TURBULENCE AND HEAT TRANSFER CHARACTERISTICS OF
LOW REYNOLDS NUMBER VISCOELASTIC FLUID FLOW IN
A SERPENTINE CHANNELKAZUYA TATSUMI
Department of Mechanical Engineering and Science, Kyoto University, andAdvanced Research Institute of Fluid Science and Engineering, Kyoto University
Japantatsumi@me.kyoto-u.ac.jp
CHEE LEONG HEONG
Department of Mechanical Engineering and Science, Kyoto University
Japan
KAZUHIKO SUGA
Department of Mechanical Engineering, Osaka Prefecture University
Japan
suga@me.osakafu-u.ac.jp
KAZUYOSHI NAKABE
Department of Mechanical Engineering and Science, Kyoto University, andAdvanced Research Institute of Fluid Science and Engineering, Kyoto University
Japannakabe@me.kyoto-u.ac.jp
Abstract
This paper describes the study on the flow pattern and heat transfer characteristic ofviscoelastic fluid in a serpentine channel under low Reynolds number condition. The
Reynolds number was varied in the range of 0.1< Re < 5 that was found in this study to cover
the regime from steady laminar flow to unsteady elastic turbulent flow. Average heat transfer
coefficients of the wall were measured under isothermal wall condition. And pressure loss
and flow visualization measurements were also carried out. Two types of fluids were used asworking fluids: one was 64.4wt% sucrose solution that was a Newtonian fluid, and the otherwas a 64.4wt% sucrose solution with which 500ppm polyacrylamide was mixed that was aviscoelastic fluid. In the visualization measurement, strong flow fluctuations were observed in
the serpentine channel as Re increased in the case of the viscoelastic fluid, while the flow
remained steady in the case of sucrose solution. An effective heat transfer enhancementcompared with the sucrose solution case was observed as Re increased due to this flow
fluctuation. An increase of the pressure loss was accompanied, too. However, a reasonableenhancement was obtained considering the total performances.
INTRODUCTION
Enhancement technologies for convection heat transfer of laminar flows in channels havebeen keenly studied in the past half century. Various techniques have been developed inresult, as typified by those such as using fins and corrugated channels to increase the heattransfer area and generating secondary flows, or applying jet flows, heat pipes, porous media
and microchannels.

An alternative solution to the above geometrical and mechanical improvements of thechannel is changing the properties of the operating fluid. The so-called "Nano-fluids" is a
fluid in which nano-meter size particles are mixed in water or other solvents (Das et al.,2007). Although the physical reason is not fully understood, it is known that the heat transfer
coefficient in channels can be increased compared with water in this case. Another working
fluid that is of significant interest is viscoelastic fluids, for example polyacrylamide water
solutions. Such fluid possesses not only viscous property but also elastic property. Providing
high shear stress to the fluid by the shear flow will cause the polymers solved in the fluid tocontract, expand and rotate leading to an appearance of the elasticity that produces additional
normal stresses in the flow. Further, in combination with a certain flow structure, theelasticity of the fluid and the accompanied normal stress will increase the flow instability.
The increase of the flow instability evokes unsteady flow with large fluctuation in the channeleven in very low Reynolds number regime. Such unsteady flow is believed to possess a highpotential to enhance the fluid mixing and wall heat transfer.
The influences of the rheological properties of the viscoelastic fluid on laminar heat transfer
have been keenly studied in the past several decades, particularly in the case of straightchannel. Hartnett et al. (1985) and Chunbo & Hartnet (1992) have reported from theirexperiment that the heat transfer coefficient in the case of viscoelastic fluid flow was higher
than the Newtonian fluid flow case. The main cause was believed to be attributed to thesecondary flow which was generated and enhanced by the normal force difference. Naccacheet al. (1996) conducted a numerical simulation on the non-Newtonian laminar heat transfer ina rectangular duct and reported that the heat transfer was strongly enhanced due to thesecondary flow. Norouzi et al. (2009) have numerically investigated the heat transfer
characteristics of viscoelastic flow in a curved duct. The results showed that the first normal
stress differences encourage the secondary flow generation and enhance the heat transfer
while the second normal stress differences suppress heat transfer efficiency.
Flow instability of viscoelastic fluid flow has also been studied with strong interest. Larson etal. (1990) studied the flow instability of Taylor-Cuette flows in rotating coaxial cylinders. Joo& Shaqfeh (1994) investigated the mechanism of the increase in the flow instability of
Taylor-Dean flows in coaxial cylinder.
Recently, several studies on the mixing characteristics of viscoelastic fluid in a serpentinecurved channel under low Reynolds number condition have been published in the literatures.
Groisman & Steinberg (2001) carried out flow visualization and point measurement of theflow velocities using laser Doppler anemometer. Flow fluctuation was observed even in theorder of Re ~ 10, accompanying the enhancement of the fluid mixing. Teodor et al (2004) andFeng et al. (2010) conducted a visualization experiment in serpentine microchannels using
fluorescent dye. Even in this scale, production of the unsteady flow and enhancement in thefluid mixing was observed. Tamano et al. (2009) made PIV measurements in curved channels
with circular cross-section and found a pair of vortices similar to the Taylor vortices in thetime-mean velocity fields. All these results indicate the possibility of effectively enhancing
the mixing performance or moreover mass and heat transfer at the channel wall. To theknowledge of the authors, however, there are no experiments conducted to link such unsteadyflow and the fluid mixing with the heat/mass transfer performance. It is essential tounderstand the heat transfer characteristic of viscoelastic fluid flows in serpentine channels inorder to provide an insight for the feasibility of using the viscoelastic fluids as working fluids,
and to aid the designing of an effective channel shape and conditions in order to apply thistechnique to the fields of thermal engineering, chemical engineering and medicine.

