Streamlines in Two-Dimensional Laminar Branch Flows

A study is made of two-dimensional laminar flow in a long channel when fluid is extracted or allowed to leak from a small side channel. The full Navier Stokes equations are linearized about a plane Poiseuille flow (which exists in the main channel in the absence of the branch) and solved for the entire flow by a relaxation procedure. It is found that, in the neighborhood of the branch entrance, there are a number of regions of closed circulation. Their presence is accounted for by the viscous nature of the fluid and eludes any previous inviscid calculations. This problem is of interest in the fields of hydraulics and blood flow where the flows are somewhat more complicated and therefore mathematically less tractable. However, the present study, which includes a physical interpretation of the flow characteristics, should provide a basis for predicting the flow behavior in these more complex branching systems.