One-to-few and one-to-many branching tube flows

Branching tube flows are examined, for one mother to two, three or more dau ghter tubes. The case of many daughters (abrupt multi-branching) models blo od flow through an arteriovenous malformation in the brain, while that of v ery few daughters (gradual branching) applies elsewhere in physiology and s urgical grafting, as well as other applications including industrial ones. Theory and computation are presented for two- and three-dimensional motions , under the viscous and inviscid effects of small changes in mass flux betw een the daughter tubes, area expansion and turning of the flow. Specific co nfigurations for which flow solutions are obtained are (a) with two large d aughters, (b) with one small daughter/side branch, and (c) with multiple sm all daughters. The numerous physical mechanisms acting concern overall upstream influence and through-flow, and how separation and criteria for its avoidance, as wel l as criteria for the amount of turning and area expansion possible without energy loss and other factors associated with separation, and the role of the branching geometry versus that of the mass-flux distribution in the dau ghters. In particular, configuration (a) allows substantial separation-free turning and expansion only with certain shaping of the outer wall and an a rea expansion ratio typically less than 1.2, whereas more daughters involve a balance between geometry and mass flux. In (b), an abrupt pressure jump is induced at the mouth of the small daughter, near which mass-flux effects tend to dominate over geometrical shaping effects. In (c), as the number o f daughters increases, the amount of separation-free turning and expansion is found to increase substantially, and the distributed mass-flux influence readily overrides the geometrical influence throughout the branching; ther e is also an integrated upstream effect of the multi-branching on the incid ent mother flow even though each daughter flow acts as if independent. Tent ative designs based on wall shaping, flux distributions and divider placeme nt are considered for flow improvement/surgery.