Formation and dynamics of finite amplitude localized pulses in elastic tubes

We present theoretical and simulation studies of the formation and dynamics of finite-amplitude localized pulses (solitary waves) of an incompressible fluid in an elastic tube. Starting from a set of hydrodynamic equations, we derive a Hamiltonian which represents the energy integral of our system. The energy integral is analyzed to obtain explicit profiles of finite-amplitude solitary pulses. Also studied are the excitation and dynamics of solitary pulses by using computer simulations. It is found that a train of solitary pulses can be excited by the nonlinear self-steepening at shock fronts. The relevance of our investigation to blood solitary waves in arteries is discussed.

Figure 1

The Sagdeev potential, describing the amplitude of the blood solitary waves (upper panel), and the corresponding profile of the solitary pulses, for different sets of parameters: $\alpha =2$ and $v_0=1.2$ (solid lines), $\alpha =2$ and $v_0=1.1$ (dashed lines), and $\alpha =2.5$ and $v_0=1.2$ (dash-dotted lines).

Figure 2

The profile of the tube cross section $S$ (upper panel) and velocity $u$ (lower panel) at different times, for $\alpha =2.0$.

Figure 3

The profile of the flow velocity (left panels) and pressure (right panel) at different times for $\alpha =2.0$.