DSpace Collection:https://dspace.lboro.ac.uk/2134/46422015-10-10T16:14:08Z2015-10-10T16:14:08ZMathematical modelling of nonlinear ring waves in a stratified fluidZhang, Xizhenghttps://dspace.lboro.ac.uk/2134/185872015-09-14T15:28:06Z2015-01-01T00:00:00ZTitle: Mathematical modelling of nonlinear ring waves in a stratified fluid
Authors: Zhang, Xizheng
Abstract: Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this thesis, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition, which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1-dimensional cylindrical Korteweg-de Vries (cKdV)-type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant shear flow, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts: the wavefront of the surface wave is elongated in the shear flow direction while the wavefront of the interfacial wave is squeezed in this direction. We solve the derived 2+1-dimensional cKdV-type equation numerically using a finite-difference scheme. The effects of nonlinearity and dispersion are studied by considering numerical results for surface and interfacial ring waves generated from a localised source with and without shear flow and the 2D dam break problem. In these examples, the linear and nonlinear surface waves are faster than interfacial waves, the wave height decreases faster at the surface, the shear flow leads to the wave height decreasing slower downstream and faster upstream, and the effect becomes more prominent as the shear flow strengthens.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZRadiation damage and inert gas bubbles in metalsGai, Xiaohttps://dspace.lboro.ac.uk/2134/179272015-07-09T14:26:41Z2015-01-01T00:00:00ZTitle: Radiation damage and inert gas bubbles in metals
Authors: Gai, Xiao
Abstract: Inert gases in metals can occur due to ion implantation, from a plasma in a magnetron device or as a result of being by-products of nuclear reactions. Mainly because of the nuclear applications, the properties of the inert gases, helium, argon and xenon in the body centred cubic (bcc) iron crystal are examined theoretically using a combination of molecular dynamics, static energy minimisation and long time scale techniques using empirical potential functions. The same techniques are also used to investigate argon and xenon in aluminium.
The primary interest of the work occurred because of He produced in nuclear fission and its effect on the structural materials of a fission reactor. This structure is modelled with perfectly crystalline bcc Fe. In bcc iron, helium is shown to diffuse rapidly forming small bubbles over picosecond time scales, which reach a certain optimum size. In the initial phase of He accumulation, Fe interstitials are ejected. This occurs instantaneously for bubbles containing 5 He atoms and as the more He accumulates, more Fe interstitials are ejected. The most energetically favourable He to vacancy ratios at 0 K, vary from 1 : 1 for 5 vacancies up to about 4 : 1 for larger numbers of vacancies. An existing He bubble can be enlarged by a nearby collision cascade through the ejection of Fe interstitials, allowing more He to be trapped.
Ar and Xe in bcc Fe prefer to be substitutional rather than interstitial and there are large barriers to be overcome for the inert gas atoms to diffuse from a substitutional site. Bubbles that form can again be enlarged by the presence of a nearby collision cascade or at very high temperatures. In this case the most energetically favourable vacancy ratios in the bubbles is 1: 1 for Ar and from 0.6: 1 to 0.8: 1 for Xe. For Ar and Xe, bubble formation is more likely as a direct result of radiation or radiation enhanced diffusion rather than diffusion from a substitutional site.
Ar in aluminium is also studied. Ar atoms in fcc Al prefer to be substitutional rather than interstitial and evolution into substitutional occurs over picosecond time scales at room temperature. Bubble formation can occur more easily than in bcc iron, mainly because the barriers for vacancy diffusion are much lower but the time scales for bubble accumulation are much longer than those for He. A vacancy assisted mechanism is found which allows Ar to diffuse through the lattice. Finally some preliminary results on the energetics of different geometrical structures of larger Xe bubbles in Al are investigated since experiment has indicated that these can become facetted.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZDispositional factors affecting children's early numerical developmentBatchelor, Sophiehttps://dspace.lboro.ac.uk/2134/174742015-05-28T13:55:19Z2014-01-01T00:00:00ZTitle: Dispositional factors affecting children's early numerical development
Authors: Batchelor, Sophie
Abstract: Children show large individual differences in numerical skills, even before they begin formal education. These early differences have significant and long-lasting effects, with numerical knowledge before school predicting mathematical achievement throughout the primary and secondary school years. Currently, little is known about the dispositional factors influencing children's numerical development. Why do some children engage with and succeed in mathematics from an early age, whilst others avoid mathematics and struggle to acquire even basic symbolic number skills?
This thesis examines the role of two dispositional factors: First, spontaneous focusing on numerosity (SFON), a recently developed construct which refers to an individual's tendency to focus on the numerical aspects of their environment; and second, mathematics anxiety (MA), a phenomenon long recognised by educators and researchers but one which is relatively unexplored in young children. These factors are found to have independent effects on children's numerical skills, thus the empirical work is presented in two separate parts.
The SFON studies start by addressing methodological issues. It is shown that the current measures used to assess children's SFON vary in their psychometric properties and subsequently a new and reliable picture-based task is introduced. Next, the studies turn to theoretical questions, investigating the causes, consequences and mechanisms of SFON. The findings give rise to three main conclusions. First, children's SFON shows little influence from parental SFON and home numeracy factors. Second, high SFON children show a symbolic number advantage. Third, the relationship between SFON and arithmetic can be explained, in part, by individual differences in children's ability to map between nonsymbolic and symbolic representations of number.
The MA studies focus primarily on gender issues. The results reveal no significant differences between boys' and girls' overall levels of MA; however, there are gender differences in the correlates of MA. Specifically, boys' (but not girls') MA is related to parents' MA. Moreover, the relationship between MA and mathematical outcomes is stronger for boys than it is for girls. Possible causal explanations for these gender differences are explored in two ways: First, by examining the reliability of the scales used to assess MA in boys and girls. Second, by investigating the relationship between girls' (and boys') mathematics anxiety and their societal math-gender stereotypes.
The findings from both sets of studies draw a link between children's emerging dispositions towards mathematics and their early numerical skills. Future research needs to examine how these dispositional factors interact with other (cognitive and non-cognitive) predictors of mathematics achievement.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2014-01-01T00:00:00ZQuasilinear PDEs and forward-backward stochastic differential equationsWang, Xincehttps://dspace.lboro.ac.uk/2134/173832015-07-09T15:22:21Z2015-01-01T00:00:00ZTitle: Quasilinear PDEs and forward-backward stochastic differential equations
Authors: Wang, Xince
Abstract: In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}(
\mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d};
\mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00Z