Proc● edings of ASCHT09

2nd Asiari SympOs:um On cOITlputational Heat Transfer and Fluid F:ow
October 20‐23,2009,Jeju,Korea

Paper No.ASCHT09‐ C94
NUMIERiCAL STUDY ON FLOW AND HEAT TRANSFER CHARACTERISTiCS OFPERISTALT:C PUMP
R.Kurokil,K.■ latsurnit,2,M.Nakalnural and K.:N● kabeL2.●
1.D"armcnt ofM∝ hanical Enanccnng am Science,Kyoto Univ∝ siり ,Kyoto 606‑8501,Jゆ an
2.Advanccd Research lnstimte ofFluid Scie c and Engmeermg,Kyoto Universiけ*Corrcっ onding autor,E‐ maili nよ abe@me.kyoto‐ ■ac」 p

V

The presert study numerically investigates the flow andheat transfer cha.acteristics of a micro peristaltic purnp, thebottom wall of which fluctuates in the form of a sinusoi&lly
progressive wave. This pump can be applied effectively to
micro fluidic devices owing to its compactness, low fabrication
cost, and high applicability to various biological fluids. Twodimensional unsteady lumerical simulation using the lmmersed

虐

ゥ
﹁
Ｗ
ツ

ABSTRACT

‐→
;仁
｀
鴫登
中こ
≡ゴ

Boundary method is canied out for fluid motions
accompanying the peristaltically movable bottom wall. The

仁

attention is paid to, in particular, the heat transfer
characteristics of the immovable target top wall urder the

PZT elements

constant temperature condition, following the authors' previous

)
)

1

く」

Flcxiblc nlln

study that the effocts of the movable wall amplitude andvelocity on the mechanical work of the pump were examined.
A pair of large-scale recirculation flow is observed in the caseof relatively larger amplitude. This recirculation flow enhances
the coovection heat transfer and reduces dre size of reverse
flow region near the target wall.

against contamination, compared to the oth tlpes of micrc
pumps. Kanno et al. 0l and Suzuki et al. [2, 3] manufactured
several protot)?e peristaltic micro pumps and experimentally

INTRODUCTION

examined them.
In the authors' previous study [4], the flow characteristics

Peristaltic pumps are one of the most suitable candidates of

micro pumps for fluidic MEMS (Micro Eleckomechanical
Systems) and ;r,-TAS (Micro Total Analysis Systems) becauseof their compactness, low fabrication cost, and highapplicability to various biological fluids []. As schematicallyshown in Fig. l, fluids can be aaNported in the channel fromthe Ieft side to the right side in rcspoNe to the peristaltic

motion ofthe channel wall. The main mechanical movable panof the pump is consisted of piezoelectric (PZT) elements aod
attached to a micrometer-size flexible film of the channel wallfrom outside. Thus, the main part is not wetted to by biologicalfluids flowing inside the pumping channel so that the part iseasily detached from the channel. This structural feature leads
to a superior chemical or medical maneuverability of thedisposable channel and retumable main part for the prevention

Fig. l: Schematic of micro peristaltic pump.

in this q?e of micro peristaltic pumps, the watl of which
fluctuates in the form of a siousoidal-, saMooth-, ortrapezoidal-shaped progressive wave, were numericallyexamined to evaluate the pump performances. No remarkableincrease of the flow rate was obtained when the wave shape
was changed. The pump performance, however, was found tobe effectively improved by the inclease of the wave amplitude
rather than by the increase ofthe wave velocity.
In the present study, the numerical simulatiol was carriedout to understand for furthor details how the flow pattem in thc
channel changes with an increase in the wave amplitude. The
goveming equation ofenergy was also calculated this time. The

Immersed Boundary GB) method [5, 6] was employed in thesame computational manner as the previous study [4], solving
the twodimcnsional time-dependent goveming equations. The‑314‑

