Pulsatile Viscous Flows in Elliptical Vessels and Annuli: Solution to the Inverse Problem, with Application to Blood and Cerebrospinal Fluid Flow

摘要

We consider the fully-developed flow of an incompressible Newtonian fluid in a cylindrical vessel with elliptical cross-section and in an annulus between two confocal ellipses. Since flow rate is the main physical quantity which can be actually be derived from measurements, we address the \textit{inverse problem} to compute the velocity field associated with a given, time-periodic flow rate. We propose a novel numerical strategy, which is nonetheless grounded on several analytical relations and which leads to the solution of systems of ordinary differential equations. Our method holds promise to be more amenable to implementation than previous ones based on challenging computation of Mathieu functions. We report numerical results based on measured data for human blood flow in the internal carotid artery, and cerebrospinal fluid flow in the upper cervical region of the human spine. Computational efficiency is shown, but the main goal of the present study is to provide an improved source of initial/boundary data, as well as a benchmark solution for pulsatile flows in elliptical sections and the proposed method has potential applications to bio-fluid dynamics investigations (e.g. to address aspects of relevant diseases), to biomedical applications (e.g. targeted drug delivery and energy harvesting for implantable devices), up to longer-term medical microrobotics applications.