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.
A vapor fills the gap between two vertical plates, one hot and one cold. The temperatures are adjusted so that condensate forms on the cold wall. It is the dynamics of the system that is examined. The paper extends the one-sided model of evaporation–condensation to account the heat conduction in the vapor phase, which turns out to be important in many condensation problems. For the considered flow, both vapor recoil and Marangoni effect are stabilizing; as a result, the condensate becomes unstable at nonzero Reynolds numbers in contrast to the usual film flow down a vertical wall. A nonlinear evolution equation is derived and analyzed for the interaction of viscous shear and evaporation–condensation. It turns out that the one-sided model of heat and mass transfer gives a very good description of the initial stage of thin-film growth; in later stages, however, the heat conduction through the vapor becomes important when the film is sufficiently thick.
The coupled Kuramoto–Sivashinsky (CKS) equations for multilayer downflowing
films are derived and explored. The CKS equations exhibit a wealth of dynamical
behaviour, displaying travelling periodic waves, regular and chaotic-like patterns,
coexistence of different attractors, and perfect and imperfect synchronization of the
interfaces. New physical effects are found, such as suppression of the Rayleigh–Taylor
instability for heavy-top stratified films, and new surface-tension-driven instability.
The nonlinear theory of long Marangoni waves in systems with two interfaces is developed by means of asymptotic expansions. The self-consistent three-layer approach is used. In the case where the thickness of one of the layers is small, the system of coupled equations governing the deformations of both interfaces has been derived. Traveling wave solutions of this system are investigated analytically and numerically.
Experiments and theory on the rupture of a free plane viscous film are reported. The relatively thick film, with a typical thickness of the order of 0.1–0.6 mm, rests between two long parallel needles. When the film is punctured, a hole is formed with the rim on the front. The hole expands, reaches the needles, and propagates along them with a constant velocity of the order of 2–50 cm s−1. The Reynolds numbers for the present experiments are relatively small, 0.002 ≤ Re ≤ 0.34. A crude theory for propagation velocity of the rim is proposed; the theory compares well with the experimental data. The rupture profile is visually similar to a U-shaped curve. Crude equations for the rupture profile are derived, and their solutions are consistent with the experimental observations. A theory for propagation velocity and profile of the rupture, applicable to all Reynolds numbers, is proposed.
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