In this paper, we analyse the asymptotic system corresponding to a thin film flow with two different (immiscible) fluids, from theoretical and numerical points of view. We also compare this model to the Elrod-Adams one, which is the reference model in tribology, when cavitation phenomena occur.

We present a theory that enables us to construct heteroclinic connections in closed form for $2\bf{u}_{xx}=W_{\bf u}({\bf u})$, where $x\in\mathbb{R},\;{\bf u}(x)\in \mathbb{R}^2$ and $W$ is a smooth potential with multiple global minima. In particular, multiple connections between global minima are constructed for a class of potentials. With these potentials, numerical simulations for the vector Allen-Cahn equation ${\bf u}_t= 2\epsilon^2 \Delta {\bf u}-W_{\bf u}({\bf u})$ in two space dimensions with small $\epsilon>0$, show that between any fixed pair of phase regions, interfaces are partitioned into segments of different energy densities, where the proportions of the length of these segments are changing with time. Our results imply that for the case of triple-well potentials the usual Plateau angle conditions at the triple junction are generally violated.

Waves on a neutrally buoyant intrusion layer moving into otherwise stationary fluid are studied. There are two interfacial free surfaces, above and below the moving layer, and a train of waves is present. A small amplitude linearized theory shows that there are two different flow types, in which the two interfaces are either in phase or else move oppositely. The former flow type occurs at high phase speed and the latter is a low-speed solution. Nonlinear solutions are computed for large amplitude waves, using a spectral type numerical method. They extend the results of the linearized analysis, and reveal the presence of limiting flow types in some circumstances.

The propagation of a solitary wave in a horizontal fluid layer is studied. There is an interfacial free surface above and below this intrusion layer, which is moving at constant speed through a stationary density-stratified fluid system. A weakly nonlinear asymptotic theory is presented, leading to a Korteweg–de Vries equation in which the two fluid interfaces move oppositely. The intrusion layer solitary wave system thus forms a widening bulge that propagates without change of form. These results are confirmed and extended by a fully nonlinear solution, in which a boundary-integral formulation is used to solve the problem numerically. Limiting profiles are approached, for which a corner forms at the crest of the solitary wave, on one or both of the interfaces.

New results concerning Lie symmetries of nonlinear reaction-diffusion-convection equations, which supplement in a natural way the results published in the European Journal of Applied Mathematics (9(1998) 527–542) are presented.

We study surface tension effects for two-dimensional Darcy flow with a free boundary in a corner between two non-parallel walls. The analytic solution is based on two governing expressions constructed in an auxiliary parameter domain, namely a complex velocity and a derivative of the complex potential. These expressions admit a general solution for the problem in a corner geometry for the flow generated by a source/sink at the corner vertex or at infinity. We derive an integral equation in terms of the velocity modulus and angle at the free surface, determined by the dynamic boundary condition. A numerical procedure, used to solve the obtained system of equations, and numerical results concerning the effect of surface tension on the time evolution of the free boundary, are discussed.