Nonlinear resonance in viscous films on inclined wavy planes

We study nonlinear resonance in viscous gravity-driven films flowing over undulated substrates. Numerical solution of the full, steady Navier–Stokes equations is used to follow the emergence of the first few free-surface harmonics with increasing wall amplitude, and to study their parametric dependence on film thickness, inertia and capillarity. Bistable resonance is computed for steep enough bottom undulations. As an analytic approach, we apply the integral boundary-layer method and derive an asymptotic equation valid for rather thin films. The analysis recovers the key numerical findings and provides qualitative understanding. It shows that higher harmonics are generated by a nonlinear coupling of the wall with lower-order harmonics of the free surface. It also accounts for bistable resonance, and produces a minimum model whose solution is similar to that of the Duffing oscillator.     

An extended study of the hydrodynamics of gravity-driven film flow of power-law fluids

A new similarity transformation has been devised for extensive studies of accelerating non-Newtonian film flow. The partial differential equations governing the hydrodynamics of the flow of a power-law fluid down along an inclined plane surface are transformed into a set of two ordinary differential equations by means of the dimensionless velocity component approach. Although the analysis is applicable for any angle of inclination (0<π/2), the resulting one-parameter problem involves only the power-law index n. Nevertheless, physically essential quantities, like the velocity components and the skin-friction coefficient, do depend on and relevant relationships are deduced between the vertical and inclined cases. Accurate numerical similarity solutions are provided for n in the range from 0.1 to 2.0. The present method enables solutions to be obtained also for highly pseudo-plastic films, i.e. for n below 0.5. The mass flow rate entrained into the momentum boundary layer from the inviscid freestream is expressed in terms of a dimensionless mass flux parameter Φ, which depends on the dimensionless boundary layer thickness and the velocity components at the edge of the viscous boundary layer. Φ, which is thus an integral part of the similarity solution, turns out to decrease monotonically with n. This parameter is of particular relevance in the determination of the streamwise position at which the entire freestream has been entrained and viscous stresses prevail all the way to the free surface of the film. A short-cut method to facilitate rapid and yet accurate estimates of the mass flux parameter is developed to this end.     

Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline

We consider a thin film of a power-law liquid flowing down an inclined wall with sinusoidal topography. Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on the steady free surface deformation. For weakly undulated walls we solve the governing equation analytically by a perturbation approach and find a resonant interaction of the free surface with the wavy bottom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in importance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an amplification of the steady free surface. A linear stability analysis of the steady state shows that the bottom undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear stability analysis. Finally, we show the important role of capillarity and discuss its influence on the steady solution and on the stability.     

Stability of flow of a thin layer of viscous magnetic liquid

The laminar flow of a thin layer of heavy viscous magnetic liquid down an inclined wall is examined. The stability and control of the flow of an ordinary liquid are affected only by alteration of the angle of inclination of the solid wall and the velocity of the adjacent gas flow. When magnetic liquids are used [1, 2], an effective method of flow control may be control of the magnetic field. By using magnetic fields of various configurations it is possible to control the flow of a thin film of viscous liquid, modify the stability of laminar film flow, and change the shape of the free surface of the laminarly flowing thin film, a factor which plays a role in mass transfer, whose rate depends on the phase contact surface area. The magnetic field significantly affects the shape of the free surface of a magnetic liquid [3, 4]. In this paper the velocity profile of a layer of viscous magnetic liquid adjoining a gas flow and flowing down an inclined solid wall in a uniform magnetic field is found. It is shown that the flow can be controlled by the magnetic field. The problem of stability of the flow is solved in a linear formulation in which perturbations of the magnetic field are taken into account. The stability condition is found. The flow stability is affected by the nonuniform nature of the field and also by its direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–65, September–October, 1977.     

Measurements of velocity,temperature and velocity fluctuation distributions in falling liquid films

The velocity, temperature and velocity fluctuation distributions within falling spindle oil films in an inclined rectangular channel were measured using hot-wire techniques and thin thermocouples. The interfacial shear was caused by cocurrent air flow.The results indicate that the liquid films are as a whole much more laminar-like than turbulent in a range of Reynolds numbers (4γ/μ) up to the experimental limit of 6000. Mixing motion occurs in the vicinity of the interface; however, the flow near the wall surface exhibits no sign of such eddy motions, as predicted by the wall law for single phase turbulent flow. Although velocity fluctuation is observed within films with interfacial shear, mean velocity profiles are approximately the same as those obtained by the laminar film prediction.     

