Determination of velocity profiles of Bird-Carreau fluids in curvilinear microchannels using random sample consensus
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Korea-Australia Rheology Journal, 32(2), 159-164 (May 2020) www.springer.com/13367
DOI: 10.1007/s13367-020-0015-4
Short Communication
Determination of velocity profiles of Bird-Carreau fluids in curvilinear
microchannels using random sample consensus
Kyu Yoon1,2, Hyun Wook Jung1,* and Myung-Suk Chun2,3,*
1
Department of Chemical and Biological Engineering, Korea University, Seongbuk-gu, Seoul 02841, Republic of Korea
2
Complex Fluids Laboratory, National Agenda Research Division, Korea Institute of Science and Technology (KIST),
Seongbuk-gu, Seoul 02792, Republic of Korea
3
Bio-Med Department, KIST School, Korea University of Science and Technology, Seoul 02792, Republic of Korea
(Received October 17, 2019; final revision received November 18, 2019; accepted November 19, 2019)
Flow features of rheologically complex fluids inside curved microchannels should be meaningfully scru-
tinized for effective mixing, sorting, and manipulation of nano- and micro-sized colloids or particles. In this
study, a particle streak velocimetry method with coordinate transformation is incorporated to depict exper-
imentally the axial velocity profiles of Newtonian and non-Newtonian (Bird-Carreau, BC) fluids in a cur-
vilinear microchannel under constant flow rate conditions. Theoretical velocity distributions for both fluids
are favorably substantiated from experimental observations that employ a random sample consensus
(RanSAC) algorithm under various channel geometric conditions, demonstrating the good agreement
between experiments and simulations previously developed. It is confirmed that the BC fluid showed blunt
and non-parabolic profiles in comparison to the Newtonian case at a low Dean number. The suggested algo-
rithm and method for accurately observing microscale flow fields provide useful insights into the elaborate
manipulation and processing of non-Newtonian fluids in curved channel devices.
Keywords: curved microchannel, Bird-Carreau fluid, particle streak velocimetry, micro flows, RanSAC
1. Introduction streaks, and correlations of particle images (Chun and
Lee, 2005; Chun et al., 2005; Khodaparast et al., 2013;
For the purpose of chemical synthesis, biological assays, Lima et al., 2006; Lochab et al., 2019; Yang, 1989).
and medical diagnostics, both systematic analysis and pre- Recent studies (Yoon et al., 2017; 2020; Nekoubin,
cise control of fluids or colloidal particles in lab-on-chips 2018) have theoretically investigated the dynamics of sec-
(i.e., micro mixing, size-based sorting, and focusing) have ondary Dean flow, which efficiently reflect the non-New-
attracted considerable interest because of the many bene- tonian nature of fluids. However, computational velocity
fits these methods offer, such as miniaturization, portabil- profiles of rheologically different fluids in a curvilinear
ity, and micro-scale sample volumes (McClain et al., confined geometry must be verified using a wide range of
2003; Shen et al., 2018; Stone et al., 2004; Volpe et al., process parameters such as aspect ratio, curvature ratio,
2017). The curvilinear microchannel is one of the most fluid properties, and flow rate. In this study, axial flow
common device designs, providing not only compactness fields of both Newtonian and non-Newtonian fluids in a
but also delicate manipulation of particulate suspensions curved microchannel are experimentally measured based
(Di Carlo et al., 2007; Nivedita et al., 2017). Thus, many on the particle streak velocimetry (PSV) method with
researchers have conducted both numerical and experi- coordinate transformation and the random sample consen-
mental studies to elucidate the corresponding Dean flow sus (RanSAC) (Fischler and Bolles, 1981) algorithm under
dynamics and their applications (Bayat and Rezai, 2017; constant flow rate conditions. Then, simulation results
Dean, 1927; Garcia and Pennathur, 2019; Thangam and through a numerical framework previously developed by
Hur, 1990). Particle image velocimetry (PIV) is a practical Yoon et al. (2017; 2020) are checked by comparing them
measurement technique that enables flows to be effec- with observations from experiments. In addition, the
tively characterized inside these microfluidic devices, cap- RanSAC is verified to be an effective tool in favorably
turing the velocity of fluid elements flowing in confined delineating flow fields in contrast to the conventional least
channels (Adrian, 1991; Degré et al., 2006; Paul et al., square (LS) fitting method, particularly during unavoid-
1998). Since its initial development in the 1980s, flow able disturbances.
