Solitary waves in fluid-filled elastic tubes: existence, persistence, and the role of axial displacement

Abstract

We re-examine the problem of solitary wave propagation in a fluid-filled elastic membrane tube using a much simplified procedure. It is shown that there may exist four families of solitary waves with speeds close to those given by the linear dispersion relation, whether the fluid is initially stationary or not, and that it is not asymptotically consistent to neglect the axial displacement even in a long-wave approximation. It is also shown that the solitary wave solutions obtained by neglecting higher-order terms persist for the full system of equations in the sense that the full system has solutions of the solitary-wave type and each exact solution is uniformly approximated by the corresponding leading-order solution.

Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow

“…Of course, this persistence result can easily be verified numerically, as was done in [21] for the case when r ∞ is held fixed and c is viewed as a bifurcation parameter.…”

“…We note that the notations here are the same as those in [21] except that the c and v f∞ here correspond to √ mc and √ mv f∞ there. In the limit k → ∞, the four branches of the dispersion relation tend to c = ± √ α 0 and c = ± √ γ 1 , respectively, which are independent of v f∞ and m. Furthermore, when c = ± √ γ 1 , the left hand side of (3.1) reduces to −m(α 1 −β 0 ) 2 , which is in general non-zero (in particular it is non-zero in the stress-free configuration where β 0 ≡ 0).…”

Stability of an inflated hyperelastic membrane tube with localized wall thinning

“…It is further postulated that it is after the initiation of such a localized bulge that biological processes such as remodelling take over, which in turn leads to further growth and final rupture of the aneurysm. The present study is also closely related to studies of solitary waves in hyperelastic membrane tubes; for a review of the relevant literature we refer to [5]. On the one hand, a static localized bulge can be viewed as a solitary wave that has zero propagation speed, the zero speed being induced by the internal pressure in the membrane tube.…”

“…where r 1 is defined by r ∞ = r cr + ǫr 1 , with r cr being the critical value of r ∞ at which a bulge will initiate without any imperfections, ω ′ cr = dω(r cr )/dr cr , γ cr = γ(r cr ), and explicit expressions for ω(r ∞ ), γ(r ∞ ) and ζ in terms of the strain-energy function can be found in [7], [5] and [10], respectively. In [10] several classes of a(ξ) are considered for which (3.1) has closed-form solutions.…”

“…and the material constants α 0 , α 1 , β 0 , β 1 , γ 1 are defined in [5]. It can easily be seen that the four eigenvalues of M ∞ take the form ±k 1 , ±k 2 .…”