Swirling Fluid Flow in Flexible, Expandable Elastic Tubes: Variational Approach, Reductions and Integrability

Abstract

Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In real-life applications like blood flow, a swirl in the fluid often plays an important role, presenting an additional complexity not described by previous theoretical models. We present a theory for the dynamics of the interaction between elastic tubes and swirling fluid flow. The equations are derived using a variational principle, with the incompressibility constraint of the fluid giving rise to a pressure-like term. In order to connect this work with the previous literature, we consider the case of inextensible and unshearable tube with a straight centerline. In the absence of vorticity, our model reduces to previous models considered in the literature, yielding the equations of conservation of fluid momentum, wall momentum and the fluid volume. We pay special attention to the case when the vorticity is present but kept at a constant value. We show the conservation of energy-like quality and find an additional momentum-like conserved quantity. Next, we develop an alternative formulation, reducing the system of three conservation equations to a single compact equation for the back-to-labels map. That single equation shows interesting instability in solutions when the velocity exceeds a critical value. Furthermore, the equation in stable regime can be reduced to Boussinesq-type, KdV and Monge–Ampère equations in several appropriate limits, namely, the first two in the limit of a long time and length scales and the third one in the additional limit of the small cross-sectional area. For the unstable regime, the numerical solutions demonstrate the spontaneous appearance of large oscillations in the cross-sectional area