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Focusing patterns of spherical particles in a viscoelastic flow through square channels

Published online by Cambridge University Press:  06 November 2025

Hiroshi Yamashita Open the ORCID record for Hiroshi Yamashita [Opens in a new window] ,
Yuta Miki ,
Daisuke Yoneyama ,
Kento Higashi ,
Takuma Tsukasa ,
Yoshiyuki Tagawa Open the ORCID record for Yoshiyuki Tagawa [Opens in a new window] ,
Naoto Yokoyama Open the ORCID record for Naoto Yokoyama [Opens in a new window] ,
Tomoaki Itano Open the ORCID record for Tomoaki Itano [Opens in a new window]  and
Masako Sugihara-Seki Open the ORCID record for Masako Sugihara-Seki [Opens in a new window]
Hiroshi Yamashita*
Affiliation:
Graduate School of Integrated Sciences for Life, Hiroshima University , Higashihiroshima, Hiroshima 739-8521, Japan
Yuta Miki
Affiliation:
Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564-8680, Japan
Daisuke Yoneyama
Affiliation:
Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564-8680, Japan
Kento Higashi
Affiliation:
Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564-8680, Japan
Takuma Tsukasa
Affiliation:
Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564-8680, Japan
Yoshiyuki Tagawa
Affiliation:
Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan
Naoto Yokoyama
Affiliation:
Department of Mechanical Engineering, Tokyo Denki University, Adachi, Tokyo 120-8551, Japan
Tomoaki Itano
Affiliation:
Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564-8680, Japan
Masako Sugihara-Seki
Affiliation:
Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564-8680, Japan
*
Corresponding author: Hiroshi Yamashita, hyamashi@hiroshima-u.ac.jp
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Abstract

The inertial migration of neutrally buoyant spherical particles in viscoelastic fluids flowing through square channels is experimentally and numerically studied. In the experiments, using dilute aqueous solutions of polymers with various concentrations that have nearly constant viscosities, we measured the distribution of suspended particles in downstream cross-sections for the Reynolds number ($\textit{Re}$) up to 100 and the elasticity number ($El$) up to 0.07. There are several focusing patterns of the particles, such as four-point focusing near the centre of the channel faces on the midlines for low $\textit{El}$ and/or high $\textit{Re}$, four-point focusing on the diagonals for medium $\textit{El}$, single-point focusing at the channel centre for relatively high $\textit{El}$ and low $\textit{Re}$, and five-point focusing near the four corners and the channel centre for high $\textit{El}$ and very low $\textit{Re}$. Among these focusing patterns, various types of particle distributions suggesting the presence of a new equilibrium position located between the midline and the diagonal, and multistable states of different equilibrium positions were observed. In general, as $\textit{El}$ increases from 0 at a constant $\textit{Re}$, the particle focusing positions shift from the midline to the diagonal in the azimuthal direction first, and then inward in the radial direction to the channel centre. These focusing patterns and their transitions were numerically well reproduced based on a FENE-P model with measured values of viscosity and relaxation time. Using the numerical results, the experimentally observed focusing patterns of particles are elucidated in terms of the fluid elasticity-induced lift and the wall-induced elastic lift.

