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

Published online by Cambridge University Press:  16 October 2008

R. I. BOWLES
Affiliation:
Department of Mathematics, UCL, Gower Street, London, WC1E 6BT, UK
N. C. OVENDEN
Affiliation:
Department of Mathematics, UCL, Gower Street, London, WC1E 6BT, UK
F. T. SMITH
Affiliation:
Department of Mathematics, UCL, Gower Street, London, WC1E 6BT, UK

Abstract

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.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Bennett, J. M. 1987 Theoretical properties of three-dimensional interactive boundary layers. PhD thesis, University of London.Google Scholar
Bertram, C. D. & Godbole, S. A. 1997 LDA measurements of velocities in a simulated collapsed tube. Trans. ASME J. Biomech. Engng 119, 357363.CrossRefGoogle Scholar
Blyth, M. G. & Mestel, A. J. 1999 Steady flow in a dividing pipe. J. Fluid Mech. 401, 339364.CrossRefGoogle Scholar
Cassot, F., Zagzoule, M. & Marc-Vergnes, J. P. 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.CrossRefGoogle ScholarPubMed
Comer, J. K., Kleinstreuer, C. & Zhang, Z. 2001 a Flow structures and particle deposition patterns in double-bifurcation airway models. Part 1. Air flow fields. J. Fluid Mech. 435, 2554.Google Scholar
Comer, J. K., Kleinstreuer, C. & Zhang, Z. 2001 b Flow structures and particle deposition patterns in double-bifurcation airway models. Part 2. Aerosol transport and deposition. J. Fluid Mech. 435, 5580.Google Scholar
Cummings, L. J., Wattis, S. L. & Graham, J. A. D. 2004 The effect of ureteric stents on urine flow: reflux. J. Math. Biol. 49, 5682.CrossRefGoogle ScholarPubMed
Ferrandez, A., David, T., Bamford, J. & Guthrie, A. 2000 Computational models of blood flow in the circle of Willis. Computer Meth. Biomech. Biomed. Engng 4, 126.Google ScholarPubMed
Gao, E., Young, W. L., Ornstein, E., Pile-Spellman, J. & Ma, Q. 1997 A theoretical model of cerebral hemodynamics: application to the study of arteriovenous malformations. J. Cerebral Blood Flow Metab. 17, 905918.CrossRefGoogle Scholar
Gnanalingham, K., Taylor, W. & Watkin, L. 2002 Dual technique for obliteration of small arteriovenous malformations. Brit. J. Neurosurg. 16 (4), 376380.CrossRefGoogle ScholarPubMed
Griffiths, D. J. 1971 Hydrodynamics of male micturition – I: theory of steady flow through elastic-walled tubes. Med. Biol. Engng 9, 581588.CrossRefGoogle ScholarPubMed
Griffiths, D. J. 1987 Dynamics of the upper urinary tract: I. peristaltic flow through a distensible tube of limited length. Phys. Med. Biol. 32, 813822.CrossRefGoogle ScholarPubMed
Griffiths, D. J., Constantinou, C. E., Mortensen, J. & Djurhuus, J. C. 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.CrossRefGoogle ScholarPubMed
Grotberg, J. B. & Jensen, O. E. 2004 Biofluidmechanics of flexible tubes. Annu. Rev. Fluid Mech. 36, 121147.CrossRefGoogle Scholar
Hademenos, G. J., Massoud, T. F. & Vinuela, F. 1996 A biomathematical model of intracranial arteriovenous malformations based on electrical network analysis. theory and hemodynamics. Neurosurgery 38, 10051015.CrossRefGoogle ScholarPubMed
Handa, T., Negoro, M., Miyachi, S. & Sugita, K. 1993 Evaluation of pressure changes in feeding arteries during embolization of intracerebral arteriovenous malformations. J. Neurosurg. 79, 383389.CrossRefGoogle ScholarPubMed
Hillen, B., Drinkenburg, B. A. H., Hoogstraten, H. W. & Post, L. 1988 Analysis of flow and vascular resistance in a model of the circle of Willis. J. Biomech. 21, 807814.CrossRefGoogle Scholar
Hillen, B., Hoogstraten, H. W. & Post, L. 1986 A mathematical model of the flow in the circle of Willis. J. Biomech. 19, 187194.CrossRefGoogle ScholarPubMed
Kufahl, R. H. & Clark, M. E. 1985 A circle of Willis simulation using distensible vessels and pulsatile flow. J. Biomech. Engng 107, 112122.CrossRefGoogle ScholarPubMed
