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Multi-branching three-dimensional flow with substantial changes in vessel shapes

  • R. I. BOWLES (a1), N. C. OVENDEN (a1) and F. T. SMITH (a1)

This theoretical investigation of steady fluid flow through a rigid three-dimensional branching geometry is motivated by applications to haemodynamics in the brain especially, while the flow through a tube with a blockage or through a collapsed tube provides another motivation with a biomedical background. Three-dimensional motion without symmetry is addressed through one mother vessel to two or several daughters. A comparatively long axial length scale of the geometry leads to a longitudinal vortex system providing a slender-flow model for the complete mother-and-daughters flow response. Computational studies and subsequent analysis, along with comparisons, are presented. The relative flow rate varies in terms of an effective Reynolds number dependence, allowing a wide range of flow rates to be examined theoretically; also any rigid cross-sectional shape and ratio of cross-sectional area expansion or contraction from the mother vessel to the daughters can be accommodated in principle in both the computations and the analysis. Swirl production with substantial crossflows is found. The analysis shows that close to any carina (the ridge separating daughter vessels) or carinas at a branch junction either forward or reversed motion can be observed locally at the saddle point even though the bulk of the motion is driven forward into the daughters. The local forward or reversed motion is controlled, however, by global properties of the geometry and incident conditions, a feature which applies to any of the flow rates examined.

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C. D. Bertram & S. A. Godbole 1997 LDA measurements of velocities in a simulated collapsed tube. Trans. ASME J. Biomech. Engng 119, 357363.

F. Cassot , M. Zagzoule & J. P. Marc-Vergnes 2000 Hemodynamics role of the circle of Willis in stenoses of internal carotid arteries. an analytical solution of a linear model. J. Biomech 33, 395405.

J. K. Comer , C. Kleinstreuer & Z. Zhang 2001 aFlow structures and particle deposition patterns in double-bifurcation airway models. Part 1. Air flow fields. J. Fluid Mech. 435, 2554.

L. J. Cummings , S. L. Wattis & J. A. D. Graham 2004 The effect of ureteric stents on urine flow: reflux. J. Math. Biol. 49, 5682.

A. Ferrandez , T. David , J. Bamford & A. Guthrie 2000 Computational models of blood flow in the circle of Willis. Computer Meth. Biomech. Biomed. Engng 4, 126.

E. Gao , W. L. Young , E. Ornstein , J. Pile-Spellman & Q. Ma 1997 A theoretical model of cerebral hemodynamics: application to the study of arteriovenous malformations. J. Cerebral Blood Flow Metab. 17, 905918.

K. Gnanalingham , W. Taylor & L. Watkin 2002 Dual technique for obliteration of small arteriovenous malformations. Brit. J. Neurosurg. 16 (4), 376380.

D. J. Griffiths 1971 Hydrodynamics of male micturition – I: theory of steady flow through elastic-walled tubes. Med. Biol. Engng 9, 581588.

D. J. Griffiths 1987 Dynamics of the upper urinary tract: I. peristaltic flow through a distensible tube of limited length. Phys. Med. Biol. 32, 813822.

D. J. Griffiths , C. E. Constantinou , J. Mortensen & J. C. Djurhuus 1987 Dynamics of the upper urinary tract: II. the effect of variations or peristaltic frequency and bladder pressure on pyeloureteral pressure/flow relations. Phys. Med. Biol. 32, 823833.

J. B. Grotberg & O. E. Jensen 2004 Biofluidmechanics of flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.

G. J. Hademenos , T. F. Massoud & F. Vinuela 1996 A biomathematical model of intracranial arteriovenous malformations based on electrical network analysis. theory and hemodynamics. Neurosurgery 38, 10051015.

T. Handa , M. Negoro , S. Miyachi & K. Sugita 1993 Evaluation of pressure changes in feeding arteries during embolization of intracerebral arteriovenous malformations. J. Neurosurg. 79, 383389.

B. Hillen , B. A. H. Drinkenburg , H. W. Hoogstraten & L. Post 1988 Analysis of flow and vascular resistance in a model of the circle of Willis. J. Biomech. 21, 807814.

B. Hillen , H. W. Hoogstraten & L. Post 1986 A mathematical model of the flow in the circle of Willis. J. Biomech. 19, 187194.

R. H. Kufahl & M. E. Clark 1985 A circle of Willis simulation using distensible vessels and pulsatile flow. J. Biomech. Engng 107, 112122.

A. Marzo , X. Y. Luo & C. D. Bertram 2005 Three-dimensional collapse and steady flow in thick-walled flexible tubes. J. Fluids Struct. 20, 817835.

R. Mittal & G. Iaccarino 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239261.

T. J. Pedley 1997 Pulmonary fluid dynamics. Annu. Rev. Fluid Mech. 9, 229274.

C. V. Seal & C. R. Smith 1997 Intertwining laminar necklace vortices. Phys. Fluids 9 (9).

F. T. Smith 1977 Steady motion through a branching tube. Proc. R. Soc. Lond. A 355, 167187.

F. T. Smith 1978 Flow through symmetrically constricted tubes. J. Inst. Maths Applics. 21, 145156.

F. T. Smith , S. C. R. Dennis , M. A. Jones , N. C. Ovenden , R. Purvis & M. Tadjfar 2003 aFluid flow through various branching tubes. J. Engng Maths 47, 277298.

F. T. Smith & S. N. Timoshin 1996 Blade-wake interactions and rotary boundary layers. Proc. R. Soc. Lond. A 452, 13011329.

M. Zagzoule & J. P. Marc-Vergnes 1986 A global mathematical model of the cerebral circulation in man. J. Biomech. 19 (12), 10151022.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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