fields have been determined through laser speckle pho-
tography related to pulsed light, streak length, areas of 2. Materials and Methods
*Corresponding authors; E-mail: M.-S. Chun (mschun@kist.re.kr) and Incompressible flow of Newtonian or shear-thinning flu-
H.W. Jung (hwjung@grtrkr.korea.ac.kr) ids through a uniformly curved microchannel with a rect-
© 2020 The Korean Society of Rheology and Springer pISSN 1226-119X eISSN 2093-7660 159
Kyu Yoon, Hyun Wook Jung and Myung-Suk Chun
angular cross-section is the focus of this study. The brief
experimental setup is schematically depicted in Fig. 1,
showing a left turn with width W and height H with a
hydraulic diameter of the channel cross-section dh (= 2HW/
(H+W)). In this study, three types of channel geometries
with different aspect ratios and curvature ratios were con-
sidered: 1) W = 62.9 μm, H = 40.2 μm, RC = 125.3 μm for
H/W ≈ 2/3 and W/RC ≈ 0.5; 2) W = 63.8 μm, H = 40.2 μm,
RC = 248 μm for H/W ≈ 2/3 and W/RC ≈ 0.25; and 3) W
= 41 μm, H = 62.7 μm, RC = 82.7 μm for H/W ≈ 3/2 and
W/RC ≈ 0.5. Through the efficient demonstration of exper-
imental observations, we tried to substantiate theoretical
axial velocities from a simulation framework of the inelas- Fig. 1. Schematic experimental setup for flow analysis in the
tic Dean flow described by Yoon et al. (2017; 2020). Gov- curved rectangular channel.
erning equations and boundary conditions in local Cartesian
coordinates are given by:
2.1. Experimental setup and procedure
v v v2 p After determining channel dimensions to verify simula-
x: vx x v y x z
x y RC x x tion results, we fabricated polydimethylsiloxane (PDMS)/
glass microfluidic chips using a standard soft lithography
1 vx 2vx 2vx vx
2 2 technique. To manufacture the master mold, we applied
2
RC x x x y ( RC x) photolithography to the negative photoresist (PR) SU-8
, (1a) 2050 (Microchem, MA) that was uniformly coated onto
v vx v y
2 x the cleaned wafer by spin coating. After UV patterning by
x x y x y
PR on the mask for 10 s using the mask aligner, we care-
v y v y fully performed the post-exposure bake stage and then the
p
y: v x vy unexposed PR was removed by dissolving with the SU-8
x y y
developer (Microchem, MA). A PDMS replica was man-
1 v y 2 v y 2 v y ufactured by filling the mixture of the base and curing
agent of the PDMS (Sylgard 184, Dow Corning, MI) at a
RC x x x 2 y 2 volume ratio of 10:1 in the master mold, which was suf-
, (1b)
v y v x v y ficiently degassed. The mixture was then cured at 80°C
2 for 1 h. Following the post-bake stage, the replica was
y y y x x
punched to generate holes for the inlet and outlet and
bonded to a slide glass using an O2 plasma generator
vz v vv R p
z: vx vy z x z C (CUTE-1MP, FemtoScience, Korea). Teflon tubing (ID:
x y RC x RC x z 0.8 mm, OD: 1.5 mm) was connected to both the inlet and
1 vz 2vz 2vz
outlet.
