Small- and waiting-time behaviour of the thin-film-equation

We consider the small-time behavior of interfaces of zero contact angle solutions to the thin-film equation. For a certain class of initial data, through asymptotic analyses, we deduce a wide variety of behavior for the free boundary point. These are supported by extensive numerical simulations. © 2007 Society for Industrial and Applied Mathematics

An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

Understanding mixing efficiency in the oceans. Do the nonlinearities of the equation of state matter?

There exist two central measures of turbulent mixing in turbulent stratified fluids that are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the turbulent rate of change Wr, turbulent of background gravitational potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to the well known exact equality D(APE)=Wr, turbulent for a Boussinesq fluid with a linear equation of state. Recently, however, Tailleux (2009) pointed out that the above equality no longer holds for a thermally-stratified compressible, with the ratio ξ=Wr, turbulent/D(APE) being generally lower than unity and sometimes even negative for water or seawater, and argued that D(APE) and Wr, turbulent actually represent two distinct types of energy conversion, respectively the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE0, and the conversion between GPEr and a different subcomponent of internal energy called "exergy" IEexergy. In this paper, the behaviour of the ratio ξ is examined for different stratifications having all the same buoyancy frequency N vertical profile, but different vertical profiles of the parameter Υ=α P/(ρCp), where α is the thermal expansion coefficient, P the hydrostatic pressure, ρ the density, and Cp the specific heat capacity at constant pressure, the equation of state being that for seawater for different particular constant values of salinity. It is found that ξ and Wr, turbulent depend critically on the sign and magnitude of dΥ/dz, in contrast with D(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured in practice, which are discussed.

Diversity and difference in the everyday lives of Ugandan street children: The significance of age and gender for understanding the use of space

Childhood is characterised by diversity and difference across and within societies. Street children have a unique relationship to the urban environment evident through their use of the city. The everyday geographies that street children produce are diversified through the spaces they frequent and the activities they engage in. Drawing on a range of children-centred qualitative methods, this article focuses on street children's use of urban space in Kampala, Uganda. The article demonstrates the importance of considering variables such as gender and age in the analysis of street children's socio-spatial experiences, which, to date, have rarely been considered in other accounts of street children's lives. In addition the article highlights the need for also including street children's individuality and agency into understanding their use of space. The article concludes by arguing for policies to be sensitive to the diversity that characterises street children's lives and calls for a more nuanced approach where policies are designed to accommodate street children's age and gender differences, and their individual needs, interests and abilities.

Factors influencing the dissolution of phosphate rock in a range of high P-fixing soils from Cameroon

A laboratory incubation experiment was conducted to evaluate the soil factors that influence the dissolution of two phosphate rocks (PRs) of different reactivity (Gafsa, GPR, reactive PR; and Togo-Hahotoe, HPR, low reactivity PR) in seven agricultural soils from Cameroon having variable phosphorus (P)- sorption capacities, organic carbon (C) contents, and exchangeable acidities. Ground PR was mixed with the soils at a rate of 500 mg P kg 21 soil and incubated at 30 degrees C for 85 days. Dissolution of the PRs was determined at various intervals using the Delta NaOH-P method ( the difference of the amount of P extracted by 0.5 M NaOH between the PR-treated soils and the control). Between 4 and 27% of HPR and 33 and 50% of GPR were dissolved in the soils. Calcium (Ca) saturation of cation exchange sites and proton supply strongly affected PR dissolution in these soils. Acid soils with pH-(H2O), < 5 (NKL, ODJ, NSM, MTF) dissolved more phosphate rock than those with pH-(H2O) > 5 (DSC, FGT, BAF). However, the lack of a sufficient Ca sink in the former constrained the dissolution of both PRs. The dissolution of GPR in the slightly acidic soils was limited by increase in Ca saturation and that of HPR was constrained by limited supply in protons. Generally, the dissolution of GPR was higher than that of HPR for each soil. The kinetics of dissolution of PR in the soils was best described by the power function equation P At B. More efficient use of PR in these soils can be achieved by raising the soil cation exchange capacity, thereby increasing the Ca sink size. This could be done by amending such soils with organic materials.

Diversity and difference in the everyday lives of Ugandan street children: The significance of age and gender for understanding the use of space

Childhood is characterised by diversity and difference across and within societies. Street children have a unique relationship to the urban environment evident through their use of the city. The everyday geographies that street children produce are diversified through the spaces they frequent and the activities they engage in. Drawing on a range of children-centred qualitative methods, this article focuses on street children's use of urban space in Kampala, Uganda. The article demonstrates the importance of considering variables such as gender and age in the analysis of street children's socio-spatial experiences, which, to date, have rarely been considered in other accounts of street children's lives. In addition the article highlights the need for also including street children's individuality and agency into understanding their use of space. The article concludes by arguing for policies to be sensitive to the diversity that characterises street children's lives and calls for a more nuanced approach where policies are designed to accommodate street children's age and gender differences, and their individual needs, interests and abilities.

A well-posed integral equation formulation for three-dimensional rough surface scattering

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space L-2 (Gamma) when the scattering surface G does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, kappa, for kappa > 0, if the coupling parameter h is chosen proportional to the wave number. In the case when G is a plane, we show that the choice eta = kappa/2 is nearly optimal in terms of minimizing the condition number.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

A comparison of conservative upwind difference schemes for the Euler equations

Numerical results are presented and compared for three conservative upwind difference schemes for the Euler equations when applied to two standard test problems. This includes consideration of the effect of treating part of the flux balance as a source, and a comparison of different averaging of the flow variables. Two of the schemes are also shown to be equivalent in their implementation, while being different in construction and having different approximate Jacobians. (C) 2006 Elsevier Ltd. All rights reserved.