A model for inertial particles in curvilinear flows

  • Mike Garcia
  • Sumita PennathurEmail author
Research Paper


The recent popularity of inertial microfluidic devices has driven attention towards the behavior of particles in flows within curvilinear microchannels for particle separation and concentration applications. The robust technique of inertial focusing is particularly advantageous in such applications, not only because curved geometries can greatly reduce the footprint of a lab-on-chip devices, but also because the coupling of secondary Dean flows to inertial forces allows for exquisite particle manipulations. However, the ability to accurately predict the behavior of inertial particles in a curvilinear channel is often based on empirical results, as the commonly used theoretical models treat the effects of Dean flow on an inertial particle to be a consequence of the undisturbed velocity field. In particular, this simplification is problematic when the size of the particle is comparable to that of the confining channel. Here we present the first complete direct numerical model that directly simulates a particle within a confined curvilinear flow. This numerical model allows us to not only investigate the three-dimensional focusing behavior of inertial particles but also determine the applicability of the point particle assumptions previous researchers have proposed. Finally, we propose a more computationally efficient second-order model that takes into account the full physics by relying on a perturbation expansion of the lateral forces, where the perturbation parameter is the curvature ratio of the channel. This simpler model can be used to efficiently and accurately predict the behavior of particles in complex channel geometries where the curvature may not be constant.


Inertial microfluidics Particle separations Microfluidic design 



The authors would like to thank Professor Paolo Luzzato-Fegiz for insightful discussion regarding rotating reference frames. Mike Garcia was supported by the Institute for Collaborative Biotechnologies through Grants no. W911NF-09-D-0001 and no. W911NF-12-1-0031 through the U.S. Army Research Office. The content of the information does not necessarily reflect the position of the policy of the government, and no official endorsement should be inferred

