Wave propagation and induced steady streaming in viscous fluid contained in a prestressed viscoelastic tube

Abstract

The oscillatory and time-mean motions induced by a propagating wave of small amplitude through a viscous incompressible fluid contained in a prestressed and viscoelastic (modeled as a Voigt material) tube are studied by a perturbation analysis based on equations of motion in the Lagrangian system. The classical problem of oscillatory viscous flow in a flexible tube is re-examined in the contexts of blood flow in arteries or pulmonary gas flow in airways. The wave kinematics and dynamics, including wavenumber, wave attenuation, velocity, and stress fields, are found as analytical functions of the wall and fluid properties, prestress, and the Womersley number for the cases of a free or tethered tube. On extending the analysis to the second order in terms of the small wave steepness, it is shown that the time-mean motion of the viscoelastic tube with sufficient strength is short lived and dies out quickly as a limit of finite deformation is approached. Once the tube has attained its steady deformation, the steady streaming in the fluid can be solved analytically. Results are generated to illustrate the combined effects on the first-order oscillatory flow and the second-order steady streaming due to elasticity, viscosity, and initial stresses of the wall. The present model as applied to blood flow in arteries and gas flow in pulmonary airways during high-frequency ventilation is examined in detail through comparison with models in the literature.