Closed-Form Solution for Oscillatory Flow and Wall Shear Stress in Axisymmetric Corrugated Tubes

Abstract

Pulsatile flow in corrugated tubes arises in hemodynamics and microfluidics, where oscillatory forcing and geometric constrictions jointly determine transport and wall loading. We extend Womersley's classical solution for oscillatory flow in a rigid circular tube to rigid axisymmetric tubes of slowly varying radius. Within the lubrication approximation, we derive closed-form expressions for the axial velocity profile, volumetric flow rate, phase lag, and wall shear stress at an arbitrary Womersley number. Three-dimensional lattice Boltzmann simulations are used to assess the regime of validity of the theory. The analysis shows that the local velocity profile varies strongly along the tube, ranging from plug-like in wide sections to more parabolic near bottlenecks. The cycle-maximum flow rate decreases with increasing corrugation, but this reduction weakens as pulsatility increases, reflecting a crossover from the quasi-steady scaling $\langle R^{-4}\rangle^{-1}$ to the high-frequency scaling $\langle R^{-2}\rangle^{-1}$. The wall shear stress is maximal at the bottleneck and decreases with Womersley number. For sinusoidal corrugations, the bottleneck wall shear stress depends non-monotonically on the corrugation because of the competition between local shear amplification and global hydraulic resistance. Closed-form expressions for the time-averaged wall shear stress and oscillatory shear index further connect the theory to standard hemodynamic metrics.