In this paper, we investigate the behaviour of a wave as it climbs a sloping beach. Explicit solutions of the equations of the non-linear inviscid shallow-water theory are obtained for several physically interesting wave-forms. In particular it is shown that waves can climb a sloping beach without breaking. Formulae for the motions of the instantaneous shoreline as well as the time histories of specific wave-forms are presented.

A model for the movement of a small viscous droplet on a surface is constructed that is based on the lubrication equations and uses the dynamic contact angle to describe the forces acting on the fluid at the contact line. The problems analysed are: the spreading or retraction of a circular droplet; the advance of a thin two-dimensional layer; the creeping of a droplet or cell on a coated surface to a region of greater adhesion; the distortion of droplet shape owing to surface contamination. Relevant biological problems concerning cell movement and adhesion are described.

A simple mathematical model of tumor growth by diffusion is constructed in order to examine and evaluate different hypotheses concerning the evolution of a solid carcinoma. A primary objective is to infer the chemical source of growth inhibition from the most easily obtained data, namely, the outer radius of the nodule as a function of time and a histological examination of the final dormant state. In section 6 some of the conclusions of this study relating to a prototype experiment and described with as little mathematics as possible.

No abstract