JFM classification

Non-Newtonian Flows: Viscoelasticity Micro-/Nano-fluid dynamics: Microfluidics

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JFM Papers
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Journal of Fluid Mechanics , Volume 1022 , 10 November 2025 , A44
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© The Author(s), 2025. Published by Cambridge University Press

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References

Abbas, M., Magaud, P., Gao, Y. & Geoffroy, S. 2014 Migration of finite sized particles in a laminar square channel flow from low to high Reynolds numbers. Phys. Fluids 26, 123301.10.1063/1.4902952CrossRefGoogle Scholar
Abdel Azim, A.-A.A., Tenhu, H., Maerta, J. & Sundholm, F. 1992 Flexibility and hydrodynamic properties of poly(vinylpyrrolidone) in non-ideal solvents. Polym. Bull. 29, 461–467.10.1007/BF00944845CrossRefGoogle Scholar
Asmolov, E.S. 1999 The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J. Fluid Mech. 381, 63–87.10.1017/S0022112098003474CrossRefGoogle Scholar
Balci, N., Thomases, B., Renardy, M. & Doering, C.R. 2011 Symmetric factorization of the conformation tensor in viscoelastic fluid models. J. Nonnewton. Fluid Mech. 166, 546–553.10.1016/j.jnnfm.2011.02.008CrossRefGoogle Scholar
Bhagat, A.A.S., Kuntaegowdanahalli, S.S. & Papautsky, I. 2008 Enhanced particle filtration in straight microchannels using shear-modulated inertial migration. Phys. Fluids 20, 101702.10.1063/1.2998844CrossRefGoogle Scholar
Choi, Y., Seo, K. & Lee, S. 2011 Lateral and cross-lateral focusing of spherical particles in a square microchannel. Lab Chip 11, 460–465.10.1039/C0LC00212GCrossRefGoogle Scholar
Clasen, C., Plog, J.P., Kulicke, W.-M., Owens, M., Macosko, C., Scriven, L.E., Verani, M. & McKinley, G.H. 2006 How dilute are dilute solutions in extensional flows? J. Rheol. 50, 849–881.10.1122/1.2357595CrossRefGoogle Scholar
Del Guidice, F., Romeo, G., D’Avino, G., Greco, F., Netti, P. & Maffettone, P.L. 2013 Particle alignment in a viscoelastic liquid flowing in a square-shaped microchannel. Lab Chip 13, 4263–4271.10.1039/c3lc50679gCrossRefGoogle Scholar
Del Giudice, F., D’Avino, G., Greco, F., Maffettone, P.L. & Shen, A.Q. 2018 Fluid viscoelasticity drives self-assembly of particle trains in a straight microfluidic channel. Phys. Rev. Appl 10, 064058.10.1103/PhysRevApplied.10.064058CrossRefGoogle Scholar
Di Carlo, D. 2009 Inertial microfluidics. Lab Chip 9, 3038–3046.10.1039/b912547gCrossRefGoogle ScholarPubMed
Di Carlo, D., Edd, J.F., Humphry, K.J., Stone, A.H. & Toner, M. 2009 Particle segregation and dynamics in confined flows. Phys. Rev. Lett. 102, 094503.10.1103/PhysRevLett.102.094503CrossRefGoogle ScholarPubMed
Di Carlo, D., Irima, D., Tompkins, R.G. & Toner, M. 2007 Continuous inertial focusing, ordering, and separation of particles in microchannels. Proc. Natl. Acad. Sci. USA 104, 18892–18897.10.1073/pnas.0704958104CrossRefGoogle ScholarPubMed
D’Avino, G. & Maffettone, P.L. 2015 Particle dynamics in viscoelastic liquids. J. Nonnewton. Fluid Mech. 215, 80–104.10.1016/j.jnnfm.2014.09.014CrossRefGoogle Scholar
