Motivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.

A system of differential equations describing the joint motion of thermo-elastic porous body and slightly compressible viscous thermofluid occupying pore space is considered. Although the problem is correct in an appropriate functional space, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification under various conditions imposed on physical parameters is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As a result, we derive Biot's system of equations of thermo-poroelasticity, a similar system, consisting of anisotropic Lamé equations for a thermoelastic solid coupled with acoustic equations for a thermofluid, Darcy's system of filtration, or acoustic equations for a thermofluid, according to ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.

We consider a single-species structured population with distributed maturity and spatial diffusion in a cylindrical domain subject to Neumann and Robin boundary conditions. We first establish the threshold property of the reaction–diffusion system with distributed delay and non-local interaction in a corresponding lower-dimensional domain, so that the system approaches either an extinction state or a stable spatially varying pattern. We then investigate the transition from the extinction state to the stable pattern of the system in the cylindrical domain.

We study the coupled surface and grain boundary motion in a bi-crystal in the context of the ‘quarter loop’ geometry. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. Three curves meet at points where junction conditions are given. A formulation that describes the coupled normal motion of the curves and preserves arc length parametrisation up to scaling is proposed. The formulation is shown to be well-posed in a simple, linear setting. Equations and junction conditions are approximated by finite difference methods. Numerical convergence to exact travelling wave solutions is shown. The method is applied to other problems of physical interest.

Linear stability theory is developed for an activator–inhibitor model where fractional derivative operators of generally different exponents act both on diffusion and reaction terms. It is shown that in the short wave limit the growth rate is a power law of the wave number with decoupled time scales for distinct anomaly exponents of the different species. With equal anomaly exponents an exact formula for the anomalous critical value of reactants diffusion coefficients' ratio is obtained.