On the exact solution of the Riemann problem for blood flow in human veins, including collapse

We solve exactly the Riemann problem for the non-linear hyperbolic system governing blood flow in human veins and note that, as modeled here, veins do not admit complete collapse, that is zero cross-sectional area A. This means that the Cauchy problem will not admit zero cross-sectional areas as initial condition. In particular, rarefactions and shock waves (elastic jumps), classical waves in the conventional Riemann problem, cannot be connected to the zero state with A=0. Moreover, we show that the area A* between two rarefaction waves in the solution of the Riemann problem can never attain the value zero, unless the data velocity difference uR−uL tends to infinity. This is in sharp contrast to analogous systems such as blood flow in arteries, gas dynamics and shallow water flows, all of which admitting a vacuum state. We discuss the implications of these findings in the modelling of the human circulation system that includes the venous system.