O. Yu. Tsvelodub

Nonlinear waves on the surface of a falling liquid film. Part 2. Bifurcations of the first-family waves and other types of nonlinear waves

The paper is devoted to a theoretical analysis of nonlinear two-dimensional waves on the surface of a liquid film freely falling down a vertical plane. A bifurcation analysis of the wave regimes found in Part 1 of this work (Tsvelodub & Trifonov 1991), and of the new wave families obtained here in Part 2, has been carried out. It is demonstrated that there is a great number of different steady-state travelling wave classes which are parameterized by wavenumber at a fixed Reynolds number for a given liquid. It is shown that some of them quantitatively agree with experimental results. The question of stability of various wave regimes with respect to two-dimensional infinitesimal disturbances is examined and it is shown that one particular wave family is found. The most amplified disturbances are evaluated.

On steady-state travelling solutions of an evolution equation describing the behaviour of disturbances in active dissipative media

Abstract A nonlinear evolution equation is considered which is often encountered in modelling the behaviour of perturbations in various active dissipative media, e.g. in problems of fluid film flow hydrodynamics. Periodic steady-state travelling solutions have been found numerically for it. Stability of these solutions has been investigated and bifurcation analysis has been carried out. The analysis has demonstrated that decrease of the wave number causes more and more new families of steady-state travelling solutions. A countable set of such solutions is formed in the limit when the wave number tends to zero. It is also shown that time-oscillating solutions can be generated from steady-state ones due to bifurcation of the Landau-Hopf type.

Spatial wave regimes on a surface of thin viscous liquid film

Abstract A nonlinear equation describing the evolution of spatial long-wave perturbations on the surface of a viscous liquid film is considered. The equation is valid for the case of small flow rates and slightly nonlinear perturbations. Steady-state travelling periodic solutions of the equation have been found numerically and analytically. Bifurcation analysis of solutions has been carried out using the methods of stability theory. As a result some new families of spatially periodic solutions have been constructed. Complicated mutual transitions of these families are demonstrated. It is worth to distinguishing the family generating an isolated wave of the horseshoe-like shape in the small wave vector limit.

Wave regimes of viscous film flow down a vertical cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.

Gravity-capillary waves on the surface of a liquid dielectric

Waves on the surface of a liquid dielectric layer in an alternating electric field are studied both experimentally and theoretically. The experiments reveal the characteristic patterns of standing waves forming rectangular and hexagonal cells—quasicrystalline wave structures. The shapes and sizes of these cells are studied as functions of the intensity and frequency of the electric field. The theoretical analysis is performed in the linear approximation. Equations are derived that describe the dynamics of standing waves in arbitrarily thick layers of both ideal and viscous fluids. Theory is in good quantitative and qualitative agreement with experiments.

Wave regimes on a ferromagnetic viscous‐fluid film flowing down a vertical cylinder

Axially symmetrical waves on the surface of a ferromagnetic viscous‐fluid film flowing down a cylindrical conductor with alternating current are considered. In this case, in addition to the gravitational force, the film is affected by a spatially nonuniform time‐dependent magnetic field. The film thickness was assumed to be small compared to the radius of the conductor. In the long‐wave approximation, a model equation for the deviation of the film thickness from its undisturbed value is obtained. Some numerical solutions of this equations are reported.

Simulation of nonlinear waves on the surface of a thin fluid film moving under the action of turbulent gas flow

This paper derives a new system of equations for the simulation of the long-wave perturbation dynamics on the surface of a thin horizontal layer of heavy viscous fluid moving under the action of turbulent gas flow. In the case of small Reynolds numbers of the fluid, this system of equations is used to derive an evolution equation for the value of deviation of the film thickness from the unperturbed level. Some solutions of this equation are given.

Numerical simulation of plane and spatial nonlinear stationary waves in a two-layer fluid of arbitrary depth

The solution of a model differential equation for the three-dimensional perturbations of the interface between two immiscible fluids of different densities lying between a stationary nondeformable bottom and cover is presented. It is assumed that the waves have an arbitrary length and small, though finite, amplitude. The shapes of stationary traveling internal waves, both periodic in the two horizontal coordinates and soliton-like, are presented. These shapes depend on different parameters of the problem: the direction of the perturbation wave vector and the fluid layer depth and density ratios.

Wave regimes on a film of generalized Newtonian fluid flowing down a vertical plane

The flow of a thin film of generalized Newtonian fluid down a vertical wall in the gravity field is considered. For small flow-rates, in the long-wave approximation, an equation describing the evolution of the surface perturbations is obtained. Depending on the signs of the coefficients, this equation is equivalent to one of four equations with solutions significantly different in evolutionary behavior. For the most interesting case, soliton solutions are numerically found.

Resonant interactions of two waves in a model of a two-layer fluid

The behavior of long-wave perturbations on the interface between two layers of different fluids with interfacial interaction taken into account, which can be described by the quasiperiodic solutions of a pseudodifferential equation, is considered.