Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology

In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism taken from quantum theory. We also provide the vectorial expression of the secular tidal torque expanded in Fourier series. Applying this model to close-in exoplanets, we observe that if the relaxation time is longer than the revolution period, the phase space of the system is characterized by the presence of several spin-orbit resonances, even in the circular case. As the system evolves, the planet spin can visit different spin-orbit configurations. The obliquity is decreasing along most of these resonances, but we observe a case where the planet tilt is instead growing. These conclusions derived from the secular torque are successfully tested with numerical integrations of the instantaneous equations of motion on HD 80606 b. Our formalism is also well adapted to close-in super-Earths in multiplanet systems which are known to have non-zero mutual inclinations.

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.

Passive control of a bi-ventricular assist device : an experimental and numerical investigation

For the last two decades heart disease has been the highest single cause of death for the human population. With an alarming number of patients requiring heart transplant, and donations not able to satisfy the demand, treatment looks to mechanical alternatives. Rotary Ventricular Assist Devices, VADs, are miniature pumps which can be implanted alongside the heart to assist its pumping function. These constant flow devices are smaller, more efficient and promise a longer operational life than more traditional pulsatile VADs. The development of rotary VADs has focused on single pumps assisting the left ventricle only to supply blood for the body. In many patients however, failure of both ventricles demands that an additional pulsatile device be used to support the failing right ventricle. This condition renders them hospital bound while they wait for an unlikely heart donation. Reported attempts to use two rotary pumps to support both ventricles concurrently have warned of inherent haemodynamic instability. Poor balancing of the pumps’ flow rates quickly leads to vascular congestion increasing the risk of oedema and ventricular ‘suckdown’ occluding the inlet to the pump. This thesis introduces a novel Bi-Ventricular Assist Device (BiVAD) configuration where the pump outputs are passively balanced by vascular pressure. The BiVAD consists of two rotary pumps straddling the mechanical passive controller. Fluctuations in vascular pressure induce small deflections within both pumps adjusting their outputs allowing them to maintain arterial pressure. To optimise the passive controller’s interaction with the circulation, the controller’s dynamic response is optimised with a spring, mass, damper arrangement. This two part study presents a comprehensive assessment of the prototype’s ‘viability’ as a support device. Its ‘viability’ was considered based on its sensitivity to pathogenic haemodynamics and the ability of the passive response to maintain healthy circulation. The first part of the study is an experimental investigation where a prototype device was designed and built, and then tested in a pulsatile mock circulation loop. The BiVAD was subjected to a range of haemodynamic imbalances as well as a dynamic analysis to assess the functionality of the mechanical damper. The second part introduces the development of a numerical program to simulate human circulation supported by the passively controlled BiVAD. Both investigations showed that the prototype was able to mimic the native baroreceptor response. Simulating hypertension, poor flow balancing and subsequent ventricular failure during BiVAD support allowed the passive controller’s response to be assessed. Triggered by the resulting pressure imbalance, the controller responded by passively adjusting the VAD outputs in order to maintain healthy arterial pressures. This baroreceptor-like response demonstrated the inherent stability of the auto regulating BiVAD prototype. Simulating pulmonary hypertension in the more observable numerical model, however, revealed a serious issue with the passive response. The subsequent decrease in venous return into the left heart went unnoticed by the passive controller. Meanwhile the coupled nature of the passive response not only decreased RVAD output to reduce pulmonary arterial pressure, but it also increased LVAD output. Consequently, the LVAD increased fluid evacuation from the left ventricle, LV, and so actually accelerated the onset of LV collapse. It was concluded that despite the inherently stable baroreceptor-like response of the passive controller, its lack of sensitivity to venous return made it unviable in its present configuration. The study revealed a number of other important findings. Perhaps the most significant was that the reduced pulse experienced during constant flow support unbalanced the ratio of effective resistances of both vascular circuits. Even during steady rotary support therefore, the resulting ventricle volume imbalance increased the likelihood of suckdown. Additionally, mechanical damping of the passive controller’s response successfully filtered out pressure fluctuations from residual ventricular function. Finally, the importance of recognising inertial contributions to blood flow in the atria and ventricles in a numerical simulation were highlighted. This thesis documents the first attempt to create a fully auto regulated rotary cardiac assist device. Initial results encourage development of an inlet configuration sensitive to low flow such as collapsible inlet cannulae. Combining this with the existing baroreceptor-like response of the passive controller will render a highly stable passively controlled BiVAD configuration. The prototype controller’s passive interaction with the vasculature is a significant step towards a highly stable new generation of artificial heart.