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Spheres in the vicinity of a bifurcation: elucidating the Zweifach–Fung effect

Published online by Cambridge University Press:  17 March 2011

V. DOYEUX ,
T. PODGORSKI ,
S. PEPONAS ,
M. ISMAIL  and
G. COUPIER
V. DOYEUX
Affiliation:
Université Grenoble 1 / CNRS, Laboratoire Interdisciplinaire de Physique / UMR 5588, Grenoble, F-38041, France
T. PODGORSKI
Affiliation:
Université Grenoble 1 / CNRS, Laboratoire Interdisciplinaire de Physique / UMR 5588, Grenoble, F-38041, France
S. PEPONAS
Affiliation:
Université Grenoble 1 / CNRS, Laboratoire Interdisciplinaire de Physique / UMR 5588, Grenoble, F-38041, France
M. ISMAIL
Affiliation:
Université Grenoble 1 / CNRS, Laboratoire Interdisciplinaire de Physique / UMR 5588, Grenoble, F-38041, France
G. COUPIER*
Affiliation:
Université Grenoble 1 / CNRS, Laboratoire Interdisciplinaire de Physique / UMR 5588, Grenoble, F-38041, France
*
†Email address for correspondence: gwennou.coupier@ujf-grenoble.fr
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Abstract

The problem of the splitting of a suspension in bifurcating channels divided into two branches of non-equal flow rates is addressed. As has long been observed, in particular in blood flow studies, the volume fraction of particles generally increases in the high-flow-rate branch and decreases in the low-flow-rate branch. In the literature, this phenomenon is sometimes interpreted as the result of some attraction of the particles towards this high-flow-rate branch. In this paper, we focus on the existence of such an attraction through microfluidic experiments and two-dimensional simulations and show clearly that such an attraction does not occur but is, on the contrary, directed towards the low-flow-rate branch. Arguments for this attraction are given and a discussion on the sometimes misleading arguments found in the literature is given. Finally, the enrichment of particles in the high-flow-rate branch is shown to be mainly a consequence of the initial distribution in the inlet branch, which shows necessarily some depletion near the walls.

JFM classification

Biological Fluid Dynamics: Blood flow Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 359 - 388
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98, 188302.CrossRefGoogle ScholarPubMed
Abkarian, M., Lartigue, C. & Viallat, A. 2002 Tank treading and unbinding of deformable vesicles in shear flow: determination of the lift force. Phys. Rev. Lett. 88, 068103.CrossRefGoogle ScholarPubMed
Angelova, M. I., Soleau, S., Meleard, P., Faucon, J. F. & Bothorel, P. 1992 Preparation of giant vesicles by external AC electric fields: kinetics and applications. Prog. Colloid. Polym. Sci. 89, 127–131.CrossRefGoogle Scholar
Asmolov, E. S. 2002 The inertial lift on a small particle in a weak-shear parabolic flow. Phys. Fluids 14, 15–28.CrossRefGoogle Scholar
Audet, D. M. & Olbricht, W. L. 1987 The motion of model cells at capillary bifurcations. Microvasc. Res. 33, 377–396.CrossRefGoogle ScholarPubMed
Bagchi, P. 2007 Mesoscale simulation of blood flow in small vessels. Biophys. J. 92, 1858–1877.CrossRefGoogle ScholarPubMed
Balvin, M., Sohn, E., Iracki, T., Drazer, G. & Frechette, J. 2009 Directional locking and the role of irreversible interactions in deterministic hydrodynamics separations in microfluidic devices. Phys. Rev. Lett. 103, 078301.CrossRefGoogle ScholarPubMed
Barber, J. O., Alberding, J. P., Restrepo, J. M. & Secomb, T. W. 2008 Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations. Ann. Biomech. Engng 36, 1690–1698.CrossRefGoogle ScholarPubMed
Beaucourt, J., Rioual, F., Séon, T., Biben, T. & Misbah, C. 2004 Steady to unsteady dynamics of a vesicle in a flow. Phys. Rev. E 69, 011906.CrossRefGoogle ScholarPubMed