D’Avino, G., Greco, F. & Maffettone, P.L. 2017 Particle migration due to viscoelasticity of the suspending liquid and its relevance in microfluidic devices. Annu. Rev. Fluid Mech. 49, 341–360.10.1146/annurev-fluid-010816-060150CrossRefGoogle Scholar
D’Avino, G. & Maffettone, P.L. 2020 Numerical simulations on the dynamics of trains of particles in a viscoelastic fluid flowing in a microchannel. Meccanica 55, 317–330.10.1007/s11012-019-00985-6CrossRefGoogle Scholar
Dinic, J., Jimenez, L.N. & Sharma, V. 2017 Pinch-off dynamics and dripping-onto-substrate (DoS) rheometry of complex fluids. Lab Chip 17, 460–473.10.1039/C6LC01155ACrossRefGoogle ScholarPubMed
Dubief, Y., Terrapon, V.E. & Hof, B. 2023 Elasto-inertial turbulence. Annu. Rev. Fluid Mech. 55, 675–705.10.1146/annurev-fluid-032822-025933CrossRefGoogle Scholar
Franco-Gómez, A., Onuki, H., Yokoyama, Y., Nagatsu, Y. & Tagawa, Y. 2021 Effect of liquid elasticity on the behaviour of high-speed focused jets. Exp. Fluids 62, 41.10.1007/s00348-020-03128-wCrossRefGoogle Scholar
Gauthier, F., Goldsmith, H.L. & Mason, S.G. 1971 Particle motions in non-Newtonian media. II. Poiseuille flow. Trans. Soc. Rheol 15, 297–330.10.1122/1.549212CrossRefGoogle Scholar
Herrchen, M. & Öttinger, H.C. 1997 A detailed comparison of various FENE dumbbell models. J. Non-Newton. Fluid Mech. 68, 17–42.10.1016/S0377-0257(96)01498-XCrossRefGoogle Scholar
Ho, B.P. & Leal, L.G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flow. J. Fluid Mech. 65, 365–400.10.1017/S0022112074001431CrossRefGoogle Scholar
Ho, B.P. & Leal, L.G. 1976 Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid. J. Fluid Mech. 76, 783–700.Google Scholar
Hu, X., Lin, P., Lin, J., Zhu, Z. & Yu, Z. 2022 On the polydisperse particle migration and formation of chains in a square channel flow of non-Newtonian fluids. J. Fluid Mech. 936, A5.10.1017/jfm.2022.38CrossRefGoogle Scholar
Hu, X., Lin, J., Zhu, Z., Yu, Z., Lin, Z. & Li, X. 2024 Elasto-inertial focusing and rotating characteristics of ellipsoidal particles in a square channel flow of Oldroyd-B viscoelasitc fluids. J. Fluid Mech. 997, A32.10.1017/jfm.2024.479CrossRefGoogle Scholar
Humphry, K.J., Kulkarni, P.M., Weitz, D.A., Morris, J.F. & Stone, H.A. 2010 Axial and lateral particle ordering in finite Reynolds number channel flows. Phys. Fluids 22, 081703.10.1063/1.3478311CrossRefGoogle Scholar
Hur, S.C., Tse, H.T.K. & Di Carlo, D. 2010 Sheathless inertial cell ordering for extreme throughput flow cytometry. Lab Chip 10, 274–280.10.1039/B919495ACrossRefGoogle ScholarPubMed
James, D.F. 2009 Boger fluids. Annu. Rev. Fluid Mech. 41, 129–142.10.1146/annurev.fluid.010908.165125CrossRefGoogle Scholar
John, T.P., Poole, R.J., Kowalski, A.J. & Fonte, C.P. 2024 A comparison between the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear. J. Fluid Mech. 979, A10.10.1017/jfm.2023.977CrossRefGoogle Scholar
Kahkeshani, S., Haddadi, H. & Di Carlo, D. 2016 Preferred interparticle spacings in trains of particles in inertial microchannel flows. J. Fluid Mech. 786, R3.10.1017/jfm.2015.678CrossRefGoogle Scholar
Kajishima, T., Takiguchi, S., Hamasaki, H. & Miyake, Y. 2001 Turbulence structure of particle-laden flow in a vertical plane channel due to vortex shedding. JSME J. B 44, 526–535.Google Scholar