vz
2 2 Deionized water (Newtonian fluid) and a mixture of
2
RC x x x y ( RC x) 2000 ppm of schizophyllan (GlucanREAL, Mw = 3,500,000
, (1c)
v vz vz
g/mol, Quegen Biotech Co., Ltd., Korea) with deionized
z water (non-Newtonian fluid) were prepared as working
y y x RC x y fluids. The rheological properties of non-Newtonian fluids
with different concentrations were measured using a rota-
with no-slip boundary conditions
tional rheometer (MCR-301, Anton Paar) at 25oC, and
v(W/2, y) = v(W/2, y) = v(x, 0) = v(x, H) = 0, (1d) exhibited a shear-thinning nature (Fig. 2). Their viscosi-
ties, depending on shear rate, were well fitted with the Bird-
where v is the velocity vector, vx, vy, and vz are the velocity Carreau (BC) fluid model, which can suitably describe
components in each coordinate, respectively, p is the shear thinning as a function of shear rate (Bird et al., 1987)
pressure, and is the viscosity. The modified SIMPLER
( n1) 2
algorithm and the finite volume method (FVM) were 1 ( )2 (2)
incorporated to solve Eqs. (1a)-(1d), employing the pres- 0
sure-velocity-viscosity coupling scheme (Patankar, 1980; where 0 is the zero-shear viscosity, is the infinite-
Yoon et al., 2017; 2020; Yun et al., 2010). shear viscosity, is the relaxation time constant, n is the
160 Korea-Australia Rheology J., 32(2), 2020
Determination of velocity profiles of Bird-Carreau fluids in curvilinear microchannels using RanSAC
Fig. 2. (Color online) Shear viscosity of non-Newtonian working
fluids. Experimental data are fitted using the Bird-Carreau fluid
model.
power-law index, and · is the shear rate. Model param-
eters for a non-Newtonian working fluid (2000 ppm
schizophyllan in this experiment) were 0 = 0.79 Pa·s,
= 0.025 Pa·s, n = 0.2, and = 3 s. Two working fluids
were injected into the inlet of the microchannel using a
syringe pump (Pump 11 Elite-Nanomite, Harvard Appa-
ratus, MA) with 250 L syringe (1725TLL, Hamilton,
USA) under a constant flow rate of 30 nL/min.
2.2. PSV for analyzing curvilinear channel flows
A PSV method that included image acquisition, pre-pro-
cessing, and processing treatment was employed to observe
accurate velocity fields of working fluids in a confined
channel. For better measurement of their local velocities,
fluorescent polystyrene latex particles (Thermo Fisher Fig. 3. (a) Particle streak images before (left) and after (right)
Scientific Inc., MA) of diameter 0.5 μm with 0.1 ppm and treatment for removing static and dynamic noises. (b) Schematic
Triton X-100 of 0.2 wt.% were added to the working image of numbering shell groups. (c) Procedure for image pro-
fluids as tracer and surfactant, respectively, thus guaran- cessing employed in this study.
teeing their small size in comparison to the channel width.
Note that their densities were nearly the same as thoes of
the working fluids without floating or sedimentation. A could be eliminated by subtracting the averaged intensity
microfluidic chip was positioned on an inverted micro- at each position:
scope (Eclipse Ti-E, NiKon, Japan) with a 40× objective Nf
(e.g., NA of 0.75, focal depth 1.4 μm) for monitoring and I *(t, r, ) = I(t, r, ) I(t, r, )/Nf t, (3)
1
imaging. Streak images were captured by a digital 5M
pixel sCMOS camera (Zyla, monochrome cooled, ANDOR, where I is the original fluorescence intensity, I * is the fil-
UK) at an exposure time of 200 ms using NISElements tered intensity, Nf is the number of frames, and ∆t is the
software. All streak images were analyzed by MATLAB. exposure time for each frame. In addition, dynamic noises
As shown in Fig. 3c, as the first stage to evaluate veloc- inevitably created by particle streaks, such as faint streaks
ity distribution along the channel width, the position of located outside the focal depth or temporal light glare,
each pixel was transformed to cylindrical coordinates to were effectively removed according to the threshold of
quantify the arc length of streaks at the mid-height of the particle intensity. An example of this noise treatment is
curved microchannel. After the coordinates were converted, presented in Fig. 3a. After static and dynamic noises were
noises from streak images were properly filtered out to removed, the position (r, ) of particle streaks, when I *
improve image quality. Static noises (e.g., reflection from existed, was aligned in order from r = RC W/2 to r =
