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Effect of inertial lift on a spherical particle suspended in flow through a curved duct. (English) Zbl 1419.76129

Summary: We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrate. The Navier-Stokes equations determine the hydrodynamic forces acting on a particle. A leading-order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales: specifically, the particle radius, duct height and duct bend radius. Additionally, a trapezoidal-shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76T20 Suspensions
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics

Software:

FEniCS
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abbas, M., Magaud, P., Gao, Y. & Geoffroy, S.2014Migration of finite sized particles in a laminar square channel flow from low to high Reynolds numbers. Phys. Fluids26 (12), 123301.
[2] Alnæs, M. S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M. E. & Wells, G. N.2015The FEniCS project version 1.5. Arch. Numer. Softw.3 (100), 9-23.
[3] Amini, H., Lee, W. & Di Carlo, D.2014Inertial microfluidic physics. Lab on a Chip14, 2739-2761.
[4] Asmolov, E. S.1999The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J. Fluid Mech.381, 63-87. · Zbl 0935.76025
[5] Ciftlik, A. T., Ettori, M. & Gijs, M. A. M.2013High throughput-per-footprint inertial focusing. Small9 (16), 2764-2773.
[6] Dean, W. R.1927Note on the motion of fluid in a curved pipe. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science4 (20), 208-223. · JFM 54.0909.05
[7] Dean, W. R. & Hurst, J. M.1959Note on the motion of fluid in a curved pipe. Mathematika6 (1), 77-85. · Zbl 0094.39703
[8] Di Carlo, D.2009Inertial microfluidics. Lab on a Chip9, 3038-3046.
[9] Di Carlo, D., Edd, J. F., Humphry, K. J., Stone, H. A. & Toner, M.2009Particle segregation and dynamics in confined flows. Phys. Rev. Lett.102, 094503.
[10] Geislinger, T. M. & Franke, T.2014Hydrodynamic lift of vesicles and red blood cells in flow from Fåhræus and Lindqvist to microfluidic cell sorting. Adv. Colloid Interface Sci.208, 161-176.
[11] Guan, G., Wu, L., Bhagat, A. A., Li, Z., Chen, P. C. Y., Chao, S., Ong, C. J. & Han, J.2013Spiral microchannel with rectangular and trapezoidal cross-sections for size based particle separation. Sci. Rep.3, 1475.
[12] Harding, B.2018A study of inertial particle focusing in curved microfluidic ducts with large bend radius and low flow rate. In Proceedings of the 21st Australasian Fluid Mechanics Conference, Adelaide, South Australia, Australia. Australasian Fluid Mechanics Society.
[13] Harding, B.2019A Rayleigh-Ritz method for Navier-Stokes flow through curved ducts. ANZIAM J.61 (1), 1-22. · Zbl 1409.65070
[14] Harding, B. & Stokes, Y.2018Fluid flow in a spiral microfluidic duct. Phys. Fluids30 (4), 042007.
[15] Ho, B. P. & Leal, L. G.1974Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech.65 (2), 365-400. · Zbl 0284.76076
[16] Hood, K., Lee, S. & Roper, M.2015Inertial migration of a rigid sphere in three-dimensional Poiseuille flow. J. Fluid Mech.765, 452-479. · Zbl 1331.76039
[17] Hood, K. T.2016 Theory of particle focusing in inertial microfluidic devices. PhD thesis, University of California, Los Angeles, CA.
[18] Lashgari, I., Ardekani, M. N., Banerjee, I., Russom, A. & Brandt, L.2017Inertial migration of spherical and oblate particles in straight ducts. J. Fluid Mech.819, 540-561. · Zbl 1383.76493
[19] Liu, C., Xue, C., Sun, J. & Hu, G.2016A generalized formula for inertial lift on a sphere in microchannels. Lab on a Chip16, 884-892.
[20] Logg, A., Mardal, K.-A. & Wells, G. N.2012Automated Solution of Differential Equations by the Finite Element Method. Springer. · Zbl 1247.65105