Kalyan, S., Torabi, C., Khoo, H., Sung, H.W., Choi, S.-E., Wang, W., Treutler, B., Kim, D. & Hur, S.C. 2021 Inertial microfluidics enabling clinical research. Micromachines 12, 257.10.3390/mi12030257CrossRefGoogle ScholarPubMed
Kamli, M., Guettari, M. & Tajouri, T. 2023 Structure of polyvinylpyrrolidone in water/ethanol mixture in dilute and semi-dilute regimes: roles of solvents mixture composition and polymer concentration. J. Molecular Liquids 382, 122014.10.1016/j.molliq.2023.122014CrossRefGoogle Scholar
Karnis, A. & Mason, S.G. 1966 Particle motions in sheared suspensions. XIX. Viscoelastic media. Trans. Soc. Rheol. 10, 571–592.10.1122/1.549066CrossRefGoogle Scholar
Kazerooni, H.T., Fornari, W., Hussong, J. & Brandt, L. 2017 Inertial migration in dilute and semidilute suspensions of rigid particles in laminar square duct flow. Phys. Rev. Fluids 2, 084301.10.1103/PhysRevFluids.2.084301CrossRefGoogle Scholar
Kim, B. & Kim, J.M. 2016 Elasto-inertial particle focusing under the viscoelastic flow of DNA solution in a square channel. Biomicrofluidics 10, 024111.10.1063/1.4944628CrossRefGoogle ScholarPubMed
Kim, J.Y., Ahn, S.W., Lee, S.S. & Kim, J.M. 2012 Lateral migration and focusing of colloidal particles and DNA molecules under viscoelastic flow. Lab Chip 12, 2807–2814.Google ScholarPubMed
Kim, Y.W. & Yoo, J.Y. 2008 The lateral migration of neutrally-buoyant spheres transported through square microchannels. J. Micromech. Microeng. 18, 065015.10.1088/0960-1317/18/6/065015CrossRefGoogle Scholar
Lee, S., Kim, H. & Yang, S. 2023 Microfluidic label-free hydrodynamic separation of blood cells: recent developments and future perspectives. Adv. Mater. Technol. 8, 2201425.10.1002/admt.202201425CrossRefGoogle Scholar
Leshansky, A.M., Bransky, A., Korin, N. & Dinnar, U. 2007 Tunable nonlinear viscoelastic ‘focusing’ in a microfluidic device. Phys. Rev. Lett. 98, 234501.10.1103/PhysRevLett.98.234501CrossRefGoogle Scholar
Li, G., McKinley, G.H. & Ardekani, A.M. 2015 Dynamics of particle migration in channel flow of viscoelastic fluids. J. Fluid Mech. 785, 486–505.10.1017/jfm.2015.619CrossRefGoogle Scholar
Lim, E.J., Ober, T.J., Edd, J.F., Desai, S.P., Neal, D., Bong, K.W., Doyle, P.S., McKinley, G.H. & Toner, M. 2014 Inertio-elastic focusing of bioparticles in microchannels at high throughput. Nature Comm. 5, 4120.10.1038/ncomms5120CrossRefGoogle ScholarPubMed
Lu, X., Liu, C., Hu, G. & Xuan, X. 2017 Particle manipulations in non-Newtonian microfluidics: a review. J. Colloid Interface Sci 500, 182–201.10.1016/j.jcis.2017.04.019CrossRefGoogle ScholarPubMed
Matas, J.-P., Glezer, V., Guazzelli, E. & Morris, J.F. 2004 a Trains of particles in finite-Reynolds-number pipe flow. Phys. Fluids 16, 4192–4195.10.1063/1.1791460CrossRefGoogle Scholar
Matas, J.-P., Morris, J.F. & Guazzelli, E. 2004 b Inertial migration of rigid spherical particles in Poiseuille flow. J. Fluid Mech. 515, 171–195.10.1017/S0022112004000254CrossRefGoogle Scholar
Matas, J.-P., Morris, J.F. & Guazzelli, E. 2004 c Lateral forces on a sphere. Oil Gas Sci. Technol.-Rev. JFP 59, 59–70.10.2516/ogst:2004006CrossRefGoogle Scholar
