Low-Mach number asymptotic analysis of fluid–structure-interaction (FSI) pressure waves inside an elastic tube

Abstract

We consider the pressure wave having velocity cp inside an elastic tube of internal radius R0, thickness e, length L, shear modulus G and density ρs0, filled with a fluid with kinematic viscosity νf. We theoretically analyze the fluid–structure coupling between: (i) the elastic sheath, (ii) the fluid boundary layer, and (iii) the core acoustic pressure and velocity fields. Our analysis provides an asymptotic derivation of the fluid–structure-interactions (FSI) model that recovers known pulse-wave velocities and provides a new theoretical prediction for the exponential time decay of the wave longitudinal attenuation envelope. Taking advantage of highly distinct time-scales between the viscous radial diffusion τd=R02/νf compared with wave-convective time τc=L/cp as well as the elastic relaxation time τe=eρs/G, such that τe∼τc≪τd we perform a two time-scale asymptotic analysis based on a small parameter δ=τc/τd. Said parameter is obtained by balancing the momentum acceleration and the viscous damping rate in the inner unsteady boundary layer, the thickness of which being δR0. The resulting asymptotic sequence provides a unique consistent scaling for solid deformation and velocity fields, with the secularity condition associated with the leading-order slow-time scale envelope attenuation obtained by extending the analysis to investigate the first-order corrections.On the one hand our approach reconciles both predictions for the precursive elastic wave and the pulse velocities obtained when considering solid deformation only, and, on the other hand, predictions for the longitudinal attenuation resulting from the effect of boundary layers only. Our analysis also permits the derivation of a new convoluted model for the wall shear stress, which is (FSI)–consistent. The theoretical results are successfully compared with experimental measurements.