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Inertial migration of a rigid sphere in three-dimensional Poiseuille flow

Published online by Cambridge University Press:  22 January 2015

Kaitlyn Hood ,
Sungyon Lee  and
Marcus Roper
Kaitlyn Hood*
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USA
Sungyon Lee
Affiliation:
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
Marcus Roper
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USA
*
†Email address for correspondence: [email protected]
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Abstract

Inertial lift forces are exploited within inertial microfluidic devices to position, segregate and sort particles or droplets. However, the forces and their focusing positions can currently only be predicted by numerical simulations, making rational device design very difficult. Here we develop theory for the forces on particles in microchannel geometries. We use numerical experiments to dissect the dominant balances within the Navier–Stokes equations and derive an asymptotic model to predict the lateral force on the particle as a function of particle size. Our asymptotic model is valid for a wide array of particle sizes and Reynolds numbers, and allows us to predict how focusing position depends on particle size.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 765 , 25 February 2015 , pp. 452 - 479
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Copyright
© 2015 Cambridge University Press 

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References

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.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bramble, J. H. 1981 The Lagrange multiplier method for Dirichlet’s problem. Maths Comput. 37 (155), 1–11.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.CrossRefGoogle Scholar
Chung, A. J., Gossett, D. R. & Di Carlo, D. 2013 Three dimensional, sheathless, and high-throughput microparticle inertial focusing through geometry-induced secondary flows. Small 9 (5), 685–690.CrossRefGoogle ScholarPubMed
Cox, R. G. & Brenner, H. 1968 The lateral migration of solid particles in Poiseuille flow – I. Theory. Chem. Engng Sci. 23 (2), 147–173.CrossRefGoogle Scholar
Cox, R. G. & Hsu, S. K. 1977 The lateral migration of solid particles in a laminar flow near a plane. Intl J. Multiphase Flow 3 (3), 201–222.CrossRefGoogle Scholar
Dennis, S. C. R. & Walker, J. D. A. 1971 Calculation of the steady flow past a sphere at low and moderate Reynolds numbers. J. Fluid Mech. 48 (4), 771–789.CrossRefGoogle Scholar
Di Carlo, D., Edd, J. F., Humphry, K. J., Stone, H. A. & Toner, M. 2009 Particle segregation and dynamics in confined flows. Phys. Rev. Lett. 102 (9), 094503.Google ScholarPubMed
Happel, J. & Brenner, H. 1982 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media. (Mechanics of Fluids and Transport Processes 1) , Springer.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. (Cambridge Texts in Applied Mathematics 6) , Cambridge University Press.CrossRefGoogle Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365–400.CrossRefGoogle Scholar
Hogg, A. J. 1994 The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows. J. Fluid Mech. 272, 285–318.CrossRefGoogle 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.CrossRefGoogle Scholar
Hur, S. C., Tse, H. T. K. & Di Carlo, D. 2010 Sheathless inertial cell ordering for extreme throughput flow cytometry. Lab on a Chip 10, 274–280.CrossRefGoogle ScholarPubMed
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12 (1), 435–476.CrossRefGoogle Scholar
LeClair, B. P., Hamielec, A. E., Pruppacher, H. R. & Hall, W. D. 1972 A theoretical and experimental study of the internal circulation in water drops falling at terminal velocity in air. J. Atmos. Sci. 29 (4), 728–740.2.0.CO;2>CrossRefGoogle Scholar
Lee, W., Amini, H., Stone, H. A. & Di Carlo, D. 2010 Dynamic self-assembly and control of microfluidic particle crystals. Proc. Natl Acad. Sci. USA 107 (52), 22413–22418.CrossRefGoogle ScholarPubMed
Mach, A. J., Kim, J. H., Arshi, A., Hur, S. C. & Di Carlo, D. 2011 Automated cellular sample preparation using a centrifuge-on-a-chip. Lab on a Chip 11, 2827–2834.CrossRefGoogle ScholarPubMed
Maxworthy, T. 1965 Accurate measurements of sphere drag at low Reynolds numbers. J. Fluid Mech. 23, 369–372.CrossRefGoogle Scholar
McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261–274.CrossRefGoogle Scholar
Papanastasiou, T. C., Georgiou, G. C. & Alexandrou, A. N. 1999 Viscous Fluid Flow. CRC Press.CrossRefGoogle Scholar
Perry, R. 1950 Chemical Engineers’ Handbook, 3rd edn. McGraw-Hill Education.CrossRefGoogle Scholar
Rubinow, S. I. & Keller, J. B. 1961 The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11, 447–459.CrossRefGoogle Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400.CrossRefGoogle Scholar
Schonberg, J. A. & Hinch, E. J. 1989 Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech. 203, 517–524.CrossRefGoogle Scholar
Segré, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189 (4760), 209–210.CrossRefGoogle Scholar
Segré, G. & Silberberg, A. 1962a Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams. J. Fluid Mech. 14, 115–135.CrossRefGoogle Scholar
Segré, G. & Silberberg, A. 1962b Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14, 136–157.CrossRefGoogle Scholar
Sollier, E., Go, D. E., Che, J., Gossett, D. R., O’Byrne, S., Weaver, W. M., Kummer, N., Rettig, M., Goldman, J., Nickols, N., McCloskey, S., Kulkarni, R. P. & Di Carlo, D. 2014 Size-selective collection of circulating tumor cells using vortex technology. Lab on a Chip 14, 63–77.CrossRefGoogle ScholarPubMed
Vasseur, P. & Cox, R. G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78, 385–413.CrossRefGoogle Scholar
Veysey, J. & Goldenfeld, N. 2007 Simple viscous flows: from boundary layers to the renormalization group. Rev. Mod. Phys. 79, 883–927.CrossRefGoogle Scholar
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