(a) Apparatus(b) Channel dimensions
Figure 1: Experimental apparatus used for the heat transfer measurement.
From this background, measurement on heat transfer coefficient of viscoelastic fluid flow in a
serpentine channel with a square cross-section is carried out under isothermal wall conditionin this study. The pressure loss in the channel is measured together in order to evaluate therelationship between the heat transfer enhancement and pressure loss penalty. Further,visualization experiments are carried out to measure the flow behavior and discuss the mixing
characteristic in the channel. Two types of working fluids are investigated. One is aviscoelastic fluid that is a sucrose water solution with which polyacrylamide of 500ppm ismixed. And the other is a sucrose water solution that is a Newtonian fluid used in order tomake comparison with the viscoelastic fluid case.
EXPERIMETNAL METHOD
Experimental apparatus
Figure 1 shows the schematic diagram of the experimental apparatus applied to the heattransfer measurement. The channel consists of the three parts: the inlet and outlet channel
parts, and the test section, the channel walls of which are heated under isothermal condition.
The working fluid was supplied to the channel by a pump driven by pressurized air. The mass
flow rate was measured by measuring the mass of the fluid flowing out from the outlet of thechannel using an electronic weighing machine. The average values was used to calculate theflow rate, the accuracy of which was 0.6ml/min.
For the channel, the inlet and outlet units were made of acrylic, to which a straight channel
was embedded. The test section was made of copper in order to produce an isothermal
condition to the wall. As shown in Fig. 1 (a), cylindrical channels were transversely drilled to
the wall to which constant temperature water was supplied from the thermostat bath to keep
the wall temperature of the channel constant.
Two types of channel, the schematic and dimensions of which are shown in Fig. 1 (b), were
investigated in this study. Both channels had a square cross-section of 5mm in width andheight. The serpentine channel had a periodical shape, which consisted of 10 repeating unitsof circular curved parts. The inner and outer radii were 5 and 10mm, respectively. The lengthof the straight channel was L = 0.2m, while the length of the serpentine channel measured
along the curvature was L = 0.471m.

Table 1: Flow conditions in the experiment.
Fluid
Channel shape
Q [ml/min]SucroseSerpentine
25.3 − 138
PAAm
Serpentine
20.5 − 148
PAAm
Straight
24.6 − 70.0

