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  • N. C. OVENDEN (a1) and F. T. SMITH (a1)


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.


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[1] Aidun, C. K. and Clausen, J. R., “Lattice-Boltzmann method for complex flows”, Annu. Rev. Fluid Mech. 42 (2010) 439472; doi:10.1146/annurev-fluid-121108-145519.
[2] 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) 547551; doi:10.1136/jnnp.73.5.547.
[3] 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) 156172; doi:10.1016/j.mvr.2005.02.002.
[4] 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) H1031H1037; doi:10.1152/ajpheart.00989.2006.
[5] Balta, S. and Smith, F. T., “Inviscid and low-viscosity flows in multi-branching and reconnecting networks”, J. Engrg. Math. 104 (2017) 118; doi:10.1007/s10665-016-9869-3.
[6] Bennett, J., “Theoretical properties of three-dimensional interactive boundary-layers”. Ph. D. Thesis, University College London, 1987.
[7] 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.
[8] Blyth, M. G. and Mestel, A. J., “Steady flow in a dividing pipe”, J. Fluid Mech. 401 (1999) 339364; doi:10.1017/S0022112099006904.
[9] 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) 10451055; doi:10.1098/rsta.2005.1548.
[10] 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) 329354; doi:10.1017/S0022112008003522.
[11] Cassidy, K. J., Gavriely, N. and Grotberg, J. B., “Liquid plug flow in straight and bifurcating tubes”, J. Biomech Engrg. 123 (2001) 580589; doi:10.1115/1.1406949.
[12] Chen, S. and Doolen, G. D., “Lattice Boltzmann method for fluid flows”, Annu. Rev. Fluid Mech. 30 (1998) 329364; doi:10.1146/annurev.fluid.30.1.329.
[13] 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) 2554; doi:10.1017/S0022112001003809.
[14] 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.
[15] 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) 608618; doi:10.1161/01.RES.43.4.608.
[16] Formaggia, L., Lamponi, D. and Quarteroni, A., “One-dimensional models for blood flow in arteries”, J. Engrg. Math. 47 (2003) 251276; doi:10.1023/B:ENGI.0000007980.01347.29.
[17] 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) 353365; doi:10.1007/s10665-009-9285-z.
[18] Guo, Z., Zheng, C. and Shi, B., “An extrapolation method for boundary conditions in lattice boltzmann method”, Phys. Fluids 14 (2002) 20072010; doi:10.1063/1.1471914.
[19] 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) 10051015; doi:10.1097/00006123-199605000-00030.
[20] He, X. and Luo, L.-S., “Lattice Boltzmann model for the incompressible Navier–Stokes equation”, J. Stat. Phys. 88 (1997) 927944; doi:10.1023/B:JOSS.0000015179.12689.e4.
[21] Inamuro, T., Yoshino, M. and Ogino, F., “A non-slip boundary condition for lattice Boltzmann simulations”, Phys. Fluids 7 (1995) 29282930; doi:10.1063/1.868766.
[22] 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.
[23] 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) 12811299; doi:10.1114/1.1326031.
[24] Ovenden, N. C., Smith, F. T. and Wu, G. X., “The effects of nonsymmetry in a branching flow network”, J. Engrg. Math. 63 (2009) 213239; doi:10.1007/s10665-008-9232-4.
[25] Pedley, T. J., “Mathematical modelling of arterial fluid dynamics”, J. Engrg. Math. 47 (2003) 419444; doi:10.1023/B:ENGI.0000007978.33352.59.
[26] 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) 14781481; doi:10.1161/STROKEAHA.110.603852.
[27] Pries, A. R. and Secomb, T. W., “Modeling structural adaptation of microcirculation”, Microcirculation 15 (2008) 753764; doi:10.1080/10739680802229076.
[28] 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) 197206; doi:10.1080/10623320390246289.
[29] Rieu, R. and Pelissier, R., “In vitro study of a physiological type flow in a bifurcated vascular prosthesis”, J. Biomech. 24 (1991) 923933; doi:10.1016/0021-9290(91)90170-R.
[30] Secomb, T. W. and Pries, A. R., “Basic principles of hemodynamics”, in: Handbook of hemorheology and hemodynamics (IOS Press, Amsterdam, 2007) 289306.
[31] Smith, F. T., “Steady motion through a branching tube”, Proc. R. Soc. Lond. Ser. A Math. Phys. Engrg. Sci. 355 (1977) 167187; doi:10.1098/rspa.1977.0093.
[32] Smith, F. T., “On internal fluid dynamics”, Bull. Math. Sci. 2 (2012) 125180; doi:10.1007/s13373-012-0019-6.
[33] Smith, F. T. and Jones, M. A., “One-to-few and one-to-many branching tube flows”, J. Fluid Mech. 423 (2000) 131; doi:10.1017/S0022112000002019.
[34] 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) 183204; doi:10.1093/imammb/20.2.183.
[35] 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) 231258; doi:10.1017/S002211200200366X.
[36] 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) 277298; doi:10.1023/B:ENGI.0000007981.46608.73.
[37] Tadjfar, M. and Smith, F. T., “Direct simulations and modelling of basic three-dimensional bifurcating tube flows”, J. Fluid Mech. 519 (2004) 132; doi:10.1017/S0022112004000606.
[38] Tutty, O. R., “Flow in a tube with a small side branch”, J. Fluid Mech. 191 (1988) 79109; doi:10.1017/S0022112088001521.
[39] 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) 13121324; doi:10.1016/j.mcm.2012.10.033.
[40] 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) 5965; doi:10.1115/1.2796065.
[41] Wolf-Gladrow, D. A., Lattice-gas cellular automata and lattice Boltzmann models: an introduction (Springer, Heidelberg, 2000); doi:10.1007/b72010.
[42] 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) 119; doi:10.1016/
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