Fluid rheology effect on wave propagation in an elastic tube with viscoelastic liquid, containing fine bubbles

The paper is devoted to theoretical analysis of the role of fluid rheology at sound wave propagation in thin-walled elastic tubes with viscoelastic liquid, containing fine gas bubbles. The fluid mechanics description is based on a generalized linear Maxwell model; dynamic equations of the tube wall are formulated within the Kirchhoff-Love approximation. Compressibility of liquid in the presence of free gas is related to dynamic compressibility of bubbles. It is assumed that volume gas concentration is small, bubble radii are much less than the tube diameter and bubbles interact with the pipe wall and between themselves via the pressure field. Rheological properties of the liquid are accounted for both in calculation of the transient friction in the liquid flow in the wave and in the description of bubble dynamics. The problem of wave propagation in a waveguide with viscoelastic liquid–gas mixture is solved in conjugated quasi-one-dimensional approximation; the dispersion equation is studied in the long-wave range. Results of simulations illustrate the strong influence of the liquid viscoelasticity on sound dispersion and attenuation both at frequencies close to the resonance frequency of the bubbles, and in the low frequency range, where interaction of the fluid with the tube wall plays the dominant role. The study has potential importance for the description of transient processes in pipes with flowing polymeric liquid, containing microbubbles; it can find application also in free gas diagnostics in viscoelastic fluid by acoustic means.