Experiments on the flow of a thin liquid film over a horizontal stationary and rotating disk surface

Experiments on characterization of thin liquid films flowing over stationary and rotating disk surfaces are described. The thin liquid film was created by introducing deionized water from a flow collar at the center of an aluminum disk with a known initial film thickness and uniform radial velocity. Radial film thickness distribution was measured using a non-intrusive laser light interface reflection technique that enabled the measurement of the instantaneous film thickness over a finite segment of the disk. Experiments were performed for a range of flow rates between 3.0 lpm and 15.0 lpm, corresponding to Reynolds numbers based on the liquid inlet gap height and velocity between 238 and 1,188. The angular speed of the disk was varied from 0 rpm to 300 rpm. When the disk was stationary, a circular hydraulic jump was present in the liquid film. The liquid-film thickness in the subcritical region (downstream of the hydraulic jump) was an order of magnitude greater than that in the supercritical region (upstream of the hydraulic jump) which was of the order of 0.3 mm. As the Reynolds number increased, the hydraulic jump migrated toward the edge of the disk. In the case of rotation, the liquid-film thickness exhibited a maximum on the disk surface. The liquid-film inertia and friction influenced the inner region where the film thickness progressively increased. The outer region where the film thickness decreased was primarily affected by the centrifugal forces. A flow visualization study of the thin film was also performed to determine the characteristics of the waves on the free surface. At high rotational speeds, spiral waves were observed on the liquid film. It was also determined that the angle of the waves which form on the liquid surface was a function of the ratio of local radial to tangential velocity.     

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     

Wave regimes of ascending flow of a thin layer of a viscous liquid in contact with a gas

The flow of a liquid in thin layers is one of the hydrodynamic problems of chemistry and heat engineering. The large surface area of films and their small thickness make it possible to accelerate thermal, diffusive, and chemical processes at the gas-liquid boundary.Theoretical studies of liquid flow in a vertical descending thin layer are presented in [1–4]. In this paper we study ascending wave flows of a liquid in a thin vertical layer in contact with a gas, i.e., flows in the direction opposite the action of the force due to gravity, with account for the action of the gas on the liquid surface. Such motions are encountered when oil is extracted from strata that are saturated with gas. At some distance from the stratum the oil and gas separate: the gas travels at high velocity inside the pipe, occupying a considerable portion of the pipe, and the liquid is displaced toward the pipe walls, forming a thin film. In certain cases a wave-like interface develops between the oil and gas that travels with a velocity greater than that of the liquid but less than the average gas velocity. Similar phenomena are observed in high velocity mass exchangers.We examine the effect of the gas for both laminar and turbulent flow.Studies that neglect the effect of the gas flow on the liquid show that for waves on the film surface whose lengths are considerably longer than the average thickness of the layer, the liquid motion in the film is described by boundary layer equations in which account is taken of the mass force, i.e., the force due to gravity. With some approximation, we can assume that in accounting for the effect of the gas on the liquid the liquid flow is described by these same equations.     

Doppler effects of an oscillating line source in shear flow with a free surface

The linearised water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. There is a nonzero surface velocity. The depth is infinite and the vorticity is uniform. The amplitudes radiated from the source are calculated analytically. Due to Doppler effects, there may be up to four different emitted waves, and there is resonance with zero group velocity and infinite amplitude.     

Some results obtained on studying the flow of liquid films by stroboscopic visualization

The characteristic features of stroboscopic visualization and the possibilities of using this method for studying the flow of thin films are considered. The velocity field and the field of turbulent pulsations are studied experimentally for the film flow of liquids with Reynolds numbers of R = 40–1770.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 140–143, March–April, 1972.     

Thermal radiation effects on the flow and heat transfer in a liquid film on an unsteady stretching sheet

The influence of thermal radiation on the flow and heat transfer within Newtonian liquid film over an unsteady stretching sheet with and without thermocapillarity is examined. The governing non‐linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformation, which is solved numerically for different values of the thermal radiation parameter and the thermocapillarity parameter. The results show that the dimensionless velocity, the film thickness and the local Nusselt number increase as the thermocapillarity parameter increases, while the free surface temperature decreases with increasing the thermocapillarity parameter. Also, both the dimensionless temperature and the free surface temperature increase and the local Nusselt number decreases as the thermal radiation parameter increases. Copyright © 2010 John Wiley & Sons, Ltd.     

Behaviour of liquid films and flooding in counter-current two-phase flow—Part 1. Flow in circular tubes

Experimental results are presented on the flooding gas velocity in tubes over a wide range of parameters—tube diameter, tube length, liquid flow rate, liquid viscosity and surface tension. The flooding phenomenon is caused by interaction between the waves on the liquid film and the upward gas stream. By measuring variation of the maximum height of the wavy liquid films with an increase of the gas flow rate, the complicated effects of tube length and surface tension on flooding are revealed. The data of the flooding velocity are empirically correlated in termes of nondimensional groups for each tube length.     