the light source and particles stuck on the channel wall) RC +W/2. To recognize each particle streak separately, its
Korea-Australia Rheology J., 32(2), 2020 161
Kyu Yoon, Hyun Wook Jung and Myung-Suk Chun position was identified by discrete shell groups at a par- 2.3. Refinement of experimental data using RanSAC ticle-size interval as schematically shown in Fig. 3b. Here, For the situation of small disturbances on measuring thick streak lines covering multiple shell groups were also velocity, the LS method can reliably fit the experimental excluded because they were aggregated. From r and ∆θ of data. Although this method determines the velocity profile streaks, the length of the streak Ls and the local velocity of following noise removal, it can distort actual parameter the tracer particle vz(r) were estimated as Ls = r∆θ and values because of imposed disturbances (called outliers), vz(r) = Ls/∆t, respectively. including out-of-focus, temporal fluctuations of flow rates, Fig. 4. (Color online) Comparison of axial flow velocity profiles for Newtonian ((a), (c), and (e)) and Bird-Carreau ((b), (d), and (f)) fluids from simulations, experimental data, and fitting methods (RanSAC and LS) along the spanwise direction at the mid-height of the channel (y = H/2). Variations of aspect and curvature ratios: (a) and (b) H/W = 2/3 and W/Rc = 0.5, (c) and (d) H/W = 2/3 and W/ RC = 0.25, and (e) and (f) H/W = 3/2 and W/RC = 0.5. 162 Korea-Australia Rheology J., 32(2), 2020
Determination of velocity profiles of Bird-Carreau fluids in curvilinear microchannels using RanSAC
etc. Fischler and Bolles (1981) introduced the RanSAC to investigated because both ratios are critical structural fac-
minimize the effect of outliers when fitting experimental tors that affect Dean flow vortices (Bayat and Rezai, 2017;
data to a model. This RanSAC was newly incorporated Thangam and Hur, 1990) in curved channels. Figures 4a,
into this study because this method provides the best fit- 4c, and 4e confirm that the experimentally measured
ting parameters through the following statistical treatments: velocity profiles as well as those from the RanSAC for the
(i) Perform random sampling among NS from whole Newtonian fluid are well portrayed by a slightly skewed
experimental data, where NS is the number of samples, (ii) parabola caused by dominant effects of a spanwise pres-
enable a fitting model to appropriately delineate selected sure gradient over the inertial force (De Vriend, 1981; Yun
samples, (iii) compare residuals (i) with inlier tolerance et al., 2010). As explained in Yun et al. (2010), the dec-
T to determine whether the ith value is an inlier or outlier, rement of the curvature ratio (W/RC) as shown in Figs. 4a
(iv) count the number of inliers NIn and calculate the total and 4c indicates that the velocity profile in the spanwise
residual i(i), and (v) repeat (i)-(iv) steps and revamp the direction rapidly changed from a skewed state toward the
fitting model for better parameter estimation until the inner wall of the curved channel to a symmetric-like state.
number of iterations exceeds sampling iterations Ni. The In addition, a comparison of Figs. 4a and 4e shows that
more detailed algorithm implemented herein can be found the deep channel (i.e., higher aspect ratio) produces a less
in Fischler and Bolles (1981). inward-skewed velocity profile.
In the case of non-Newtonian flow, the axial velocity of
3. Results and Discussion 2000 ppm schizophyllan solution under the same flow
conditions as in the Newtonian case tends to become con-
In a curved microchannel, tracer particle velocity at each siderably blunt along the spanwise direction by spreading
position can be reasonably evaluated by incorporating the out to both sides of the wall. This indicates that the shear-
RanSAC algorithm, particularly when handling large sparsely thinning feature of a working fluid prominently influences
scattered data. Note that experimental observations were the non-parabolic velocity profile. Furthermore, the local
performed at a low Dean number, where the Dean number velocity data of a non-Newtonian fluid from the flow visu-
is defined as Dn = dnv W/RC / and is the fluid density. alization are not sparsely scattered near the center (Figs.