attention was paid, ia particular, to the hcat transfer rate fromthe immovable target top wall under the constant temperah[econdition.
Couceming the thermal boundary conditions configurated
in the present study, the constant temperature condition wasgiven to a small restricted region on the immovable top wall,
the rest of which had the adiabatic condition. The movablebottom wall was assumed to be also adiabatic. The restricted
region which was a constalt temperature higher than the inlet
fluid can be considered as a micro heater or bio-sensor. In
micro channels, the Fourier number will generally becomelarge because of small channel size, which means that thethennal conductio! maioly govems the tempeBture field ofwalls and fluids. Then, the preseat boundary conditiol might bean irrlpractical one for micro fluidic pheoomena, but, it can b€ooe of the useful examples to understa[d fundamental heattsansfer characte stics of peristaltic fluid motion. Cotrtrol ofthermal field in fluid flow channels of MEMS and p.-TAS isextemely itDportant for chemical- or bio-reactors. wheDfurther improvement of micro matorial fabrication techniques
can yield better thermal insulation treatments in thc near futue,
local heat generation due to chemical or biochemical reactions
will be more prccisely controlled using such feated materials
regardless of the channel size. As aoother aspect of the results
obtaiDed under the preseot thermal boundary condition, thetomperature distribution corresponds to the mass ftaction o[e,based on the similarity between heat and mass t[ansfer. Non-channet leogth, L = lrnrn. The chaonel height, ll, was fixed at aconsta[t value, A= l00pm.
The twodimensional and incompressible time-dependent
Navier-Stokcs equatioD and energy equation were solved

corresponding mass fluxes of effirsion from biosensor
embedded in micro bio{evices.

fortogether with the corresponding continuity equation, as
described below:

9!L =n
ot,

Dび

・n̲̲一
Dι

cyclically shaped wall pattems ofthe pump, shown in the 6gure
represents the computational domain adopting the twodimensional Cartesia[ coordinate system. The abscissa is thestreamwise distance along the t-axis and the ordinate is thechannel height along the faxis with the origin situated at the
time-averaged height of the movable bottom wall. A simple
sinusoidal pro$essive wave was assumed to simulate theperistaltic motion ofthe bottom wall as follows:
)

(l)

)

where .4 and c are the wall amplitude and progressive wave
velocity, respectively. D,,a is the initial phas€ of the peristaltic
wall with the hxed value of t/2 rad. a is the wavelength of the
progressive wave, which was equal to the computational″

utational domain

￨′

(1′

2)

1上 望 ￨(j=■｀ ・′tPr∂ ■り

=12)

(3)
)

conection for the velocity field in order to solve the linite
difference equations [7]. Time increment with the fully-implicit
method was adopted fol the timedepeDdent terms.
For the boundary of the velocity field, no-slip condition
was employed on both the top and bottom wall surfaces. For
the boundary of the thermal field, the isothemal cofidition wasapplied to the reskicted region from dII = 3.75 to 6.25 on theimmovable top wall where the temperahre lvas fixed at 50"C,
4eC higher than the inlet fluid temperature. Adiabatic
condition was imposed upon the rest of the top wall atrd also
the movable bottom wall. The computational domain drawn in
Fig. 2 was one of the pump constituent elemeots periodicallyaranged along the streamwise direction. Thus, the periodic
boundary condition proposed by Patankar et al. [8] was applied
to the present computation fo. the flow field, while the
Neumann's condition was applied for the thermal field,
assuming that the pump will be situated at a sufficiently large
distance from both the inlet and outlet of micro devices.
ln order to examine these flow and thermal boundary
conditions for their validity, the computation with removing theperiodic boundary was performed in the double domai[ casethat two computational domains were sequcntially connectedalong the ,-direction, comparing to the single domain case. The
maximum relative differences betweeo these two cases were
only 0-25% a,.rd l.lYo, respectively, for the estimations of
rclative streamrMise velocity, Ufc, and mean Nusselt number.
This irnplies that the computational rcsults are little affected by

ノi″ 3 ,5Heated力>′∂■.∂ %t″ ∂%一summation convention.
As mentiored in the previous section, the IB method was
employ€d solying the above equations. Dstail cornputational
procedures on the IB method were explained in the prcvious
study [4]. To calculate Eqs. (2) - (4) by using the conventional
finite differerce mcthod including the IB mctho4 the lst-oder
upwind scheme was adopted for the convection terms and 2ndorder centsal{ifference scheme w.as for the diffusion tcdns,
avoiding the increase of computational time atrd cost. The
Pressure-Linked Equations
Semilmplicit Method
Complemented (SIMPLEC) algorithm was applied to pressule

NUMERICAL PROCEDURES AND CONDITIONS
Figure 2 shows the geometrical configuration of aperistaltic pump and the coordinate syslem of numericalcomputation. The boldlined rectangular region, one of thelr-∂
P +― ∂ ■∂仏f―where U, is.4teciona vd∝ iり ∞ m"nett P sふ cpressure, and 9 is the fluid temperah[e. p, p, and Pr ari,
respectively, the fluid density, dynamic viscosity and Prandtl
number, each ofwhich has the value ofpurc water al lo"C as a
NeMonian fluid. Subscripts i ard j obey the Einstein's

dimeDsionally nomBlized heat fluxes can mimic the

Asinljl(c, -.) + 4-n |?(r,t):
'
to.