Re
1.01 − 4.29
0.33 − 1.92
0.28 − 0.88

De
0.59 − 2.48
0.19 − 1.11−Wi−
7.0 − 50
8.0 − 24

The inlet and outlet parts were attached to the test section and sealed with silicon sealant.
5mm holes were drilled through the top plate at the end of the channels. Tubes wereconnected to these holes supplying fluids to the channel from the pump, and leading theoutlet fluid to a container to measure the mass flow rate.
Heat transfer measurement
In order to measure the fluid temperature for bulk mean temperature calculations,temperature measurement units were placed at the inlet and outlet of the cupper test section.
The fluid temperature at the inlet, Ti, was measured by a sheath K-type thermocouple of
0.5mm in diameter having its end point positioned at the center of the channel cross section.
Temperature at a single point only was measured since a uniform temperature distribution
was expected for the inlet flow. The fluid temperature at the outlet To was measured by 7 Ktype thermocouples located evenly in the height direction in the middle of the channel width.
Since the available temperature distributions obtained from this measurement are only in onedimensional form, expansion of the temperature distribution from one- to two-dimensional
directions is necessary. In this study, the fluid temperature was assumed to have a similar
distribution to the measured one at z = 0 in the y and z directions. The temperature of theheated cupper part was measured by K-type thermocouples, which were embedded in thecupper surface. The variation of the temperature was 0.5 °C affirming that a constant wall
temperature condition was obtained at the channel.
Streamwise velocity distributions of a square cross-section in a laminar flow case (Shah,1978) were used in the sucrose solution case. In the PAAm solution case also, we haveassumed that the velocity distribution is similar to the laminar flow of Newtonian fluid case
based on the discussion on the friction factor, which will be shown in the next section. The
temperatures and velocities obtained from these processes were employed to calculate thebulk mean temperatures.
Pressure loss measurement
To measure the pressure loss in the test section, electronic pressure gauges (Keyence Co.,
AP-C30), the measurement resolution of which was 0.1kPa, were connected to the inlet andoutlet of the test section through a 1mm hole drilled at the top wall. The static pressure ateach position was measured and used to calculate the fanning friction factor, f , defined as Eq.
(1) .
f =

2ΔP DhρU m 2 4 L(1)

Dh is the hydraulic diameter of the channel. Um is the mean velocity.
Flow Visualization
Flow visualization using dye is carried out in this study to understand the flow behavior in thechannel. Syringe pump (Nihon Konden Co., CFV-3200) was used to supply the ink though a

4mm inner diameter tube to a syringe needle, which was inserted to the channel through the
top wall. The images of the visualization was recorded using a digital video camera (Sony,
HDR-CX370V) measured from above the channel. The configurations of the test section are
nearly the same as those of the heat transfer experiment, except that the top wall was replacedby transparent acrylic plate. Therefore, the sidewalls and bottom walls only were heated in
the visualization measurement.
Working fluid preparation
Two kinds of working fluids were prepared for the experiment. One was a sucrose water
solution (sucrose 64.4wt%), which was a Newtonian fluid. The other was the polyacrylamide(PAAm) water solution (sucrose 64,4wt%, NaCl 1wt%, PAAm 500ppm), which was aviscoelastic fluid. The fluids were prepared mixing the solute to water using a pot rotating
mill. NaCl was mixed in the fluid to keep the polymer structure and fluid properties stable.
The fluid viscosity μ , density ρ and relaxation time γ were measured prior to the experiment.
μ measured by the rheometer were μ =1.0×105 and 1.0×105 for the sucrose and PAAm
solutions, respectively. The relaxation time was 2~3s which was calculated by appying the
Powell-Eyring (Hartnetl et al., 1985) model to the measured results using the rheometer. The
heat capacity Cp and thermal conductivity λ, were derived from the empirical equation shownin the references of Gucker and Ayers (1937) and Werner et al. (2007) for the fluid
temperature of 25°C.
Experimental conditions
The flow rate was change in the experiments, in order to examine the influence of the
Reynolds number, Re, on the heat transfer coefficient, pressure loss and flow behavior in thechannel. The conditions are summarized in Table 1. De shown in the table is the Dean
number, which is defined as Eq. (2).

ρU m Dh ⎛ 2W ⎞⎜⎟De =μ ⎜⎝ (Ri + Ro ) ⎟⎠

0.5

(2)

W is the channel width, and Ri and Ro are the inner and outer radii of the curved channel.
Dean number can be used to assess the magnitude of the curvature effect on the flow
structure, for example the generation of the Taylor-Dean vortices. Wi is the Weissenberg
number and is defined as follows.
⎛U ⎞Wi = γ ⎜⎜ m ⎟⎟⎝ Dh ⎠(3)

RESULTS AND DISCUSSION

Figure 2 shows the relationship between the average Nusselt number, Num, and the Reynolds
number, Re, for the sucrose and PAAm solutions. The analytical results for the Nusselt
number of laminar flow in a straight channel with square cross-section is depicted in the
figure for comparison. This analysis also considering the entrance region effect, and isderived by Chandrupatla & Sastri (1977).

