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Pressure-driven flow past spheres moving in a circular tube

  • G. J. SHEARD (a1) (a2) and K. RYAN (a1)


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.



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Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.
Blackburn, H. M. & Lopez, J. M. 2003 On three-dimensional quasi-periodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15 (8), L57L60.
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, 759778.
Cai, X. & Wallis, G. B. 1992 Potential flow around a row of spheres in a circular tube. Phys. Fluids A 4 (5), 904912.
Charm, S. & Kurland, G. 1965 Viscometry of human blood for shear rates of 0–100000 sec−1. Nature 206, 617618.
Charm, S. & Kurland, G. S. 1968 Discrepancy in measuring blood in a Couette cone and plate, and capillary tube viscometers. J. Appl. Physiol. 25, 786789.
Charm, S. E., McComis, W. & Kurland, G. 1964 Rheology and structure of blood suspensions. J. Appl. Physiol. 19, 127133.
Fitz-Gerald, J. M. 1969 Mechanics of red-cell motion through very narrow capillaries. Proc. R. Soc. Lond. B. 174, 193227.
Ghidersa, B. & Dušek, J. 2000 Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere. J. Fluid Mech. 423, 3369.
Happell, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8, 16831685.
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 1970.
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press.
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414443.
Leshansky, A. M. & Brady, J. F. 2004 Force on a sphere via the generalized reciprocal theorem. Phys. Fluids 16 (3), 843844.
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34, 113143.
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, 1006010065.
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, 561605.
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, 115142.
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323344.
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297330.
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, 32923301.
Piercy, N. A. V., Hooper, M. S. & Winney, H. F. 1933 Viscous flow through pipes with cores. Phil. Mag. 15 (7), 647676.
Pozirikidis, C. 2005 Numerical simulation of cell motion in tube flow. Ann. Biomed. Engng 33 (2), 165178.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
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), 22752283.
Rosenson, R. S., McCormick, A. & Uretz, E. F. 1996 Distribution of blood viscosity values and biochemical correlates in healthy adults. Clin. Chem. 42, 11891195.
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, 10161022.
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, 629636.
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, 147180.
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, 4578.
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, 257275.
Sheard, G. J., Leweke, T., Thompson, M. C. & Hourigan, K. 2007 Flow around an impulsively arrested circular cylinder. Phys. Fluids 19 (8), 083601.
Sherwin, S. J. & Blackburn, H. M. 2005 Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows. J. Fluid Mech. 533, 297327.
Skalak, R. & Branemark, P. I. 1969 Deformation of red blood cells in capillaries. Science 164, 717719.
Smythe, W. R. 1961 Flow around a sphere in a circular tube. Phys. Fluids 4 (6), 756759.
Smythe, W. R. 1964 Flow around a spheroid in a circular tube. Phys. Fluids 7 (5), 633638.
Thompson, M. C., Leweke, T. & Provansal, M. 2001 Kinematics and dynamics of sphere wake transition. J. Fluids Struct. 15, 575585.
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.
Tözeren, H. 1983 Drag on eccentrically positioned spheres translating and rotating in tubes. J. Fluid Mech. 129, 7790.
Tözeren, H. & Skalak, R. 1978 The steady flow of closely fitting incompressible elastic spheres in a tube. J. Fluid Mech. 87, 116.
Tözeren, H. & Skalak, R. 1979 Flow of elastic compressible spheres in tubes. J. Fluid Mech. 95, 743760.
Vink, H. & Duling, B. R. 1996 Identification of distinct luminal domains for macromolecules, erythrocytes and leukocytes within mammalian capillaries. Circ. Res. 79 (3), 581589.
Wang, H. & Skalak, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 7596.
Wang, W. & Parker, K. H. 1998 Movement of spherical particles in capillaries using a boundary singularity method. J. Biomech. 31, 347354.
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, 27422744.
Wu, J. Z. & Wu, J. M. 1996 Vorticity dynamics on boundaries. Adv. Appl. Mech. 32, 119275.
Zang, T. A. 1991 On the rotation and skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7, 2740.
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Pressure-driven flow past spheres moving in a circular tube

  • G. J. SHEARD (a1) (a2) and K. RYAN (a1)


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