A mathematical analysis of nonlinear waves in a fluid filled visco-elastic tube

Viscoelastic Modulation of Solitary Pressure Pulses in Nonlinear Fluid-Filled Distensible Tubes

“…It is possible to show that these terms are at least two orders of magnitude smaller than the leading terms in (2.5) and consequently will play no role in our analysis. The viscoelastic shell-wall model (2.5) differs from the model assumed by Ravinchan and Prasad (19) in several respects. Besides the choice of the Kelvin-Voigt viscoelastic model (which we shall comment on momentarily), the model (2.5) retains terms corresponding to the rotatory inertia, flexural rigidity and circumferential stiffness of the tube wall.…”

“…Besides the choice of the Kelvin-Voigt viscoelastic model (which we shall comment on momentarily), the model (2.5) retains terms corresponding to the rotatory inertia, flexural rigidity and circumferential stiffness of the tube wall. These terms do not appear in (19). With respect to the choice of the viscoelastic model, it is well known that for the relatively low wavenumber and frequency motions examined here, the Kelvin-Voigt model is the most appropriate choice for a material in which the response to a stress change after a sufficiently long time is elastic rather than viscous (28) as opposed to the Maxwell model in (19).…”

“…These terms do not appear in (19). With respect to the choice of the viscoelastic model, it is well known that for the relatively low wavenumber and frequency motions examined here, the Kelvin-Voigt model is the most appropriate choice for a material in which the response to a stress change after a sufficiently long time is elastic rather than viscous (28) as opposed to the Maxwell model in (19). Perhaps of equal importance to the above remarks is the fact that (2.5) has been used to model many aspects of pulse propagation in viscoelastic tubes (11, 12) and therefore an analysis of its predictions in the weakly-nonlinear and weakly-dispersive limit is of interest.…”

“…The Kelvin-Voigt viscoelastic tube-wall model adopted here is essentially that derived by Moodie et al (10,27) and successfully tested experimentally (11, 12). Our derivation of the KdVB equation will be relatively brief since the asymptotic balances are well known and can be found in, for example, (5, 19).…”

“…Previously, Ravinchan and Prasad (19) had shown for a Maxwell viscoelastic tube-wall model that long small-amplitude pressure pulses were described by a Korteweg-de Vries-Burgers equation. However, no analysis of any kind was presented detailing the propagation characteristics of solitary or periodic pressure pulses.…”