Yu. Ya. Trifonov

Viscous liquid film flows over a periodic surface

Abstract The paper is devoted to a theoretical analysis of a viscous liquid film flowing down a vertical one-dimensional periodic surface. The investigation is based on both Navier–Stokes and integral equations and performed over a wide range of Reynolds number and surface geometry characteristics taking into account viscosity, inertia and surface tension. Shape of the film free surface and streamline function are calculated. It is shown that there are two ranges of parameters where the film flow is controlled by surface tension or inertia forces and where qualitatively different behavior of the flow main characteristics is obtained. Stagnation zones are found and their transformation with increasing Reynolds number is investigated. Comparison with experimental data is carried out.

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.

Two-periodical and quasi-periodical wave solutions of the Kuramoto-Sivashinsky equation and their stability and bifurcations

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. Wave solutions regular in space and both periodical and quasi-periodical in time generating from steady-state and steady-state travelling waves have been found numerically. Stability of some solutions has been investigated and bifurcation analysis has been carried out. The analysis has demonstrated that there is a sequence of bifurcations of solutions stable with respect to disturbances of the same spatial period and it has been shown that the bifurcations are related to the loss of some symmetries of the initial solution. It is also shown that the regions of “quiet” time behaviour and those of intensive variations are separated in the temporal period of solutions with the bifurcation parameter increasing.

BIFURCATIONS OF TWO-DIMENSIONAL INTO THREE-DIMENSIONAL WAVE REGIMES FOR A VERTICALLY FLOWING LIQUID FILM

The three-dimensional steady traveling wave regimes of a viscous liquid film flowing down a vertical wall which branch off from two-dimensional nonlinear waves are investigated. The numerical calculations are based on a model system of equations valid for intermediate Reynolds numbers. It is shown that there exist two fundamentally different types of three-dimensional steady traveling waves branching off from plane waves. One of these possesses checkerboard symmetry in the distribution of the maxima of the wave profile thickness and is the more interesting. An important difference in the “breakdown” of plane waves of the first and second families is also demonstrated. The wave characteristics of certain three-dimensional regimes are calculated as functions of the bifurcation parameter.

Viscous liquid film flow down an inclined corrugated surface. Calculation of the flow stability to arbitrary perturbations using an integral method

Viscous liquid film flow along an inclined corrugated (sinusoidal) surface has been studied. Calculations were performed using an integral model. The stability of nonlinear steady-state flows to arbitrary perturbations was examined using the Floquet theory. It has been shown that for each type of corrugation there is a critical Reynolds number for which unstable perturbations occur. It has been found that this value greatly depends on the physical properties of the liquid and geometric parameters of the flow. In particular, in the case of film flow down a smooth wall, the critical waveformation parameter depends only on the angle of inclination of the flow surface. The values of the corrugation parameters (amplitude and period) were obtained for which the film flow down a wavy wall is stable to arbitrary perturbations up to moderate Reynolds numbers. Such parameter values exist for all investigated angles of inclination of the flow surface.

Wavy flow of a liquid film in the presence of a cocurrent turbulent gas flow

Wavy downflow of viscous liquid films in the presence of a cocurrent turbulent gas flow is analyzed theoretically. The parameters of two-dimensional steady-state traveling waves are calculated for wide ranges of liquid Reynolds number and gas flow velocity. The hydrodynamic characteristics of the liquid flow are computed using the full Navier-Stokes equations. The wavy interface is regarded as a small perturbation, and the equations for the gas are linearized in the vicinity of the main turbulent flow. Various optimal film flow regimes are obtained for the calculated nonlinear waves branching from the plane-parallel flow. It is shown that for high velocities of the cocurrent gas flow, the calculated wave characteristics correspond to those of ripple waves observed in experiments.

Spreading of a viscous liquid film over the corrugated surface and the heat and mass transfer calculations

There are a lot of industrial applications of structured packing. Distillation columns are one of the examples where the liquid flows over the corrugated surface as a thin film to provide a good mass-transfer surface between the liquid and vapor phase. The purpose of the present paper is to study the hydrodynamics and the heat-mass transfer of the liquid film spreading down the corrugated surfaces when the corrugation amplitude is comparable with Nusselt’s film thickness (the amplitude corresponds to a small texture of the structured packing). As a result, a nonlinear type diffusion equation is obtained to describe the evolution of the film thickness profile. The nonlinear diffusion coefficient is obtained for three cases: a smooth inclined plate, a corrugated plate with large ribs, and an inclined corrugated plate with small ribs. The equations are solved numerically. As a result, it has been obtained that the small texture significantly increases the rate of the film thickness evolution in comparison with a smooth plate. To obtain the nonlinear diffusion coefficient in the case of a small texture, the hydrodynamics of the film flow over an inclined corrugated surface are studied. The viscosity, inertia, and surface tension forces are taken into account. The calculations were carried out on the basis of the Navier-Stokes equations. The influence of the microcorrugations on both the heat transfer from the wall and the mass transfer through the free surface was investigated.

Calculation of Linear Stability of a Stratified Gas–Liquid Flow in an Inclined Plane Channel

Linear stability of liquid and gas counterflows in an inclined channel is considered. The full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode (corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase. The wave length and the phase velocity of neutral disturbances of both modes are calculated as functions of the Reynolds number for the liquid. It is shown that these dependences for the surface mode are significantly affected by the gas velocity.

Waves on down-flowing fluid films: Calculation of resistance to arbitrary two-dimensional perturbations and optimal downflow conditions

Wavy downflow of viscous fluid films is studied. The full Navier-Stokes equations are used to calculate the hydrodynamic characteristics of the flow. The stability of calculated nonlinear waves to arbitrary two-dimensional perturbations is considered within the framework of the Floquet theory. It is shown that, for small values of the Kapitza number, the waves are stable over a wide range of wavelengths and values of the Reynolds number. It is found that, as the Kapitza number increases, the parameter range where nonlinear waves are calculated is divided into a series of alternating zones of stable and unstable solutions. A large number of narrow zones where the solutions are stable are revealed on the wavelength-Reynolds number parameter plane for large values of the Kapitza number. Optimal regimes of film downflow that correspond to the minimum value of average film thickness for nonlinear waves with different wavelengths are determined. The basic characteristics of these waves are calculated in a wide range of Reynolds and Kapitza numbers.