Finite-amplitude capillary waves, which can accompany the axisymmetric flow of a thin viscous film over a rotating disk, are considered. A system of approximate evolution equations for the film thickness and volumetric flow rates in the radial and azimuthal directions is derived, which contains two similarity parameters. In order to inspire confidence in this model, its steady solutions and their linear stability characteristics are compared to those of the full Navier–Stokes equations. Localized equations, which account for the presence of inertial, capillary, centrifugal and Coriolis forces, are obtained via truncation of the approximate system. Periodic solutions of these equations are then determined and found to be similar to those observed experimentally. Our results suggest that the steady quasi-periodic waves with largest amplitude compare well with experimentally observed wave profiles.

An intriguing, dramatic and, at present, not fully understood instability often accompanies surfactant driven flows on thin films. This paper investigates a candidate mechanism that could create and drive this instability, van der Waals rupture, via numerical simulations coupled with analytical techniques. The spreading process itself is modelled with a pair of coupled evolution equations for the fluid film thickness and surfactant concentration that are derived in the lubrication approximation. These equations are then linearized about a base state that corresponds to the one-dimensional rupturing solution, and equations for the evolution of the transverse disturbances are derived. These linearized equations are investigated in several ways: numerical simulations where the perturbations are driven by the time evolving base state, or where the base state is frozen at a time tf close to the rupture event. The quasistatic initial value problem is also investigated as an eigenvalue problem, where the eigenvalue represents the quasistatic growth rate. We also take advantage of recent similarity scalings and results deduced for rupture, in the absence of surfactant, to motivate some of our numerical investigations. Additionally, we investigate the fully nonlinear equations including the transverse components. Perhaps interestingly, three-dimensional reconstructions of the film profile using the most dangerous mode from linear theory, as well as profiles from direct numerical simulations of the full nonlinear governing equations, that is, including interactions in the transverse direction, assume the form of finger-like patterns.

The dynamics and breakup of electrified viscous jets in the presence of ionic surfactants at the interface are investigated theoretically. Axisymmetric configurations are considered and the jet is surrounded by a concentrically placed cylindrical electrode, which is held at a constant voltage potential. The annular region between the jet and the electrode is taken to be a hydrodynamically passive dielectric medium and an electric field is set up there and drives the flow, along with other physical mechanisms including capillary instability and viscous effects. The jet fluid is taken to be a symmetric electrolyte and proper modeling of the cationic and anionic species is used by considering the Nernst–Planck equations in order to find the volume charge density that influences the electric field in the jet. A positively charged insoluble surfactant is present at the interface, and its evolution, as well as the resulting value of the local surface tension coefficient, is coupled with the voltage potential at the interface. The resulting coupled nonlinear systems are derived and analytical progress is made by carrying out a nonlinear slender jet approximation. The reduced model is described by a number of hydrodynamic, electrical, and electrokinetic parameters, and an extensive computational study is undertaken to elucidate the dynamics along with allied linear properties. It is established that the jet ruptures in finite time provided the outer electrode is sufficiently far away, and numerous examples are given where the dimensionless parameters can be used to control the size of the satellite drops that form beyond the topological transition, as well as the time to break up. It is also shown that pinching solutions follow the self-similar dynamics of clean viscous jets at times close to the breakup time. Finally, a further asymptotic theory is developed for large Debye layers to produce an additional model that incorporates the effects of surface charge diffusion. Numerical solutions establish that the presence of electrostatic and electrokinetic effects increases the sizes of satellites but have a rather weak influence on the time to rupture.

We study the dynamics of an interface separating two immiscible layers in an inclined channel. Lubrication theory is used to derive an evolution equation for the interface position that models the two-dimensional flow in both co-and countercurrent configurations. This equation is parameterized by viscosity and density ratios, and a total dimensionless flow rate; the system is further characterized by the height of the interface at the channel inlet and outlet, which are treated as additional parameters. In the present work, which corresponds to part I of a two-part paper, we focus on one-dimensional flows. We use an entropy-flux analysis to delineate the existence of various types of shocklike solutions, which include compressive Lax shocks, pairs of Lax and under-compressive shocks, and rarefaction waves. Flows characterized by unstably stratified layers are accompanied by the formation of propagating, large-amplitude interfacial waves, which are not shocklike in nature. The results of our transient numerical simulations accord with our analytical predictions and elucidate the mechanisms underlying spatio-temporal development of the various types of waves; the stability of these waves to spanwise perturbations is investigated in part II of this work.