20

20
PAAm case
Sucrose case[1]
Chandrupatla et al.

Serpentine (PAAm)
Straight (PAAm)

15

Nu

Nu

15

10

5

10

5

0.0

0.5

1.0

1.5

2.0

2.5

0

20

40

60

Re

80

100

Re

(a) Sucrose vs PAAm solutions(b) Serpentine vs straight
Figure 2: Average Nusselt number distributions: (a) comparison of the sucrose and
PAAm solutions in serpentine channel, and (b) comparison of the serpentine and straightchannels in the PAAm solution case.

100

10

f

f

100

10

PAAm case
Sucrose case
14.2/Re

PAAm case
16/Re

1

1
0.1

1

10

Re

0.1

1

10

Re*

(a) Re vs f(b) Re* vs f
Figure 3: Friction factor distributions: (a) relationship with Reynolds number, and (b)relationship with modified Reynolds number based on the power law.
The Nusselt number Num of the sucrose solution case slightly increases with Re due to thedevelopment of the thermal boundary layer at the inlet. Further, the result agrees well withthe analytical one showing the validation of the measurement. Therefore, the flow in theserpentine channel is flowing along the curved channel without accompanying any secondary
flows, which is consistent with the small value of the Dean number shown in Table 1.
In the PAAm solution case, Num increases markedly as Re increases, and shows aconsiderable heat transfer enhancement compared to the sucrose solution case. Num appears
to approach to the value of sucrose solution case as Re decreases. Therefore, it is expectedthat the increase of Num is related more to the production of secondary flow than the change
in the velocity distribution due to the non-Newtonian behavior such as shear thinning.
In order to see whether the heat transfer enhancement is achieved by the viscoelasticity of thefluid only, or if it is a combinational effect of the viscoelastic fluid and the curvature of theserpentine channel, heat transfer measurement was carried out for a straight channel using thesame PAAm solution. The results are shown in Fig. 2 (b). Since the channel length isdifferent between the two cases, Graetz number Gz, defined by Eq. (4), is taken as theabscissas axis in this case.

Gz = Re

Dh ρμ
L λ

(4)

In the figure, we can see that the Num does not increase in the straight channel case for the Re
conditions investigated in this study as it did in the serpentine channel case. It remains nearly
constant at a value close to Num ~ 3, which is the value of fully developed flow and thermalboundary layers in a square duct. This shows that the increase of heat transfer coefficient canbe obtained only by the combination of the viscoelastic fluid flow and the serpentine channel,
at least under the conditions of the Reynolds number considered in this study.
Figure 3 shows the relationship between the fanning friction factor, f, and the Re in both
solution. The solid line in the figure presents the value of 14.2/Re, which shows the friction
factor of a Newtonian fluid flow in a square duct. Although the results in the sucrose solution
case shows a relatively smaller value compared with the solid line, the scattering plots show areasonable agreement with the theoretical value. Since the Reynolds number and Dean
number are both very small, it is believed that no noticeable secondary flow is generated in
the channel. Therefore, same as in the case of Num shown in Fig. 2, the results indicate thatthe flow in the serpentine channel in the sucrose solution case can be considered to be alaminar flow in a straight duct with square cross-section.
On the other hand, f in the PAAm solution case shows a different tedemcy. In the lower Re
area, f shows a slightly smaller value than the solid line similar to the sucrose solution case.
However, a large increase of f is observed in the region of larger Re , in which f shows agreater value than 14.2/Re. The reason for this increase can be two-fold. One is the change inthe velocity distribution due to the viscoelastic properties of the fluid. In this case, thevelocity gradient increases in the area adjacent to the channel wall, and a more flat
distribution is obtained at the channel center leading to a larger friction at the channel wall.
The other reason is the generation of the flow fluctuation or secondary flow in the channel.
In order to clarify each influence in the friction factor shown in Fig. 3 (a), the relationshipbetween f and Re is replotted by using the modified Reynolds number, Re*. Re* is calculated
by Eq. (5).

Re =*

ρU m 2−n Dh n

8 n −1 K (b + a / n )n(5)This equation considers the influence of the viscoelasticity of the fluid on the velocity profile,and shows the relationship between the friction factor and the modified Reynolds number. In
this case, the relationship between f and Re* can be shown as f = 16/Re* for viscoelastic fluid
flows. The parameters a and b appearing in Eq. (5) are used to describe the effect of thecross-sectional shape of the channel. As a square cross-section, a = 0.21 and b = 0.68 are
employed in this study. K and n are calculated from the logarithm law described in thereferences of Hartnetl et al. (1985) and Kozicki et al. (1966). The results are shown in Fig. 3(b).
In Fig. 3 (b), the present results decreases and separatures further from the solid line
(theoretical value) in the regions of small Re*. Then, f increases, crosses the line, anddeviates from the solid line in the positive direction as Re* increases. Therefore, under thesmall Re* conditions, the flow is believed to be steady and also having a velocity profile

