This paper investigates the coupled motion between the dynamics of N vessels
coupled together in a one-dimensional array by springs, and the motion of the
inviscid fluid sloshing within each vessel. We develop a fully-nonlinear model for
the system relative to a moving frame such that the fluid in each vessel is governed
by the Euler equations and the motion of each vessel is modelled by a forced spring
equation. By considering a linearization of the model, the characteristic equation
for the natural frequencies of the system is derived, and analysed for a variety of
non-dimensional parameter regimes. It is found that the problem can exhibit a
variety of resonance situations from the 1 : 1 resonance to (N + 1)-fold 1 : · · · : 1
resonance, as well as more general r : s : · · · : t resonances for natural numbers
r, s, t. This paper focuses in particular on determining the existence of regions of
parameter space where the (N + 1)-fold 1 : · · · : 1 resonance can be found.
Advances in monitoring technology allow blood pressure waveforms to be collected at sampling frequencies of 250-1000Hz for long time periods. However, much of the raw data are under analysed. Heart rate variability (HRV) methods, in which beat-to-beat interval lengths are extracted and analysed, have been extensively studied, However, this approach discards the majority of the raw data. Objective: Our aim is to detect changes in the shape of the waveform in long streams of blood pressure data. Approach: Our approach involves extracting key features from large complex datasets by generating a reconstructed attractor in a three-dimensional phase space using delay coordinates from a window of the entire raw waveform data. The naturally occurring baseline variation is removed by projecting the attractor onto a plane from which new quantitative measures are obtained. The time window is moved through the data to give a collection of signals which relate to various aspects of the waveform shape. Main results: This approach enables visualisation and quantification of changes in the waveform shape and has been applied to blood pressure data collected from conscious unrestrained mice and to human blood pressure data. The interpretation of the attractor measures is aided by the analysis of simple artificial waveforms. Significance: We have developed and analysed a new method for analysing blood pressure data that uses all of the waveform data and hence can detect changes in the waveform shape that HRV methods cannot, which is confirmed with an example, and hence our method goes "beyond HRV".
A variational principle is derived for two-dimensional
incompressible rotational fluid flow with a free
surface in a moving vessel when both the vessel
and fluid motion are to be determined. The fluid
is represented by a stream function and the vessel
motion is represented by a path in the planar
Euclidean group. Novelties in the formulation include
how the pressure boundary condition is treated, the
introduction of a stream function into the Euler-
Poincaré variations, the derivation of free surface
variations, and how the equations for the vessel path
in the Euclidean group, coupled to the fluid motion,
are generated automatically.
In this thesis we investigate the dynamics of coupled liquid sloshing systems, which consist
of a one-dimensional array of vessels, partially filled with fluid, being connected together
by nonlinear springs. The fluid motion induces a hydrodynamic force on the side walls of
the vessel, which induces the vessel to move. The vessel movement is also controlled via a
restoring force of the attached springs, which in turn causes the fluid to alter its motion.
The simplest coupled liquid sloshing system consists of one vessel connected to a side wall via
a spring. We review this example and investigate the techniques documented in the literature
to study the linear problem of such a system. Then we extend this linear theory to multivessel systems, in particular focussing on the 2-vessel system which introduces the notion of
modes being in-phase or out-of-phase with each other. In the general N-vessel system we
also identify the existence in parameter space of internal resonances, where different modes
oscillates at the same frequency. Such a resonance provides a mechanism for energy exchange
between modes in the nonlinear system.
The nonlinear dynamics of the coupled liquid sloshing system are studied by employing a
symplectic integration scheme based on a variational principle, to the shallow-water form of
the governing equations. We present results for the shallow-water scheme in the linear, weakly
nonlinear and fully nonlinear regimes. The shallow-water numerical scheme has trouble
dealing with breaking waves, so we perform a feasibility study where we attempt to use a
similar approach for a non-shallow fluid system. The governing equations are derived and
results presented in the linear amplitude regime for the 1-vessel system.
Finally, we develop variational principles for the coupled liquid sloshing system in the Eulerian
framework based on the principle of constrained variations to derive the governing fluid
equations and free-surface boundary conditions, from a natural Lagrangian functional. We
use this constrained variational approach to derive the fully rotational 2D Euler equations
and its stream function formulation.