ρ

O x

+ρ

τ
∂
∂Pτ
∂ ⎧⎪ ⎛ ∂U iτ ∂U j ⎞ ⎫⎪−+U τj U iτ ) +⎟⎬(⎨ μ ⎜⎜∂x j∂xi ∂x j ⎩⎪ ⎝ ∂x j
∂xi ⎟⎠ ⎭⎪where Δτ denotes the computational time interval. UIB,iτ, Uiτ
and Pτ represent the target velocity for IB method, the fluid
velocity in the IB-represented boundary region and thecorresponding pressure, respectively. Fluid velocities on thegrids closest to the wall region are estimated with linearinterpolation of area-weighted average velocities against the
distance from the wall surface along a normal line to thesurface.
In the preliminary computation, flow velocity distributionsof flows in a bent channel or flows over a backward-facing step
were computed. The predicted reattachment length in the caseof the backward-facing step flows showed a good agreementwith other researchers’ data, the figure of which is omitted on
account of limited space in the present paper.
The computational domain and coordinate system for thechannel in the peristaltic pump are shown in Fig. 3. The
channel height, H, and the streamwise wavelength, L, were
fixed at a constant value as H = 100μm and L/H = 10,
respectively. These parameters including other geometricparameters, the bottom wall amplitude A1 (the top wall one A2,also) and the wall speed c, are tabulated in Table 1 with theircase labels.
The cases categorized in Table 1 were some representative
cases examined in the present study. Θ denotes a phasedifference between the top and bottom walls, in the cases
A05c01b and A05c01c of which both the top and bottom walls

Computational domain.

Table 1 Geometric parameters.
Label
A1c01
A1c1
A1c10
A05c01bA05c01cA1c01s
A1c01t

A1/H
0.1
0.1
0.1
0.05
0.05
0.1
0.1

A2/H
0
0
0
0.05
0.05
0
0

c [m/s]
0.1
1.0
10
0.1
0.1
0.1
0.1

Grid

Θ [rad]200×140—200×220200×140300×1400–π——

RESULTS AND DISCUSSION
Cases of one-side sinusoidal wall motion(a) Pressure distribution(b) Streamwise velocity distribution
Fig. 4 Instantaneous contours of pressure and streamwise
velocity (Case A1c01).
Figure 4 (a) and (b) shows typical results of instantaneous
fluid pressure and streamwise velocity contours. At thisinstance, the region of 2 < x/H < 4 retains higher pressures
above the surrounding pressure, where coincides with thenegative-gradient slope region of the wall forming a sinusoidal
traveling wave from the left side to the right side. Relatively
lower pressure regions are located in the front and in the rear of2relatively lower wall speed range, c = 0.01 ~ 0.1m/s. As thewall speed is increased to c = 1.0 and 1.5m/s, Um/c is gradually
decreased, which implies that Um is not linearly increased withthe wall moving speed, c.
-2

1.6x10-3

0.0
0.00 0.02 0.04 0.06 0.08 0.10

A1/H(a) Effect of A1/H
-2

1.6x10-2

1.2x10Um /c(x/H, y/H)=(5, 0.8) 0.9 0.7
t/T=0.1 0.3
0.5

0.8
0.6

y/H

-3

8.0x104.0x101.0(5, 0.4)

