Turbulence characteristics and mixing performancesof viscoelastic ﬂuid ﬂow in a serpentine microchannel
K. Tatsumi1,2 , Y. Takeda1 , K. Suga3 and K. Nakabe1,2
1

Department of Mechanical Engineering and Science, Kyoto University, Yoshida-honmachi,
Sakyo-ku, Kyoto, Japan
2
Advanced Research Institute of Fluid Science and Engineering, Kyoto University,
Yoshida-honmachi, Sakyo-ku, Kyoto, Japan3
Department of Mechanical Engineering, Osaka Prefecture University, Gakuen-cho, Sakai,
Osaka, Japan
E-mail: tatsumi@me.kyoto-u.ac.jp
Abstract. Flow velocity measurement and visualization using particle image velocimetry andﬂuorescent dye were carried out for a viscoelastic ﬂuid ﬂow in a serpentine microchannel for the
purpose to quantitatively evaluate the unsteady ﬂow characteristics that is observed even undervery low Reynolds number regime due to the combined eﬀect of the viscoelastic ﬂuid properties
and the channel shape. Sucrose water solution (Newtonian ﬂuid) and the polyacrylamide-sucrose
water solution (viscoelastic ﬂuid) were used as working ﬂuids. The mixing performance markedly
increased when the Reynolds number exceeded a certain value in the polyacrylamide solution
case. The single-point, cross-sectional and two-dimensional velocity distributions showed thatlow frequency ﬂuctuation was produced in the polyacrylamide solution case. Particularly large
ﬂuctuation in the channel spanwise direction was observed in the upstream area of the serpentine
channel. On the other hand, the amplitude of the ﬂuctuation decreased in the downstream
region. The ﬂuctuation in the upstream region is believed to be generated by the ﬂow instability
at the curved part of the channel, while the ﬂuctuations in the downstream area were attributed
to the local instability and the vortices provided from the upstream region.

1. Introduction
Viscoelastic ﬂuids are now well known to produce a signiﬁcant diﬀerence in the ﬂow
characteristics from Newtonian ﬂuids in terms of reducing turbulence and producing dragreduction eﬀects in high Reynolds number ﬂow regime (James , 2009), or increasing the ﬂow
instability in low Re regime (Burghelea et al. , 2004; Larson et al. , 1990). Recently, Groisman,
A. & Steinberg, V. (2001), and Burghelea et al. (2004) reported that by adding polymers ofseveral hundreds of ppm to the ﬂuid, the ﬂow instability is enhanced and ﬂow ﬂuctuation is
produced in a mm- or µm-scale serpentine channels due to the elasticity of the ﬂuid.
Instability study and experimental measurements have been keenly carried out in the pastfew decades on the ﬂow characteristics of viscoelastic ﬂuids in Taylor-Couette and Taylor-Deanﬂows (Larson et al. , 1990; Joo, Y. L. & Shaqfeh, E. S. G. , 1994). However, the ﬂow structure,especially the turbulence characteristics of the viscoelastic ﬂuid ﬂow in serpentine channels havenot been studied and understood yet. The study of such ﬂow structures can give us an insightfor designing an eﬀective channel shape to be used in micro ﬂuidic devices or compact mixing
and heat transfer equipments, and provide large contributions to the ﬁelds of chemistry, heattransfer and medicine.