(2)

一■ ｅ
９
ｔ 一∂ 血D@'Dt(,=1r\

Movi.g wlu opplied by 6ing IB mclnod

the present inlet and outlet conditions.

Fig. 2: Geometrical conirguration and coordinate systcm.

‑315‑

The number densify of computational g d was rcfued

1.0

betw€en two different cases; 402 x 230 and 202 x 140. The
maximum relative differences between thess cases were only
O.l3% efi 032yo fot the estirnations of average velocity andNusselt number, respectively. Thus, the latter number density
was used in the present computation.

0.8
・

′
＾

﹃・

0.2
0.0

progressive wave velocity, c = 0.1 m,/s, and the other is theeffect of the velocity, c = 0.1
l0 m/s, under the constant
amplitude, A / H = 0 .l . The lafter case is omitted on account oflimited space in this article.

40

-

こ■ヽ
='1[1)
0
(ν ぅ

"0

¨︱ユ
ぃ︱
■
ヽ一
●Ｏ﹂Ｌ¨
ｒヽ 一

｀ 04

the bottom wall, AIH = 0.1 - 0.9, under the constant

4.5

.

̲:´ ヽ
̲

5.O x′,/ 5.5

6.0(→ Absolute coordinatcs―

Pュthunc.―― wau:"31●。
ǹ0Tablc ltional conditions
(″

Labcl

ス/″

C[WS〕

´、
01o01〜 A09o01

0.1〜 0.9

0.1

A01c01〜 A01c10

0.1

0.1〜 100.5
ミヽ

‐), 00,,S‐
3C)

Flow patterns in the chonnel
Figure 3 (a) shows the path lines of fluid particles in one
cycte ofperistaltic motiotr within the regiot of 4.0 < c/ H < 6.0
in the case of Aolcol. The path lines drawn with bold lines
have three differert starting heighs: (z/Il, UlI4 = $.0, 0.1),(5.0,0.4), and (5.0,0.8). Chain lines cxtended from the boldlines indicate the path lines of the second and third cycles. The
other types of the lines drawn in the figule indicate the 1,all
positions at five different time phases of /?= 0.1, 0.3, 0.5, 0.7,
and 0.9. 7 indicates a time period of the cycle. Open circles
plotted on each bold line are the conesponding positions of thefluid palticles at each time phase. In Fig. 3 (b), solid lines wereobtained by the transformation of the coordinate system o[tothe ro-axis relative to the streamwise progressive wave

difference between these opposite flows causes large shear
stresses iD the channel.

TemDerchre distributions in the channel

Figurc 5(ぅ 〜(o shows thc isothemial contouls at eveW
ノ7'=・ 0.1 〜 0.9,in thc snlallcr

,5

)(

̲̲

‐
10

‐
15

‐
5
/177

Fig. 3: Path lines (Case A01c0l)
1.0

″●o
ヽ

「

′

0.5 l〈

′
′
′

̲ ヽ

■

ヽ

0・

̲

0

̲

‐
0.5 ̲
‐
1.0

0

ヽ
ヽ
ヽ
ヽ

̀

F)̀‐
ヽ

‐ ‐ ■●■,o0… Abo

,,

1.0

′

)。

ヽ ′

′

a,
'47

̲ 2,

30

螂 血●
は

1

ヽ0
.

‐
1

2‐ 101̀̀●‑― Wall,08饉(m2″

peristaltic wall motion of large amplitude. The velocity

0,

′
7 ),

・︶ ■

(5.0, -{.4), (5.0,0.0), (5.0,0.2), (5.0,0.5), (5.0,0.8), and (5.0,0.9). The four path lines in the upper half of the channel shifftheir positions upside dowq and the rest in the lower half also
shift in the same marmer. Viewed from the rclative coordinate
system in Fig. 4 @), the path lines are formed into a pair oflarge-scale counter-rotating recirculation almost thrcughout thechannel height. The path lines closc to the top and bonomwalls, on the contrary, follow the wall shapes with the reverse
flow direction against the flow direction in the middle of therccirculation flows. These flow fields are genemted by the

