Nonlinear dynamics of elastic filaments conveying a fluid and numerical applications to the static Kirchhoff equations

Two problems in the study of elastic filaments are considered. First, a reliable numerical algorithm is developed that can determine the shape of a static elastic rod under a variety of conditions. In this algorithm the governing equations are written entirely in terms of local coordinates and are discretized using finite differences. The algorithm has two significant advantages: firstly, it can be implemented for a wide variety of the boundary conditions and, secondly, it enables the user to work with general constitutive relationships with only minor changes to the algorithm. In the second problem a model is presented describing the dynamics of an elastic tube conveying a fluid. First we analyze instabilities that are present in a straight rod or tube under tension subject to increasing twist in the absence of a fluid. As the twist is increased beyond a critical value, the filament undergoes a twist-to-writhe bifurcation. A multiple scales expansion is used to derive nonlinear amplitude equations to examine the dynamics of the elastic rod beyond the bifurcation threshold. This problem is then reinvestigated for an elastic tube conveying a fluid to study the effect of fluid flow on the twist-to-writhe instability. A linear stability analysis demonstrates that for an infinite rod the twist-to-writhe threshold is lowered by the presence of a fluid flow. Amplitude equations are then derived from which the delay of bifurcation due to finite tube length is determined. It is shown that the delayed bifurcation threshold depends delicately on the length of the tube and that it can be either raised or lowered relative to the fluid-free case. The amplitude equations derived for the case of a constant average fluid flux are compared to the case where the flux depends on the curvature. In this latter case it is shown that inclusion of curvature results in small changes in some of the coefficients in the amplitude equations and has only a small effect on the post-bifurcation dynamics.