Gravity-driven fingering simulations for a thin liquid film flowing down the outside of a vertical cylinder

A numerical study is presented to examine the fingering instability of a gravity-driven thin liquid film flowing down the outer wall of a vertical cylinder. The lubrication approximation is employed to derive an evolution equation for the height of the film, which is dependent on a single parameter, the dimensionless cylinder radius. This equation is identified as a special case of that which describes thin film flow down an inclined plane. Fully three-dimensional simulations of the film depict a fingering pattern at the advancing contact line. We find the number of fingers observed in our simulations to be in excellent agreement with experimental observations and a linear stability analysis reported recently by Smolka & SeGall (Phys Fluids 23, 092103 (2011)). As the radius of the cylinder decreases, the modes of perturbation have an increased growth rate, thus increasing cylinder curvature partially acts to encourage the contact line instability. In direct competition with this behaviour, a decrease in cylinder radius means that fewer fingers are able to form around the circumference of the cylinder. Indeed, for a sufficiently small radius, a transition is observed, at which point the contact line is stable to transverse perturbations of all wavenumbers. In this regime, free surface instabilities lead to the development of wave patterns in the axial direction, and the flow features become perfectly analogous to the two-dimensional flow of a thin film down an inverted plane as studied by Lin & Kondic (Phys Fluids 22, 052105 (2010)). Finally, we simulate the flow of a single drop down the outside of the cylinder. Our results show that for drops with low volume, the cylinder curvature has the effect of increasing drop speed and hence promoting the phenomenon of pearling. On the other hand, drops with much larger volume evolve to form single long rivulets with a similar shape to a finger formed in the aforementioned simulations.

Numerical Characterization of Nanowires

In this chapter, we will present a contemporary review of the hitherto numerical characterization of nanowires (NWs). The bulk of the research reported in the literatures concern metallic NWs including Al, Cu, Au, Ag, Ni, and their alloys NWs. Research has also been reported for the investigation of some nonmetallic NWs, such as ZnO, GaN, SiC, SiO2. A plenty of researches have been conducted regarding the numerical investigation of NWs. Issues analyzed include structural changes under different loading situations, the formation and propagation of dislocations, and the effect of the magnitude of applied loading on deformation mechanics. Efforts have also been made to correlate simulation results with experimental measurements. However, direct comparisons are difficult since most simulations are carried out under conditions of extremely high strain/loading rates and small simulation samples due to computational limitations. Despite of the immense numerical studies of NWs, a significant work still lies ahead in terms of problem formulation, interpretation of results, identification and delineation of deformation mechanisms, and constitutive characterization of behavior. In this chapter, we present an introduction of the commonly adopted experimental and numerical approaches in studies of the deformation of NWs in Section 1. An overview of findings concerning perfect NWs under different loading situations, such as tension, compression, torsion, and bending are presented in Section 2. In Section 3, we will detail some recent results from the authors’ own work with an emphasis on the study of influences from different pre-existing defect on NWs. Some thoughts on future directions of the computational mechanics of NWs together with Conclusions will be given in the last section.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.

Spectral approximations to the fractional integral and derivative

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

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.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

A new optimal control model of the interactions between a growing tumour and the host immune system along with an immunotherapy treatment strategy is presented. The model is based on an ordinary differential equation model of interactions between the growing tu- mour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treat- ment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum al- lowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour.

Thermal performance of load bearing cold-formed steel walls under fire conditions using numerical studies

Cold–formed Light gauge Steel Frame (LSF) wall systems are increasingly used in low-rise and multi-storey buildings and hence their fire safety has become important in the design of buildings. A composite LSF wall panel system was developed recently, where a thin insulation was sandwiched between two plasterboards to improve the fire performance of LSF walls. Many experimental and numerical studies have been undertaken to investigate the fire performance of non-load bearing LSF wall under standard conditions. However, only limited research has been undertaken to investigate the fire performance of load bearing LSF walls under standard and realistic design fire conditions. Therefore in this research, finite element thermal models of both the conventional load bearing LSF wall panels with cavity insulation and the innovative LSF composite wall panel were developed to simulate their thermal behaviour under standard and realistic design fire conditions. Suitable thermal properties were proposed for plasterboards and insulations based on laboratory tests and available literature. The developed models were then validated by comparing their results with available fire test results of load bearing LSF wall. This paper presents the details of the developed finite element models of load bearing LSF wall panels and the thermal analysis results. It shows that finite element models can be used to simulate the thermal behaviour of load bearing LSF walls with varying configurations of insulations and plasterboards. Failure times of load bearing LSF walls were also predicted based on the results from finite element thermal analyses. Finite element analysis results show that the use of cavity insulation was detrimental to the fire rating of LSF walls while the use of external insulation offered superior thermal protection to them. Effects of realistic design fire conditions are also presented in this paper.

Final-state effects in neutron Compton scattering measurements on zirconium deuteride and beryllium

We report inelastic neutron scattering measurements of the neutron Compton profile, J(y), for Be and for D in polycrystalline ZrD2 over a range of momentum transfers, q between 27 and 178 °A−1. The measurements were performed using the inverse geometry spectrometer eVS which is situated at the UK pulsed spallation neutron source ISIS. We have investigated deviations from impulse approximation (IA) scattering which are generically referred to as final state effects (FSEs) using a method described by Sears. This method allows both the magnitude and the q dependence of the FSE to be studied. Analysis of the measured data was compared with analysis of numerical simulations based on the harmonic approximation and good agreement was found for both ZrD2 and Be. Finally we have shown how (∇2V), where V is the interatomic potential, can be extracted from the antisymmetric component of J(y).

Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is to study the effect that the curvature of the cylinder has on the gravity-driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane.