[21] Martel, J. M. & Toner, M.2012Inertial focusing dynamics in spiral microchannels. Phys. Fluids24 (3), 032001.
[22] Martel, J. M. & Toner, M.2013Particle focusing in curved microfluidic channels. Sci. Rep.3, 3340.
[23] Martel, J. M. & Toner, M.2014Inertial focusing in microfluidics. Annu. Rev. Biomed. Engng16 (1), 371-396.
[24] Matas, J.-P., Morris, J. F. & Guazzelli, É.2004Inertial migration of rigid spherical particles in Poiseuille flow. J. Fluid Mech.515, 171-195. · Zbl 1130.76301
[25] Matas, J.-P., Morris, J. F. & Guazzelli, É.2009Lateral force on a rigid sphere in large-inertia laminar pipe flow. J. Fluid Mech.621, 59-67. · Zbl 1171.76348
[26] Miura, K., Itano, T. & Sugihara-Seki, M.2014Inertial migration of neutrally buoyant spheres in a pressure-driven flow through square channels. J. Fluid Mech.749, 320-330.
[27] Moloudi, R., Oh, S., Yang, C., Warkiani, M. E. & Naing, M. W.2018Inertial particle focusing dynamics in a trapezoidal straight microchannel: application to particle filtration. Microfluid. Nanofluid.22 (33), 1-14.
[28] Nakagawa, N., Yabu, T., Otomo, R., Kase, A., Makino, M., Itano, T. & Sugihara-Seki, M.2015Inertial migration of a spherical particle in laminar square channel flows from low to high Reynolds numbers. J. Fluid Mech.779, 776-793. · Zbl 1360.76342
[29] Nivedita, N., Ligrani, P. & Papautsky, I.2017Dean flow dynamics in low-aspect ratio spiral microchannels. Sci. Rep.7, 44072.
[30] Ookawara, S., Street, D. & Ogawa, K.2006Numerical study on development of particle concentration profiles in a curved microchannel. Chem. Engng Sci.61 (11), 3714-3724.
[31] Prohm, C. & Stark, H.2014Feedback control of inertial microfluidics using axial control forces. Lab on a Chip14, 2115-2123.
[32] Ramachandraiah, H., Ardabili, S., Faridi, A. M., Gantelius, J., Kowalewski, J. M., Mårtensson, G. & Russom, A.2014Dean flow-coupled inertial focusing in curved channels. Biomicrofluidics8 (3), 034117.
[33] Russom, A., Gupta, A. K., Nagrath, S., Di Carlo, D., Edd, J. F. & Toner, M.2009Differential inertial focusing of particles in curved low-aspect-ratio microchannels. New J. Phys.11 (7), 075025.
[34] Schonberg, J. A. & Hinch, E. J.1989Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech.203, 517-524. · Zbl 0675.76038
[35] Segre, G. & Silberberg, A.1961Radial particle displacements in Poiseuille flow of suspensions. Nature189 (4760), 209-210.
[36] Sofela, S., Sahloul, S., Rafeie, M., Kwon, T., Han, J., Warkiani, M. E. & Song, Y.-A.2018High-throughput sorting of eggs for synchronization of C. Elegans in a microfluidic spiral chip. Lab on a Chip18, 679-687.
[37] Warkiani, M. E., Guan, G., Luan, K. B., Lee, W. C., Bhagat, A. A. S., Kant Chaudhuri, P., Tan, D. S.-W., Lim, W. T., Lee, S. C., Chen, P. C. Y.2014Slanted spiral microfluidics for the ultra-fast, label-free isolation of circulating tumor cells. Lab on a Chip14, 128-137.
[38] Winters, K. H.1987A bifurcation study of laminar flow in a curved tube of rectangular cross-section. J. Fluid Mech.180, 343-369. · Zbl 0632.76032
[39] Wu, L., Guan, G., Hou, H. W., Bhagat, A. A. S. & Han, J.2012Separation of leukocytes from blood using spiral channel with trapezoid cross-section. Analyt. Chem.84 (21), 9324-9331.
[40] Yamamoto, K., Wu, X., Hyakutake, T. & Yanase, S.2004Taylor-Dean flow through a curved duct of square cross section. Fluid Dyn. Res.35 (2), 67-86. · Zbl 1072.76029
[41] Yanase, S., Goto, N. & Yamamoto, K.1989Dual solutions of the flow through a curved tube. Fluid Dyn. Res.5 (3), 191-201.
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