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Steady motion through a branching tube

The fluid motion through a straight pipe that bifurcates symmetrically into two semi-circular pipes is considered at high Reynolds numbers, with the aim of deriving some of the effects of the three-dimensionality per se. In a corresponding planar problem the long-scale boundary layer approach yields the initial development of the induced pressure gradient fairly directly, the problem being linear. By contrast, in the three-dimensional situation the favourable pressure gradient is determined by a nonlinear eigenvalue relation, because the pressure gradient is much stronger here and interacts critically with the inviscid coreflow to produce nonlinear behaviour near the outer walls. The proposed structure of the initial part of the boundary layer is apparently self-consistent, with a two-tiered development near the outer walls where the secondary motion is of a relatively fast vortex type and the axial skin friction suffers a sharp increase. The secondary coreflow is a displacement-induced streaming combined with a vortex provoked by the pressure gradient. The analysis applies for any realistic oncoming flow in the single tube and can be extended to treat other tube cross-sections.