(′

T), ‐
0012,

.....(′

‐
20

‐
25

(″

(b)Rclative coordinates

velocity. A dotted line indicates the wall position.
As obviously seen in Fig. 3 (a), the fluid particles move
forward almost in the first and last quarte$ of the cycle, andbackward in the second and third quarters, coEesponding to thewall motion. These fluid motions, which are 8s almost same as
the analltical results obtained by Fung et al. [9, l0], produce a
net flow mte in one cycle.
The case of relatively larger wave amplitude (A08c0l) is
shown in Fig. 4 (a) and (b). In Fig.4 (a), there are sevendifferent paflicle-staning heights: ("/II, glID = (5.0, -{.7),

O.2 T tilnc phase of onc cyclc,

7, ‐0024,.コ くヽ

RESULTS AND DISCUSSIONヽ″--- (?o,,t

E「The computational conditions ale summa zed in Table L
Two seri6 were examined: one is the effect ofthe amplihrd€ of‐′′
9'‐ 00‑l.0

一

̲ 0.6

(b)Rela● ve coordinatcs

Fig. 4: Path lines (Case A08c0l).

amplitude case of A0lc0l. lt was observed that the fluid
temperature distribution changes periodically because of theperistaltic wave motion. Sharp changes of the fluid tempemture
were observed at aroud the leading edge of the heated region,d H = 3.15, while almost no changes were in the downstleamof the trailing edge, a/H 2 6.25, at any time phase. As
mentioned in the previous section, the thermal conduction
mainly govems the temperature field in micro channel flows
due to large Fourier number, lo (=arllf , q is a thermal
dimrsivity of water, ca. l.4x l0-7 mls.). ln this case, the valueof Fo is the order of l0', much larger than unity. since thecharacteristic time , based on the channel length, -L, and thecrcss-sectional averaged velocity, U*, can be estimated to be
the order of 100s. Thus, it is appropriate that the qoss-sectional‑316‑

uniformity of fluid temperature is achieved at a considerably
short distance in the downstream ofthe hoated region.
Figure 6 (a) - (e) shows the isothermal contours in thelarger arrplitude case of A08c0l. Comparing to the smaller
amplitude casc shown in Fig. 5, non-uniformity of fluid
temperature distributions was observed above the heated region
and in its downstreaq regaldless of the time phase. According
to the authors' pleyious study, the cross-sectional averaged
velocity ofthe fluid [/. reaches the progressive wave velocity cwhen the wave amplitude approaches the channel height. ln thecase of A08c0l, the averaged velocity was almost 0.7c n/s.
Thus, Fo can be the order of l0-r. Convection heat transfer,

heated in the heated region upward. Then, the higher
temperature region gadually moves downward along the slopeof the bottom wall and mixes well with the fiesh incoming
fluid. the substantial scalar transfer of heat and rnass is
expected in the peristaltic wall motion, ascribable to a pair ofrecirculation predicted in Fig. 4 (b). Based oo when therecirculation is geDeratcd with inqeasing the wave amplitude,
r〔 c￢

1.0

05

00
ヽ0.5
‐
1.0

therefore, has a co$iderable conkibution to the fluid

temperature and also velocity vector, lhe narrow regions
situated between the top ard bottom walls are magnified in Fig.
7 (a) and (b). Thc two time phases, (a) t/T = 0.3 md (b) tl T
0.7 in Fig. 7, correspond to the timings when the peak of thebottom wall approaches close to the leading edge of the heatedregion and recedes from its trailing edge, respectively. At ,/ 7=
0.3, there exists the flow bound for the top wall arcwrd tfH =3.7 due to the upward moving of the bottom wall while there
2.3 dueexists the flow bound for the bottom wall aloundto the downward moving ofthe bottom wall. This fluid motion
results irl the reverse flow near the top wall. At ,/ 7= 0.7, also
the reyerse flow is produced in the same manner within theregion, 6.3 < af H < 7 .7 . These reverse flows convect the fluids￢ 書 ≡霧多 多 ″0r [co] ro

4

2

6

x/″

10

8

(b)ィ T=0.3

10
06

ミ‐0.0
0.5

V

‐
1.0

r/Ii-

￨‐

x/77

TやヽL珍鰤
ノロ

-

ェ′
″ 3,

6

(→ ″T=0・ 1

１ ０ ０ ０・﹁ヽ ヽヽ

more wavelengths will be needed to achieve uniform fluid
temperahle as expected from the time-sequential pattems of
fluid temperati,ue in Fig. 6.
For a further detail observation on the distributions of

4

2

0

temperahrre distlibution. Some streamwise distance of one or

4

2

0

6

χ/〃

8
̀0

(c)イ r=0.5

10

05

ミ‐0.0
0.5
‐
■0

ェ′
=f=62'

