This study presents a semi-analytical model for flow in a tapered compliant tube, incorporating both fluid viscosity and wall viscoelasticity (modelled as a Kelvin–Voigt material) while capturing downstream wave reflections through an impedance boundary condition. A linearised fluid–structure interaction (FSI) system is formulated in the frequency domain under the long-wave approximation. A new semi-analytical solution procedure is developed, allowing for efficient and accurate computation of the frequency-domain response, with time-domain solutions reconstructed using the inverse Fourier transform. The model is validated against three-dimensional FSI simulations, showing excellent agreement in pressure, flow rate, velocity profile and wall shear stress (WSS). Theoretical analysis reveals that the influences of fluid and wall viscosities on the wave speed and transmission coefficient are complex. Wall viscosity has a dominant effect on wave speed and damping at high frequencies, with wave speed increasing with both wall viscosity and frequency. In contrast, fluid viscosity has minimal effect on wave speed. The study further shows that viscosities significantly alter the characteristics of flow quantities, such as pressure, flow rate and pressure power. Compared with the inviscid case, solid viscosity leads to strong attenuation of these quantities above approximately 10 Hz, with their amplitudes monotonically decreasing as frequency rises. Fluid viscosity causes a moderate reduction in amplitudes. Notably, wall viscosity plays a critical role in regulating WSS. Its absence causes WSS to increase substantially with frequency, highlighting the importance of including solid viscous effects in theoretical analysis. In summary, this work provides an efficient and accurate framework for studying the combined effects of fluid and solid viscosities in a more physiologically accurate setting, offering clear insight into their roles in pulse-wave propagation.
Wu et al. (Tue,) studied this question.