This paper summarizes and extends some mathematical results for a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected. Explicit solutions for the pressure distributions are given in the case of two-dimensional flow and a variational formulation is described which provides a simple numerical algorithm for three-dimensional flows. We also pose some open questions concerning the well-posedness and physical relevance of the model for exit problems or when there is an air gap between the impacting body and the water.

The world around us, natural or man-made, is built and held together by solid materials. Understanding their behaviour is the task of solid mechanics, which is in turn applied to many areas, from earthquake mechanics to industry, construction to biomechanics. The variety of materials (metals, rocks, glasses, sand, flesh and bone) and their properties (porosity, viscosity, elasticity, plasticity) is reflected by the concepts and techniques needed to understand them: a rich mixture of mathematics, physics and experiment. These are all combined in this unique book, based on years of experience in research and teaching. Starting from the simplest situations, models of increasing sophistication are derived and applied. The emphasis is on problem-solving and building intuition, rather than a technical presentation of theory. The text is complemented by over 100 carefully-chosen exercises, making this an ideal companion for students taking advanced courses, or those undertaking research in this or related disciplines.

Abstract. This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions.

The method of matched asymptotic expansions is applied to the fluid model of the low-pressure positive column. The expansion of the eigenvalue in the plasma balance equation is obtained to second order in plane and in cylindrical geometry, and uniformly valid expressions for charged particle densities and fluid velocity in two separate regions are indicated.The free-fall model is also examined and the scales of the transition layer and sheath layer found. Comparison is made with the results of direct numerical integration of the equations involved for both models.

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