A transverse picoNewton force revealed in anisotropic Womersley flow

Abstract

Abstract The fluid-dynamic quantity that regulates endothelial mechanotransduction remains unsettled. Wall Shear Stress (WSS) characterizes boundary traction but does not represent the volumetric inertial structure of arterial blood flow. Meanwhile, blood exhibits direction-dependent stress under physiological shear, suggesting that the classical Womersley solution of the Navier-Stokes equation may omit constitutive mechanisms relevant to near-wall dynamics. Here, I derive an anisotropic extension of Womersley flow by introducing a tensorial viscosity into the incompressible Navier–Stokes equations. By evaluating the nonlinear interaction of velocity and vorticity within a near-wall control volume, I demonstrate that anisotropic viscosity produces a non-trivial spectral signature in the transverse forcing, maintaining power across higher-order harmonics. While macroscopic geometric drivers dominate the bulk flow at the fundamental cardiac frequency, they are subject to significant inertial damping as the harmonic frequency increases. In contrast, the anisotropy-induced Lamb vector, sustained by the sharp gradients of the oscillatory boundary layer, evades this macroscopic attenuation. These findings define a geometry-independent baseline for multidirectional pulsatile dynamics and provide a theoretical basis for future spectral investigations of endothelial mechanobiology under high-frequency, near-wall inertial stimuli.