Hostname: page-component-6bb9c88b65-znhjv Total loading time: 0 Render date: 2025-07-24T04:21:58.682Z Has data issue: false hasContentIssue false
  • English
  • Français

The steady motion of a closely fitting vesicle in a tube

Published online by Cambridge University Press:  28 November 2017

Joseph M. Barakat Open the ORCID record for Joseph M. Barakat [Opens in a new window]  and
Eric S. G. Shaqfeh
Joseph M. Barakat
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
Eric S. G. Shaqfeh*
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: esgs@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A singular perturbation theory is developed for the steady, inertialess motion of a lipid-bilayer vesicle flowing through a narrow tube. The vesicle is treated as a sac of fluid enclosed by an inextensible membrane that admits a bending stiffness. Matched asymptotic expansions are developed in terms of a clearance parameter $\unicode[STIX]{x1D716}\ll 1$ in order to calculate the flow field and vesicle shape. Mild restrictions are applied to the ratio of viscosities $\unicode[STIX]{x1D705}$ and the ratio of bending stresses to viscous stresses $\unicode[STIX]{x1D6FD}$; in particular, the theory holds for $\unicode[STIX]{x1D705}=o(\unicode[STIX]{x1D716}^{-1/2})$ and $\unicode[STIX]{x1D6FD}=O(\unicode[STIX]{x1D716}^{-1})$. The ratio of the vesicle length to the tube radius $\ell$ is included as a parameter and asymptotic solutions in the limit of negligible bending stiffness are developed for long, cylindrical vesicles and short, spherical vesicles. The main result of the theory is a prediction for the vesicle speed and extra pressure drop due to the presence of the vesicle in the tube. The effects of confinement, vesicle length, and membrane bending elasticity are examined. The theoretical predictions show good agreement with experimental measurements reported for vesicles and red blood cells in highly confined channel flow. Previously reported models for red blood cells (Secomb et al., J. Fluid Mech., vol. 163, 1986, pp. 405–423; Halpern & Secomb, J. Fluid Mech., vol. 203, 1989, pp. 381–400) are clarified and extended in light of the new theory.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 721 - 761
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abkarian, M., Faivre, M., Horton, R., Smistrup, K., Best-Popescu, C. A. & Stone, H. A. 2008 Cellular-scale hydrodynamics. Biomed. Mater. 3 (3), 034011.CrossRefGoogle ScholarPubMed
Abkarian, M., Faivre, M. & Stone, H. A. 2006 High-speed microfluidic differential manometer for cellular-scale hydrodynamics. Proc. Natl Acad. Sci. USA 103 (3), 538–542.CrossRefGoogle ScholarPubMed
Ahmmed, S., Suteria, N. S., Garbin, V. & Vanapalli, S. A.2017 Hydrodynamic mobility of confined polymeric particles, vesicles, and cancer cells in a square microchannel. arXiv:1711.02187.Google Scholar
Albrecht, K. H., Gaehtgens, P., Pries, A. & Heuser, M. 1979 The Fahraeus effect in narrow capillaries (i.d. 3. 3–11. 0 μm). Microvasc. Res. 18 (1), 33–47.CrossRefGoogle Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Bruinsma, R. 1996 Rheology and shape transitions of vesicles under capillary flow. Physica A 234 (1–2), 249–270.CrossRefGoogle Scholar
Buckingham, E. 1914 On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4 (4), 345–376.CrossRefGoogle Scholar
Bungay, P. M. & Brenner, H. 1973 The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiphase Flow 1 (1), 25–56.CrossRefGoogle Scholar
Canham, P. B. & Burton, A. C. 1968 Distribution of size and shape in populations of normal human red cells. Circulat. Res 22 (3), 405–422.CrossRefGoogle ScholarPubMed
Cantat, I. 2013 Liquid meniscus friction on a wet plate: bubbles, lamellae, and foams. Phys. Fluids 25 (3), 031303.CrossRefGoogle Scholar
Dixit, H. N. & Homsy, G. M. 2013a The elastic Landau–Levich problem. J. Fluid Mech. 732, 5–28.CrossRefGoogle Scholar
Dixit, H. N. & Homsy, G. M. 2013b The elastocapillary Landau–Levich problem. J. Fluid Mech. 735, 1–28.CrossRefGoogle Scholar
