Viscous flowing film instability down an inclined plane in the presence of constant electromagnetic field

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of constant electromagnetic field. Using the Kármán approximation, the problem is reduced to the study of the evolution equation for the free surface of the liquid film derived through a long-wave approximation. A linear stability analysis of the base flow is performed. Also, the solutions of stationary waves and Shkadov waves are introduced and discussed analytically by analyzing the linearized instability of the fixed points and Hopf bifurcation.     

Stability of conducting liquid flowing down an inclined plane at moderate Reynolds number in the presence of constant electromagnetic field

The stability of a conducting viscous film flowing down an inclined plane at moderate Reynolds number in the presence of electromagnetic field is investigated under induction-free approximation. Using momentum integral method a non-linear evolution equation for the development of the free surface is derived. The linear stability analysis of the evolution equation shows that the magnetic field stabilizes the flow whereas the electric field stabilizes or destabilizes the flow depending on its orientation with the flow. The weakly non-linear study reveals that both the supercritical stability and subcritical instability are possible for this type of thin film flow. The influence of magnetic field on the different zones is very significant, while the impact of electric field is very feeble in comparison.     

Flow stability of a conducting liquid flowing down an inclined plane in the presence of a magnetic field

The stability of a steady flow of incompressible, conducting liquid down an inclined plane in the presence of longitudinal and transverse magnetic fields is studied. Solutions of the linearized magnetohydrodynamic equations with corresponding boundary conditions are found on the assumption that the Reynolds number R_g and the wave number are small. It is shown that the longitudinal magnetic field plays a stabilizing role. It is known [1] that the flow of a viscous liquid over a vertical wall is always unstable. In this article it is shown that the instability effect at small wave numbers may be eliminated if the longitudinal magnetic field satisfies the conditions found. The case when the Alfvén number and the wave number are small and the Reynolds number is finite is also examined.     

Stationary travelling waves on a film flowing down an inclined plane

At small flow rates, the study of long-wavelength perturbations reduces to the solution of an approximate nonlinear equation that describes the change in the film thickness [1–3]. Steady waves can be obtained analytically only for values of the wave numbers close to the wave number _n that is neutral in accordance with the linear theory [1, 2]. Periodic solutions were constructed numerically for the finite interval of wave numbers 0.5_{n n} in [4]. In the present paper, these solutions are found in almost the complete range of wave numbers 0 _n that are unstable in the linear theory. In particular, soliton solutions of this equation are obtained. The results were partly published in [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–146, July–August, 1980.     

Stability of wavy conditions in a film flowing down an inclined plane

The nonlinear theory of motion in a film of liquid flowing down an inclined plane predicts the existence of an interval k₀m, inside of which the wave number of periodic wave motion may lie [1]. The condition of the stability of experimentally attained motions imposes a limitation on their wave numbers. In [2] a numerical investigation of the stability of wavy motions was made; in the investigated range of change in the Galileo number and the wave number all the motions were found to be unstable; however, the fastest growing were perturbations imposed on a motion with a determined wave number (“optimal” conditions). In [3] the instability of motions with a wavelength exceeding some limiting value was established in a long-wave approximation. In the present work, within the framework of the two-dimensional problem, an investigation was made of the stability of periodic wavy motions, based on expansion in terms of the small parameter k_m. It is established that, within the interval k₀m, there lies a finite subinterval of wave numbers for which wavy motions are stable. The narrowness of this interval (δk≈0.07 k_m) may be the reason why, in the experiment, with not too great Galileo numbers for fully established periodic wavy motions, no substantial differences in the wave-length are observed [4].     

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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1989.     

On thin film flow of a third-grade fluid down an inclined plane

An exact solution for the thin film flow of a third-grade fluid on an inclined plane is presented. This is a corrected version of the solution obtained by Hayat et al. (Chaos Solitons Fractals 38:1336–1341, 2008). An alternative parametric form for the solution is also derived. The variation of the dimensionless velocity and average velocity is given for a wide range of parameter values. An asymptotic solution for large parameter values is obtained giving rise to a boundary-layer structure at the free surface.     

An initial-value problem for a heavy viscous liquid flowing down an inclined plane

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–98, September–October, 1991.     

Meandering of jets flowing down over an inclined plane

Experiments on the stability of rectilinear jets flowing down over an inclined plane against meandering are described and a technique for processing the experimental results is proposed.     

