Dynamics of miscible flows on networks
Networks of fluids and energy are ubiquitous in our societies. A natural approach is to describe the network topology using a graph. The fluid like quantity follows conservation laws and constitutive equations such as Kirchoff and Ohm's laws. I will derive and review the main ODE model for these systems, the graph wave equation where the usual (continuous) Laplacian is replaced by the graph Laplacian. The model can be extended by vertex nonlinearities and I will present two classes of nonlinear periodic solutions. Finally I will examine two applications to the electrical grid : the detection of a defect and an approximation to the solutions of the load flow equations.