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

Mathematical modelling of elastoplasticity at high stress

This paper describes a simple mathematical model for one-dimensional elastoplastic wave propagation in a metal in the regime where the applied stress greatly exceeds the yield stress. Attention is focussed on the increasing ductility that occurs in the over-driven limit when the plastic wave speed approaches the elastic wave speed. Our model predicts that a plastic compression wave is unable to travel faster than the elastic wave speed, and instead splits into a compressive elastoplastic shock followed by a plastic expansion wave

Elastic-Plastic Modelling of Shaped Charge Jet Penetration

This paper concerns the mathematical modelling of high rate penetration of a metaltarget by a shaped charge device that produces a high velocity jet. A key objective is to predict the penetration velocity, be it subsonic, transonic, or supersonic. We do this by considering, on the local scale near the tip of the penetrated cavity, an elastic-plastic free boundary problem that takes into account the residual stresses produced by the moving plasticized region of the target. It is the self-consistency of this elastic-plastic model that dictates predictions for the penetration velocity

Continuum and discrete models of dislocation pile-ups. I Pile-up at a lock

A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip pane under the action of an external force in the direction of a locked dislocation in that plane is considered. As $n \rightarrow \infty$ there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away

Geoid Anomalies and the Near-Surface Dipole Distribution of Mass

Although geoid or surface gravity anomalies cannot be uniquely related to an interior distribution of mass, they can be related to a surface mass distribution. However, over horizontal distances greater than about 100 km, the condition of isostatic equilibrium above the asthenosphere is a good approximation and the total mass per unit column is zero. Thus the surface distribution of mass is also zero. For this case we show that the surface gravitational potential anomaly can be uniquely related to a surface dipole distribution of mass. Variations in the thickness of the crust and lithosphere can be expected to produce undulations in the geoid

Asymptotic analysis of a system of algebraic equations arising in dislocation theory

The system of algebraic equations given by\ud \ud $\sum_{j=0, j \neq i}^n sgn(x_i - x_j) / |x_i - x_j|^a = 1, i = 1, 2, \ldots n, x_0 = 0,$\ud \ud appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole.\ud \ud We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n -> ∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment but, up to corrections of logarithmic order, it also leads to a differential equation.\ud \ud The continuum approximation is only valid for i not too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem

Ray theory for high-Péclet-number convection-diffusion

Asymptotic methods based on those of geometrical optics are applied to some steady convection-diffusion streamed flows at a high Péclet number. Even with the assumption of inviscid, irrotational flow past a body with uniform ambient conditions, the rays from which the solution is constructed can only be found after local analyses have been carried out near the stagnation points. In simple cases, the temperature away from the body is the sum of contributions from each stagnation point

Air-cushioning in impact problems

This paper concerns the displacement potential formulation to study the post-impact influence of an aircushioning layer on the two-dimensional impact of a liquid half-space by a rigid body. The liquid and air are both ideal and incompressible and attention is focussed on cases when the density ratio between the air and liquid is small. In particular, the correction to classical Wagner theory is analysed in detail for the impact of circular cylinders and wedges

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

Inertial levitation

We consider the steady levitation of a rigid plate on a thin air cushion with prescribed injection velocity. This injection velocity is assumed to be much larger than that in a conventional Prandtl boundary layer, so that inertial effects dominate. After applying the classical ‘blowhard’ theory of Cole & Aroesty (1968) to the two-dimensional version of the problem, it is shown that in three dimensions the flow may be foliated into streamline surfaces using Lagrangian variables. An example is given of how this may be exploited to solve the three-dimensional problem when the injection pressure distribution is known