Exact solution to the inverse Womersley problem for pulsatile flows in cylindrical vessels, with application to magnetic particle targeting
An exact solution to the inverse Womersley problem was derived for the fully-developed, laminar pulsatile flow of a viscous Newtonian fluid, within a circular cylindrical vessel with rigid walls. In particular, given an arbitrary, time-periodic flow rate, the axisymmetric velocity profile was obtained by means of two neat and computable maps relating the corresponding Fourier coefficients. The study of such an inverse problem is motivated by the fact that flow rate is the main physical quantity which can be actually measured in many practical situations. The hypothesis of a fully-developed flow was deliberately introduced, in order to obtain an analytical solution (otherwise hardly achievable). Despite the intrinsic simplifications associated with the adopted position (which restrict the applicability of our results to 3D finite-length complex domains, and non-Newtonian fluids), the obtained solution provides a benchmark - and at the same time an approximation - for the inverse problem of pulsatile flows, it may serve as a debugging tool for more ambitious numerical approaches based on realistic data, and can also be used as an improved source of boundary data. As expected, the main advantage of our analytical solutions (compared to fully numerical approaches) resides in computational efficiency; this was quantitatively assessed through numerical tests. Moreover, the proposed solution was applied in the context of magnetic particle targeting, to highlight some peculiar effects on particle trajectories and capture efficiency due to pulsatility. Such a transport problem is increasingly drawing the attention of an interdisciplinary community, ranging from physicians to biomedical engineers, physicists and roboticists, thanks to its potential for targeted therapy, up to remote guidance of intravascular devices. More in general, the obtained benchmark solution holds potential for effectively exploitation in an interdisciplinary context.