A model is presented for the diffusion-driven drying of a polymeric solution such as liquid paint. Included is a stress build-up and relaxation in the polymer network of the viscoelastic material, which influences the diffusion process. The behaviour of the (one-dimensional) model is analysed by means of the maximum principle and illustrated with numerical calculations.

A semi-infinite body, modelling the leading edge of a cutting tool or submerged hydrofoil, lies beneath a free surface in a uniform stream of infinitely-deep inviscid incompressible fluid flowing steadily under gravity. The body has horizontal upper and lower surfaces. The oncoming flow is partly diverted over, and partly under the body. The flow in that portion that travels over the body can be supercritical or subcritical. When it is supercritical it approaches a stream of some (to be determined) speed in a channel of some (to be determined) depth. When it is subcritical, there is also a train of waves on that stream, whose amplitude is also to be determined. Semi-analytic high-speed and low-speed solutions are obtained, and a numerical solution for finite speeds. There is a ‘forbidden’ intermediate speed range, within which steady flow may not be possible.

We study a unilateral equilibrium problem for the energy functional of a lipid tubule subject to an external field. These tubules, which constitute many biological systems, may form assemblies when they are brought in contact, and so made to adhere to one another along at interstices. The contact energy is taken to be proportional to the area of contact through a constant, which is called the adhesion potential. This competes against the external field in determining the stability of patterns with flat interstices. Though the equilibrium problem is highly nonlinear, we determine explicitly the stability diagram for the adhesion between tubules. We conclude that the higher the field, the lower the adhesion potential needed to make at interstices energetically favourable, though its critical value depends also on the surface tension of the interface between the tubules and the isotropic fluid around them.

The paper is devoted to convective instability of reaction fronts. New approaches are developed to study some eigenvalue problems arising in chemical hydrodynamics. For gaseous combustion in the case of equality of transport coefficients, a linear stability analysis of an upward propagating front is carried out. A minimax representation of the stability boundary is obtained.

New Lie and conditional symmetries of nonlinear diffusion equations with convection terms are constructed. Examples of new ansätze and exact solutions for some nonlinear equations (in particular, for the Murray and the Newell–Whitehead equations) are presented.