Figure 1. Schematic of the microchannel and sample images of the ﬂuorescence.
In this study, velocity measurements using micro particle image velocimetry (µPIV) andvisualization using ﬂuorescent dye are carried out for a viscoelastic ﬂuid ﬂow in a serpentine
microchannel in order to investigate the turbulent ﬂow characteristics and mixing performances
quantitatively. Polyacrylamide solution is used as viscoelastic ﬂuids and is compared with asucrose solution which is a Newtonian ﬂuid. The inﬂuence of the curvature and streamwise
position in the serpentine channel on the ﬂuctuation characteristics, and the two-dimensional
distributions are particularly discussed in this paper.
2. Experimental procedure and conditions
2.1. Experimental apparatus
The visualization and PIV measurement, which will be described shortly, were carried out using
a microchannel and invert-type microscope. The objective lens was a long working distance lenswith ×40 magniﬁcation and numerical aperture N A=0.75 (Olympus Co.; UPLFLN40XPH). The
images were recorded by a high speed digital video camera (Vision Research; Phantom V7.3).
The CCD resolution was 800×650 pixels, and the optical resolution of the image in combination
with the lens was 0.542µm/pixel. The focusing depth of the image was 1.7µm.
The schematic of the microchannel is shown in Fig. 1. The channel had two inlets which
join at the Y-shape junction. At the downstream of the junction was positioned the serpentine
channel where the measurement was carried out. The serpentine part of the channel consistsof 14 periodical circular curve units. The inner and outer radials Ri and Ro of the curve partwere 100 and 180µm. The channel width and height W and H were 84 and 60µm, respectively.
The microchannel was made of PDMS (poly-dimethylsiloxane) and was prepared by using SU-8
(MicroChem Co.) as a casting mold that was fabricated through a lithography process. The
PDMS channel removed from the SU-8 was attached to a cover glass. The working ﬂuids
were supplied to the inlet of the microchannel by syringe (Hamilton) and syringe pump (Nihon
Kohden Co.: CFV-3200) through a Teﬂon tube.
Two types of working ﬂuids were used in this study. The ﬁrst one was a 64.4wt% sucrose water
solution. The sucrose solution was prepared by adding sucrose (nacalai tesque Co.) to ultrapure
water puriﬁed by Direct-Q UV (Millipore Co.), and was mixed by rotating the container(As One
Co.: AV-1) at 250rmp for 12hours. This ﬂuid was regarded as a Newtonian ﬂuid and is referred
to as sucrose solution hereafter. The other one was a water solution of 64.4wt% sucrose,
1wt% NaCl, 100ppm Polyacrylamide (Polysciences Inc.: Polyacrylamide 18522-100, molecularweight=18,000,000). The polymer solution was mixed under the same condition of the sucrose
solution. This ﬂuid is regarded as a viscoelastic ﬂuid and is referred to as PAAm solution
hereafter.
2.2. Visualization measurement
Visualization measurement was carried out by mixing ﬂuorescent dye in the working ﬂuid andmeasuring the ﬂuorescent intensity distributions. 3wt% RhodamineB (Kishida chemical Co.:
E27036) was mixed with the solution and was used as ﬂuorescent dye. The ﬂuorescent solution
was supplied from one of the inlet, and a solution without ﬂuorescent mixture was supplied from

Table 1. Measurement conditions.
Q [µl/min]

Um [m/s]ReWi

0.08∼1.6

2.7 × 10−4 ∼ 5.3 × 10−3

1.8 × 10−4 ∼ 3.6 × 10−3

17.6∼352

the other side. Mercury lamp (Olympus Co.: U-LH100HGAPO) was used as the light source
to produce the excitation light that was guided to the microchannel through a band-pass ﬁlter(Olympus Co.: 25BP530), dichroic mirror (Olympus Co. BA575IF), and the objective lens. The
ﬂuorescence images of the dye was recorded by the high speed camera through another band
pass ﬁlter (Olympus Co.: BA575IF) in order to ﬁlter out the excitation and background lights.
The sampling rate and exposure time of the camera were set as 100Hz and 70µs, respectively.
2.3. PIV measurement
Fluorescent micro-particles (Molecular Probes Co.: F8819, nominal diameter= 1.0µm) weremixed in the ﬂuid as tracer particles in the PIV measurement. The same optical system employedin the visualization measurement was used in the PIV measurement. In order to measure a pair ofimages with appropriate time diﬀerence ∆t that is required for the PIV analysis, the high speed
camera was triggered by pulse signals generated by a function generator (NF Co.: WF1974).
Further, the sample rate of these pair images fs was appropriately selected to capture the low
frequency ﬂuctuation of the ﬂow ﬁeld. ∆t, fs and the sample number of pair images were set
as, 250µs, 100Hz and 1250, respectively.
2.4. Experimental conditions
The experimental conditions for the visualization and PIV measurements are summarized inTable 1. The ﬂow rate was changed in the visualization measurement, so that the Reynolds
number Re = Um Dh /ν varied in the range of 1.8 × 10−4 ∼ 3.6 × 10−3 . Um and Dh are the crosssectional mean velocity and the hydraulic diameter. The Weissenberg number, W i = 4λUm /Dh ,in this case ranges from 17.6 to 352. In the PIV measurement, on the other hand, the condition
of Re = 2.2 × 10−3 only was considered following the results obtained by the visualization.
3. Results and Discussion
3.1. Visualization results
To provide some information on the visualization pattern recorded in the measurement,
instantaneous photographs of the ﬂuorescence distributions measured at the streamwise locationsof curve N = 2, 7 and 13 in the case of PAAm solution is shown in Fig. 1. In the sucrose case,the boundary between the ﬂuid with and without ﬂuorescent dye formed along the streamwise
direction at the middle of the channel was clearly observed at the N =2 curve. This boundary
continuously remained in the downstream area to the N = 13 curve. On the other hand, in thecase of PAAm solution shown in Fig. 1, mixing of the two ﬂuids begin at the N = 2 curve where
the movement of the area with ﬂuorescence in the spanwise direction is observed. In the area
further downstream, the two ﬂuid will create multiple layers, and the ﬂuorescence distribution
becomes uniform presenting the enhancement in the ﬂuid mixing. Such tendency agrees withthe results of Burghelea et al. (2004).
The mixing enhancement observed in the case of PAAm solution is attributed to the unsteadyﬂow that is produced only in the PAAm case and not in the sucrose case. During the visualization
carried out at diﬀerent streamwise positions, several types of behaviors in the ﬂuid mixing wereobserved. One of them is the ”ﬂapping” motion that is observed mainly in the upstream area ofthe serpentine channel where the ﬂow begins to be unsteady. Figure 2 shows the ﬂapping motions

