Analytical frequency response of pulsatile viscoelastic channel flow

Flows in many biological and engineering applications are viscoelastic and subject to pulsatile disturbances. In this work, we analyze a pulsatile viscoelastic channel flow with the Oldroyd-B model. Using a Fourier framework, we obtain closed-form expressions for the velocity, polymer shear stress, pressure gradient, and wall shear stress for retained modes of a general periodic forcing. The polymer normal stress is evaluated through a spectral convolution that captures coupling between frequency modes. The frequency response is organized by a complex transfer function that relates the imposed flow rate waveform to the required pressure-gradient response. At low frequency, Oldroyd-B fluids require a smaller pressure-gradient pulsation than Newtonian fluids to deliver the same flow rate modulation, which defines a dynamic flow enhancement regime dominated by elasticity. A low frequency expansion yields an explicit critical elasticity for the onset of enhancement and its dependence on polymer concentration through the solvent viscosity ratio, β. As the forcing frequency and fluid elasticity increase, oscillations of the polymer shear stress weaken, and the dynamics approach a Newtonian limit. In this regime, the Oldroyd-B and Newtonian momentum equations become asymptotically equivalent under an effective Womersley number based on the solvent viscosity. Numerical evaluations confirm that this mapping accurately predicts the velocity distribution, critical pulsation amplitude, and reversal regions near the wall. These findings suggest a simplified way to account for pulsatile viscoelastic flows at high frequency, with potential implications that can guide the design and optimization of related biological and industrial applications.