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NONSYMMETRIC BRANCHING OF FLUID FLOWS IN 3D VESSELS

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  08 June 2018

N. C. OVENDEN Open the ORCID record for N. C. OVENDEN [Opens in a new window]  and
F. T. SMITH Open the ORCID record for F. T. SMITH [Opens in a new window]
N. C. OVENDEN*
Affiliation:
Department of Mathematics, University College London, Gower St, London WC1E 6BT, UK email [email protected], [email protected]
F. T. SMITH
Affiliation:
Department of Mathematics, University College London, Gower St, London WC1E 6BT, UK email [email protected], [email protected]
*
✉[email protected]
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Abstract

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Nonsymmetric branching flow through a three-dimensional (3D) vessel is considered at medium-to-high flow rates. The branching is from one mother vessel to two or more daughter vessels downstream, with laminar steady or unsteady conditions assumed. The inherent 3D nonsymmetry is due to the branching shapes themselves, or the differences in the end pressures in the daughter vessels, or the incident velocity profiles in the mother. Computations based on lattice-Boltzmann methodology are described first. A subsequent analysis focuses on small 3D disturbances and increased Reynolds numbers. This reduces the 3D problem to a two-dimensional one at the outer wall in all pressure-driven cases. As well as having broader implications for feeding into a network of vessels, the findings enable predictions of how much swirling motion in the cross-plane is generated in a daughter vessel downstream of a 3D branch junction, and the significant alterations provoked locally in the shear stresses and pressures at the walls. Nonuniform incident wall-shear and unsteady effects are examined. A universal asymptotic form is found for the flux change into each daughter vessel in a 3D branching of arbitrary cross-section with a thin divider.

Keywords

branchingnonsymmetry

MSC classification

Primary: 76D09: Viscous-inviscid interaction
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 4 , April 2018 , pp. 533 - 561
Copyright
© 2018 Australian Mathematical Society 