Figure 2. Photographs of the instantaneous ﬂuorescence distributions at the N =2 curve.
1.0

1.0

2.0

0.8

1.0

0.4

0.2

(r-R i)/W(r-R i)/W

0.8

0.6

0.8

0.6

0.6

0.0
2

4

6

8 10 12 14 16 18 20 [Iy/Im]

case PAAm
case Sucrose

0.4

0.4

0.2

0.0
0

1.0

0.2

0.0

0.0
0

2

4

6

0

8 10 12 14 16 18 20

1

t [s]

t [s]

(a) sucrose solution(b) PAAm solution

2

3

Re×10

4
3

(c) mixing performance

Figure 3. Time variation of the ﬂuorescence distributions, and the mixing index φ distributions
in relation to Reynolds number Re.
presented by the instantaneous ﬂuorescence distributions in the time period of 0 ≤ t ≤ 700ms
at the N = 2 curve. At t = 0ms shown in Fig. 2 (a), a clear boundary of the ﬂuorescence similar
to the one observed in the sucrose case is formed. As the time elapse, the area with ﬂuorescence
expands outwards in the radial direction, and then move inwards returning to the ﬁrst position(t=0ms). This motion in the radial direction was observed periodically, and is attributed to theradial ﬂow generated by the viscoelastic ﬂuid which will be discussed shortly.
To evaluate the ﬂuctuating characteristics of the ﬂuorescence distribution in the sucrose and
PAAm cases, the time variation of the ﬂuorescence intensity Iy distributions measured in thechannel spanwise direction is shown in Figs. 3(a) and (b). The values are measured at the cornerof the N = 13 curve, i.e. θ = 90◦ . Iy is normalized by the spatial average value Im .
The Iy /Im distribution in the sucrose case shown in Fig. 3(a) remains nearly constant against
time t, and the areas of larger Iy and smaller Iy are clearly divided at the center of the channel.
This shows that the mixing at the boundary is mainly attributed to the molecular diﬀusion, andthere is no secondary ﬂow that is strong enough to enhance the mixing. On the other hand,
Iy /Im distributions in the PAAm case shown in Fig. 3(b) varies as the time t passes presenting ahighly unsteady behavior of the ﬂuorescence pattern. Further, Iy /Im is nearly unity across thechannel width and also the time axis, which shows that the ﬂuorescence distribution is uniform
and is well mixed.
In order to discuss the inﬂuence of the ﬂow Reynolds number Re on the time mean mixing
performance of PAAm and sucrose cases at N = 13, the relationship between the Re and themixing index φ is shown in Fig. 3(c). φ presents the deviation of the ﬂuorescence intensity Iy
from the average value Im and is deﬁned as Eq. (1). σ is the standard deviation of I/Im .
√

∑m
φ=1−

j=1 σj

m

where

σj =

∑n

t=1 (Ii,j /Im,j − 1)

n

2

(1)

