Abstract
The problem of fluid flow in a compliant-walled channel which branches into two or more daughters is considered with the aim of understanding blood flow through arterio-venous malformations (AVMs) in the brain. The outer walls of the channel are assumed for definiteness to behave as spring-back plates, whilst the divider is taken as rigid. The fluid is assumed to be incompressible and inviscid. When the Strouhal number is small (as occurs in practice in the brain), there are two main axial length scales, one much longer than the vessel width and the other comparable with the vessel width. Also, in the case of small wall displacements, one can analyse the local flow-structure interaction problem using a complex variable method. The flow shows markedly different qualitative features downstream of the branching, depending on the wall stiffness.
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References
Smith FT, Jones MA (2000) One-to-few and one-to-many branching tube flows. J Fluid Mech 423: 1–31
Smith FT, Jones MA (2003) AVM modelling by multi-branching tube flow: large flow rates and dual solutions. Math Med Biol 20: 183–204
Smith FT, Ovenden NC, Franke PT, Doorly DJ (2003) What happens to pressure when a flow enters a side branch?. J Fluid Mech 479: 231–258
Smith FT, Purvis R, Dennis SCR, Jones MA, Ovenden NC, Tadjfar M (2003) Fluid flow through various branching tubes. J Eng Math 47: 277–298
Tadjfar M, Smith FT (2004) Direct simulations and modelling of basic three-dimensional bifurcating tube flows. J Fluid Mech 519: 1–32
Bowles RI, Dennis SCR, Purvis R, Smith FT (2005) Multi-branching flows from one mother tube to many daughters or to a network. Philos Trans R Soc A 363: 1045–1055
Pedley TJ (2003) Mathematical modelling of arterial fluid dynamics. J Eng Math 47: 419–444
Smith FT (1977) Steady motion through a branching tube. Proc R Soc A 355(1681): 167–187
Blyth MG, Mestel AJ (1999) Steady flow in a dividing pipe. J Fluid Mech 401: 339–364
Grotberg JB, Jensen OE (2004) Biofluid mechanics in flexible tubes. Annu Rev Fluid Mech 36: 121–147
Guneratne JC, Pedley TJ (2006) High-Reynolds-number steady flow in a collapsible channel. J Fluid Mech 569: 151–184
Jensen OE, Heil M (2003) High-frequency self-excited oscillations in a collapsible-channel flow. J Fluid Mech 481: 235–268
Pedley TJ, Luo XY (1998) Modelling flow and oscillations in collapsible tubes. Theor Comput Fluid Dyn 10: 277–294
Davies C, Carpenter PW (1997) Instabilities in a plane channel flow between compliant walls. J Fluid Mech 352: 205–243
Gajjar JSB, Sibanda P (1996) The hydrodynamic stability of channel flow with compliant boundaries. Theor Comput Fluid Dyn 8: 105–129
Heil M, Waters SL (2006) Transverse flows in rapidly oscillating cylindrical shells. J Fluid Mech 547: 185–214
Larose PG, Grotberg JB (1997) Flutter and long-wave instabilities in compliant channels conveying developing flows. J Fluid Mech 331: 37–58
Vito RP, Dixon SA (2003) Blood vessel constitutive models 1995–2002. Annu Rev Biomed Eng 5: 413–439
Gaver DP, Halpern D, Jensen OE, Grotberg JB (1996) The steady motion of a semi-infinite bubble through a flexible-walled channel. J Fluid Mech 319: 25–65
Jensen OE, Horsburgh MK, Halpern D, Gaver DP (2002) The steady propagation of a bubble in a flexible-walled channel: asymptotic and computational models. Phys Fluids 14(2): 443–457
Canic S, Tambaca J, Guidoboni G, Mikelic A, Hartley CJ, Rosenstrauch D (2006) Modelling viscoelastic behaviour of arterial walls and their interaction with pulsatile blood flow. SIAM J Appl Math 67(1): 164–193
Sherwin SJ, Franke V, Piero J, Parker K (2003) One-dimensional modelling of a vascular network in space-time variables. J Eng Math 47: 217–250
Pott F, Ray CA, Olesen HL, Ide K, Secher NH (1997) Middle cerebral artery blood velocity, arterial diameter and muscle sympathetic nerve activity during post-exercise muscle ischaemia. Acta Physiol Scand 160: 43–47
Leutin VP, Pystina EA, Yarosh SV (2004) Linear blood flow velocity in arteries of the brain hemispheres in left-handers and right-handers during hypoxia. Hum Physiol 30(3): 290–292
Berger SA (1993) Flow in large blood vessels. Contemp Math 141: 479–518
Bloor MIG (1978) Large amplitude surface waves. J Fluid Mech 84(1): 167–179
King AC, Bloor MIG (1987) Free-surface flow over a step. J Fluid Mech 182: 193–208
King AC, Bloor MIG (1990) Free-surface flow of a stream obstructed by an arbitrary bed topography. Q J Mech Appl Math 43(1): 87–106
Baroud CN, Tsikata S, Heil M (2006) The propagation of low-viscosity fingers into fluid-filled branching networks. J Fluid Mech 546: 285–294
Bowles RI, Ovenden NC, Smith FT (2008) Multi-branching three-dimensional flow with substantial changes in vessel shapes. J Fluid Mech 614: 329–354
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Green, J.E.F., Smith, F.T. & Ovenden, N.C. Flow in a multi-branching vessel with compliant walls. J Eng Math 64, 353–365 (2009). https://doi.org/10.1007/s10665-009-9285-z
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DOI: https://doi.org/10.1007/s10665-009-9285-z