ADVANCES ON NUMERICAL MODELLING OF BLOOD FLOW PROBLEMS

In these notes we will review some recent contributions to the mathematical and numerical modelling of vascular flows. In particular, we will address a couple of issues: boundary treatment (and multiscale models) and advection-diffusion of solutes. The goal is to show that numerical solutions of the blood flow in the human cardiovascular system may be obtained by coupling models having different physical dimensions. One of the aspects of circulatory system is indeed its multiscale nature; local flow features may have a global effect on circulation. For that, we set up a numerical device adequate to represent accurately both local and systemic features. It encompasses lumped models (which are described by means of an electrical network analogue), local models based on 2D or 3D Navier-Stokes equations coupled with advection-diffusion equations for the mass transfer, as well as intermediate 1D models. We present the different assumptions underlying these models, some mathematical issues related to their coupling and numerical results illustrating their effectiveness on the problem at hand.