At the smallest Re considered in this study, i.e. Re = 1.8 × 10−4 , φ of the PAAm cases

9instantaneous velocity , umean velocity , u76mean velocity , ur

1
0
-1

5
4

instantaneous velocity , ur2u r [mm/s]

u [mm/s]

8

-2

0

4

8

t [s]

12

(a) streamwise velocity (case sucrose at N =2)

0

4

8

t [s]

12

(b) spanwise velocity (case sucrose at N =2)28u r [mm/s]

u [mm/s]

9
7
6
5
4

1
0
-1
-2

0

4

8

t [s]

12

(c) streamwise velocity (case PAAm at N =2)

0

4

8

t [s]

12

(d) spanwise velocity (case PAAm at N =2)

Figure 4. Time distributions of the streamwise and spanwise velocities in the cases of sucrose
and PAAm at the N = 2 curve.
approaches to the value of the sucrose case. In the measurement, the ﬂuorescence distributionsof the two cases were very similar, namely, the clear boundary of the ﬂuorescence remained
steady. As Re increases φ increases markedly, and becomes 0.82 at Re = 6.7 × 10−4 . Above thisRe, φ then remains constant around 0.82. It is expected that the ﬂow transits from steady tounsteady state in the vicinity of this Re.
In the sucrose solution case, φ becomes maximum at the minimum Re considered in thisstudy, Re = 1.8 × 10−4 . φ then decreases as Re increases that is an opposite tendency to the
PAAm case. During the measurement although the Re was varied, the boundary position of theﬂuorescence remained steady and no ﬂuctuation was observed in the visualization measurement.
The reason why φ decreases with Re is attributed to the decrease in the Peclet number, thatindicates the inﬂuence of the molecular diﬀusion during the ﬂow ﬂows along the serpentine
channel decreases as the ﬂow velocity decreases. Comparing the results of sucrose solutions with
PAAm case, the enhancement in the mixing performance is obvious. At Re = 3.6 × 10−3 , φ in
the PAAm case shows 8 times greater value compared with the sucrose case. On the contrary,for smaller Re both values are expected to continuously approach and match with each other.
3.2. Flow velocity measurements
3.2.1. Point measurement and analysis To ﬁrst evaluate the ﬂuctuation characteristic of theunsteady ﬂow in the serpentine channel, the velocity variation at a point will be discussed.
Figure 4 shows the instantaneous velocity u distributions in the cases of sucrose and PAAm
solutions. The measurement position is the channel center at the corner ((r − Ri )/W = 0.5 andθ = 90◦ ) of the N = 2 curve. The subscripts θ and r for each value present the tangential andradial velocity components in the curved area. The Reynolds number is Re = 2.2 × 10−3 that isthe condition at which the ﬂow is steady in the sucrose case and unsteady in the PAAm case,respectively.
First, the distributions in the sucrose case shown in Figs. 4(a) and (b) contain high frequency
ﬂuctuations for both uθ and ur values. The amplitude and frequency are nearly identical in thetwo cases showing that ﬂuctuation is isotropic. As previously mentioned, the ﬂow is steady in

360

10

300

2

2

P cross [mm /s ]

100

240

[ ]

1
0.1

120

0.01

60

1E-3

0

0.1

1

10

0.01

f [Hz](a) power (case sucrose at N =2)
10

300

[ ]

360

1

0.1

1

10

f [Hz](b) phase (case sucrose at N =2)

100

2

2

Pcross [mm /s ]

180

240
180

0.1

120

0.01

60

1E-3

0

0.1

1

10

f [Hz](c) power (case PAAm at N =2)

0.01

0.1

1

10

f [Hz](d) phase (case PAAm at N =2)

