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.