van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Evans, E. & Needham, D. 1987 Physical properties of surfactant bilayer membranes: thermal transitions, elasticity, rigidity, cohesion and colloidal interactions. J. Phys. Chem. 91 (16), 4219–4228.CrossRefGoogle Scholar
Halpern, D. & Secomb, T. W. 1989 The squeezing of red blood cells through capillaries with near-minimal diameters. J. Fluid Mech. 203, 381–400.CrossRefGoogle Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. 28c (11), 693–703.CrossRefGoogle Scholar
Hochmuth, R. M. 2000 Micropipette aspiration of living cells. J. Biomech. 33 (1), 15–22.CrossRefGoogle ScholarPubMed
Hochmuth, R. M. & Sutera, S. P. 1970 Spherical caps in low Reynolds-number tube flow. Chem. Engng Sci. 25 (4), 593–604.CrossRefGoogle Scholar
Jenkins, J. T. 1977 Static equilibrium configurations of a model red blood cell. J. Math. Biol. 4 (2), 149–169.CrossRefGoogle ScholarPubMed
Landau, L. D. & Levich, V. G. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34 (1), 113–143.CrossRefGoogle Scholar
Mohandas, N. & Evans, E. 1994 Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annu. Rev. Biophys. Biomol. Struct. 23, 787–818.CrossRefGoogle ScholarPubMed
Morrison, D. D., Riley, J. D. & Zancanaro, J. F. 1962 Multiple shooting method for two-point boundary value problems. Comm. ACM 5 (12), 613–614.CrossRefGoogle Scholar
Noguchi, H. & Gompper, G. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. USA 102 (40), 14159–14164.CrossRefGoogle ScholarPubMed
O’Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705–724.CrossRefGoogle Scholar
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139, 291–308.CrossRefGoogle Scholar
Pozrikidis, C. 2005 Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17 (3), 1503.CrossRefGoogle Scholar
Pries, A. R., Neuhaus, D. & Gaehtgens, P. 1992 Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Phys. 263 (6), H1770–H1778.Google ScholarPubMed
Quéguiner, C. & Barthès-Biesel, D. 1997 Axisymmetric motion of capsules through cylindrical channels. J. Fluid Mech. 348, 349–376.CrossRefGoogle Scholar
Ratulowski, J. & Chang, H. C. 1990 Marangoni effects of trace impurities on the motion of long gas bubbles in capillaries. J. Fluid Mech. 210, 303–328.CrossRefGoogle Scholar
Savin, T., Bandi, M. M. & Mahadevan, L. 2016 Pressure-driven occlusive flow of a confined red blood cell. Soft Matt. 12 (2), 562–573.CrossRefGoogle ScholarPubMed
Schwartz, L. W., Princen, H. M. & Kiss, A. D. 1986 On the motion of bubbles in capillary tubes. J. Fluid Mech. 172, 259–275.CrossRefGoogle Scholar
Secomb, T. W. 1988 Interaction between bending and tension forces in bilayer membranes. Biophys. J. 54 (4), 743–746.CrossRefGoogle ScholarPubMed
Secomb, T. W., Skalak, R., Oozkaya, N. & Gross, J. F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.CrossRefGoogle Scholar
Sharei, A., Zoldan, J., Adamo, A., Sim, W. Y., Cho, N., Jackson, E., Mao, S., Schneider, S., Han, M.-J., Lytton-Jean, A. et al. 2013 A vector-free microfluidic platform for intracellular delivery. Proc. Natl Acad. Sci. USA 110 (6), 2082–2087.CrossRefGoogle ScholarPubMed
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77 (3), 977–1026.CrossRefGoogle Scholar
Stoer, J. & Bulirsch, R. 2002 Introduction to Numerical Analysis, vol. 12. Springer Science & Business Media.CrossRefGoogle Scholar
Tomaiuolo, G., Simeone, M., Martinelli, V., Rotoli, B. & Guido, S. 2009 Red blood cell deformation in microconfined flow. Soft Matt. 5 (19), 3736.CrossRefGoogle Scholar
Trozzo, R., Boedec, G., Leonetti, M. & Jaeger, M. 2015 Axisymmetric boundary element method for vesicles in a capillary. J. Comput. Phys. 289, 62–82.CrossRefGoogle Scholar
Vitkova, V., Mader, M. & Podgorski, T. 2004 Deformation of vesicles flowing through capillaries. Europhys. Lett. 68 (3), 398–404.CrossRefGoogle Scholar
Wilson, S. K. 1995 The effect of an axial temperature gradient on the steady motion of a large droplet in a tube. J. Engng Maths 29 (3), 205–217.CrossRefGoogle Scholar
Wong, H., Radke, C. J. & Morris, S. 1995 The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95–110.CrossRefGoogle Scholar
Zhong-can, O.-Y. & Helfrich, W. 1989 Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39 (10), 5280–5288.CrossRefGoogle Scholar