Accurate series solutions for gravity-driven Stokes waves

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that higher order behaviour of series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Pade approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Pade approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Exponential asymptotics with multiple stokes lines

When asymptotic series methods are applied in order to solve problems that arise in applied mathematics in the limit that some parameter becomes small, they are unable to demonstrate behaviour that occurs on a scale that is exponentially small compared to the algebraic terms of the asymptotic series. There are many examples of physical systems where behaviour on this scale has important effects and, as such, a range of techniques known as exponential asymptotic techniques were developed that may be used to examinine behaviour on this exponentially small scale. Many problems in applied mathematics may be represented by behaviour within the complex plane, which may subsequently be examined using asymptotic methods. These problems frequently demonstrate behaviour known as Stokes phenomenon, which involves the rapid switches of behaviour on an exponentially small scale in the neighbourhood of some curve known as a Stokes line. Exponential asymptotic techniques have been applied in order to obtain an expression for this exponentially small switching behaviour in the solutions to orginary and partial differential equations. The problem of potential flow over a submerged obstacle has been previously considered in this manner by Chapman & Vanden-Broeck (2006). By representing the problem in the complex plane and applying an exponential asymptotic technique, they were able to detect the switching, and subsequent behaviour, of exponentially small waves on the free surface of the flow in the limit of small Froude number, specifically considering the case of flow over a step with one Stokes line present in the complex plane. We consider an extension of this work to flow configurations with multiple Stokes lines, such as flow over an inclined step, or flow over a bump or trench. The resultant expressions are analysed, and demonstrate interesting implications, such as the presence of exponentially sub-subdominant intermediate waves and the possibility of trapped surface waves for flow over a bump or trench. We then consider the effect of multiple Stokes lines in higher order equations, particu- larly investigating the behaviour of higher-order Stokes lines in the solutions to partial differential equations. These higher-order Stokes lines switch off the ordinary Stokes lines themselves, adding a layer of complexity to the overall Stokes structure of the solution. Specifically, we consider the different approaches taken by Howls et al. (2004) and Chap- man & Mortimer (2005) in applying exponential asymptotic techniques to determine the higher-order Stokes phenomenon behaviour in the solution to a particular partial differ- ential equation.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Water uptake of aerosols with a focus on seeded aerosols and instrumentation techniques

This thesis focuses on the volatile and hygroscopic properties of mixed aerosol species. In particular, the influence organic species of varying solubility have upon seed aerosols. Aerosol studies were conducted at the Paul Scherrer Institut Laboratory for Atmospheric Chemistry (PSI-LAC, Villigen, Switzerland) and at the Queensland University of Technology International Laboratory for Air Quality and Health (QUT-ILAQH, Brisbane, Australia). The primary measurement tool employed in this program was the Volatilisation and Hygroscopicity Tandem Differential Mobility Analyser (VHTDMA - Johnson et al. 2004). This system was initially developed at QUT within the ILAQH and was completely re-developed as part of this project (see Section 1.4 for a description of this process). The new VHTDMA was deployed to the PSI-LAC where an analysis of the volatile and hygroscopic properties of ammonium sulphate seeds coated with organic species formed from the photo-oxidation of á-pinene was conducted. This investigation was driven by a desire to understand the influence of atmospherically prevalent organics upon water uptake by material with cloud forming capabilities. Of particular note from this campaign were observed influences of partially soluble organic coatings upon inorganic ammonium sulphate seeds above and below their deliquescence relative humidity (DRH). Above the DRH of the seed increasing the volume fraction of the organic component was shown to reduce the water uptake of the mixed particle. Below the DRH the organic was shown to activate the water uptake of the seed. This was the first time this effect had been observed for á-pinene derived SOA. In contrast with the simulated aerosols generated at the PSI-LAC a case study of the volatile and hygroscopic properties of diesel emissions was undertaken. During this stage of the project ternary nucleation was shown, for the first time, to be one of the processes involved in formation of diesel particulate matter. Furthermore, these particles were shown to be coated with a volatile hydrophobic material which prevented the water uptake of the highly hygroscopic material below. This result was a first and indicated that previous studies into the hygroscopicity of diesel emission had erroneously reported the particles to be hydrophobic. Both of these results contradict the previously upheld Zdanovksii-Stokes-Robinson (ZSR) additive rule for water uptake by mixed species. This is an important contribution as it adds to the weight of evidence that limits the validity of this rule.