Fluid flow through a network of thin pipes

We study the fluid flow through a network of intersected thin pipes with prescribed pressure at their ends. Pipes are either thin or long and the ratio between the length and the cross-section is considered as the small parameter. Using the asymptotic analysis with respect to that small parameter the effective behaviour of the flow is found. At each junction an explicit formula for computing the value of the pressure is found. The interior layer phenomenon in vicinity of the junction is studied. We generalize the junction formula on the case of adiabatic compressible flow.