The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

The linear and nonlinear spatio-temporal stability of an interface separating two Newtonian fluids in pressure-driven channel flow at moderate Reynolds numbers is analysed both theoretically and numerically. A linear, Orr–Sommerfeld-type analysis shows that most of such systems are unstable. The transition to an absolutely unstable regime is investigated, and is shown to occur in an intermediate range of Reynolds numbers and ratios of the thicknesses of the two layers, for near-density matched fluids with a viscosity contrast. A critical Reynolds number is found for transition from convective to absolute instability of relatively thin films. Results obtained from direct numerical simulations (DNSs) of the Navier–Stokes equations for long channels using a diffuse-interface method elucidate that waves generated by random noise at the inlet show that, near the inlet, waves are formed and amplified strongly, leading to ligament formation. Successive waves coalesce with each other further downstream, resulting in longer larger-amplitude waves further downstream. In the linearly absolute regime, the characteristics of the spatially growing wave near the inlet agree with that of the saddle point as predicted by the linear theory. The transition point from a convective to an absolute regime predicted by linear theory is also in agreement with a sharp change in the value of a healing length obtained from the DNSs.

The advective Cahn-Hilliard equation describes the competing processes of stirring and separation in a two-phase fluid. Intuition suggests that bubbles will form on a certain scale, and previous studies of Cahn-Hilliard dynamics seem to suggest the presence of one dominant length scale. However, the Cahn-Hilliard phase-separation mechanism contains a hyperdiffusion term and we show that, by stirring the mixture at a sufficiently large amplitude, we excite the diffusion and overwhelm the segregation to create a homogeneous liquid. At intermediate amplitudes we see regions of bubbles coexisting with regions of hyperdiffusive filaments. Thus, the problem possesses two dominant length scales, associated with the bubbles and filaments. For simplicity, we use a chaotic flow that mimics turbulent stirring at large Prandtl number. We compare our results with the case of variable mobility, in which growth of bubble size is dominated by interfacial rather than bulk effects, and find qualitatively similar results.

We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard equations to describe phase separation in thin films. The equations we derive underscore the coupled behaviour of free-surface variations and phase separation. We introduce a repulsive substrate-film interaction potential and analyse the resulting fourth-order equations by constructing a Lyapunov functional, which, combined with the regularizing repulsive potential, gives rise to a positive lower bound for the freesurface height. The value of this lower bound depends on the parameters of the problem, a result which we compare with numerical simulations. While the theoretical lower bound is an obstacle to the rupture of a film that initially is everywhere of finite height, it is not sufficiently sharp to represent accurately the parametric dependence of the observed dips or 'valleys' in free-surface height. We observe these valleys across zones where the concentration of the binary mixture changes sharply, indicating the formation of bubbles. Finally, we carry out numerical simulations without the repulsive interaction, and find that the film ruptures in finite time, while the gradient of the Cahn-Hilliard concentration develops a singularity.