Matas, J.-P., Morris, J.F. & Guazzelli, E. 2009 Lateral force on a rigid sphere in large-inertia laminar pipe flow. J. Fluid Mech. 621, 59–67.10.1017/S0022112008004977CrossRefGoogle Scholar
Matel, J.M. & Toner, M. 2014 Inertial focusing in microfluidics. Annu. Rev. Biomed. Eng 16, 371–396.10.1146/annurev-bioeng-121813-120704CrossRefGoogle Scholar
Miura, K., Itano, T. & Sugihara-Seki, M. 2014 Inertial migration of neutrally buoyant spheres in a pressure-driven flow through square channels. J. Fluid Mech. 749, 320–330.10.1017/jfm.2014.232CrossRefGoogle Scholar
Nakagawa, N., Yabu, T., Otomo, R., Kase, A., Makino, M., Itano, T. & Sugihara-Seki, M. 2015 Inertial migration of a spherical particle in laminar square channel flows from low to high Reynolds numbers. J. Fluid Mech. 779, 776–793.10.1017/jfm.2015.456CrossRefGoogle Scholar
Peng, S., Tang, T., Li, J., Zhang, M. & Yu, P. 2023 Numerical study of viscoelastic upstream instability. J. Fluid Mech. 959, A16.10.1017/jfm.2023.92CrossRefGoogle Scholar
Prohm, C. & Stark, H. 2014 Feedback control of inertial microfluidics using axial control forces. Lab Chip 14, 2115.10.1039/c4lc00145aCrossRefGoogle ScholarPubMed
Purnode, B. & Crochet, M.J. 1998 Polymer solution characterization with the FENE-P model. J. Non-Newton. Fluid Mech. 77, 1–20.10.1016/S0377-0257(97)00096-7CrossRefGoogle Scholar
Raoufi, M.A., Mashhadian, A., Niazmand, H., Asadnia, M., Razmjou, A. & Warkiani, M.E. 2019 Experimental and numerical study of elasto-inertial focusing in straight channels. Biomicrofluidics 13, 034103.10.1063/1.5093345CrossRefGoogle ScholarPubMed
Razavi Bazaz, S., Mashhadian, A., Ehsani, A., Saha, S.C., Krüger, T. & Warkiani, M.E. 2020 Computational inertial microfluidics: a review. Lab Chip 20, 1023.10.1039/C9LC01022JCrossRefGoogle ScholarPubMed
Rubio, A., Vega, E.J., Gaán-Calvo, A.M. & Montanero, J.M. 2022 Unexpected stability of micrometer weakly viscoelastic jets. Phys. Fluids 34, 062014.10.1063/5.0091095CrossRefGoogle Scholar
Schonberg, J.A. & Hinch, E.J. 1989 Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech. 203, 517–524.10.1017/S0022112089001564CrossRefGoogle Scholar
Segré, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189, 209–210.10.1038/189209a0CrossRefGoogle Scholar
Segré, G. & Silberberg, A. 1962 Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14, 136–157.10.1017/S0022112062001111CrossRefGoogle Scholar
Seo, K.W., Kang, Y.J. & Lee, S.J. 2014 Lateral migration and focusing of microspheres in a microchannel flow of viscoelastic fluids. Phys. Fluids 26, 063301.10.1063/1.4882265CrossRefGoogle Scholar
Shichi, H., Yamashita, H., Seki, J., Itano, T. & Sugihara-Seki, M. 2017 Inertial migration regimes of spherical particles suspended in submillimeter square tube flows. Phys. Rev. Fluids 2, 044201.10.1103/PhysRevFluids.2.044201CrossRefGoogle Scholar
Steinberg, Z. 2021 Elastic turbulence: an experimental view on inertialess random flow. Ann. Rev. Fluid Mech. 53, 27–58.10.1146/annurev-fluid-010719-060129CrossRefGoogle Scholar
Stoecklein, D. & Di Carlo, D. 2019 Nonlinear microfluidics. Anal. Chem. 91, 296–314.10.1021/acs.analchem.8b05042CrossRefGoogle ScholarPubMed
