Branching tube flows are examined, for one mother to two, three or more daughter tubes. The case of many daughters (abrupt multi-branching) models blood flow through an arteriovenous malformation in the brain, while that of very few daughters (gradual branching) applies elsewhere in physiology and surgical 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 between the daughter tubes, area expansion and turning of the flow. Specific configurations for which flow solutions are obtained are (a) with two large daughters, (b) with one small daughter/side branch, and (c) with multiple small daughters.

The numerous physical mechanisms acting concern overall upstream influence and through-flow, and flow separation and criteria for its avoidance, as well 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 daughters. In particular, configuration (a) allows substantial separation-free turning and expansion only with certain shaping of the outer wall and an area 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 of 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; there is also an integrated upstream effect of the multi-branching on the incident mother flow even though each daughter flow acts as if independent. Tentative designs based on wall shaping, flux distributions and divider placement are considered for flow improvement/surgery.