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.

Building bridges between classical results and contemporary nonstandard problems, Mathematical Bridges embraces important topics in analysis and algebra from a problem-solving perspective. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors, motivated mathematics students from high school juniors to college seniors, and students interested in mathematical competitions should have This volume-the first comprehensive treatment of all the major types of systems models-emphasizes both theory and applications throughout the text; the applicability of the developed theory is demonstrated by means of many specific examples and applications to important classes of systems in areas such as power and energy, feedback control, artifical neural networks, digital signal processing and control, manufacturing, computer networks, and socioeconomics. The book may be used as a graduate textbook or as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, physics, chemistry, biology, and economics.

We report the results of physical experiments that demonstrate the strong influence of the thermal conductivity of the substrate on the evaporation of a pinned droplet. We show that this behaviour can be captured by a mathematical model including the variation of the saturation concentration with temperature, and hence coupling the problems for the vapour concentration in the atmosphere and the temperature in the liquid and the substrate. Furthermore, we show that including two ad hoc improvements to the model, namely a Newton's law of cooling on the unwetted surface of the substrate and the buoyancy of water vapour in the atmosphere, give excellent quantitative agreement for all of the combinations of liquid and substrate considered

The complete description of the lifetime of a droplet on a solid substrate evaporating in a 'stick-slide' mode is obtained. The unexpectedly subtle relationship between the lifetime of such a droplet and the lifetimes of initially identical droplets evaporating in the extreme modes (namely the constant contact radius and constant contact angle modes) is described and summarised in an appropriate master diagram. In particular, it is shown that the lifetime of a droplet is not, in general, constrained by the lifetimes of the extreme modes.

A mathematical model for the quasi-steady diffusion-limited evaporation of a thin axisymmetric sessile droplet of liquid with a pinned contact line is formulated and solved. The model generalises the theoretical model proposed by Deegan et al. [Phys. Rev. E, 62 (2000) 756-765] to include the effect of evaporative cooling on the saturation concentration of vapour at the free surface of the droplet, and the dependence of the coefficient of diffusion of vapour in the atmosphere on the atmospheric pressure. The predictions of the model are in good qualitative, and in some cases also quantitative, agreement with recent experimental results. In particular, they capture the experimentally observed dependence of the total evaporation rate on the thermal conductivities of the liquid and the substrate, and on the atmospheric pressure.