Figure 4 compares the axial velocity profiles of Newto- 4b, 4d, and 4f) as compared to the Newtonian case. This
nian and BC fluids along the channel width at the mid- is closely related to the non-parabolic velocity pattern with
height using statistical regressions from the RanSAC and respect to the longitudinal position. It can be seen that the
LS methods and numerical simulation (Yoon et al., 2017; position of the maximum axial velocity of a non-Newto-
2020) under different aspect ratios and curvature ratios. nian fluid appears at the outer side of the curved channel
The velocity profile fitted by the RanSAC was determined under the same conditions as in the Newtonian case (Yoon
with number of samples NS = 50, sampling iteration Ni = et al., 2017).
3000, and inlier tolerance T = 0.1% of vz,max. In the case of
the LS method, fitting model parameters for the velocity 4. Conclusion
profile were obtained under the criterion of maximum iter-
ation Ni,max = 2000 or total residuals < 1012. An LS-fitting Experimentally well-defined images for particle streaks
model typically predicts local velocities slower than expect- are useful in characterizing and analyzing micro flows
ed, because it includes all of the outliers by slowly moving because flow fields for complex fluids in confined geom-
particles. The RanSAC method, by contrast, describes more etries can be easily determined, thus favorably guiding the
realistic velocity profiles than those of the LS method, optimal designs of micro-devices. The RanSAC method
thus offering better agreement with simulation results and for meaningfully refining streak imaging was developed
scattered experimental data. This is because the RanSAC and implemented to realize the flow of Newtonian and
method encompasses as many inliers as possible in the shear-thinning fluids in a curved microchannel. Velocity
absence of outliers. This method supports an effective profiles under low Dean flow conditions through the
analysis under a sparsely scattered data set of local veloc- RanSAC method from experimental streak images were
ities. Except the non-Newtonian case near the channel compared with simulation results by changing the aspect
wall, it was found that the RanSAC could be a fine fitting and curvature ratios. It was clearly demonstrated that both
model for the curved channel flow regardless of aspect experimental and simulation results are in good agree-
ratio, curvature ratio, or fluid properties. Even in the cir- ment, exhibiting a blunt velocity distribution along the
cumstance of disturbances causing the data to become channel width for non-Newtonian working fluid as com-
scattered, the image analysis applied in this study improved pared to the Newtonian flow case. The RanSAC applied in
the accuracy of the somewhat skewed and non-parabolic this study is found to be an effective and robust tool for
flow fields in the curved channel. representing sparsely scattered experimental flow data in
Here, the effects of aspect ratio and curvature ratio were micro-devices.
Korea-Australia Rheology J., 32(2), 2020 163Kyu Yoon, Hyun Wook Jung and Myung-Suk Chun
Acknowledgements suspension flow in a square microchannel, Meas. Sci. Technol.
17, 797-808.
This research was supported by the KIST Institutional Lochab, V., A. Yee, M. Yoda, A.T. Conlisk, and S. Prakash, 2019,
Program (project No. 2E29720 and No. 2E30580) provided Dynamics of colloidal particles in microchannels under com-
to M.-S. Chun and by the National Research Foundation bined pressure and electric potential gradients, Microfluid.
of Korea (NRF) of Korea grant (No. 2016R1A5A1009592 Nanofluid. 23, 134.
McClain, M.A., C.T. Culbertson, S.C. Jacobson, N.L. Allbritton,
and No. 2017R1E1A1A01075107) provided to H.W. Jung.
C.E. Sims, and J.M. Ramsey, 2003, Microfluidic devices for
the high-throughput chemical analysis of cells, Anal. Chem.
References 75, 5646-5655.
Nekoubin, N., 2018, Electroosmotic flow of power-law fluids in
Adrian, R.J., 1991, Particle-imaging techniques for experimental curved rectangular microchannel with high zeta potentials, J.
fluid mechanics, Annu. Rev. Fluid Mech. 23, 261-304. Non-Newtonian Fluid Mech. 260, 54-68.
Bayat, P. and P. Rezai, 2017, Semi-empirical estimation of dean Nivedita, N., P. Ligrani, and I. Papautsky, 2017, Dean flow
flow velocity in curved microchannels, Sci. Rep. 7, 13655. dynamics in low-aspect ratio spiral microchannels, Sci. Rep. 7,
Bird, R.B., R.C. Armstrong, and O. Hassager, 1987, Dynamics of 44072.
