On the propagation of solitary waves in a prestressed thin elastic tube filled with an inviscid fluid

In this work, we studied the derivation of the Korteweg-deVries equation in a prestressed elastic tube filled with an inviscid fluid. In the analysis, considering the physiological conditions of the arteries, the tube is assumed to be subjected to a uniform inner pressure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $P_0$\end{document} and a constant axial stretch ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda_z$\end{document}. In the course of blood flow in arteries, it is assumed that a finite time dependent displacement field is superimposed on this static field but, due to axial tethering, the effect of axial displacement is neglected. The governing nonlinear equation for the radial motion of the tube under the effect of fluid pressure is obtained. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the longwave approximation is investigated. Treating the blood as an incompressible inviscid fluid, two cases are investigated separately. In the first case, a set of approximate fluid equations is used; namely, the field variables are assumed to be independent of the radial coordinate. Further, the momentum equation of the fluid in the radial direction is neglected. In the second case, the exact equations of an incompressible fluid are employed. It is shown that in both cases the governing equations reduce to the Korteweg-deVries equation but with different propagation speeds. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. The result is numerically discussed for some elastic materials, existing in the current literature.