References

Aidun, C. K. and Clausen, J. R., “Lattice-Boltzmann method for complex flows”, Annu. Rev. Fluid Mech. 42 (2010) 439–472; doi:10.1146/annurev-fluid-121108-145519.CrossRefGoogle Scholar
Al-Shahi, R., Fang, J. S. Y., Lewis, S. C. and Warlow, C. P., “Prevalence of adults with brain arteriovenous malformations: a community based study in Scotland using capture-recapture analysis”, J. Neurol. Neurosurg. Psychiatry 73 (2002) 547–551; doi:10.1136/jnnp.73.5.547.CrossRefGoogle ScholarPubMed
Alarcón, T., Byrne, H. M. and Maini, P. K., “A design principle for vascular beds: the effects of complex blood rheology”, Microvascular Res. 69 (2005) 156–172; doi:10.1016/j.mvr.2005.02.002.CrossRefGoogle ScholarPubMed
Augst, A. D., Ariff, B., McG. Thom, S. A. G., Xu, X. Y. and Hughes, A. D., “Analysis of complex flow and the relationship between blood pressure, wall shear stress, and intima-media thickness in the human carotid artery”, Am. J. Physiology-heart Circulatory Physiol. 293 (2007) H1031–H1037; doi:10.1152/ajpheart.00989.2006.CrossRefGoogle ScholarPubMed
Balta, S. and Smith, F. T., “Inviscid and low-viscosity flows in multi-branching and reconnecting networks”, J. Engrg. Math. 104 (2017) 1–18; doi:10.1007/s10665-016-9869-3.CrossRefGoogle Scholar
Bennett, J., “Theoretical properties of three-dimensional interactive boundary-layers”. Ph. D. Thesis, University College London, 1987.Google Scholar
Bhatnagar, P. L., Gross, E. P. and Krook, M., “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems”, Phys. Rev. 94 (1954) 511; doi:10.1103/PhysRev.94.511.CrossRefGoogle Scholar
Blyth, M. G. and Mestel, A. J., “Steady flow in a dividing pipe”, J. Fluid Mech. 401 (1999) 339–364; doi:10.1017/S0022112099006904.CrossRefGoogle Scholar
Bowles, R. I., Dennis, S. C. R., Purvis, R. and Smith, F. T., “Multi-branching flows from one mother tube to many daughters or to a network”, Philos. Trans. R. Soc. Lond. A Math. Phys. Engrg. Sci. 363 (2005) 1045–1055; doi:10.1098/rsta.2005.1548.Google ScholarPubMed
Bowles, R. I., Ovenden, N. C. and Smith, F. T., “Multi-branching three-dimensional flow with substantial changes in vessel shapes”, J. Fluid Mech. 614 (2008) 329–354; doi:10.1017/S0022112008003522.CrossRefGoogle Scholar
Cassidy, K. J., Gavriely, N. and Grotberg, J. B., “Liquid plug flow in straight and bifurcating tubes”, J. Biomech Engrg. 123 (2001) 580–589; doi:10.1115/1.1406949.CrossRefGoogle ScholarPubMed
Chen, S. and Doolen, G. D., “Lattice Boltzmann method for fluid flows”, Annu. Rev. Fluid Mech. 30 (1998) 329–364; doi:10.1146/annurev.fluid.30.1.329.CrossRefGoogle Scholar
Comer, J. K., Kleinstreuer, C. and Zhang, Z., “Flow structures and particle deposition patterns in double-bifurcation airway models. Part 1. Air flow fields”, J. Fluid Mech. 435 (2001) 25–54; doi:10.1017/S0022112001003809.CrossRefGoogle Scholar
Denisenko, N. S. et al. , “Experimental measurements and visualisation of a viscous fluid flow in y-branching modelling the common carotid artery bifurcation with mr and doppler ultrasound velocimetry”, J. Phys.: Conf. Ser. 722 (2016) 012013; doi:10.1088/1742-6596/722/1/012013.Google Scholar
El-Masry, O. A., Feuerstein, I. A. and Round, G. F., “Experimental evaluation of streamline patterns and separated flows in a series of branching vessels with implications for atherosclerosis and thrombosis”, Circ. Res. 43 (1978) 608–618; doi:10.1161/01.RES.43.4.608.CrossRefGoogle Scholar
Formaggia, L., Lamponi, D. and Quarteroni, A., “One-dimensional models for blood flow in arteries”, J. Engrg. Math. 47 (2003) 251–276; doi:10.1023/B:ENGI.0000007980.01347.29.CrossRefGoogle Scholar
Green, J. E. F., Smith, F. T. and Ovenden, N. C., “Flow in a multi-branching vessel with compliant walls”, J. Engrg. Math. 64 (2009) 353–365; doi:10.1007/s10665-009-9285-z.CrossRefGoogle Scholar
Guo, Z., Zheng, C. and Shi, B., “An extrapolation method for boundary conditions in lattice boltzmann method”, Phys. Fluids 14 (2002) 2007–2010; doi:10.1063/1.1471914.CrossRefGoogle Scholar
Hademenos, G. J., Massoud, T. F. and Viñuela, F., “A biomathematical model of intracranial arteriovenous malformations based on electrical network analysis: theory and hemodynamics”, Neurosurgery 38 (1996) 1005–1015; doi:10.1097/00006123-199605000-00030.CrossRefGoogle ScholarPubMed
He, X. and Luo, L.-S., “Lattice Boltzmann model for the incompressible Navier–Stokes equation”, J. Stat. Phys. 88 (1997) 927–944; doi:10.1023/B:JOSS.0000015179.12689.e4.CrossRefGoogle Scholar
