Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrability

Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In the real-life applications like blood flow, there is often an additional complexity of vorticity being present in the fluid. We present a theory for the dynamics of interaction of fluids and structures. The equations are derived using the 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 show that even when the vorticity is present, but is kept at a constant value, the case of an inextensible, unshearable and straight tube with elastics walls carrying a fluid allows an alternative formulation, reducing to a single compact equation for the back-to-labels map instead of three conservation equations. That single equation shows interesting instability in solutions when the vorticity exceeds a certain threshold. Furthermore, the equation in stable regime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equations equations in several appropriate limits, namely, the first two in the limit of long time and length scales and the third one in the additional limit of the small cross-sectional area. For the unstable regime, we numerical solutions demonstrate the spontaneous appearance of large oscillations in the cross-sectional area.Comment: 57 pages, 11 figures. arXiv admin note: text overlap with arXiv:1805.1102

Geometric theory of flexible and expandable tubes conveying fluid: equations, solutions and shock waves

We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive the conservation laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the tubes, and compute several examples of particular solutions. The theory is derived from a variational treatment of Cosserat rod theory extended to incorporate expandable walls and moving flow inside the tube. The results presented here are useful for biological flows and industrial applications involving high speed motion of gas in flexible tubes

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

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