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.     

Wave regimes on a nonisothermal film of a viscous liquid flowing down a vertical plane

A thin film flow of a viscous liquid flowing down a vertical wall in the field of the gravity force is studied. The values of temperatures on the solid wall and on the free surface are constant. The viscosity and thermal diffusivity are functions of temperature. An equation that describes the evolution of surface disturbances is derived for small flow rates in the long-wave approximation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 89–97, March–April, 2008.     

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.     

Steady wave regimes in a film of viscous liquid flowing down a vertical wall

A study is made of the steady wave solutions of the nonlinear third-order differential equation [1] that describes the behavior of the wave boundary of a thin film of viscous liquid flowing down a vertical wall. It is shown that for long waves of small amplitude the general equation can be reduced [2] to a form containing a unique dimensionless parameter. A qualitative investigation is made of the behavior of the integral curves and the types of the singular points in the phase space. It is shown that a solitary wave exists for discrete values of the dimensionless parameter. A numerical solution is obtained. The structure of the jump in the thickness of the film is investigated qualitatively. Numerical solutions of nonmonotonic structure are obtained for different parameters.     

Stability of a viscous liquid film flowing down a periodic surface

The paper is devoted to a theoretical analysis of linear stability of the viscous liquid film flowing down a wavy surface. The study is based on the Navier–Stokes equations in their full statement. The developed numerical algorithm allows us to obtain pioneer results in the stability of the film flow down a corrugated surface without asymptotic approximations in a wide range over Reynolds and Kapitsa’s numbers. It is shown that in the case of moderate Reynolds numbers there is a region of the corrugation parameters (amplitude and period) where all disturbances decay in time and the wall corrugation demonstrates a stabilizing effect. At the same time, there exist corrugation parameters at which the steady-state solution is unstable with respect to perturbations of the same period as the period of corrugation. In this case the waveless solution cannot be observed in reality and the wall corrugation demonstrates a destabilizing effect.     

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.     

Waves on a viscous-fluid film flowing down a vibrating vertical plate

A thin film of a viscous fluid flowing down a vertical plane in a gravitational field is considered. The plane executes harmonic oscillations in the direction normal to itself. An equation that describes the evolution of surface disturbances at small fluid flow rates is obtained. Some solutions of this equation are found. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 90–98, July–August, 1999.     

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

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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.     

Waves on the surface of a falling power-law fluid film

Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.     

Possible classification of the self-oscillatory spouting regimes of plane vertical submerged jets of a heavy fluid

New results of an experimental investigation of self-oscillatory regimes of plane vertical jet spouting from beneath the free surface of a heavy incompressible fluid are discussed. The experiments were performed on a setup with discharge over a weir. The range of dimensionless jet submergence values on which bifurcation change of spouting regime is observable is studied. It is established that on the Froude number and dimensionless jet submergence ranges considered in the study six characteristic spouting regimes differing in free surface shape and self-oscillation frequency can exist. It is shown that these regimes can be subdivided into three typical groups with respect to the dependence of the self-oscillation period on the jet flow rate. A dimensionless parameter that makes it possible to identify the boundaries of the bifurcation change in spouting regimes is obtained for each of these groups. For certain spouting regimes without the formation of free jets numerical calculations are carried out using the STAR-CD software package; the calculated results are in good agreement with experimental data.     

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.     

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.     

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     

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.     

Thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined plane

This study is devoted to the investigation of thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined isothermal plane. It is assumed that the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. The governing non-linear equations for conservation of momentum and energy are obtained and solved by using a new computational approach based on a special type of Hermite-Padé approximation technique implemented in MAPLE. This semi-numerical scheme offers some advantages over solutions obtained with traditional methods such as finite differences, spectral method, and shooting method. It reveals the analytical structure of the solution function. Important properties of overall flow structure including velocity field, temperature field, thermal criticality, and bifurcations are discussed.     

Mixed convection of magneto hydrodynamic and viscous fluid in a vertical channel

Mixed convective flow and heat transfer in a vertical channel with one region filled with conducting fluid and another region with non-conducting fluid is analyzed. The viscous and Ohmic dissipation terms are included in the energy equation. Three types of thermal boundary conditions such as isothermal-isothermal, isoflux-isothermal and isothermal-isoflux for the left-right walls of the channel are prescribed. Analytical solutions are found for the governing equations using the regular perturbation method. A selected set of graphical results illustrating the effects of various parameters involved in the problem are presented and discussed.     

Stability of stationary traveling waves on the surface of a vertical film of viscous fluid

The stability of stationary traveling waves of the first and second families with respect to infinitesimal perturbations of arbitrary wavelength is subjected to a detailed numerical investigation. The existence of a unique region of stability of the first family is established for wave numbers (₁, ₁) corresponding to the optimal wave regime. There are several regions of stability of the second family ( _k , _k),k=2,3,..., lying close to the local flow rate maxima. In the regions of instability the growth rates of perturbations of the first family are several times greater than for the second family. This difference increases with increase in the Reynolds number. The calculations make it possible to explain a number of experimental observations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–41, May–June, 1989.The authors are grateful to V. Ya. Shkadov for his constant interest, and to A. G. Kulikovskii, A. A. Barmin and their seminar participants for useful discussions and suggestions.