Steady-state Traveling Solutions Research Articles

Waves on the surface of a thin liquid film entrained by a turbulent gas flow in a narrow channel

The paper considers the problem of a joint flow in a narrow vertical channel of a turbulent gas flow and a wave film of liquid flowing down its walls. The calculation of tangential stresses on the interface is performed. The components of the Reynolds stress tensor were determined within the framework of the Boussinesq hypothesis. The turbulent viscosity coefficient was taken into account both when calculating the average velocity profile and when considering disturbances caused by wavy interface. The main difference this case and the semi-infinite channel case is observed in the longwave region. Nonlinear waves on a liquid film flowing down under the action of gravity are studied in a known stress field at the interface. For the case of low Reynolds numbers of the liquid (Re ∼1), the problem is reduced to a nonlinear integro-differential equation for the deviation of the layer thickness from the unperturbed level. For this equation, the character of branching from the unperturbed solution of weakly nonlinear steady-state traveling solutions, whose wave numbers lie in the neighborhood of neutral points, is studied. A numerical study of the evolution of periodic perturbations, whose wave numbers noticeably differ from neutral wave numbers, has been carried out. Several characteristic scenarios for the development of such disturbances are presented.

Simulating waves on a horizontal liquid film entrained by a gas flow

Nonlinear waves on a horizontal liquid film surface are considered. The effect of adjacent gas flow is taken into account through the data on shear stress at the film-gas interface obtained by the Boussinesq model of turbulence. A model nonlinear equation for the film thickness deviation from the undisturbed level is used to simulate nonlinear wave modes. Weakly nonlinear steady-state travelling solutions of this equation with wave numbers located in the vicinity of neutral wave numbers are constructed analytically. The evolution of periodic perturbations with wave numbers lying in the depth of the linear instability region is also considered numerically. Several typical scenarios of their evolution have been identified.

Studying the topological structure of steady-state travelling solutions for the model of film flow of a viscous fluid entrained by a gas flow

The article studies nonlinear waves on a liquid film, flowing under the action of gravity in a known stress field at the interface. In the case of small flow rates for the long-wave modes, the problem is reduced to solving a nonlinear equation for the film thickness deviation from the undisturbed level. The paper presents the results of calculations for this model equation of families of steady-state travelling periodic solutions. For these families, the limiting solutions, solitary waves, have been found. It is also investigated how the topological reorganization of such families occurs with a smooth change in the degree of influence of the gas flow. It is shown that although the eigenform of specific solitons changes smoothly, for certain values of the problem parameter for a particular family an abrupt change in the shape of its limiting soliton occurs.

Singular surface points of steady-state traveling solutions of mathematical models of a falling fluid film

The solutions to a divergent system of equations were studied numerically for a freely flowing fluid film. Various families of steady-state traveling solutions in a wide range of wave numbers were found. The shapes of the fluid film surface corresponding to these solutions are given. Specific ranges of parameter values, where re-closing of different families occurs, were determined. It is shown that they are the singular points of the saddle type.

Modeling nonlinear wave regimes in a falling liquid film entrained by a gas flow

The article studies nonlinear waves on a liquid film, flowing under the action of gravity in a known stress field at the interface. In the case of small Reynolds numbers, the problem is reduced to solving a nonlinear integro-differential equation for the film thickness deviation from the undisturbed level. The nature of branching of wave modes of the unperturbed flow with a flat interface has been investigated. The steady-state traveling solutions with wave numbers that are far enough from the neutral ones, have been numerically found. Using methods of stability theory, the analysis of branching of new families of steady-state traveling solutions has been performed. In particular, it is shown that, similarly to the case of the falling film, this model equation has solutions in the form of solitons-humps.

Simulating nonlinear steady-state traveling waves on the falling liquid film entrained by a gas flow

The article is devoted to the simulation of nonlinear waves on a liquid film flowing under gravity in the known stress field at the interface. The paper studies nonlinear waves on a liquid film, flowing under the action of gravity in a known stress field at the interface. In the case of small Reynolds numbers the problem is reduced to the consideration of solutions of the nonlinear integral-differential equation for film thickness deviation from the undisturbed level. The periodic and soliton steady-state traveling solutions of this equation have been numerically found. The analysis of branching of new families of steady-state traveling solutions has been performed. In particular, it is shown that this model equation has solutions in the form of solitons-humps.

Simulating the nonlinear wave mode evolution in the problem of the falling thin layer of a viscous liquid

The work is devoted to numerical simulation of the dynamics of nonlinear periodic waves on the surface of freely flowing liquid film realized with the use of the previously discovered symmetry of model equations. The calculation results have been presented for the dynamics of amplitudes of significant harmonics and for the wave surface evolution. They prove to be in good agreement with previously calculated steady-state traveling solutions obtained on the full and the reduced bases. In addition, the work has identified intervals of wavenumbers in which the evolution of perturbations with small initial amplitude leads to a wave signal periodically changing in time rather than to the steady-state traveling solution.

Simulating nonlinear steady-state traveling waves on the falling liquid film entrained by a gas flow

The article is devoted to the simulation of nonlinear waves on a liquid film flowing under gravity in the known stress field at the interface. In the case of small Reynolds numbers the problem is reduced to the consideration of solutions of the nonlinear integral-differential equation for film thickness deviation from the undisturbed level. Weakly nonlinear steady-state traveling solutions of the equation with wave numbers in a vicinity of neutral wave numbers are constructed analytically. The nature of the wave branching from the undisturbed solution is investigated. Steady-state traveling solutions, whose wave numbers within the instability area are far from neutral wave numbers, are found numerically.

Symmetry in the problem of wave modes of thin viscous liquid layer flow

The equations in conservative form for modelling nonlinear waves on a liquid film flowing down a vertical plane have been investigated. It has been found out that the equations with boundary conditions are invariant under parity transformation in the extended computational domain. The steady-state travelling solutions are numerically shown to have the detected symmetry for moderate Reynolds numbers. The use of this symmetry for the numerical solution of the problem by Galerkin methods significantly increases the efficiency of calculations.

Modeling of wave regimes on the film of viscous liquid falling down a vertical plate

Evolution of disturbances on the surface of liquid film falling free over a vertical plane is considered. In the case of low Reynolds numbers, the problem is reduced to investigation of a single equation describing evolution of film thickness disturbances. With the use of information about the structure of the family of steady-state traveling solutions to this equation, the typical scenarios of evolution for periodic disturbances were identified.

Wave regimes on a nonlinearly viscous fluid film flowing down a vertical plane

The flow of a thin film of a nonlinearly viscous fluid whose stress tensor is modeled by a power law, flowing down a vertical plane in the field of gravity, is considered. For the case of low flow rates, an equation that describes the evolution of surface disturbances is derived in the long-wave approximation. The domain of linear stability of the trivial solution is found, and weakly nonlinear, steady-state travelling solutions of this equation are obtained. The mechanism of branching of solution families at the singular point of the neutral curve is described.

Set of steady-state traveling waves on the surface of a liquid film flowing down an inclined plane

The nonlinear evolution equation often encountered in modeling the behavior of perturbations in various nonconservative media, for example, in problems of the hydrodynamics of film flow, is examined. Steady-state traveling periodic solutions of this equation are found numerically. The stability of the solutions is investigated and a bifurcation analysis is carried out. It is shown how as the wave number decreases ever new families of steady-state traveling solutions are generated. In the limit as the wave number tends to zero a denumerable set of these solutions is formed. It is noted that solutions which also oscillate in time may be generated from the steadystate solutions as a result of a bifurcation of the Landau-Hopf type.

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

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.