0.4

0.9

0.0
4.2

4.4

4.6

t/T=0.1

0.3

4.8

5.2

5.0

5.4

-3

8.0x10-3

4.0x10

0.7

t/T=0.1 0.3
t/T=0.3 t/T=0.7
0.9 0.7
0.2 t/T=0.1 t/T=0.5 t/T=0.9(5, 0.1)
0.5

-0.2
4.0

2

Um/c = 1.51x(A1/H)-2

1.2x10Um /c

the higher pressure region. Putting the pressure contours andthe velocity contours for comparison, it is noticed that theregions of positive and negative streamwise velocities exist
respectively downstream and upstream of the higher pressure
region. Also, as shown in Fig. 4 (b), the positive maximum of
forward-directional velocity is found to be a bit larger than thenegative maximum of backward-directional velocity, which
implies that the net transportation of the working fluid occurs
as the time passes by, during one cycle.
Figure 5 (a) shows the path lines of one cycle in the case of
A1c01, starting from three different spanwise positions (x/H,
y/H) = (5, 0.1), (5, 0.4) and (5, 0.8), while Fig. 5 (b) shows thepath lines analytically predicted by Yin and Fung [9]. What isapparently obvious on comparing both figures (a) and (b) is thatthe patterns of the path lines are similar to each other. The fluid
moves forward almost in the first half of one cycle and thenbackward in the latter half, corresponding to the wall motion.
These fluid motions produce a pulsating flow with a net flowrate during one cycle.

0.5

5.6

0.0
0.0

5.8

x/H

(b) Results of Ref. [9]Fig. 5 Comparison of the numerically-obtained path lines toother researchers’ data.
Figure 6 (a) and (b) shows how the amplitude, A, andvelocity, c, of the moving wall have influences on the timeaveraged streamwise fluid velocity, Um. The ratio of Um to themoving wall speed, c, was assigned to the scale of the ordinate.
As shown in Fig. 6 (a), the relation between Um/c and A/Hshows a good agreement with the fitting function of a parabolic
curve. This trend also coincides with the analytically-obtained
results shown in Ref. [9]. In Fig. 6 (b), Um/c keeps constant in

1.0

1.5

c [m/s](b) Effect of c

6.0(a) Present results (Case A1c01)

0.5

Fig. 6

Relations among Um/c, A1/H and c.

To understand this non-linear trend of the bulk fluid
velocity against the relatively higher wall moving speed, phaseaveraged velocity distributions were calculated. Figure 7 (a)and (b) shows the spanwise distributions of streamwise
velocity, U, at the location of x/H = 5 in the two cases of A1c01
and A1c1. ˆt/T represents the dimensionless time, i.e., theratio of phase-averaged time to a time-period of one cycle. In
the relatively-low wall speed case of (a) A1c01, the velocitychanges its sign periodically in one cycle, and the velocity
distributions maintain almost parabolic patterns. In the higher
wall speed case of (b) A1c1, on the other hand, the velocity
distribution pattern deforms from a parabolic shape to a quarticlike shape possessing a depression around the channel center.
This deformation of Case A1c1 is considered to be
attributable to a phase difference between the motions of thefluid core and the fluids adjacent to both the top stationary walland bottom moving wall. As clearly seen in the velocity
distribution of Fig. 7 (b), the fluids in the vicinity of the bothwalls at ˆt/T = 0.3 can act in accordance with the bottom wallmotion, while the fluids around the channel center do not
obediently follow the wall motion. At ˆt/T = 0.7, the fluids inthe vicinity of the wall already have positive velocities while
the fluid core has still negative velocities, the fact of whichsuggests the occurrence of the phase difference and may cause3the aforementioned non-linear relation between the bulk fluid
velocity and the wall speed.
(a) Case A1c01

Cases of other wall motion patterns
In the previous section, the flow characteristics of the fluid
driven by the one-side sinusoidal wall motion were clearly
reveals through the present numerical simulation using IB
method. In this section, some other different wall motion
patterns were numerically simulated to examine how the flow
patterns or the net flow rates change.
h

A2
Flow direction

H
y

h

O x

(b) Case A1c1

A1

Wave direction

L(a) Sinusoidal wall on both sides (Case A05c01b, Θ = 0)

Fig. 7(b) Sinusoidal wall on both sides (Case A05c01c, Θ = -π)u/c distributions at x/H = 5.0.

■ A1c01 (c=0.1m/s)• A1c1 (c=1.0m/s)▲ A1c10 (c=10m/s)(c) Sawtooth-shape wall (Case A1c01s)
Flow direction

HyFig. 8hTime-averaged streamwise velocity distributions.