Doubly periodic and quasiperiodic wave regimes in a liquid film flowing down an inclined plane,their stability and bifurcations

A nonlinear evolution equation frequently encountered in modeling the behavior of disturbances in various nonconservative media, for example, in problems of the hydrodynamics of liquid film flow, is considered. Wave solutions of this equation, regular in space and both periodic and quasiperiodic in time, branching off from steady and steady-state traveling waves are found numerically. The stability and bifurcations are analyzed for some of the solutions obtained. As a result, a bifurcation chain is found for solutions stable with respect to disturbances of the same spatial period. It is shown that the bifurcations are related to the loss of certain symmetries of the initial solution. It is demonstrated that as the bifurcation parameter increases it is possible to distinguish in the structure of the solutions intervals of quiet behavior and intervals of intense outbursts.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 98–107, July–August, 1992.     

Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder

PIV technique is applied for measurements of instant velocity distributions in a liquid film flowing down an inclined tube in the form of a wavy rivulet. An application of special optical calibration is applied to correct distortion effects caused by the curvature of the interface. A vortex flow of liquid is observed inside a wave hump in the reference system moving with wave phase velocity. Conditionally averaged profiles of longitudinal and transverse components of liquid velocity are obtained for different cross-sections of developed non-linear waves. It is shown that the increase in wave amplitude slightly changes the location of the vortex center. The analysis of modification of vortex motion character due to wavy flow conditions, such as tube inclination angle, film Reynolds number, wave excitation frequency, is fulfilled.     

Solution to the orr–sommerfeld equation for liquid film flowing down an inclined plane: An optimal approach

The Orr–Sommerfeld equation is solved numerically for a layer of liquid film flowing down an inclined plane under the action of gravity using the sequential gradient-restoration algorithm (SGRA) The method consists of solving the governing equation as it is a Bolza problem in the calculus of variations. The neutral stability curves, eigenvalues and eigenfunctions to the stability problem can be determined simultaneously during the process.     

Wavy wall influence on the hydrodynamic instability of a liquid film flowing along an inclined plane

The film dynamic of a thin liquid along an inclined and wavy wall was numerically depicted in a weighted-residual integral boundary layer equation. A qualitative and quantitative analysis was initially carried out and accurate comparisons were obtained from experimental data on film instability along a flat and inclined as well as a wavy wall. To pinpoint the effect of waviness on film instability, 20 wavy wall periods in the computational CFD domain were considered. Several waviness parameters were studied and shown to have taken on a major role in the film instability process. Finally, a wide range of main wall inclination angles was taken into account, and consequent numerical data permitted identification of a threshold angle value. For wall angles higher than the threshold angle, the film behaved as though no corrugations were present. For lower angles, the film was repeatedly altered during the acceleration and deceleration phases.     

Effect of bottom topography on the flow of a non-Newtonian liquid film down an inclined plane

The problem of flow of a nonlinear viscous liquid film down an inclined surface with local microtopography is considered. Numerical and approximate analytic solutions are obtained for steady flows of power-law liquid films down inclined surfaces with topography. Steps, hills, and periodic structures are considered as local topography. Basic properties of flows are found.     

Solitary waves in a viscous liquid film flowing down a thin vertical cylinder

Several equations to describe the flow of a viscous liquid film on a thin cylinder are derived. The solitary-wave solutions to these equations are studied. The families of solutions are constructed for the first two eigenvalues that correspond to single-humped and double-humped waves. It is found that these families become similar as the similarity parameter increases. The dependencies of phase velocities and wave amplitudes on the free parameters of the problem are analyzed. The resulting solutions are compared with solitary waves in films on a flat surface.     

On the stability of the flow of a viscoelastic fluid down an inclined plane

The paper presents an analysis of laminar flow of a film of viscoelastic fluid flowing under gravity down an infinite inclined plane. It is assumed that the mechanical behavior of the fluid can be represented by a generalized Maxwell model, whose constitutive equation contains a time derivative of the deviator of the stress tensor in the Jaumann sense [1. 2]. The equations of motion of the viscoelastic fluid considered here admit an exact solution for the case of rectilinear laminar flow with a plane free boundary. The stability of this flow with respect to surface waves is investigated by the method of successive approximations described in [3, 4].     

The analysis of stability of Bingham fluid flowing down an inclined plane

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Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.