Mathematical Formulations of Lumped Parameter Models for Unsteady Flow and Comparison of the Models Derived From Womersley's Solution and Paraboloid Velocity Profile

ABSTRACT Mathematical expressions for resistance and inductance that correspond to Womersley's well‐known solution for periodic unsteady flow in a cylindrical tube are rigorously formulated. A formal mathematical analysis is also formulated for periodic unsteady flow by assuming the paraboloid velocity profile. The volumetric flow rate given by such unsteady analysis is compared here with Womersley's solution; Womersley himself had compared his solution with Poiseuille's steady solution (without mathematical justification). Analytical expressions for the phase difference between the flow rate and pressure are derived, both for the paraboloid velocity profile and Womersley's solution. The formulations of the equivalent electrical circuits are mathematically exact. The equivalent electrical circuit corresponding to Womersley's solution and that for paraboloid velocity are shown to be different at an arbitrary Womersley number. When the Womersley number approaches zero, the resistances in the two analyses are the same, but the inductances are not. When the Womersley number approaches infinity, the inductances in the two analyses are the same, but the resistances are not. The validity (or inappropriateness) of the expressions for the resistance and the inductance that are routinely used, on a rather ad hoc basis, in the vast literature on the lumped parameter analysis of pulsatile physiological flows is thus rigorously assessed.