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.     

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.     

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.     

Wave regimes on power-law fluid film flowing down a porous plane

Non-linear waves on the surface of a falling film of power-law fluid on a vertical porous plane are investigated. The waves are described by evolution equations generalising equations previously derived in the case of solid plane. It is shown that the slip condition on the interface between pure liquid and the porous substrate drastically changes structure of the steady waves travelling in the film.     

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.     

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].     

Multi-layer film flow down an inclined plane: experimental investigation

We report the results from an experimental study of the flow of a film down an inclined plane where the film itself is comprised of up to three layers of different liquids. By measuring the total film thickness for a broad range of parameters including flow rates and liquid physical properties, we provide a thorough and systematic test of the single-layer approximation for multi-layer films for Reynolds numbers \(Re = \rho Q/\mu \approx 0.03 - 60\) . In addition, we also measure the change in film thickness of individual layers as a function of flow rates for a variety of experimental configurations. With the aid of high-speed particle tracking, we derive the velocity fields and free-surface velocities to compare to the single-layer approximation. Furthermore, we provide experimental evidence of small capillary ridge formations close to the point where two layers merge and compare our experimental parameter range for the occurrence of this phenomenon to those previously reported.     

Influence of the capillarity on a creeping film flow down an inclined plane with an edge

Summary In creeping flows of thin films, the capillarity can play a dominant role. In this paper, the creeping film flow down an inclined plane with an edge is considered. The influence of the capillarity on the velocity and the film surface is studied analytically, numerically and experimentally. Received 12 April 1999; accepted for publication 9 May 1999     

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.     

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.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Long waves on a layer of a visco-elastic fluid down an inclined plane

Summary In this paper the finite amplitude stability of long waves on a layer of a second-order fluid flowing down an inclined plane is discussed. A systematic expansion procedure in terms of a parameterµ, which is the ratio of the undisturbed layer thickness to a representative length down the plane, is developed and solutions are obtained toO(µ ³). It is found that weakly non-linear monochromatic waves tend to attain equilibrium states for Weber numbers ofO(µ ^–2). This equilibrium amplitude first increases with increase in the elastic parameterM, reaches a maximum and then decreases withM. It is also shown that the second fluid behaves like a Newtonian fluid with its viscosity reduced through division by the factor 1 + (5M/2).

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Zusammenfassung In dieser Veröffentlichung wird die Stabilität von langen Wellen mit endlicher Amplitude in der Schicht einer Flüssigkeit zweiter Ordnung diskutiert, die längs einer geneigten Ebene abfließt. Es wird ein systematisches Entwicklungsverfahren nach einem Parameterµ angegeben, der das Verhältnis der ungestörten Schichtdicke zu einer repräsentativen Länge längs der Ebene beschreibt, und es werden Lösungen bis zur OrdnungO(µ ³) erhalten. Man findet, daß schwach nicht-lineare monochromatische Wellen für Weber-Zahlen der OrdnungO(µ ^–2) einem Gleichgewichtszustand zustreben. Die Gleichgewichtsamplitude nimmt mit wachsendem elastischem ParameterM zuerst zu, erreicht ein Maximum und fällt dann mitM wieder ab. Es wird schließlich noch gezeigt, daß sich die Flüssigkeit zweiter Ordnung wie eine newtonsche Flüssigkeit verhält, deren Viskosität jedoch durch Division durch einen Faktor 1 + (5M/2) reduziert ist.

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Instabilities of a thin viscoelastic liquid film flowing down an inclined plane in the presence of a uniform electromagnetic field

The effect of a uniform electromagnetic field on the stability of a thin layer of an electrically conducting viscoelastic liquid flowing down on a nonconducting inclined plane is studied under the induction-free approximation. Long-wave expansion method is used to obtain the surface evolution equation. The stabilizing role of the magnetic parameter M and the destabilizing role of the viscoelastic parameter Γ as well as the electric parameter E on this flow field are established. A novel result which emerges from our analysis is that the stabilizing effect of M holds no longer true for both viscous and viscoelastic fluids in the presence of electromagnetic field. It is found that when E exceeds a certain critical value depending on Γ, magnetic field exhibits the destabilizing effect on this flow field. Indeed, this critical value decreases with the increase of the viscoelastic parameter Γ since it has a destabilizing effect inherently. Another noteworthy result which arises from the weakly nonlinear stability analysis is that both the subcritical unstable and supercritical stable zones are possible together with the unconditional stable and explosive zones for different values of Γ depending on the wave number k.     

Anomalous rolling of spheres down an inclined plane

A sphere in air will roll down a plane that is tilted away from the vertical. The only couple acting about the point of contact between the sphere and the plane is due to the component of the weight of the sphere along the plane, provided that air friction is negligible. If on the other hand the sphere is immersed in a liquid, hydrodynamic forces will enter into the couples that turn the sphere, and the rotation of the sphere can be anomalous, i.e., as if rolling up the plane while it falls. In this paper we shall show that anomalous rolling is a characteristic phenomenon that can be observed in every viscoelastic liquid tested so far. Anomalous rolling is normal for hydrodynamically levitated spheres, both in Newtonian and viscoelastic liquids. Normal and anomalous rolling are different names for dry and hydrodynamic rolling. Spheres dropped at a vertical wall in Newtonian liquids are forced into anomalous rotation and are pushed away from the wall while in viscoelastic liquids, they are forced into anomalous rotation, but are pushed toward the wall. If the wall is inclined and the fluid is Newtonian, the spheres will rotate normally for dry rolling, but the same spheres rotate anomalously in viscoelastic liquids when the angle of inclination from the vertical is less than some critical value. The hydrodynamic mechanisms underway in the settling of circular particles in a Newtonian fluid at a vertical wall are revealed by an exact numerical simulation based on a finite-element solution of the Navier-Stokes equations and Newton's equations of motion for a rigid body.     

Stability of flow of a viscous liquid down an inclined plane

Using the Navier-Stokes equation the stability of a layer of viscous liquid flowing down a solid surface under gravity is studied in the linear formulation. The effect of surface tension and the inclination of the solid surface on the limits of stability are examined also. Curves are calculated for the neutral stability with respect to two types of perturbations — surface waves and shear waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskol Fiziki, No. 2, pp. 172–176, March–April, 1975.     

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.     

Motion of a sphere down an inclined plane in a viscous flow

The motion of a rigid particle near a wall in a fluid flow is an important element of particle transport by fluids. The aim of this study was to carry out an experimental and theoretical investigation of the gravity-induced motion of a rigid sphere in a viscous fluid in the presence of a transverse flow. The experimental study of this configuration is a way of understanding the specific features of the hydrodynamically constrained particle motion. It is established that the transverse motion of the fluid substantially increases the particle settling velocity, which grows with increase in the transverse flow velocity. This effect is most pronounced for small angles of inclination of the plane. The difference in the particle settling velocities in the presence and absence of the transverse flow could reach a factor of two.     

Effect of inclined temperature gradient on thermal instability in an anisotropic porous medium

Effect of anisotropy on thermal instability in a fluid saturated porous medium subjected to an inclined temperature gradient of finite magnitude is analysed using Galerkin technique. Results are compared with those of isotropic and horizontally isotropic cases. It is observed that anisotropic medium is the most stable while either isotropic situation or the horizontally isotropic situation is the most unstable one depending on the horizontal Rayleigh number (R _H ), anisotropy parametersk ₁(=k _y /k _x ), and ?₂(=?_γ/? _z ).