The author discusses the development of mathematical models for the flow of blood in the human circulatory system. By necessity, such a model of a living system must include several simplifications, ignore some factors, and make several assumptions. This is not a simple fluid flow problem with a routine application of Navier-Stoke equations. There are several chemical reactions going on, there is a coupling between the fluid flow and the motion of the vessel wall. Moreover, the material of the vessel is not easily described. It would be a good experimental project to come up with even approximate constitutive equations for the wall region of a human artery. Thus we have a moving boundary type of flow, with mechanical interactions between the boundary and the fluid not really defined. The resulting compromise ignores nonlinearities by making the small displacement assumption''. The author uses a compromise between the Eulerian and Lagrangian description of the flow. A geomerically conforming finite element grid is reconstructed at each finite step i.e. at each time interval of the flow. A linear variation accompanies each such small'' time step. An electrical circuit analogue describes the flow based on a lumped parameter model.par Probably the next step shall be to coordinate the measurements predicted by this model by using a computerized feedback control identification loop, thus clinically checking the validity of these models of cardiovascular systems, and perhaps revising some assumptions.