Inamuro, T., Yoshino, M. and Ogino, F., “A non-slip boundary condition for lattice Boltzmann simulations”, Phys. Fluids 7 (1995) 2928–2930; doi:10.1063/1.868766.CrossRefGoogle Scholar
McEvoy, A. W., Kitchen, N. D. and Thomas, D. G. T., “Intracerebral haemorrhage in young adults: the emerging importance of drug misuse”, Br. Med. J. 320 (2000) 1322; doi:10.1136/bmj.320.7245.1322.CrossRefGoogle ScholarPubMed
Olufsen, M. S., Peskin, C. S., Kim, W. Y., Pedersen, E. M., Nadim, A. and Larsen, J., “Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions”, Ann. Biomed. Engrg. 28 (2000) 1281–1299; doi:10.1114/1.1326031.CrossRefGoogle ScholarPubMed
Ovenden, N. C., Smith, F. T. and Wu, G. X., “The effects of nonsymmetry in a branching flow network”, J. Engrg. Math. 63 (2009) 213–239; doi:10.1007/s10665-008-9232-4.CrossRefGoogle Scholar
Pedley, T. J., “Mathematical modelling of arterial fluid dynamics”, J. Engrg. Math. 47 (2003) 419–444; doi:10.1023/B:ENGI.0000007978.33352.59.CrossRefGoogle Scholar
Pranevicius, O., Pranevicius, M. and Liebeskind, D. S., “Partial aortic occlusion and cerebral venous steal: venous effects of arterial manipulation in acute stroke”, Stroke 42 (2011) 1478–1481; doi:10.1161/STROKEAHA.110.603852.CrossRefGoogle ScholarPubMed
Pries, A. R. and Secomb, T. W., “Modeling structural adaptation of microcirculation”, Microcirculation 15 (2008) 753–764; doi:10.1080/10739680802229076.CrossRefGoogle ScholarPubMed
Resnick, N., Einav, S., Chen-Konak, L., Zilberman, M., Yahav, H. and Shay-Salit, A., “Hemodynamic forces as a stimulus for arteriogenesis”, Endothelium 10 (2003) 197–206; doi:10.1080/10623320390246289.CrossRefGoogle ScholarPubMed
Rieu, R. and Pelissier, R., “In vitro study of a physiological type flow in a bifurcated vascular prosthesis”, J. Biomech. 24 (1991) 923–933; doi:10.1016/0021-9290(91)90170-R.CrossRefGoogle Scholar
Secomb, T. W. and Pries, A. R., “Basic principles of hemodynamics”, in: Handbook of hemorheology and hemodynamics (IOS Press, Amsterdam, 2007) 289–306.Google Scholar
Smith, F. T., “Steady motion through a branching tube”, Proc. R. Soc. Lond. Ser. A Math. Phys. Engrg. Sci. 355 (1977) 167–187; doi:10.1098/rspa.1977.0093.Google Scholar
Smith, F. T., “On internal fluid dynamics”, Bull. Math. Sci. 2 (2012) 125–180; doi:10.1007/s13373-012-0019-6.CrossRefGoogle Scholar
Smith, F. T. and Jones, M. A., “One-to-few and one-to-many branching tube flows”, J. Fluid Mech. 423 (2000) 1–31; doi:10.1017/S0022112000002019.CrossRefGoogle Scholar
Smith, F. T. and Jones, M. A., “AVM modelling by multi-branching tube flow: large flow rates and dual solutions”, Math. Med. Biol. 20 (2003) 183–204; doi:10.1093/imammb/20.2.183.CrossRefGoogle ScholarPubMed
Smith, F. T., Ovenden, N. C., Franke, P. T. and Doorly, D. J., “What happens to pressure when a flow enters a side branch?”, J. Fluid Mech. 479 (2003) 231–258; doi:10.1017/S002211200200366X.CrossRefGoogle Scholar
Smith, F. T., Purvis, R., Dennis, S. C. R., Jones, M. A., Ovenden, N. C. and Tadjfar, M., “Fluid flow through various branching tubes”, J. Engrg. Math. 47 (2003) 277–298; doi:10.1023/B:ENGI.0000007981.46608.73.CrossRefGoogle Scholar
Tadjfar, M. and Smith, F. T., “Direct simulations and modelling of basic three-dimensional bifurcating tube flows”, J. Fluid Mech. 519 (2004) 1–32; doi:10.1017/S0022112004000606.CrossRefGoogle Scholar
Tutty, O. R., “Flow in a tube with a small side branch”, J. Fluid Mech. 191 (1988) 79–109; doi:10.1017/S0022112088001521.CrossRefGoogle Scholar
White, A. H. and Smith, F. T., “Computational modelling of the embolization process for the treatment of arteriovenous malformations (AVMs)”, Math. Comput. Model. 57 (2013) 1312–1324; doi:10.1016/j.mcm.2012.10.033.CrossRefGoogle Scholar
Wilquem, F. and Degrez, G., “Numerical modeling of steady inspiratory airflow through a three-generation model of the human central airways”, J. Biomech. Engrg. 119 (1997) 59–65; doi:10.1115/1.2796065.CrossRefGoogle ScholarPubMed
Wolf-Gladrow, D. A., Lattice-gas cellular automata and lattice Boltzmann models: an introduction (Springer, Heidelberg, 2000); doi:10.1007/b72010.CrossRefGoogle Scholar
Yokoi, K., Xiao, F., Liu, H. and Fukasaku, K., “Three-dimensional numerical simulation of flows with complex geometries in a regular cartesian grid and its application to blood flow in cerebral artery with multiple aneurysms”, J. Comput. Phys. 202 (2005) 1–19; doi:10.1016/j.jcp.2004.06.018.CrossRefGoogle Scholar