(b) Sucrose solution case (Re = 2.49, De = 1.44)(b) PAAm solution case (Re = 0.369, De = 0.213 )(c) PAAm solution case (Re = 1.64, De = 0.945)
Figure 4: Flow visualization using dye in the cases of (a) sucrose solution at steady state,(b) PAAm solution at steady state, and (c) PAAm solution at unsteady state.

similar to a Newtonian fully developed laminar flow in a square duct. On the other hand, as
Re* increases, the flow becomes unsteady. And, thus, additional increase of the friction factor
is obtained due to the generations of the flow fluctuation and secondary flow. It should bemarked that the transition region in which the Num and f start to increase matches with eachother reasonably well showing a change in the flow structure is occurring.
Finally, the visualization results are shown in Fig. 4. The figures show the snapshots taken atseveral streamwise positions in the cases of sucrose and PAAm solutions. In the PAAm
solution case, the results of Re = 0.369 and 1.65, lower and higher Re conditions, are shownto discuss the influence of the Re on the flow characteristics.
In Fig. 4 (a), the streamlines of the ink flows along the curvature of the serpentine channel.
This pattern remained steady throughout the movie showing that a steady flow without any
noticeable secondary flow is formed in the channel. In Fig. 4 (b) showing the PAAm solution
case of Re = 0.369, a similar pattern is observed. Although some disturbance in thestreamlines of the dye was observed instantaneously in other periods of the movie, the flow
could be considered to be in steady state.
On the other hand, in Fig. 4 (c), one can see that the ink patterns are greatly disturbed in all
streamwise positions, showing that unsteady flow with flow fluctuation is produced in thechannel. In the movie, the fluctuation of the flow shedding downstream of the channel was
clearly observed. It is clear now that this flow fluctuation causes the heat transfer
enhancement and increase of the pressure loss in the serpentine channel shown in theprevious figures. It is difficult to estimate the exact flow pattern from the visualization results
since the flow appeared to have a three-dimensional structure and the camera time resultion
was not high enough to follow the unsteady flow behaviour in detail. We believe that,
however, in the early stages of the fluid viscoelasticity influencing the flow, a deviation of
the spanwise velocity distribution is produced at the curved part of the channel due to theradial flow heading outwards. This generation of radial flow should be based on a similarreason described in the instability studies of Joo and Shaqfeh (1994). That is, the hoop-stressproduced by the polymer motion increases the flow instability leading the generate the
Taylor-Dean voctices or a similar secondary flow. Once such asymmetric flow or vortices are
generated, then the flow fluctuation could be enhanced at the inflection point of theserpentince channel due to the change of the flow directions.
CONCLUSION
Heat transfer and pressure loss measurements, and flow visualization were carried out for
viscoelastic fluid flow in serpentine channels. The conclusions obtained from the experiments
are summarized as follows:

1. Enhancement of heat transfer performance was observed in the PAAm solution(viscoelastic fluid) case compared with the sucrose solution (Newtonian fluid) case. In
lower Reynolds number region (Re ~ 0.5), the average Nusselt number in the PAAM
case showed a similar value with the sucrose case, and then increased markedly as the Re
increased implying that a transition of the flow from steady to unsteady state occurred.
Further, average Nusselt number Num in the case of the PAAm solution in a straightchannel did not show a large difference with the sucrose case indicating that theenhancement effect could be obtained only by the combination of the channel curvatureand the fluid viscoelasticity.
2. Pressure loss increased markedly as the Re increased. By discussing the relationshipbetween Re and the friction factor f, using the modified Re based on the power law fluid,
showed that the increase of the pressure loss is mainly attributed to the flow fluctuationand not to the power law velocity distributions.
3. The tendency of the flow behavior shown by the visualization results corresponded wellwith heat transfer and pressure loss measurements mentioned above. A large scale vortex
shedding and a deviation of the streamline pattern were observed, which appeared tohave a strong three-dimensional structure.
ACKNOWLEDGEMENTS

This study was supported by the ministry of education, culture, sports, science andtechnology of Japan, and the authors acknowledge their financial support. We would also like

to send our sincerely thanks to Prof. H. Suzuki of Kobe University for his advice on theviscoelastic fluid flow, and providing their facility to measure the fluid properties.
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