Time-averaged streamwise velocity distributions along the
spanwise direction are shown in Fig. 8. In the case of A1c01,
the maximum streamwise fluid flow velocity is found to belocated at y/H ~ 0.1. As the moving speed of the wall isincreased, the velocities near-wall regions are correspondingly
increased. The velocity distributions in the fluid core region,
however, show the opposite trend that the fluid velocity isdecreased with an increase in the wall speed. Thus, it will bealmost impossible to increase the net flow rate linearly byincreasing the wall speed, because of the cancellation
superimposed between the velocity decrease in the core region
and the velocity increase in the near-wall region.

o x

A1
A1

Wave direction

L(d) Trapezoidal-shape wall (Case A1c01t)
Fig. 9Computational domains of other wall moving
patterns.

There are three types of motion pattern adopted here: thefirst one is the type of sinusoidal-shape on both top and bottom
walls, the second one sawtooth-shape and the last one
trapezoidal-shape, which are shown in Fig. 9 (a) ~ (d).
Geometric parameters were summarized previously in Table 1.
The total amplitude of both the top and bottom walls in the4cases of (a) A05c01b and (b) A05c01c is equal to the amplitudeof the one-side wall motion case, i.e. the reference case, A1c01.
Only a difference of the computational condition between thesetwo cases of A05c01b and A05c01c is the parameter of a phasedifference, Θ ; zero (no delay) or -π (half-cycle delay). The
sawtooth-shape of (c) A1c01s was described using the
combination of two different wave-length sinusoidal waves asfollows:

Table 2 Wall shape effect on Um/c.
A1c01

A05c01bA05c01c

1.51×10-2

1.67×10-5

1.51×10-2

Label

Um/c

Label

A1c01s(Sawtooth)

A1c01t(Trapezoid)

Um/c

1.51×10-2

1.60×10-2

Lastly, the mechanical work given to the fluid by theperistaltic movement of the pump itself during one cycle will
be evaluated using the following equation.
L

W=

∫ ∫
0

0

ΔT

4

10

⎛
3⎞
⎜0 ≤ x / L ≤ ⎟⎟
⎝⎜
4 ⎠⎟
⎛3
⎞
⎜ ≤ x / L ≤ 1⎟⎟⎝⎜ 4⎠⎟

The trapezoidal-shape of (d) A1c01t was composed with asinusoidal wave and a rectangular wave.
The effects of the above-mentioned patterns of the wallmotion on the time-averaged streamwise velocity aresummarized in Table 2. The time-averaged streamwise velocity
Um/c in the case of the wall motion on both sides with the half
amplitude and the same phase (Case A05c01b) is found to be
extraordinarily small by the order of 3, compared to thereference case, A1c01, which indicates that almost no fluid
transportation occurs in the channel. In the case of the halfperiod phase shift (Case A05c01c) with the half amplitude, noincrease of Um/c was obtained, compared to the reference case,
A1c01. Even if the wall motion changes from the sinusoidal
pattern to the sawtooth pattern (Case A1c01s), no increase of
Um/c was obtained, also. The case of the combined trapezoidalsinusoidal wall motion pattern (Case A1c01t) was found toachieve 6% increase of Um/c, compared to the reference case,
A1c01.

LΔT

where ΔT is a time period of one cycle. Pw, Vw and τw,y indicate,respectively, a fluid pressure, a y-component fluid velocity andy-component shear stress on the wall surface.

3

10

2

10

1

10

A1c1
A1c01
A1c01s

0

10

A9c01
A8c01

A3c01
A2c01

A7c01

-1

10 -1
10

0

10

1

10

2

10

Um/Um A1c01

3

10

Fig. 10 Mechanical work, W vs. Um.
The relation between the evaluated mechanical work perunit area and the time-averaged streamwise velocity is shown in
Figure 10. The scales of both coordinate axes are normalized,
based on the reference case, A1c01. With respect to the wallmotion speed c (Cases A1c01, A1c1 and A1c10), both Um andW are increased as plotted with a chain line. In case that anotherset of computational condition was executed by changing thewave amplitude, A/H = 0.1 ~ 0.9 under the wall speed kept
constant at c = 0.1m/s (Cases A1c01 ~ A9c01), a similar trendof the increases concerning both Um and W was obtained asplotted with a broken line. In comparison with these two lines,
the gradient of the chain line is larger than the one of thebroken line. This implies that the peristaltic pump requires
more external energy to obtain the same flow rate by the wave
speed increase than by the wave amplitude increase.
In the case of sawtooth-shape pattern, there is no apparent
difference of both Um and W from the reference case A1c01.
3% increase of W in the case of A1c01s was found with carefulinspection, compared to the reference case A1c01. This
indicates that the pump efficiency is influenced moresignificantly by the wave amplitude and speed than by the wallshape pattern.