Biben, T., Farutin, A. & Misbah, C. 2011 Numerical study of 3D vesicles under flow: discovery of new peculiar behaviors. Phys. Rev. E (in press) arXiv:0912.4702.Google Scholar
Bugliarello, G. & Hsiao, G. C. C. 1964 Phase separation in suspensions flowing through bifurcations: a simplified hemodynamic model. Science 143, 469–471.CrossRefGoogle ScholarPubMed
Callens, N., Minetti, C., Coupier, G., Mader, M.-A., Dubois, F., Misbah, C. & Podgorski, T. 2008 Hydrodynamic lift of vesicles under shear flow in microgravity. Europhys. Lett. 83, 24002.CrossRefGoogle Scholar
Carr, R. T. & Wickham, L. L. 1990 Plasma skimming in serial microvascular bifurcations. Microvasc. Res. 40, 179–190.CrossRefGoogle ScholarPubMed
Chesnutt, J. K. W. & Marshall, J. S. 2009 Effect of particle collisions and aggregation on red blood cell passage through a bifurcation. Microvasc. Res. 78, 301–313.CrossRefGoogle ScholarPubMed
Chien, S., Tvetenstrand, C. D., Epstein, M. A. & Schmid-Schonbein, G. W. 1985 Model studies on distributions of blood cells at microvascular bifurcations. Am. J. Physiol. Heart Circ. Physiol. 248, H568–H576.CrossRefGoogle ScholarPubMed
Coupier, G., Kaoui, B., Podgorski, T. & Misbah, C. 2008 Noninertial lateral migration of vesicles in bounded Poiseuille flow. Phys. Fluids 20, 111702.CrossRefGoogle Scholar
Danker, G. & Misbah, C. 2007 Rheology of a dilute suspension of vesicles. Phys. Rev. Lett. 98, 088104.CrossRefGoogle ScholarPubMed
Danker, G., Vlahovska, P. M. & Misbah, C. 2009 Vesicles in Poiseuille flow. Phys. Rev. Lett. 102, 148102.CrossRefGoogle ScholarPubMed
Davis, J. A., Inglis, D. W., Morton, K. J., Lawrence, D. A., Huang, L. R., Chou, S. Y., Sturm, J. C. & Austin, R. H. 2006 Deterministic hydrodynamics: taking blood apart. Proc. Natl Acad. Sci. USA 103, 14779–14784.CrossRefGoogle ScholarPubMed
Dellimore, J. W., Dunlop, M. J. & Canham, P. B. 1983 Ratio of cells and plasma in blood flowing past branches in small plastic channels. Am. J. Physiol. Heart Circ. Physiol. 244, H635–H643.CrossRefGoogle ScholarPubMed
Deschamps, J., Kantsler, V., Segre, E. & Steinberg, V. 2009 Dynamics of a vesicle in general flow. Proc. Natl Acad. Sci. USA 106, 11444.CrossRefGoogle ScholarPubMed
Ditchfield, R. & Olbricht, W. L. 1996 Effects of particle concentration on the partitioning of suspensions at small divergent bifurcations. J. Biomech. Engng 118, 287–294.CrossRefGoogle ScholarPubMed
Dupire, J., Abkarian, M. & Viallat, A. 2010 Chaotic dynamics of red blood cells in a sinusoidal flow. Phys. Rev. Lett. 104, 168101.CrossRefGoogle Scholar
El-Kareh, A. W. & Secomb, T. W. 2000 A model for red blood cell motion in bifurcating microvessels. Intl J. Multiphase Flow 26, 1545–1564.CrossRefGoogle Scholar
Eloot, S., De Bisschop, F. & Verdonck, P. 2004 Experimental evaluation of the migration of spherical particles in three-dimensional Poiseuille flow. Phys. Fluids 16, 2282–2293.CrossRefGoogle Scholar
Enden, G. & Popel, A. S. 1992 A numerical study of the shape of the surface separating flow into branches in microvascular bifurcations. J. Biomech. Engng 114, 398–405.CrossRefGoogle ScholarPubMed
Engl, W., Roche, M., Colin, A., Panizza, P. & Ajdari, A. 2005 Droplet traffic at a simple junction at low capillary numbers. Phys. Rev. Lett. 95, 208304.CrossRefGoogle Scholar
Fan, R., Vermesh, O., Srivastava, A., Yen, B., Qin, L., Ahmad, H., Kwong, G. A., Liu, C.-C., Gould, J., Hood, L. & Heath, J. R. 2008 Integrated barcode chips for rapid, multiplexed analysis of proteins in microliter quantities of blood. Nature Biotech. 26, 1373–1378.CrossRefGoogle ScholarPubMed
Farutin, A., Biben, T. & Misbah, C. 2010 Analytical progress in the theory of vesicles under linear flow. Phys. Rev. E 81, 061904.CrossRefGoogle ScholarPubMed
Fenton, B. M., Carr, R. T. & Cokelet, G. R. 1985 Nonuniform red cell distribution in 20 to 100 μm bifurcations. Microvasc. Res. 29, 103–126.CrossRefGoogle Scholar
