Formulas for Phase Velocity and Damping of Longitudinal Waves in Thick-Walled Viscoelastic Tubes

Abstract

The frequency equation for longitudinal waves in a viscoelastic tube of infinite length filled with a viscous fluid is solved for the complex wavenumber, k(=k1+ik2). Thus for two modes, Young's mode and Lamb's mode, an explicit expression for k is obtained which yields the phase velocity c(=ω/k1, ω=radial frequency) and the damping constant k2. The formulas for k, c, and k2 are valid for wall thicknesses up to the size of the inner radius of the tube. The approximation as compared with results obtained numerically with an IBM 7040 digital computer from the original frequency equation involves an error which in almost all cases is considerably less than 1.5%. For Lamb's mode, as it was found earlier for the torsional mode, an increase of the fluid viscosity increases the damping at the higher frequencies but decreases the damping at the lower frequencies. The theoretical results are compared with those of experiments in which the longitudinal waves are generated in the fluid and the fluid in its turn moves the wall. The experimental curves thus obtained show the pattern of Young waves, and the agreement with the Young curves obtained from the formulas mentioned above is satisfactory. For torsional waves, formulas for k, c, and k2 analogous to these for longitudinal waves are given.

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

“…A Young wave is generated by an oscillatory pressure forcing in the fluid, while a Lamb wave can be expected when the tube wall is excited harmonically in the direction of the axis. 3 Most of our discussions below will be on the Young wave mode, which is more important in the present study.…”

“…For blood flow in large arteries 3,4 or gas flow in the upper airways, 5 a tube of lumen radius of O͑1͒ cm and longitudinal length of O͑10͒ cm is normally considered. The ratio of wall thickness to lumen radius is approximately 15% in the case of blood vessels and 5% in the case of pulmonary airways.…”

“…Since Womersley, [6][7][8] who pioneered a linearized theory of oscillatory flow in a straight, uniform, isotropic, thinwalled, and elastic tube, the problem has been advanced through numerous developments. In view of the nature of human tissues, the model has been extended to tubes which are thick walled, 3,4,9,10 viscoelastic, 4,[11][12][13] orthotropic, 14,15 tethered by the surrounding tissues, 14,16,17 or with initial stresses. 18,19 Conventionally, a tube is modeled to be a thin-walled structure ͑like a shell or membrane͒ when its wall thickness is supposed to be infinitesimally small compared with the tube radius.…”

An experimental test of the theory of waves in fluid-filled deformable tubes

“…The scatter of the results was partly due to the motion of the tube not being purely longitudinal. Klip, van Loon & Klip (1967) found that E,/E, was 'frequency dependent and close to 0.02 '. The difference between Klip's value and those of figure 2 will be shown not to have much effect in the application to a tethered tube but to be significant in the case of an unconstrained tube.…”

Transmission Characteristics of Fluidborne Pressure Ripples in Finite-Length Hydraulic Flexible Hoses

“…(3) Longitudinal resonance of the hose wall may have a significant influence on the coupling motion of the hose and the fluid; (4) For a finite-length hose, the hose end support conditions exert direct effect on longitudinal vibrations of the hose wall and subsequently wave propagation in the fluid. Some efforts have been contributed to wave propagation characteristics in viscoelastic fluid pipes [3][4][5][6][7][8][9][10][11]. But in view of the four facts mentioned above, they can not satisfy the application of actual hoses with finite length owing to infinite-length assumption and other imperfection in theory and experiment.…”