Figure 5. Cross spectrum in the cases of sucrose and PAAm at the N = 2 curve.
the sucrose solution case, and the ﬂuctuation is attributed to other reasons. It is, actually, due
to the Brownian motion of the tracer particles that becomes noticeable for the measurement in
the microchannel owing to the scale eﬀect.
In the PAAm case shown in Figs. 4(c) and (d), a ﬂuctuation with frequency and amplitude
equivalent to those obtained in the sucrose case is observed for uθ and ur distributions. Therefore,
this is considered to be attributed mainly to the Brownian motion of the particles. On the other
hand, additional ﬂuctuation of relatively low frequency and large amplitude is observed in the
distribution. This ﬂuctuation presents the unsteady behavior of the ﬂow that leads to theenhancement in the ﬂuid mixing.
The results of the cross spectral analysis carried out for the previously discussed instantaneous
data is shown in Fig. 5. Comparing the results in the cases of sucrose and PAAm solutions, nopeak is observed in the sucrose case and the distribution is ﬂat. In the case of PAAm solution,
however, a peak is clearly observed at the frequency of approximately 0.4Hz, and the ﬂuctuationsof uθ and ur are in a coherent state. The two results diﬀer in the area of a frequency smaller
than 5Hz. Above this frequency, the spectrum shows identical distribution indicating that thehigh frequency random motion due to the Brownian motion are nearly the same in the sucrose
and PAAm cases. Therefore, the low frequency ﬂuctuation is the major diﬀerence in the twocases.
As shown in Fig. 5(b), the scattering data is uniformly distributed and there is no correlation
between ϕ and the frequency f in the sucrose case. On the other hand, in the PAAm case shownin Fig. (d), the scattering distributions are concentrated in the area adjacent to ϕ = 0 and 360◦
at the low frequency range showing that ﬂuctuating of uθ and ur are in phase.
3.3. Cross-sectional distributions
Figure 6 shows the cross-sectional distributions of the streamwise and spanwise componentsof the time mean velocities and the ﬂuctuation components ũθ and ũr measured at thecorner(θ = 90◦ ) of the N = 2 curve. The values with ˜ are the average data of the adjacentpoints for the time period of 0.2s. The ﬂuctuation due to the Brownian motions are considered
to be subtracted by this process and the values present the low frequency ﬂuctuations only.

1.0

PAAm

2

Sucrose

m

0.0

/

1

PAAm

0.5

ur U

/

u U

m

Sucrose

-0.5
-1.0
0
0.0

0.2

0.4

0.6

0.8

(r-Ri)/W

0.0

1.0

PAAm

m

0.3

0.8

1.0

PAAm
Sucroseur U

mSucroseU/

0.6(b) spanwise velocity

0.2

~ 2

/

0.2

u

~ 2

0.4

(r-Ri)/W(a) streamwise velocity
0.3

0.2

0.1

0.1

0.0
0.0

0.0
0.2

0.4

0.6

0.8

1.0

(r-Ri )/W(c) streamwise ﬂuctuation intensity

0.0

0.2

0.4

0.6

0.8

1.0

(r-Ri)/W(d) spanwise ﬂuctuation intensity

Figure 6. Mean velocities and ﬂuctuation intensities in the cases of sucrose and PAAm at theN = 2 curve.
First, focusing on the ũ distributions in the case of sucrose solution, ũθ /Um and ũr /Um areboth 0 at all spanwise locations showing that no ﬂow ﬂuctuation exists except the Brownian
motion in this case. In the PAAm case, ũθ /Um and ũr /Um increase markedly taking a maximum
values of 0.26 and 0.1, respectively. Further, the fact that the values of ũθ are larger comparedwith ũr shows that the dominant ﬂuctuation appears in the streamwise direction rather than in
the spanwise direction. This is can be attributed to the similar reason explained by Larson etal. (1990) for viscoelastic ﬂuids in Taylor-Couette ﬂows.
These results inﬂuence the time mean velocity distributions shown in Figs. 6(a) and (b).
The inﬂuence is particularly large in the streamwise velocity distribution where the distribution
becomes more uniform and deviates in the spanwise direction. This is considered to be dueto the large spanwise ﬂow ﬂuctuation that results in a greater ﬂuid transport in the spanwise
direction.
Next, comparison is made with the results of the N = 2 and 13 curves in order to consider
the inﬂuences of the streamwise locations in the serpentine channel. Figure 7 shows the crossspectrum in the N = 13 case of Re = 2.2 × 10−3 . Figure 8 shows the u and ũ distributions.
In the N = 13 case, the power increases as the frequency decreases and shows a relatively ﬂat
distribution in the range of 0.1 ∼ 0.5Hz. However, the apparent peak observed at f =0.4Hz in
N = 2 case cannot be found in N = 13 case. Further, the power itself shows a smaller value in
N = 13 case than in N = 2 case showing that the ﬂuctuation amplitude decreases.
In the ϕ distribution, the scattering distribution in the lower frequency range shows a greater
dispersion in the N = 13 case indicating that the coherence is slightly smaller than in the N = 2
case. This tendency can be observed also in Figure 8. Although the ũθ and ũr are much larger
than the sucrose case, both values decreases compared with N = 2 case. Further, the diﬀerence
between ũθ and ũr in N = 13 case becomes smaller. The decrease of the ﬂuctuation amplitudes
is considered to inﬂuence the time mean distribution also. As shown in Figures 8(a) and (b), notmuch diﬀerence is observed in the u distributions compared with the sucrose case. Although notshown here, the ﬂuorescence images taken by the high speed camera indicated that ﬂapping
motion observed at the N = 2 curve does not exists at N = 13. This will be discussed in the