Frechette, J. & Drazer, G. 2009 Directional locking and deterministic separation in periodic arrays. J. Fluid Mech. 627, 379–401.CrossRefGoogle Scholar
Fung, Y. C. 1973 Stochastic flow in capillary blood vessels. Microvasc. Res. 5, 34–48.CrossRefGoogle ScholarPubMed
Fung, Y. C. 1993 Biomechanics: Mechanical Properties of Living Tissues. Springer.CrossRefGoogle Scholar
Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.CrossRefGoogle Scholar
Girault, V. & Raviart, P.A. 1986 Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer.CrossRefGoogle Scholar
Griggs, A. J., Zinchenko, A. Z. & Davis, R. H. 2007 Low-Reynolds-number motion of a deformable drop between two parallel plane walls. Intl J. Multiphase Flow 33, 182–206.CrossRefGoogle Scholar
Guibert, R., Fonta, C. & Plouraboue, F. 2010 A new approach to model confined suspensions flows in complex networks: application to blood flow. Trans. Porous Med. 83, 171–194.CrossRefGoogle Scholar
Hecht, F. & Pironneau, O. 2010 A finite element software for PDEs: freeFEM++. Available at: http://www.freefem.org/.Google Scholar
Inglis, D. W. 2009 Efficient microfluidic particle separation arrays. Appl. Phys. Lett. 94, 013510.CrossRefGoogle Scholar
Jäggi, R. D., Sandoz, R. & Effenhauser, C. S. 2007 Microfluidic depletion of red blood cells from whole blood in high-aspect-ratio microchannels. Microfluid Nanofluid 3, 47–53.CrossRefGoogle Scholar
Janela, J., Lefebvre, A. & Maury, B. 2005 A penalty method for the simulation of fluid-rigid body interaction. In ESAIM: Proc (ed. Cancès, E. & Gerbeau, J.-F.), vol. 14, pp. 115–123. ESAIM.Google Scholar
Jousse, F., Farr, R., Link, D. R., Fuerstman, M. J. & Garstecki, P. 2006 Bifurcation of droplet flows within capillaries. Phys. Rev. E 74, 036311.CrossRefGoogle ScholarPubMed
Kantsler, V., Segre, E. & Steinberg, V. 2008 Dynamics of interacting vesicles and rheology of vesicle suspension in shear flow. Europhys. Lett. 82, 58005.CrossRefGoogle Scholar
Kaoui, B., Coupier, G., Misbah, C. & Podgorski, T. 2009 Lateral migration of vesicles in microchannels: effects of walls and shear gradient. Houille Blanche 5, 112–119.CrossRefGoogle Scholar
Kaoui, B., Ristow, G., Cantat, I., Misbah, C. & Zimmermann, W. 2008 Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow. Phys. Rev. E 77, 021903.CrossRefGoogle ScholarPubMed
Kersaudy-Kerhoas, M., Dhariwal, R., Desmulliez, M. P. Y. & Jouvet, L. 2010 Hydrodynamic blood plasma separation in microfluidic channels. Microfluid Nanofluid 8, 105–114.CrossRefGoogle Scholar
Kim, Y. W. & Yoo, J. Y. 2008 The lateral migration of neutrally-buoyant spheres transported through square microchannels. J. Micromech. Microengng 18, 065015.CrossRefGoogle Scholar
Lefebvre, A. 2007 Fluid–particle simulations with FreeFem++. In ESAIM: Proc., vol. 18, pp. 120–132. ESAIM.Google Scholar
Maury, B. 2009 Numerical analysis of a finite element/volume penalty method. SIAM J. Numer. Anal. 47 (2), 1126–1148.CrossRefGoogle Scholar
Mayrovitz, H. N. & Roy, J. 1983 Microvascular blood flow: evidence indicating a cubic dependence on arteriolar diameter. Am. J. Physiol. 255, H1031–H1038.Google Scholar
McWhirter, J. L., Noguchi, H. & Gompper, G. 2009 Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries. Proc. Natl Acad. Sci. USA 106, 6039–6043.CrossRefGoogle ScholarPubMed
Mortazavi, S. & Tryggvason, G. 2000 A numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop. J. Fluid. Mech. 411, 325–350.CrossRefGoogle Scholar
Noguchi, H. 2010 Dynamic modes of red blood cells in oscillatory shear flow. Phys. Rev. E 81, 061920.CrossRefGoogle ScholarPubMed
Obrist, D., Weber, B., Buck, A. & Jenny, P. 2010 Red blood cell distribution in simplified capillary networks. Phil. Trans. R. Soc. Lond. A 368, 2897–2918.Google ScholarPubMed
