A class of exactly solvable free-boundary inhomogeneous porous medium flows

We describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources; the free boundary dynamics can be parametrized by polynomial conformal maps

Ray methods for free boundary problems

We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black–Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods

Nonclassical shallow water flows

This paper deals with violent discontinuities in shallow water flows with large Froude number $F$. On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory we show that, over a certain timescale, this discontinuity may be described by a delta-shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step functioncomponents. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted. For flows on a sloping base, we show that for flow with an aspect ratio of \emph{O}($F^{-2}$) on a base with an \emph{O(1)} or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a delta-shock. The physical manifestation of this discontinuity is a small `tube' of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base

Three-dimensional oblique water-entry problems at small\ud deadrise angles

This paper extends Wagner theory for the ideal, incompressible normal impact of rigid bodies that are nearly parallel to the surface of a liquid half-space. The impactors considered are three-dimensional and have an oblique impact velocity. A variational formulation is used to reveal the relationship between the oblique and corresponding normal impact solutions. In the case of axisymmetric impactors, several geometries are considered in which singularities develop in the boundary of the effective wetted region. We present the corresponding pressure profiles and models for the splash sheets

A note on oblique water entry

An apparently minor error in Howison, Ockendon & Oliver (J. Eng. Math. 48:321–337, 2004) obscured the fact that the points at which the free surface turns over in the solution of the Wagner model for the oblique impact of a two-dimensional body are directly related to the turnover points in the equivalent normal impact problem. This note corrects some results given in Howison, Ockendon & Oliver (2004) and discusses the implications for the applicability of the Wagner\ud model

Droplet impact on a thin fluid layer

The initial stages of high-velocity droplet impact on a shallow water layer are described, with special emphasis given to the spray jet mechanics. Four stages of impact are delineated, with appropriate scalings, and the successively more important influence of the base is analysed. In particular, there is a finite time before which part of the water in the layer remains under the droplet and after which all of the layer is ejected in the splash jet

Minimum Particle Size for Cyclone Dust Separator

Perkins technology wish to separate small soot particles from exhaust gases, and the question posed to the study group was to determine the feasibility of using a cyclone separator to remove these particles. Soot is mostly composed of polycyclicaromatic compounds and results from the incomplete combustion of the diesel fuel in the engine. The average size of the particles formed in the engine is in the range 3 to 10 nm in diameter, but this is known to increase within the exhaust system. In the first part of this report we determine the minimum particle size that can be removed by centrifugal separation. The second part discusses the mechanisms for particle growth within the exhaust system in order to estimate the particle growth rate. In section two we estimate the minimum particle diameter that can be removed by a cyclone separator is around one micron. This estimate is consistent with current applications of hydrocyclones. The particle size measurements by Perkins Technology together with our estimates from section three, suggest that the soot particles are an order of magnitude smaller than this. Although it may be possible to remove some particles less than one micron in diameter with a well designed high-speed cyclone, we do not think it will be possible to remove a substantial proportion of 100 nm or smaller particles. The growth rate of the particles increases if the particles volume fraction or the polydispersity is increased. Therefore aggregation could be enhanced by the addition of larger particles (d > 1 µm) or water droplets (provided the water does not all vapourise) to the exhaust gas

Singular Laplacian Growth

The general equations of motion for two dimensional Laplacian growth are derived using the conformal mapping method. In the singular case, all singularities of the conformal map are on the unit circle, and the map is a degenerate Schwarz-Christoffel map. The equations of motion describe the motions of these singularities. Despite the typical fractal-like outcomes of Laplacian growth processes, the equations of motion are shown to be not particularly sensitive to initial conditions. It is argued that the sensitivity of this system derives from a novel cause, the non-uniqueness of solutions to the differential system. By a mechanism of singularity creation, every solution can become more complex, even in the absence of noise, without violating the growth law. These processes are permitted, but are not required, meaning the equation of motion does not determine the motion, even in the small.Comment: 8 pages, Latex, 4 figures, Submitted to Phys. Rev.

Large droplet impact on water layers

The impact of large droplets onto an otherwise undisturbed layer of water is considered. The work, which is motivated primarily with regard to aircraft icing, is to try and help understand the role of splashing on the formation of ice on a wing, in particular for large droplets where splash appears, to have a significant effect. Analytical and numerical approaches are used to investigate a single droplet impact onto a water layer. The flow for small times after impact is determined analytically, for both direct and oblique impacts. The impact is also examined numerically using the volume of fluid (VOF) method. At small times there are promising comparisons between the numerical results, the analytical solution and experimental work capturing the ejector sheet. At larger times there is qualitative agreement with experiments and related simulations. Various cases are considered, varying the droplet size to layer depth ratio, including surface roughness, droplet distortion and air effects. The amount of fluid splashed by such an impact is examined and is found to increase with droplet size and to be significantly influenced by surface roughness. The makeup of the splash is also considered, tracking the incoming fluid, and the splash is found to consist mostly of fluid originating in the layer

Functional traits of trees on and off termite mounds:Understanding the origin of biotically-driven heterogeneity in savannas

Questions In African savannas, Macrotermes termites contribute to small-scale heterogeneity by constructing large mounds. Operating as islands of high nutrient and water availability and low fire frequency, these mounds support distinct, diverse communities of trees that have been shown to be highly attractive to browsers. However, the distinct traits of tree species on termite mounds have hardly been studied, even though this may help to understand processes determining (1) their characteristic community structure and (2) attractiveness for browsers. Here, we compare functional trait and browser preference values between tree species on and off termite mounds. Location Hluhluwe-iMfolozi Park, Kwazulu-Natal, South Africa. Methods We recorded tree community compositions for 16 large Macrotermes natalensis mounds and 16 control plots of 100 m2 each in a paired design. For each observed tree species we measured 22 traits, related to water and nutrient use, fire tolerance, light competition and anti-herbivore defence, and compared average trait values between mound and control communities. Furthermore, we investigated the feeding preferences of ungulate browsers for the most common tree species and how this was linked to their associated traits. Results Termite mounds supported tree communities that were distinct from the surrounding savanna vegetation. Mounds hosted more evergreen and less leguminous tree species than control communities, and the dominant species were less mechanically defended, less nutritious, had larger leaves and lower wood density than the species dominating control plots. Browsers preferred leguminous tree species with high leaf N and P content, which were relatively rare on termite mounds. Conclusions Overall, we conclude that termite mounds in this savanna form small refuges for tree species that seem less adapted to fire (more evergreens), have low nutrient availability (less nitrogen fixers) and suffer from water stress (larger leaf sizes) than typical savanna trees. Surprisingly, despite their reputation as browsing hotspots, the tree species dominating mounds are less nutritious and less preferred by browsers than tree species of the surrounding savanna, which may be explained by the relatively nutrient-rich nature of this savanna or intraspecific trait differences