We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike the previously known results, the surface diffusion coefficient is not assumed constant and depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting-time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations.

In this paper, we study an adsorption model arising in the dynamics of several surfactants at the air-water interface, where the Langmuir isotherm is employed for modelling the time-dependent surface concentration, providing a nonlinear dynamical boundary condition. Existence of a weak solution is proved by using the Rothe method for a semi-discrete problem in time. After obtaining some a priori estimates and passing to the limit in the time discretization parameter, we conclude that the original Langmuir problem has a bounded solution. An uniqueness result is also given.

This paper discusses the development of an invariant finite difference scheme to simulate the microphase separation of copolymers in one-dimensional thin liquid films. The film phenomena are modelled using two-phase shallow water equations and the Ohta-Kawasaki potential, which governs the phase separation of the copolymer. Non-positive volume fractions and spurious oscillations are eventually eliminated, in simulating the one-dimensional phase separation lamellar pattern.