Olla, P. 1997 The lift on a tank-treading ellipsoidal cell in a shear flow. J. Phys. II Paris 7, 1533–1540.Google Scholar
Pamme, N. 2007 Continuous flow separations in microfluidic devices. Lab Chip 7, 1644–1659.CrossRefGoogle ScholarPubMed
Peyla, P. 2007 A deformable object in a microfluidic configuration: a numerical study. Europhys. Lett. 80, 34001.CrossRefGoogle Scholar
Podgorski, T., Callens, N., Minetti, C., Coupier, G., Dubois, F. & Misbah, C. 2010 Dynamics of vesicle suspensions in shear flow between walls. Microgravity Sci. Technol. 23, 263–270.CrossRefGoogle Scholar
Pries, A. R., Ley, K., Claassen, M. & Gaethgens, P. 1989 Red cell distribution at microvascular bifurcations. Microvasc. Res. 38, 81–101.CrossRefGoogle ScholarPubMed
Pries, A. R., Secomb, T. W. & Gaethgens, P. 1996 Biophysical aspects of blood flow in the microvasculature. Cardiovasc. Res. 32, 654–667.CrossRefGoogle ScholarPubMed
Risso, F., Collé-Paillot, F. & Zagzoule, M. 2006 Experimental investigation of a bioartificial capsule flowing in a narrow tube. J. Fluid. Mech. 547, 149–173.CrossRefGoogle Scholar
Roberts, B. W. & Olbricht, W. L. 2003 Flow-induced particulate separations. AIChE J. 49, 2842–2849.CrossRefGoogle Scholar
Roberts, B. W. & Olbricht, W. L. 2006 The distribution of freely suspended particles at microfluidic bifurcations. AIChE J. 52, 199–206.CrossRefGoogle Scholar
Schindler, M. & Ajdari, A. 2008 Droplet traffic in microfluidic networks: a simple model for understanding and designing. Phys. Rev. Lett. 100, 044501.CrossRefGoogle ScholarPubMed
Schonberg, J. A. & Hinch, E. J. 1989 Inertial migration of a sphere in a Poiseuille flow. J. Fluid Mech. 203, 517–524.CrossRefGoogle Scholar
Secomb, T. W., Styp-Rekowska, B. & Pries, A. R. 2007 Two-dimensional simulation of red blood cell deformation and lateral migration in microvessels. Ann. Biomed. Engng 35, 755–765.CrossRefGoogle ScholarPubMed
Sessoms, D. A., Belloul, M., Engl, W., Roche, M., Courbin, L. & Panizza, P. 2009 Droplet motion in microfluidic networks: hydrodynamic interactions and pressure-drop measurements. Phys. Rev. E 80, 016317.CrossRefGoogle ScholarPubMed
Svanes, K. & Zweifach, B. W. 1968 Variations in small blood vessel hematocrits produced in hypothermic rats by micro-occlusion. Microvasc. Res. 1, 210–220.CrossRefGoogle Scholar
Tanaka, H. & Araki, T. 2000 Simulation method of colloidal suspensions with hydrodynamic interactions: fluid particle dynamics. Phys. Rev. Lett. 85 (6), 1338–1341.CrossRefGoogle ScholarPubMed
Vitkova, V., Mader, M.-A., Polack, B., Misbah, C. & Podgorski, T. 2008 Micro–macro link in rheology of erythrocyte and vesicle suspensions. Biophys. J. 95, 33–35.CrossRefGoogle ScholarPubMed
Vlahovska, P. M., Podgorski, T. & Misbah, C. 2009 Vesicles and red blood cells in flow: from individual dynamics to rheology. C. R. Physique 10, 775–789.Google Scholar
White, F. M. 1991 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Yang, S., Ündar, A. & Zahn, J. D. 2006 A microfluidic device for continuous, real time blood plasma separation. Lab Chip 6, 871–880.CrossRefGoogle ScholarPubMed
Yang, S. & Zahn, J. D. 2004 Particle separation in microfluidic channels using flow rate control. In Proc. ASME Conf. International Mechanical Engineering Exp Fluids Engineering (IMECE), Anaheim, CA.Google Scholar
Yen, R. T. & Fung, Y. C. 1978 Effect of velocity distribution on red cell distribution in capillary blood vessels. Am. J. Physiol. Heart Circ. Physiol. 235, H251–H257.CrossRefGoogle ScholarPubMed
Yoo, J. Y. & Kim, Y. W. 2010 Two-phase flow laden with spherical particles in a microcapillary. Intl J. Multiphase Flow 36, 460–466.Google Scholar
Zheng, S., Liu, J.-Q. & Tai, Y.-C. 2008 Streamline-based microfluidic devices for erythrocytes and leukocytes separation. J. Microelectromech. Syst. 17, 1029–1038.CrossRefGoogle Scholar