Sur, S. & Rothstein, J. 2018 Drop breakup dynamics of dilute polymer solutions: effect of molecular weight, concentration, and viscosity. J. Rheol. 62, 1245–1259.10.1122/1.5038000CrossRefGoogle Scholar
Tang, W., Zhu, S., Jiang, D., Zhu, L., Yang, J. & Xiang, N. 2020 Channel innovations for inertial microfluidics. Lab Chip 20, 3485.10.1039/D0LC00714ECrossRefGoogle ScholarPubMed
Tehrani, M.A. 1996 An experimental study of particle migration in pipe flow of viscoelastic fluids. J. Rheol. 40, 1057–1077.10.1122/1.550773CrossRefGoogle Scholar
Udono, H. 2020 Numerical investigations of strong hydrodynamic interaction between neighboring particles inertially driven in microfluidic flows. Adv. Powder Technol. 31, 4107–4118.10.1016/j.apt.2020.08.013CrossRefGoogle Scholar
Villone, M.M., D’Avino, G., Hulsen, M.A., Greco, F. & Maffettone, P.L. 2013 Particle motion in square channel flow of a viscoelastic liquid: migration vs. secondary flow. J. Non-Newton. Fluid Mech. 195, 1–8.10.1016/j.jnnfm.2012.12.006CrossRefGoogle Scholar
Yamani, S. & McKinley, G.H. 2023 Master curves for FENE-P fluids in steady shear flow. J. Non-Newton. Fluid Mech. 313,104944, 104944.10.1016/j.jnnfm.2022.104944CrossRefGoogle Scholar
Yamashita, H., Itano, T. & Sugihara-Seki, M. 2019 Bifurcation phenomena on the inertial focusing of a neutrally buoyant spherical particle suspended in square duct flows. Phys. Rev. Fluids 4, 124307.10.1103/PhysRevFluids.4.124307CrossRefGoogle Scholar
Yang, S., Kim, J.Y., Lee, S.J., Lee, S.S. & Kim, J.M. 2011 Sheathless elasto-inertial particle focusing and continuous separation in a straight rectangular microchannel. Lab Chip 11, 266–273.10.1039/C0LC00102CCrossRefGoogle Scholar
Yang, S., Lee, S.S., Ahn, S.W., Kang, K., Shim, W., Lee, G., Hyun, K. & Kim, J.M. 2012 Deformability-selective particle entrainment and separation in a rectangular microchannel using medium viscoelasticity. Soft Matter 8, 5011–5019.10.1039/c2sm07469aCrossRefGoogle Scholar
Yokoyama, N., Yamashita, H., Higashi, K., Miki, Y., Itano, T. & Sugihara-Seki, M. 2021 Variation of focusing patterns of laterally migrating particles in a square-tube flow due to non-Newtonian elastic force. Phys. Rev. Fluids 6, L072301.10.1103/PhysRevFluids.6.L072301CrossRefGoogle Scholar
Yu, Z., Wang, P., Lin, J. & Hu, H.H. 2019 Equilibrium positions of the elasto-inertial particle migration in rectangular channel flow of Oldroyd-B viscoelastic fluids. J. Fluid Mech. 868, 316–340.10.1017/jfm.2019.188CrossRefGoogle Scholar
Yuan, D., Zhao, Q., Yan, S., Tang, S.-Y., Alici, G., Zhang, J. & Li, W. 2018 Recent progress of particle migration in viscoelastic fluids. Lab Chip 18, 551.10.1039/C7LC01076ACrossRefGoogle ScholarPubMed
Zhang, J., Yan, S., Yuan, D., Alici, G., Nguyen, N.-T., Warkiani, M.E. & Li, W. 2016 Fundamentals and applications of inertial microfluidics: a review. Lab Chip 16, 10.10.1039/C5LC01159KCrossRefGoogle Scholar
Zhang, T. et al. 2024 Passive microfluidic devices for cell separation. Biotechnol. Adv 71, 108317.10.1016/j.biotechadv.2024.108317CrossRefGoogle ScholarPubMed
Zhou, J. & Papautsky, I. 2020 Viscoelastic microfluidics: progress and challenges. Microsys. Nanoeng. 6, 113.10.1038/s41378-020-00218-xCrossRefGoogle ScholarPubMed
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