Hostname: page-component-84b7d79bbc-7nlkj Total loading time: 0 Render date: 2024-07-26T12:38:08.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Pressure-driven flow past spheres moving in a circular tube

Published online by Cambridge University Press:  14 November 2007

G. J. SHEARD  and
K. RYAN
G. J. SHEARD
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical Engineering, Monash University, Melbourne, Victoria 3800, Australia Monash University Biomedical Engineering Technology Alliance (MuBeta), Monash University, Melbourne, Victoria 3800, Australia
K. RYAN
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical Engineering, Monash University, Melbourne, Victoria 3800, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

A computational investigation, supported by a theoretical analysis, is performed to investigate a pressure-driven flow around a line of equispaced spheres moving at a prescribed velocity along the axis of a circular tube. This fundamental study underpins a range of applications including physiological circulation research. A spectral-element formulation in cylindrical coordinates is employed to solve for the incompressible fluid flow past the spheres, and the flows are computed in the reference frame of the translating spheres.

Both the volume flow rate relative to the spheres and the forces acting on each sphere are computed for specific sphere-to-tube diameter ratios and sphere spacing ratios. Conditions at which zero axial force on the spheres are identified, and a region of unsteady flow is detected at higher Reynolds numbers (based on tube diameter and sphere velocity). A regular perturbation analysis and the reciprocal theorem are employed to predict flow rate and drag coefficient trends at low Reynolds numbers. Importantly, the zero drag condition is well-described by theory, and states that at this condition, the sphere velocity is proportional to the applied pressure gradient. This result was verified for a range of spacing and diameter ratios. Theoretical approximations agree with computational results for Reynolds numbers up to O(100).

The geometry dependence of the zero axial force condition is examined, and for a particular choice of the applied dimensionless pressure gradient, it is found that this condition occurs at increasing Reynolds numbers with increasing diameter ratio, and decreasing Reynolds number with increasing sphere spacing.

Three-dimensional simulations and predictions of a Floquet linear stability analysis independently elucidate the bifurcation scenario with increasing Reynolds number for a specific diameter ratio and sphere spacing. The steady axisymmetric flow first experiences a small region of time-dependent non-axisymmetric instability, before undergoing a regular bifurcation to a steady non-axisymmetric state with azimuthal symmetry m = 1. Landau modelling verifies that both the regular non-axisymmetric mode and the axisymmetric Hopf transition occur through a supercritical (non-hysteretic) bifurcation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 592 , 10 December 2007 , pp. 233 - 262
Copyright
Copyright © Cambridge University Press 2007

