The dynamics of a thick layer of viscous liquid flowing down a thin vertical fibre is investigated. Three qualitatively different regimes of the interfacial patterns in the form of beads were observed experimentally. Two typical regimes at relatively small flow rate are described reasonably well by the creeping-flow model equation proposed here.

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

It is shown that viscosity stratified plane Poiseuille flow may exhibit a long-wavelength instability of a purely kinetic nature formally resembling the so-called alpha effect known in magnetohydrodynamics or in anisotropic three-dimensional flows of homogeneous fluids. In the absence of the alpha effect, the system may display a peculiar type of long-wavelength instability, where the latter is controlled by the surface tension. The weakly nonlinear equation for the evolving interfaces is derived and solved numerically.

A reduced nonlinear model for density stratified viscous film flowing down a slightly inclined wall is derived and explored. Under buoyancy stable stratification the system exhibits various long-wavelength instabilities of noninertial, purely kinetic origin. Unlike many existing models for the film interface evolution in the present study regularization of the pertinent long-scale dynamics is provided directly by the film viscosity rather than surface tension.