Hydrogeological framework, conceptual and numerical groundwater flow model of Laidley Creek catchment, Queensland, Australia

This thesis studies the water resources of Laidley Creek catchment within the Lockyer Valley where groundwater is used for intensive irrigation of crops. A holistic approach was used to consider groundwater within the total water cycle. The project mapped the geology, measured stream flows and groundwater levels, and analysed the chemistry of the waters. These data were integrated within a catchment-wide conceptual model, including historic and rainfall records. From this a numerical simulation was produced to test data validity and develop predictions of behaviour, which can support management decisions, particularly in times of variable climate.

Numerical simulation of anomalous diffusion with application to medical imaging

The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.

A closed-form approximation for pricing temperature-based weather derivatives

This paper develops analytical distributions of temperature indices on which temperature derivatives are written. If the deviations of daily temperatures from their expected values are modelled as an Ornstein-Uhlenbeck process with timevarying variance, then the distributions of the temperature index on which the derivative is written is the sum of truncated, correlated Gaussian deviates. The key result of this paper is to provide an analytical approximation to the distribution of this sum, thus allowing the accurate computation of payoffs without the need for any simulation. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is used to demonstrate the efficacy of this approach for estimating the payoffs to temperature derivatives. It is demonstrated that expected payoffs computed directly from historical records are a particularly poor approach to the problem when there are trends in underlying average daily temperature. It is shown that the proposed analytical approach is superior to historical pricing.

Experimental and numerical study of water-filled road safety barriers

Portable water-filled road barriers (PWFB) are roadside structures placed on temporary construction zones to separate work site from moving traffic. Recent changes in governing standards require PWFB to adhere to strict compliance in terms of lateral displacement of the road barriers and vehicle redirectionality. Actual road safety barrier test can be very costly, thus researchers resort to Finite Element Analysis (FEA) in the initial designs phase prior to real vehicle test. There has been many research conducted on concrete barriers and flexible steel barriers using FEA, however not many is done pertaining to PWFB. This research probes a new method to model joint mechanism in PWFB. Two methods to model the joining mechanism are presented and discussed in relation to its practicality and accuracy to real work applications. Moreover, the study of the physical gap and mass of the barrier was investigated. Outcome from this research will benefit PWFB research and allow road barrier designers better knowledge in developing the next generation of road safety structures.

Numerical characterization of the mechanical properties of metal nanowires

Nanowires (NWs) have attracted appealing and broad application owing to their remarkable mechanical, optical, electrical, thermal and other properties. To unlock the revolutionary characteristics of NWs, a considerable body of experimental and theoretical work has been conducted. However, due to the extremely small dimensions of NWs, the application and manipulation of the in situ experiments involve inherent complexities and huge challenges. For the same reason, the presence of defects appears as one of the most dominant factors in determining their properties. Hence, based on the experiments' deficiency and the necessity of investigating different defects' influence, the numerical simulation or modelling becomes increasingly important in the area of characterizing the properties of NWs. It has been noted that, despite the number of numerical studies of NWs, significant work still lies ahead in terms of problem formulation, interpretation of results, identification and delineation of deformation mechanisms, and constitutive characterization of behaviour. Therefore, the primary aim of this study was to characterize both perfect and defected metal NWs. Large-scale molecular dynamics (MD) simulations were utilized to assess the mechanical properties and deformation mechanisms of different NWs under diverse loading conditions including tension, compression, bending, vibration and torsion. The target samples include different FCC metal NWs (e.g., Cu, Ag, Au NWs), which were either in a perfect crystal structure or constructed with different defects (e.g. pre-existing surface/internal defects, grain/twin boundaries). It has been found from the tensile deformation that Young's modulus was insensitive to different styles of pre-existing defects, whereas the yield strength showed considerable reduction. The deformation mechanisms were found to be greatly influenced by the presence of defects, i.e., different defects acted in the role of dislocation sources, and many affluent deformation mechanisms had been triggered. Similar conclusions were also obtained from the compressive deformation, i.e., Young's modulus was insensitive to different defects, but the critical stress showed evident reduction. Results from the bending deformation revealed that the current modified beam models with the considerations of surface effect, or both surface effect and axial extension effect were still experiencing certain inaccuracy, especially for the NW with ultra small cross-sectional size. Additionally, the flexural rigidity of the NW was found to be insensitive to different pre-existing defects, while the yield strength showed an evident decrease. For the resonance study, the first-order natural frequency of the NW with pre-existing surface defects was almost the same as that from the perfect NW, whereas a lower first-order natural frequency and a significantly degraded quality factor was observed for NWs with grain boundaries. Most importantly, the <110> FCC NWs were found to exhibit a novel beat phenomenon driven by a single actuation, which was resulted from the asymmetry in the lattice spacing in the (110) plane of the NW cross-section, and expected to exert crucial impacts on the in situ nanomechanical measurements. In particular, <110> Ag NWs with rhombic, truncated rhombic, and triangular cross-sections were found to naturally possess two first-mode natural frequencies, which were envisioned with applications in NEMS that could operate in a non-planar regime. The torsion results revealed that the torsional rigidity of the NW was insensitive to the presence of pre-existing defects and twin boundaries, but received evident reduction due to grain boundaries. Meanwhile, the critical angle decreased considerably for defected NWs. This study has provided a comprehensive and deep investigation on the mechanical properties and deformation mechanisms of perfect and defected NWs, which will greatly extend and enhance the existing knowledge and understanding of the properties/performance of NWs, and eventually benefit the realization of their full potential applications. All delineated MD models and theoretical analysis techniques that were established for the target NWs in this research are also applicable to future studies on other kinds of NWs. It has been suggested that MD simulation is an effective and excellent tool, not only for the characterization of the properties of NWs, but also for the prediction of novel or unexpected properties.