Approximate solutions to nonlinear fluid networks with periodic inputs

We use Kirchhoff´s laws and pipe flow dynamics equations to describe a fluid flow network in the form of a nonlinear differential equation with a periodic right hand side. We apply the averaging method to find an approximate solution of this equation and analyze its stability properties. The approximate solution consists of three parts: a mean flow part due to the resistive effects of branches, a time-periodic part due to "inductive" effects, and a mean flow average correction due to the interaction of nonlinear and time varying effects. We present an example that may help explain the processes participating in the development of venous diseases. In particular, it is shown that the widening of a branch in a venous network leads to an increase in the AC flow and decrease in the DC flow through that branch, thus increasing the stress on venous valves, and consequently leading to further increase in the effective width of the vein.