Luo, X. Y. & Pedley, T. J. 2000 Multiple solutions and flow limitation in collapsible channel flows. J. Fluid Mech. 420, 301324.CrossRefGoogle Scholar
Marini, B. D. & Smith, C. R. 2002 The influence of impinging boundary layer vorticity packets on turbulent juncture flow behavior. In Proc. 2nd Intl Sympo. on Turbulence and Shear Flow Phenomena, Stockholm, Sweden (ed. Eaton, J. A. & Sommerfield, M.), pp. 245254. Springer.Google Scholar
Marzo, A., Luo, X. Y. & Bertram, C. D. 2005 Three-dimensional collapse and steady flow in thick-walled flexible tubes. J. Fluids Struct. 20, 817835.CrossRefGoogle Scholar
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239261.CrossRefGoogle Scholar
Olufsen, M. S. 1999 Structured tree outflow condition for blood flow in larger systemic arteries. Am. J. Physiol. Heart Circ. Physiol. 276, H257268.CrossRefGoogle ScholarPubMed
Ovenden, N. C., Smith, F. T. & Wu, G. X. 2008 The effects of nonsymmetry in a branching flow network. J. Engng Maths (to appear).Google Scholar
Pedley, T. J. 1997 Pulmonary fluid dynamics. Annu. Rev. Fluid Mech. 9, 229274.CrossRefGoogle Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica pp. 1–39.Google Scholar
Seal, C. V. & Smith, C. R. 1997 Intertwining laminar necklace vortices. Phys. Fluids 9 (9).CrossRefGoogle Scholar
Seal, C. V., Smith, C. R., Akin, O. & Rockwell, D. 1995 a Quantitative characteristics of a laminar, unsteady necklace vortex system in a rectangular block-flat plate juncture. J. Fluid Mech. 286, 117135.CrossRefGoogle Scholar
Seal, C. V., Smith, C. R. & Rockwell, D. 1995 b Vorticity distribution in endwall junctions. AIAA Paper 95-2238.Google Scholar
Smith, F. T. 1977 Steady motion through a branching tube. Proc. R. Soc. Lond. A 355, 167187.CrossRefGoogle Scholar
Smith, F. T. 1978 Flow through symmetrically constricted tubes. J. Inst. Maths Applics. 21, 145156.CrossRefGoogle Scholar
Smith, F. T., Dennis, S. C. R., Jones, M. A., Ovenden, N. C., Purvis, R. & Tadjfar, M. 2003 a Fluid flow through various branching tubes. J. Engng Maths 47, 277298.CrossRefGoogle Scholar
Smith, F. T. & Gajjar, J. S. B. 1984 Flow past wing-body junctions. J. Fluid Mech. 144, 191215.CrossRefGoogle Scholar
Smith, F. T. & Jones, M. A. 2000 One-to-few and one-to-many branching tube flows. J. Fluid Mech. 423, 131.CrossRefGoogle Scholar
Smith, F. T, Ovenden, N. C, Franke, P. & Doorly, D. J. 2003 b What happens to pressure when a flow enters a side branch? J. Fluid Mech. 479, 231258.CrossRefGoogle Scholar
Smith, F. T. & Timoshin, S. N. 1996 Blade-wake interactions and rotary boundary layers. Proc. R. Soc. Lond. A 452, 13011329.CrossRefGoogle Scholar
Stacey, R. & Kitchen, N. D. 1999 Recent advances in the management of cerebrovascular disease: the diminishing role of the surgeon? Ann. R. Coll. Surg. Engrs 81, 8689.Google ScholarPubMed
Tadjfar, M. & Himeno, R. 2001 Parallel multi-zone multi-block solver to study arterial branches in the human vascular system. In Proc. Intl Mech. Engng Congr. and Expo-2001, New York, pp. 1–7. ASME.Google Scholar
Tadjfar, M. & Smith, F. T. 2004 Direct simulations and modelling of basic three-dimensional bifurcating tube flows. J. Fluid Mech. 519, 132.CrossRefGoogle Scholar
Ursino, M. & Lodi, C. A. 1997 A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J. Appl. Physiol. 82, 12561269.CrossRefGoogle ScholarPubMed
Wilquem, F. & Degrez, G. 1997 Numerical modelling of steady inspiratory airflow through a three-generation model of the human central airways. Trans. ASME: J. Biomech. Engng 119, 5967.Google Scholar
Zagzoule, M. & Marc-Vergnes, J. P. 1986 A global mathematical model of the cerebral circulation in man. J. Biomech. 19 (12), 10151022.CrossRefGoogle ScholarPubMed
Zhao, Y. & Lieber, B. B. 1994 Steady inspiratory flow in a model symmetric bifurcation. Trans. ASME: J. Biomech. Engng 116, 488496.Google Scholar