1.0
晏 0.5

Ю

̀0

0

4

2

6

iO

χ/F′

(d)″ r=0.7

ヽ00
4

0

6

4/
χ′

8

10
05

10

00
ヽ‑05

(a)ι /r‐ o.1
1.0

ヽ05
0.0

‑1.0

0
0

2

4

6

χ′
′

8

4

2

10

6

r/″

8

̀0

(e)7r=o̲9(b)ッ r=0.3

Fig. 6: Temperature disributiorc (Case A08c0l)

10
05

ミ0.0

0

2

4

6

1/″

8

̀0

̀0 r■

(o)″ r=0.5

―
澤晉

ミ05

1.0

疑へき
■● 書 ●

2

4

6

χ″/

8

'■

10

ヽ̲′

ヨ髯雪

====̲̲̲■ ミ

i●

2

0

‐
￢

=ニ
・ =三 ″身毒妻≡≡菱ξ:菫=壼

0.0

05
ミ00

‐ ‐

tr〔 ぴ110=:￨レ●多蓄雪多電

'Ч

6

=≡

事

￨

￨

χ/′′

(→ ″T=0.3
1.0(oι /r=0.7
︑
ヽヽ

ミ0.5
0.0

0246ェ

″

8

(e)イ T=0.9
Fig.5:Temperature dis● bu● ons(Case A01c01)

10

2

4

6

″rr

e

o)″ r=0.7
Fi3.7:Mattined Чew of
A08o01).

‑317‑

mpen"

dismb血ons(Cascthe size ofthe reve.se flow region is noticeably reduced.
Ti me-depe ndent s treamw is e velocitv and heat 生空

coNCLUS|ONS
The flow and thermal characteristics in the chamel of″″α″peristaltic micro pump were numerically investigated. A simple

Figure 8 (a) - (c) shows the time histodes =たof Nusselt
number N4, and streamwise velocity U/c in the cases of thrcediferent wave amplitude, A01c0l, A05c0l, and A0gc0l. The
value of ly't6 drawn with a solid line indicates the Nusselt
number spatially-averaged over the heated region. The thrce
other q.pes of lines correspond to the loial streamwise

sinusoidal wave pattem was assumed to simulat! the peristaitic
motion of the flexible bottom wall. The constant temperatureboundary_ condition is given to the rest cted ,"giori ontt"
immovable top wall. The summa4r ofthe resuls obiained in thepresent snrdy are itemized as follows:. The fluid.moves back and fonh cyclically and produces a net

velocities. [/, obtained at the three differcnt heights, y/H - O.O,
0.5, and 0.9, but, at the same streamwise positior, cf,a = S.O.
The velocity ordinate in the figure was normalized with theprogressive wave velocity, c.
. l lat can be clearly seen in each figure is that the phase of
the Nusselt number is shifted against that of the striamwise
yelocity. The maximum and minimum values oflVu_, forexample in Fig. 8 (a), were obtained, rcspectively, zt tf T-= e.2
and 0.7, while rhose of U/c at t/T= O.O and 0.5. In ihe otheramplitude^ cases, {b) and (c), there exit several peaks, ridges andvalleys of both N4and tl/c curves. Their waveforms-appear
complicared. not following simple sinusoidal ones. Note thar ahorizonral.pan of the Il/c curve during a certain time period
ildicates the passage of time when the bottom wall peak passes

liuge amplrtude cases of the bonom wall. This recirculation
enhances the convection heat transfer 8IId the reduction ofthe

reverse flow in the narrow region where the peak of thebottom wall upwardly approaches tbe top wall.
. The phase of the Nusselt number is shilied against that ofthe
streamwise velocity in one cycle of the perisraltic wallmotion. The heat transfer from the heared region is mainly
govemed by heat conduction in the present flo-w and thermat
conditions. Most of higher heat tran;fer rates are obtained ataround the leading and trailing edge ofthe heated region.

SYMBOLSИ ａｏ ｃん ″ ん ι Ｍ Ｐ

ヽノflow rate by the peristaltic pump motion.

. A pair of counter-rotating recirculahon is generated in thethough at the corresponding heights. Another clearlv
recognizable feature is that the overall Nusselt number
calculated as the time-averaged lfir, ovcr one cycle will be
rncreased with rhe wave amplitude. This will be direcly

whlch means the heat transfer rate becomes minimum when thestreamwise flow direction changes ftom backward to forward.
After that moment, lfa gradually increases with increasing thestreamwise velocity. At the other positions, on the other hand,as the wave amplitude is increased, ly', fluctuates more withsome peaks. At the trailing edge of the heated regioo, in
particular, a peak of Nu was obseryed during the timi phasewhen the reverse flow occuls. This relatively large heat transfer
rate was ascribed to the fluid flowing back along the bottom
wall slope fiom the downstream of th; heated region which isdemonstrated in Fig. 6 (c).

