Reduced modeling of pulsatile flows in compliant microfluidic conduits at arbitrary Womersley number

We investigate the pulsatile fluid--structure interaction (FSI) between a Newtonian fluid and a slender, deformable microtube. We derive a theory for the instantaneous pressure distribution by two-way coupling the pulsatile flow and the tube deformation. The radial displacement is obtained from thin shell theory, assuming axisymmetry and negligible bending at the leading-order in slenderness. The flow rate is related to the pressure gradient (and the tube deformation) via lubrication theory, specifically via the Womersley solution for pulsatile flow. Substituting this relation into the mass conservation equation, we obtain a 1D nonlinear PDE governing the instantaneous pressure distribution along the tube, at arbitrary Womersley number. This PDE is easy to solve numerically to obtain the pressure distribution in the compliant tube. A cycled-average pressure is also computed, which deviates from the steady profile, suggesting a type of FSI-induced streaming. The instantaneous and cycle-averaged pressures are both validated against 3D direct numerical simulations performed with svFSI (part of the open-source software SimVascular). We find good agreement between theory and simulations.