Polymeric Liquids: Vol. 1. Fluid Mechanics, 2nd Ed., John Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow,
Wiley & Sons, New York. McGraw-Hill, New York.
Chun, M.-S. and S. Lee, 2005, Flow imaging of dilute colloidal Paul, P.H., M.G. Garguilo, and D.J. Rakestraw, 1998, Imaging of
suspension in PDMS-based microfluidic chip using fluores- pressure- and electrokinetically driven flows through open cap-
cence microscopy, Colloid. Surf. A 267, 86-94. illaries, Anal. Chem. 70, 2459-2467.
Chun, M.-S., T.S. Lee, and K. Lee, 2005, Microflow of dilute Shen, S., L. Kou, X. Zhang, D. Wang, Y. Niu, and J. Wang, 2018,
colloidal suspension in narrow channel of microfluidic-chip Regulating secondary flow in ultra-low aspect ratio microchan-
under Newtonian fluid slip condition, Korea-Aust. Rheol. J. 17, nels by dimensional confinement, Adv. Theory Simul. 1,
207-215. 1700034.
De Vriend, H.J., 1981, Velocity redistribution in curved rectan- Stone, H.A., A.D. Stroock, and A. Ajdari, 2004, Engineering
gular channels, J. Fluid Mech. 107, 423-439. flows in small devices: Microfluidics toward a lab-on-a-chip,
Dean, W.R., 1927, XVI. Note on the motion of fluid in a curved Annu. Rev. Fluid Mech. 36, 381-411.
pipe, Philos. Mag. 4, 208-223. Thangam, S. and N. Hur, 1990, Laminar secondary flows in
Degré, G., P. Joseph, and P. Tabeling, 2006, Rheology of complex curved rectangular ducts, J. Fluid Mech. 217, 421-440.
fluids by particle image velocimetry in microchannels, Appl. Volpe, A., P. Paiè, A. Ancona, R. Osellame, P.M. Lugarà, and G.
Phys. Lett. 89, 024104. Pascazio, 2017, A computational approach to the characteri-
Di Carlo, D., D. Irimia, R.G. Tompkins, and M. Toner, 2007, zation of a microfluidic device for continuous size-based iner-
Continuous inertial focusing, ordering, and separation of par- tial sorting, J. Phys. D: Appl. Phys. 50, 255601.
ticles in microchannels, Proc. Natl. Acad. Sci. U. S. A. 104, Yang, W.J., 1989, Handbook of Flow Visualization, Hemisphere
18892-18897. Publishing Co., New York.
Fischler, M.A. and R.C. Bolles, 1981, Random sample consen- Yoon, K., H.W. Jung, and M.-S. Chun, 2017, Secondary flow
sus: A paradigm for model fitting with applications to image behavior of electrolytic viscous fluids with Bird-Carreau model
analysis and automated cartography, Commun. ACM 24, 381- in curved microchannels, Rheol. Acta 56, 915-926.
395. Yoon, K., H.W. Jung, and M.-S. Chun, 2020, Secondary Dean
Garcia, M. and S. Pennathur, 2019, A model for inertial particles flow characteristics of inelastic Bird-Carreau fluids in curved
in curvilinear flows, Microfluid. Nanofluid. 23, 63. microchannels, Korea-Aust. Rheol. J. 32, 61-70.
Khodaparast, S., N. Borhani, G. Tagliabue, and J.R. Thome, 2013, Yun, J.H., M.-S. Chun, and H.W. Jung, 2010, The geometry
A micro particle shadow velocimetry (μPSV) technique to effect on steady electrokinetic flows in curved rectangular
measure flows in microchannels, Exp. Fluids 54, 1474. microchannels, Phys. Fluids 22, 052004.
Lima, R., S. Wada, K. Tsubota, and T. Yamaguchi, 2006, Confocal
micro-PIV measurements of three-dimensional profiles of cell
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
164 Korea-Australia Rheology J., 32(2), 2020You can also read