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

Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Blackburn, H. M. & Lopez, J. M. 2003 On three-dimensional quasi-periodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15 (8), L57–L60.CrossRefGoogle Scholar
Blackburn, H. M. & Sherwin, S. J. 2004 Formulation of a Galerkin spectral element–Fourier method for three-dimensional incompressible flow in cylindrical geometries. J. Comput. Phys. 197, 759–778.CrossRefGoogle Scholar
Cai, X. & Wallis, G. B. 1992 Potential flow around a row of spheres in a circular tube. Phys. Fluids A 4 (5), 904–912.CrossRefGoogle Scholar
Charm, S. & Kurland, G. 1965 Viscometry of human blood for shear rates of 0–100000 sec−1. Nature 206, 617–618.CrossRefGoogle Scholar
Charm, S. & Kurland, G. S. 1968 Discrepancy in measuring blood in a Couette cone and plate, and capillary tube viscometers. J. Appl. Physiol. 25, 786–789.CrossRefGoogle Scholar
Charm, S. E., McComis, W. & Kurland, G. 1964 Rheology and structure of blood suspensions. J. Appl. Physiol. 19, 127–133.CrossRefGoogle ScholarPubMed
Fitz-Gerald, J. M. 1969 Mechanics of red-cell motion through very narrow capillaries. Proc. R. Soc. Lond. B. 174, 193–227.Google ScholarPubMed
Ghidersa, B. & Dušek, J. 2000 Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere. J. Fluid Mech. 423, 33–69.CrossRefGoogle Scholar
Happell, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8, 1683–1685.CrossRefGoogle Scholar
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70.CrossRefGoogle Scholar
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–443.CrossRefGoogle Scholar
Leshansky, A. M. & Brady, J. F. 2004 Force on a sphere via the generalized reciprocal theorem. Phys. Fluids 16 (3), 843–844.CrossRefGoogle Scholar
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34, 113–143.CrossRefGoogle Scholar
Long, D. S., Smith, M. L., Pries, A. R., Ley, K. & Damiano, E. R. 2004 Microviscometry reveals reduced blood viscosity and altered shear rate and shear stress profiles in micro-vessels after hemodilution. Proc. Natl Acad. Sci. USA 101, 10060–10065.CrossRefGoogle Scholar
Lovalenti, P. M. & Brady, J. F. 1993 The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J. Fluid Mech. 256, 561–605.CrossRefGoogle Scholar
Magnaudet, J. 2003 Small inertial effects on a spherical bubble, drop or particle moving near a wall in a time-dependent linear flow. J. Fluid Mech. 485, 115–142.CrossRefGoogle Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Ortega, J. M., Bristol, R. L. & Savas, O. 1998 Flow resistance and drag forces due to multiple adherent leukocytes in postcapillary vessels. Biophys. J. 74, 3292–3301.Google Scholar
Piercy, N. A. V., Hooper, M. S. & Winney, H. F. 1933 Viscous flow through pipes with cores. Phil. Mag. 15 (7), 647–676.CrossRefGoogle Scholar
Pozirikidis, C. 2005 Numerical simulation of cell motion in tube flow. Ann. Biomed. Engng 33 (2), 165–178.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Pulley, J. W., Hussey, R. G. & Davis, A. M. J. 1996 Low nonzero Reynolds number drag on a thin disk settling axisymmetrically within a cylindrical outer boundary. Phys. Fluids 8 (9), 2275–2283.CrossRefGoogle Scholar
Rosenson, R. S., McCormick, A. & Uretz, E. F. 1996 Distribution of blood viscosity values and biochemical correlates in healthy adults. Clin. Chem. 42, 1189–1195.CrossRefGoogle ScholarPubMed
Secomb, T. W., Hsu, R. & Pries, A. R. 1998 A model for red blood cell motion in glycocalyx-lined capillaries. Am. J. Physiol. Heart Circ. Physiol. 274, 1016–1022.CrossRefGoogle Scholar
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. Physiol. Heart Circ. Physiol. 281, 629–636.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003 From spheres to circular cylinders: the stability and flow structures of bluff ring wakes. J. Fluid Mech. 492, 147–180.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2004 From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings. J. Fluid Mech. 506, 45–78.CrossRefGoogle Scholar
Sheard, G. J., Hourigan, K. & Thompson, M. C. 2005 Computations of the drag coefficients for the low-Reynolds-number flow past rings. J. Fluid Mech. 526, 257–275.CrossRefGoogle Scholar
Sheard, G. J., Leweke, T., Thompson, M. C. & Hourigan, K. 2007 Flow around an impulsively arrested circular cylinder. Phys. Fluids 19 (8), 083601.CrossRefGoogle Scholar
Sherwin, S. J. & Blackburn, H. M. 2005 Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows. J. Fluid Mech. 533, 297–327.CrossRefGoogle Scholar
Skalak, R. & Branemark, P. I. 1969 Deformation of red blood cells in capillaries. Science 164, 717–719.CrossRefGoogle ScholarPubMed
Smythe, W. R. 1961 Flow around a sphere in a circular tube. Phys. Fluids 4 (6), 756–759.CrossRefGoogle Scholar
Smythe, W. R. 1964 Flow around a spheroid in a circular tube. Phys. Fluids 7 (5), 633–638.CrossRefGoogle Scholar
Thompson, M. C., Leweke, T. & Provansal, M. 2001 Kinematics and dynamics of sphere wake transition. J. Fluids Struct. 15, 575–585.CrossRefGoogle Scholar
Tomboulides, A. G., Orszag, S. A. & Karniadakis, G. E. 1993 Direct and large-eddy simulation of the flow past a sphere. In Proc. Second Intl Conf. on Turbulence Modeling and Experiments (2nd ICTME). Florence, Italy.Google Scholar
Tözeren, H. 1983 Drag on eccentrically positioned spheres translating and rotating in tubes. J. Fluid Mech. 129, 77–90.CrossRefGoogle Scholar
Tözeren, H. & Skalak, R. 1978 The steady flow of closely fitting incompressible elastic spheres in a tube. J. Fluid Mech. 87, 1–16.CrossRefGoogle Scholar
Tözeren, H. & Skalak, R. 1979 Flow of elastic compressible spheres in tubes. J. Fluid Mech. 95, 743–760.CrossRefGoogle Scholar
Vink, H. & Duling, B. R. 1996 Identification of distinct luminal domains for macromolecules, erythrocytes and leukocytes within mammalian capillaries. Circ. Res. 79 (3), 581–589.CrossRefGoogle ScholarPubMed
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75–96.CrossRefGoogle Scholar
Wang, W. & Parker, K. H. 1998 Movement of spherical particles in capillaries using a boundary singularity method. J. Biomech. 31, 347–354.CrossRefGoogle ScholarPubMed
Williamson, C. H. K. 1988 Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids 31, 2742–2744.CrossRefGoogle Scholar
Wu, J. Z. & Wu, J. M. 1996 Vorticity dynamics on boundaries. Adv. Appl. Mech. 32, 119–275.CrossRefGoogle Scholar
Zang, T. A. 1991 On the rotation and skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7, 27–40.CrossRefGoogle Scholar