360

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240

[ ]

1

180

0.1

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0.01

60

1E-3

0

0.1

1

10

0.01

f [Hz](a) power (case sucrose at N =13)
100
10

300

2
2

1

10

f [Hz](b) phase (case sucrose at N =13)360

P cross [mm /s ]

0.1

240

[ ]

1

180

0.1

120

0.01

60

1E-3

0

0.1

1

10

0.01

f [Hz](c) power (case PAAm at N =13)

0.1

1

10

f [Hz](d) phase (case PAAm at N =13)

Figure 7. Cross spectrum in the cases of sucrose and PAAm at the N = 13 curve.
1.0

PAAm

m

0.0

/

1

Sucrose

0.5

ur U

/

u U

m

2

-0.5

PAAm
Sucrose

-1.0
0
0.0

0.2

0.4

0.6

0.8

0.0

1.0

(r-Ri)/W

m

m

0.6

0.8

1.0

(b) spanwise velocity

PAAm

PAAm

0.3

Sucrose

/

/

ur U

SucroseU

0.4(r-Ri)/W(a) streamwise velocity
0.3

0.2

~2

~

u

2

0.2

0.1

0.1

0.0
0.0

0.2

0.2

0.4

0.6

0.8

1.0

0.0
0.0

0.2

0.4

0.6

(r-Ri )/W(c) streamwise ﬂuctuation intensity

0.8

1.0

(r-Ri)/W(d) spanwise ﬂuctuation intensity

Figure 8. Mean velocities and ﬂuctuation intensities in the cases of sucrose and PAAm at theN = 13 curve.
following section in which the two-dimensional ﬂow pattern is discussed.
3.3.1. Two-dimensional velocity distribution Figure 9 shows the contour distributions of the
ﬂuctuation components of the streamwise velocity ũθ at the location of N = 2 curve. The
velocity vectors are superposed on the contour map. The ﬂow ﬁeld in the PAAm case includes low
frequency ﬂuctuation and the high frequency random ﬂuctuation due to the Brownian motion.
The low frequency ﬂuctuation shows a power peak at the frequency of 0.4Hz, however, is a
mixture of several waves, and actually consists of various vortices and ﬂuctuations. Indeed, inthe two-dimensional velocity distributions measured in this study, several types of ﬂow ﬁelds
and three-dimensional vortices were observed. Among these, those which represent the typicalﬂow pattern at the period when the velocity increases, decreases, or is equal to the time mean
velocity distribution are shown in Fig. 9.
In Fig. 9, one can see that the ﬂuctuations in the ﬂow ﬁeld is not randomly distributed but
shows some tendency. In Fig. (a), an area with positive ũθ appears and covers a certain areaof the channel. In Fig. (b), a wide area with negative ũθ is observed similarly. It should be
noted that sometimes such area covered over the area of one curve unit. Therefore, additional
experiment was carried out in the straight part located upstream of the serpentine channel to
conﬁrm that no oscillating ﬂow was generated. No such ﬂow was observed, thus it is believedthat a large scale three-dimensional vortex is generated in the channel. The area size or thevelocity magnitude varied by time and location, and are not completely periodic or regulated.
However, such ﬂow ﬁeld is considered to be the main cause for the ﬂow ﬂuctuation observed in
the previous ﬁgures.
These kinds of ﬂow ﬁelds appear in the channel at the frequency of 0.1 ∼ 0.5Hz. Thisfrequency corresponds to those of the peak observed in the cross spectrum and the ﬂapping
motion observed in the visualization measurement. Furthermore, considering the mean
streamwise velocity and the area length of this increased/decreased velocity regions, the time
period in which this region occupies a curve was approximately 150ms. Since the spanwise
velocity is in the order of 0.5mm/s, the ﬂuid will ﬂuctuation in the spanwise length of 75µm.
This corresponds to the channel width and also the variation length of the ﬂapping motion
observed in the visualization.