︲
＆Ｔ ｔ・
υ
・ ■

positions, dff = 5.0, and 6.25, lfu possesses almost zerothroughout one cycle. This trend of Nu variation implies thatthe lluid tempeEture rises within a very short distance after the
fresh fluid passing though at the leading edge mainly by theheat conduction because ofthe large Fourier number.
Much focusing on the ff?, variatiol at the leading edge, as
commonly-obsewed feature of the three cases. the minimum′
ヽ口occurs when the sign of U/c changes fiom negative to positive,︐

relevant to the generation of the counter-rotating recirculationin the large. wave amplitude cases to enhance ihe convection
heat traNfer and the reduction of the revede flow region along
the heated wall.
Figure 9 (a) - (c) shows the time histories of the local
Nusselt number, ly'r4 at the thee different streamwise positions,d = 3.75,5.0, ard 6.25, on the heated region, plotted togethei
with their corresponding local streamwisJvelocities, U/I, at aconstant heighr, /.il = 0.9. Open circles, solid circles, and solid
squares in each figure indicate the streamwise positions of dll= 3.75, 5.0,-and 6.25, respectively. 1f1, ofthe smaller amplitude
case (A0lc02) changes sinusoidally only at rhe leading;dge ofthe heared region, r/I/ = 3.75. At'the other stieam"wiseamplitude
progressive wavelength

initialphaseprogressive wave velocity

Fourier number
chaniel heightheat hansfer coemcientchannel len8h
Nusselt number
pre6sure
Prandtl number

〔
司
[m〕

[腱 d]

[″ S]

彗(r/r
[m]

三 /(■ ―■)〔 W/(r・ K)]
[可

ヨた″/λ
[P可

Rqmolds number
time periodtimevelocity component(j=10F2)[=n′ s〕

Cartesian coordinates(i‐=ρ 臨 〃 ′IS〕

Relative coordinate

郡:轟鯖

キ壁摯.II絲饉

i鮒 轟ragれ̀[SIl or2)[m]国[m2/s]W′ (K・〔o][Pa・ S][K][kg′ m̀〕=″ υm[S]lue

ACKNOWLEDGMENTS
l■iS S範
Outお a pa of Adva∝ ing Tech∞ logy
Exccncnce ̀Nano‐
MediCinC' pЮ jeCt WhiCh is under KyOtO City
"WaS Cam

COllaboratlon ofRcgiOnal Entities assigncd by JST.

REFERENCES[l]Kanno,1"Suzukl,T"Ha● ,H"shintaku,H"Kawano,S.

and Kotcla,lH"2006,̀̀I)cvelOpment and charaCtで ,12atiOn Of a
TraVeling Wavc Micropunip I)rivcn by I)iezoelcctric

Acmtors,''P″ ο ″ ,4gsグ 2″ ′ ル ″′″´′′
ο″α′シ ″POS′ ″″ ο″

‑318‑

0.55

0.08

ノ″■

0.50

・

0・

'

＼

′
″‐00

0.45

01

′
で

′″‐,7

V″

02

0.06

° ―●― 7‐ 50‑「 、 ‐0‐・ル ′‐625‐ ‐
:′

̲

′
●
´

4′

0.04
0.02

3

。
.0ヽ

ミ

。
・∞ ヽ

,
2

0.40

0.02
‐
0 〉
4

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Fig.8:Tinte history of spatially‐ averagcd Nussclt numbcr and
StrCaFnWiSC Ve10City.

Fig.9:Ti=llc history of local Nussei nu=nber and stcamvnse
velocity.

Micro & Ndno Technologt (lSMNf-2), Hsinchu, March, pp.
226 - 229.

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[3] Suzuki, T., Teramura, Y., Hata, H., Inokuma, K., Kanno, I.,
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01

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[4] Tasumi, K., Matsunaga, Y., Miwa, Y. and Nakabe, K.,
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[5] Peskin, C. S., 1977, 'Numerical Analysis of Blood Flow in
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[6] Fadlum, E. A., Verzicco, R., Orlandi, P. and Mohd-Yusof,J., 2000, "Combined lmmersed-Boundary Finete-DiIIerenc€
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