The role of the Womersley number in pulsatile blood flow: a theoretical study of the Casson model

The purpose of this Note is to clarify the meaning of the Womersley number α in pulsatile blood flow in small vessels. In particular, we explain why the use of α as aperturbation parameter to obtain approximate solutions of the Casson model (frequently used in the literature) is not appropriate. Using the techniques of dimensional analysis and scaling, we show that α is the product of the Reynolds and Strouhal numbers. Since the latter is very small for physiological flows, the result is that α<1 even at relatively high values of the Reynolds number (i.e., for non-negligible inertia) and we validate our perturbation theory results by comparison with a numerical integration of the full model. Although this analysis is based on the Casson model, our method has general validity and may be applied to other models which describe more accurately the rheological properties of blood.