CONCLUSIONSThe characteristics of fluid flows in a peristaltic micropump were numerically investigated. A simple sinusoidal wave
pattern was assumed to simulate the peristaltic motion offlexible channel wall. Some other different wave patterns werealso adopted in the present numerical simulation. The effects of
the amplitude and speed of peristaltic wave upon the flow
characteristics and pump performance were summarized asfollows:
1.

( Pw (x,t ) + τ w,y )Vw ( x,t )dtdx

A1c10

5

10

W/WA1c01⎛ 2 2π⎞y = A sin ⎜⎜(ct − x ) + θt =0 ⎟⎟⎟
⎝3 L
⎠
⎛ 2π
⎞
y = A sin ⎜⎜2(ct − x ) + θt =0 ⎟⎟⎟
⎝ L
⎠

6

10

The fluid moves back and forth and produces a net flowrate in one cycle of the peristaltic pump motion.
2.
The relation between time-averaged streamwise velocity(bulk velocity) and wave amplitude can be expressed by aquadratic equation in relatively smaller wave amplitudes.
3.
Non-linear trend of the bulk velocity against the wallmoving speed can be cause by a phase difference between

5the motions of the fluid core and the fluid adjacent to thewalls.
4.
No remarkable increase of the bulk velocity is obtainedeven if the wave shape pattern is changed.
5.
The pump efficiency is influenced more significantly by
the wave amplitude and speed than by the wall shapepattern.
Further investigations will be reported somewhere soon on
the flow and also heat transfer characteristics in detail forvarious motion patterns of the peristaltic pump.

ACKNOWLEDGMENTS
This study was carried out as a part of Advancing
Technology Excellence ‘Nano-Medicine’ project, which isunder Kyoto City Collaboration of Regional Entities assignedby JST.
REFERENCES[1]G.E. Karniadakis and A.Beskok, Micro Flows –Fundamentals and Simulation, Springer.,2002.
[2] E.Wolfgang, H.Volker and Holger Löwe, Microreactors,
Wiley-VCH.,2000[3]N.T. Nguyen, and S.T.Wereley,Fundamentals andApplications of Microfluidics, Artech House,2002.
[4] K.Tatsumi, Y.Tsukanaka, and K.Nakabe, GroovePatterns Effects on Mixing Performance in a Micro-Mixer,
Proc.17th Int'l Symp. on Transfport Phenomena,
CD-ROM (No. 2-B-I-1),2006.
[5] T. Jida, K. Tatsumi, and K. Nakabe, Measurements ofFluid Temperature in Microchannels Using EmbeddedMembrane-Sensors Proc. 82th JSME-Kansai Spring Meeting,
4-1:2-1,2007. (in Japanese).
[6] I. Kanno, T. Suzuki, H. Hata , H. Shintaku, S. Kawano,and H. Kotera, Development and Characterization of aTraveling Wave Micropump Driven by Piezoelectric
Actuators. 2nd Int'l Symp. Micro & Nano Technology(ISMNT-2),pp:226 – 229,2006.
[7] C. S. Peskin, Numerical Analysis of Blood Flow in theHeart J.Computational Physics,25:220 – 252,1977.
[8] M.C. Lai, and C. S. Peskin,An Immersed Boundary
Method with Formal Second-Order Accuracy and Reduced
Numerical Viscosity J.Computational Physics, 160:705-719,
1977.
[9] F.C.P.Yin, and Y.C.Fung, Comparison of Theory andExperiment in Peristaltic Transport J. Fluid Mechanics,
47 : 93–112,1971.

6