Numerical solution of the viscous flows in a network of thin tubes: Equations on the graph

Highlights: • Simplified models for Navier-Stokes equations in networks are important in microfluidics and blood vessels flows. • Network of thin tubes may be modelled by equation on a graph. • An efficient numerical scheme on a graph is designed, with proof and order of convergence. • Comparisons with full multidimensional model and exact solution are proposed. A non-stationary flow in a network of thin tubes is considered. Its one-dimensional approximation was proposed in a paper by G. Panasenko and K. Pileckas, Flows in a tube structure: equation on the graph (Panasenko and Pileckas, 2014 [19]). It consists of a set of equations with weakly singular kernels, on a graph, for the macroscopic pressure. A new difference scheme for this problem is proposed. Several variants are discussed. Stability and convergence are carefully investigated, theoretically and numerically. In addition, numerical results are compared to the direct numerical solution of the full dimension Navier-Stokes equations.