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

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

Integrable Systems as Fluid Models with Physical Applications

In this thesis we begin with the development and analysis of hydrodynamical models as they arise in the theory of water waves and in the modelling of blood flow within arteries. Initially we derive three models of hydrodynamical relevance, namely the KdV equation, the two component Camassa-Holm equation and the Kaup-Boussinesq equation. We develop a model of blood flowing within an artery with elastic walls, and from the principles of Newtonian mechanics we derive the two-component Burger\u27s equation as our first integrable model. We investigate the analytic properties of the system briefly, with the aim of demonstrating the phenomenon of wave breaking for the system. In addition we construct a pair of diffeomorphisms which allow us to solve the system explicitly in terms of the initial data. Finally, we show that when we consider the dynamics of the arterial walls themselves, the pressure within the fluid is seen to satisfy the KdV equation. In the following chapter we investigate the trajectories followed by individual fluid particles in a fluid, as they are subject to the effects of an extreme Stokes wave. In the case of a regular stokes wave there are no stagnation points or apparent stagnation points, i.e. locations where the fluid velocity and wave velocity are equal, however this condition does no remain true in the context of extreme Stokes waves. The result for the regular Stokes wave then have to be extended to semi-infinite regions with corners, and in doing so we show that the horizontal component of the fluid velocity field is strictly increasing along any stream line, which in turn ensures the non-closure of particle trajectories over the course of a fluid wave. Next we begin with a review of the inverse scattering transform method of solving the Kortweg-de Vries equation. We construct the one-soliton solution explicitly. We then proceed to examine the Qiao equation, a non-linear partial differential equation with cubic non-linearities. We show that by a suitable change of variables and with a change of the spectral parameter of its associated spectral problem that we transform it into the spectral problem of the KdV equation. Having already analysed this spectral problem, we then proceed to construct the 1-soliton solution of the Qiao equation with this modified spectral problem. The soliton solutions decay to a non-zero constant value asymptotically. We also investigate the peakon solutions of the Qiao equation, and construct the 1 and 2-peakon profiles, the latter being in the form of travelling M-wave profile. We then go on to the analysis of a class of equations whose spectral problem are more complicated in the sense that the spectral problem has an energy dependant potential. We develop the inverse scattering transform method for these spectral problems, and construct the one-soliton solution explicitly, which in fact turn out to be a breather type solution. The hydrodynamical relevance of this problem arises from the fact that by an appropriate choice of one of the physical parameters of the system, we obtain the Kaup-Boussinesq equation, a partial differential equation with quadratic and cubic nonlinearities which arises in the theory of water waves in shallow water

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

Development, Validation, and Clinical Application of a Numerical Model for Pulse Wave Velocity Propagation in a Cardiovascular System with Application to Noninvasive Blood Pressure Measurements

High blood pressure blood pressure is an important risk factor for cardiovascular disease and affects almost one-third of the U.S. adult population. Historical cuff-less non-invasive techniques used to monitor blood pressure are not accurate and highlight the need for first principal models. The first model is a one-dimensional model for pulse wave velocity (PWV) propagation in compliant arteries that accounts for nonlinear fluids in a linear elastic thin walled vessel. The results indicate an inverse quadratic relationship (R^2=.99) between ejection time and PWV, with ejection time dominating the PWV shifts (12%). The second model predicts the general relationship between PWV and blood pressure with a rigorous account of nonlinearities in the fluid dynamics, blood vessel elasticity, and finite dynamic deformation of a membrane type thin anisotropic wall. The nonlinear model achieves the best match with the experimental data. To retrieve individual vascular information of a patient, the inverse problem of hemodynamics is presented, calculating local orthotropic hyperelastic properties of the arterial wall. The final model examines the impact of the thick arterial wall with different material properties in the radial direction. For a hypertensive subject the thick wall model provides improved accuracy up to 8.4% in PWV prediction over its thin wall counterpart. This translates to nearly 20% improvement in blood pressure prediction based on a PWV measure. The models highlight flow velocity is additive to the classic pressure wave, suggesting flow velocity correction may be important for cuff-less, non-invasive blood pressure measures. Systolic flow correction of the measured PWV improves the R2 correlation to systolic blood pressure from 0.81 to 0.92 for the mongrel dog study, and 0.34 to 0.88 for the human subjects study. The algorithms and insight resulting from this work can enable the development of an integrated microsystem for cuff-less, non-invasive blood pressure monitoring

Quantification of Hemodynamic Pulse Wave Velocity Based on a Thick Wall Multi-Layer Model for Blood Vessels

Pulse wave velocity (PWV) is an important index of arterial hemodynamics, which lays the foundation for continuous, noninvasive blood pressure automated monitoring. The goal of this paper is to examine the accuracy of PWV prediction based on a traditional homogeneous structural model for thin-walled arterial segments. In reality arteries are described as composite heterogeneous hyperelastic structures, where the thickness dimension cannot be considered small compared to the cross section size. In this paper we present a hemodynamic fluid - structure interaction model accounting for the variation of geometry and material properties in a radial direction. The model is suitable to account for the highly nonlinear orthotropic material undergoing finite deformation for each layer. Numerical analysis of single and two layer arterial segments shows that a single thick layer model provides sufficient accuracy to predict PWV. The dependence of PWV on pressure for three vessels of different thicknesses is compared against a traditional thin wall model of a membrane shell interacting with an incompressible fluid. The presented thick wall model provides greater accuracy in the prediction of PWV, and will be important for blood pressure estimation based on PWV measurements

Viscoelasticity

This book contains a wealth of useful information on current research on viscoelasticity. By covering a broad variety of rheology, non-Newtonian fluid mechanics and viscoelasticity-related topics, this book is addressed to a wide spectrum of academic and applied researchers and scientists but it could also prove useful to industry specialists. The subject areas include, theory, simulations, biological materials and food products among others