Derivation of the Gromeka Acceleration Vector for Dimensionless Womersley Flow

This manuscript presents an analytical and theoretical investigation of the Gromeka acceleration field in a dimensionless Womersley flow, derived through the exact solution of the governing Navier-Stokes equations in phase space. By decomposing the convective acceleration into rotational and nonrotational components, the derivation highlights the dominant role of vorticity dynamics near the wall, where steep velocity gradients interact with the oscillatory axial velocity to produce localized radial accelerations. The solution reveals that the Gromeka acceleration mediates nonlinear interactions between harmonics, driving energy redistribution and boundary layer development under multi-harmonic boundary conditions. Complementary analysis of the kinetic energy gradient further delineates inertial effects, demonstrating their role in phase-dependent flow separation and reattachment. These findings provide a comprehensive framework for understanding momentum transport and instability generation in pulsatile wall-bounded flows.