Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-24T21:52:08.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Motion of a two-dimensional elastic capsule in a branching channel flow

Published online by Cambridge University Press:  05 January 2011

H. C. WOOLFENDEN  and
M. G. BLYTH
H. C. WOOLFENDEN
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
M. G. BLYTH*
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The transit of a two-dimensional elastic fluid-filled capsule through a channel with a side branch is investigated numerically. The mathematical formulation allows for a capsule carried in a pressure-driven flow of fluid of generally different viscosity to that inside the capsule. Far upstream and downstream in the main channel, and downstream in the side branch, the fluid velocity profiles are assumed to adopt those of unidirectional Poiseuille flow with prescribed flow rates. The capsule boundary is treated as a two-dimensional elastic membrane developing elastic tensions and bending moments according to simple constitutive laws. A boundary-integral formulation allows for the explicit computation of the fluid pressures upstream and downstream of the branching. The novelty of the approach is the inclusion of a notional boundary at the entrance to the side branch, which avoids the need to collocate the channel ends. The deformation experienced by the capsule in the region of the junction is found to depend strongly on the branch angle. The deformation is ameliorated by increasing the membrane stiffness or lowering the viscosity of the suspending fluid relative to the encapsulated fluid. When a capsule exits the branch region, a distance of many decades of capsule diameters is required before the capsule relaxes to an equilibrium shape. Capsule residence times in the vicinity of the branch region can be considerable, depending on the line of approach into the junction and the capsule deformability. The path selection of a cell at a branch junction can depend crucially on capsule deformability: capsules with different elastic properties may follow different routes out of the junction in otherwise identical flow conditions.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 3 - 31
Copyright
Copyright © Cambridge University Press 2011

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.)

References

REFERENCES

Atkinson, K. E. 1978 An Introduction to Numerical Analysis. John Wiley & Sons.Google Scholar
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. Biomed. Engng 36 (10), 1690–1698.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Breyiannis, G. & Pozrikidis, C. 2000 Simple shear flow of suspensions of elastic capsules. Theor. Comput. Fluid Dyn. 13 (5), 327–347.CrossRefGoogle Scholar
Caro, C. G., Pedley, T. J., Schroter, R. C. & Seed, W. A. 1978 The Mechanics of the Circulation. Oxford University Press.Google Scholar
Cokelet, G. R. 1976 Macroscopic rheology and tube flow of human blood. Microcirculation 1, 9–31.Google 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
Fung, Y. C. 1973 Stochastic flow in capillary blood vessels. Microvasc. Res. 5 (1), 34–48.CrossRefGoogle ScholarPubMed
Howell, P., Kozyreff, G. & Ockendon, J. 2009 Applied Solid Mechanics. Cambridge University Press.Google Scholar
Kiani, M. F. & Cokelet, G. R. 1994 Additional pressure drop at a bifurcation due to the passage of flexible disks in a large scale model. J. Biomech. Engng 116, 497.CrossRefGoogle Scholar
Lipowsky, H. H. & Zweifach, B. W. 1974 Network analysis of microcirculation of cat mesentery. Microvasc. Res. 7 (1), 73–83.CrossRefGoogle ScholarPubMed
Manga, M. 1996 Dynamics of drops in branched tubes. J. Fluid Mech. 315, 105–117.CrossRefGoogle Scholar
Popel, A. S. & Johnson, P. C. 2005 Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269–291.CrossRefGoogle Scholar
Pozrikidis, C. 2002 a A Practical Guide to Boundary-Element Methods with the Software Library BEMLIB. Chapman & Hall/CRC.CrossRefGoogle Scholar
Pozrikidis, C. 2002 b Buckling and collapse of open and closed cylindrical shells. J. Engng Math. 42 (2), 157–180.CrossRefGoogle Scholar
Pozrikidis, C. 2005 a Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17 (3), 031503.CrossRefGoogle Scholar
Pozrikidis, C. 2005 b Computation of Stokes flow due to the motion or presence of a particle in a tube. J. Engng Math. 53, 1–20.CrossRefGoogle Scholar
Pozrikidis, C. 2005 c Numerical simulation of cell motion in tube flow. Ann. Biomed. Engng 33 (2), 165–178.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 2009 Numerical simulation of blood flow through microvascular capillary networks. Bull. Math. Biol. 71 (6), 1520–1541.CrossRefGoogle ScholarPubMed
Pries, A. R., Ley, K. & Gaehtgens, P. 1986 Generalization of the Fahraeus principle for microvessel networks. Am. J. Physiol. 251, 1324–1332.Google ScholarPubMed
Quéguiner, C. & Barthès-Biesel, D. 1997 Axisymmetric motion of capsules through cylindrical channels. J. Fluid Mech. 348, 349–376.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
Saad, Y. 2003 Iterative Methods for Sparse Linear Systems. SIAM.CrossRefGoogle Scholar
Schmid-Schönbein, G. W., Skalak, R., Usami, S. & Chien, S. 1980 Cell distribution in capillary networks. Microvasc. Res. 19 (1), 18–44.CrossRefGoogle ScholarPubMed
Secomb, T. W., Hsu, R. & Pries, A. R. 2001 Motion of red blood cells in a capillary with an endothelial surface layer: effect of flow velocity. Am. J. Phys. 281 (2), 629–636.Google Scholar
Secomb, T. W., Skalak, R., Özkaya, N. & Gross, J. F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.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 (5), 755–765.CrossRefGoogle ScholarPubMed
Woolfenden, H. C. 2010 The transit of an elastic capsule through a bifurcation. PhD thesis, University of East Anglia.Google Scholar