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  1. Bhagat AAS, Kuntaegowdanahalli SS, Papautsky I (2008) Continuous particle separation in spiral microchannels using dean flows and differential migration. Lab Chip 8:1906–1914CrossRefGoogle Scholar
  2. Chun B, Ladd AJC (2006) Inertial migration of neutrally buoyant particles in a square duct: An investigation of multiple equilibrium positions. Phys Fluids 18:031704CrossRefGoogle Scholar
  3. Dean W (1927) Xvi. note on the motion of fluid in a curved pipe. Lond Edinb Dublin Philos Mag J Sci 4:208–223CrossRefGoogle Scholar
  4. Di Carlo D, Irimia D, Tompkins RG, Toner M (2007) Continuous inertial focusing, ordering, and separation of particles in microchannels. Proc Natl Acad Sci 104:18892–18897CrossRefGoogle Scholar
  5. Di Carlo D, Edd JF, Humphry KJ, Stone HA, Toner M (2009) Particle segregation and dynamics in confined flows. Phys Rev Lett 102:094503CrossRefGoogle Scholar
  6. Dinler A, Okumus I (2018) Inertial particle separation in curved networks: a numerical study. Chem Eng Sci 182:119–131CrossRefGoogle Scholar
  7. Garcia M, Ganapathysubramanian B, Pennathur S (2019) A linearised model for calculating inertial forces on a particle in the presence of a permeate flow. J Fluid Mech 861:253–274MathSciNetCrossRefGoogle Scholar
  8. Gossett DR, Di Carlo D (2009) Particle focusing mechanisms in curving confined flows. Anal Chem 81:8459–8465CrossRefGoogle Scholar
  9. Ho BP, Leal LG (1974) Inertial migration of rigid spheres in two-dimensional unidirectional flows. J Fluid Mech 65:365–400CrossRefGoogle Scholar
  10. Kim J et al (2016) Inertial focusing in non-rectangular cross-section microchannels and manipulation of accessible focusing positions. Lab Chip 16:992–1001CrossRefGoogle Scholar
  11. Lee DJ, Brenner H, Youn JR, Song YS (2013) Multiplex particle focusing via hydrodynamic force in viscoelastic fluids. Sci Rep 3:3258CrossRefGoogle Scholar
  12. Lim DSW, Shelby JP, Kuo JS, Chiu DT (2003) Dynamic formation of ring-shaped patterns of colloidal particles in microfluidic systems. Appl Phys Lett 83:1145–1147CrossRefGoogle Scholar
  13. Liu C, Hu G, Jiang X, Sun J (2015) Inertial focusing of spherical particles in rectangular microchannels over a wide range of Reynolds numbers. Lab Chip 15:1168–1177CrossRefGoogle Scholar
  14. Martel JM, Toner M, Elabbasi N, Quinn D, Bergstorm J (2013) Modeling inertial focusing in straight and curved microfluidic channels. COMSOL News.
  15. Martel JM et al (2015) Continuous flow microfluidic bioparticle concentrator. Sci Rep 5:11300CrossRefGoogle Scholar
  16. Martel JM, Toner M (2012) Inertial focusing dynamics in spiral microchannels. Phys Fluids 24:032001CrossRefGoogle Scholar
  17. Martel JM, Toner M (2013) Particle focusing in curved microfluidic channels. Sci Rep 3:3340CrossRefGoogle Scholar
  18. Maxey MR, Riley JJ (1998) Equation of motion for a small rigid sphere in a nonuniform flow. Phys Fluids 26:883CrossRefGoogle Scholar
  19. Nathamgari SSP et al (2015) Isolating single cells in a neurosphere assay using inertial microfluidics. Lab Chip 15:4591–4597CrossRefGoogle Scholar
  20. Nivedita N, Ligrani P, Papautsky I (2017) Dean flow dynamics in low-aspect ratio spiral microchannels. Sci Rep 7:44072CrossRefGoogle Scholar
  21. Norouzi M, Biglari N (2013) An analytical solution for Dean flow in curved ducts with rectangular cross section. Phys Fluids 25:053602CrossRefGoogle Scholar
  22. Ozbey A, Karimzadehkhouei M, Akgönül S, Gozuacik D, Koşar A (2016) Inertial focusing of microparticles in curvilinear microchannels. Sci Rep 6:38809CrossRefGoogle Scholar
  23. Pedrol E, Massons J, Díaz F, Aguiló M (2018) Two-way coupling fluid-structure interaction (FSI) approach to inertial focusing dynamics under dean flow patterns in asymmetric serpentines. Fluids 3:62CrossRefGoogle Scholar
  24. Phrom C, Stark H (2014) Feedback control of inertial microfluidics using axial control forces. Lab Chip 14:2115–2123CrossRefGoogle Scholar
  25. Rasooli R, Çetin B (2018) Assessment of Lagrangian modeling of particle motion in a spiral microchannel for inertial microfluidics. Micromachines 9:433CrossRefGoogle Scholar
  26. Sarkar A, Hou HW, Mahan AE, Han J, Alter G (2016) Multiplexed affinity-based separation of proteins and cells using inertial microfluidics. Sci Rep 6:23589CrossRefGoogle Scholar
  27. Segre G, Silberberg A (1961) Radial particle displacements in Poiseuille flow of suspensions. Nature 189:209–210CrossRefGoogle Scholar
  28. Wang L, Dandy DS (2017) High-throughput inertial focusing of micrometer- and sub-micrometer-sized particles separation. Adv Sci 4:1700153CrossRefGoogle Scholar
  29. Yuan C, Pan Z, Wu H (2018) Inertial migration of single particle in a square microchannel over wide ranges of Re and particle sizes. Microfluid Nanofluidics 22:102CrossRefGoogle Scholar
  30. Zhang J, Li W, Li M, Alici G, Nguyen N-T (2014) Particle inertial focusing and its mechanism in a serpentine microchannel. Microfluid Nanofluidics 17:305–316CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of California Santa BarbaraSanta BarbaraUSA

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