Paying attention on the velocity vectors in the ﬁgures, the vectors are directed outwards ofthe channel curve when ũθ is positive (increased velocity ﬁeld) and are directed inwards whenũθ is negative (decreased velocity ﬁeld). This is considered to be attributed to the following
reason. As often the case in viscoelastic ﬂuids, the polymers (or the solution) is stretched in
the streamwise direction due to the shear stress produced by the ﬂow. This provides normal
stresses in the spanwise and streamwise direction. Considering then that the streamlines are
curved along the serpentine channel, the normal stress particularly produced in the streamwise
direction will work in the tangential direction. If a time lag exists in these forces, the forces andthe resulting velocity ﬂuctuations will appear with a certain angle to the streamwise direction
in combination with the stress in the spanwise direction.
From the other measurement carried out in this study, it is known that these areas were mainly
generated in the N = 1 and 2 curves. In the downstream area, however, the ﬂuctuations are a
mixture of those generated at each streamwise location and those provided from the upstream.
The vortices from the upstream will enhance the mixing performance in the curved channel
located further downstream of the serpentine channel, however, is expected to also disturb thegeneration of low frequency ﬂuctuation observed in the N = 1 and 2 curves. This is considered
to be the main reason why the ﬂuctuation decreases and becomes isothermal in the case of
N = 13 compared with N = 2.
Where and how such large scale ﬂuctuation is precisely generated still remains as a question
to be solved. Further measurement and discussion is required to evaluate the vortex structure
observed at the curves of the channel. Nevertheless, the present measurement has quantitatively
conﬁrmed the velocity ﬁelds and provided some insights for understanding the mechanism of theunsteady ﬂows in serpentine channel.
Conclusions
Flow characteristics of the viscoelastic ﬂuid ﬂow in the serpentine microchannel were measured
and described quantitatively in this study. The ﬂuorescence visualization measurement showedthat the mixing performance markedly increased above the Reynolds number of 6.7 × 10−4 atu’=u-uθθ θ[mm/ s]
3 mm/ s

2

1.2

100

0.8

150

0.4

3 mm/ s50

1.6y [ µ m]y [ µ m]50

u’=u-uθθ θ[mm/ s]
1.6
1.2

100

0.8

150

0.4

0

200

0

200

-0.4
-0.8

250

-0.4
-0.8

250

-1.2

300

-1.2

300

-1.6

100

200

x [µ m]

300

400

-1.6

100

-2

(a) increased velocity region

200

x [µ m]300

400

3 mm/ s

2

1.2
0.8

150

0.4

3 mm/ s50y [ µ m]y [ µ m]u’=u-uθθ θ[mm/ s]

1.6

100

-0.4
-0.8

250

1.2
0.8

150

0.4
0

200

-0.4
-0.8

250

-1.2

300

-1.6

100

200

x [µ m]300(c) transition state 1

400

-2

2
1.6

100

0

200

-2

(b) decreased velocity regionu’=u-uθθ θ[mm/ s]

50

2

-1.2

300

-1.6

100

200

x [µ m]

300

400

-2

(d) transition state 2

Figure 9. Typical two-dimensional ﬂow patterns presenting the ﬂapping motion at the N = 2
curve in the PAAm solution case.
which transition from steady to unsteady ﬂow occurred in the polyacrylamide (PAAm) solution
case. The mixing performance enhanced signiﬁcantly compared with the sucrose solution case.
The velocity measurement showed that low frequency ﬂow ﬂuctuation in the spanwise directionof approximately 0.4Hz was generated in the upstream area in the PAAm case. In this area,
relatively large areas of high and low velocities were appeared periodically in the curved area.
On the other hand, such large scale ﬂuctuation was hardly observed in the downstream area ofthe channel.
Acknowledgments
This work was ﬁnancially supported by the Ministry of Education, Culture, Sports and Scienceand Technology of Japan.
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