Numerical study of the DLC film flow field in the ECR-PECVD reaction chamber

This paper mainly investigates the optimum parameters for the fabrication of uniform diamond-like carbon (DLC) films on the electron cyclotron resonance plasma enhanced chemical vapor deposition (ECR-PECVD) reaction chamber by analyzing the effect of the reacting gas velocity on the film properties. This work makes use of computational fluid dynamics (CFD) approach to model surface chemical reactions, flow and temperature fields, as well as heat and mass transfer phenomena. The simulation has shown that natural convection and mass transfer affect the recirculating flow within the reactor and, therefore, the distribution of material deposition. In other words, as a result of attaching an endplate (baffle) at the top of the substrate, the deposition rate of the substrate is appreciably enhanced. However, the surface uniformity of the substrate is obviously deteriorated.     

Effects of varying Reynolds number, blowing ratio, and internal geometry on trailing edge cutback film cooling

Three-dimensional mean velocity and concentration fields have been measured for a water flow in a pressure side cutback trailing edge film cooling geometry consisting of rectangular film cooling slots separated by tapered lands. Three-component mean velocities were measured with conventional magnetic resonance velocimetry, while time-averaged concentration distributions were measured with a magnetic resonance concentration technique for flow at two Reynolds numbers (Re) differing by a factor of 2, three blowing ratios, and with and without an internal pin fin array in the coolant feed channel. The results show that the flows are essentially independent of Re for the regime tested in terms of the film cooling surface effectiveness, normalized velocity profiles, and normalized mean streamwise vorticity. Blowing ratio changes had a larger effect, with higher blowing ratios resulting in surface effectiveness improvements at downstream locations. The addition of a pin fin array within the slot feed channel made the spanwise distribution of coolant at the surface more uniform. Results are compared with transonic experiments in air at realistic density ratios described by Holloway et al. (2002a).     

Effect of magnetic field and velocity slip on porous-walled rectangular squeeze films

This is a study of an electrically conducting flow in a squeeze film between two infinite strips where one of the strips has a porous bounding surface backed by a solid wall. The analysis is directed to study the interaction of a transverse magnetic field with the coupled flows in the squeeze film and the porous medium including the slip velocity at the porous bounding surface. Expressions for load capacity and thickness-time are obtained. It is observed that the magnetic field increases the load capacity and response times of squeeze films. This effect is more marked for small values of the permeability K.     

气体速度对液膜在预膜板表面流动形态影响的LBM模拟

为了探究气流剪切作用对航空燃油在气动雾化喷嘴预膜板上流动形态的影响,首先对基于相场理论的两相流格子Boltzmann模型进行修正,并通过经典算例验证了修正后模型的准确性和可靠性.随后利用该模型模拟了同向气流驱动下液膜在水平预膜板表面上的流动,分析了气流速度对液膜流动形态的影响规律.研究表明,该模型可准确追踪具有大密度比...     

Numerical study on the effect of initial flow velocity on liquid film thickness of accelerated slug flow in a micro tube

Numerical simulation of air–water slug flows accelerated from steady states with different initial velocities in a micro tube is conducted. It is shown that the liquid film formed between the gas bubble and the wall in an accelerated flow is significantly thinner than that in a steady flow at the same instantaneous capillary number. Specifically, the liquid film thickness is kept almost unchanged just after the onset of acceleration, and then gradually increases and eventually converges to that of an accelerated flow from zero initial velocity. Due to the flow acceleration, the Stokes layer is generated from the wall, and the instant velocity profile can be given by superposition of the Stokes layer and the initial parabolic velocity profile of a steady flow. It is found that the velocity profile inside a liquid slug away from the bubble can be well predicted by the analytical solution of a single-phase flow with acceleration. The change of the velocity profile in an accelerated flow changes the balance between the inertia, surface tension and viscous forces around the meniscus region, and thus the resultant liquid film thickness. By introducing the displacement thickness, the existing correlation for liquid film thickness in a steady flow (Han and Shikazono, 2009) is extended so that it can be applied to a flow with acceleration from an arbitrary initial velocity. It is demonstrated that the proposed correlation can predict liquid film thickness at Re < 4600 within the range of ±10% accuracy.     

Stability of non isothermal flow of a film of viscous liquid on an inclined plane

A study is made of the stability of nonisothermal flow of a film of viscous liquid down an inclined plane under the influence of gravity with allowance for dissipation of energy in the flow. It is assumed that the liquid is incompressible, and that its physical properties do not depend on the temperature. On the free surface of the film, allowance is made for evaporation and condensation effects. The treatment is in the long-wavelength approximation of the method proposed by Yih Chia-shun [1]. The expression obtained for the critical Reynolds number at which the flow becomes unstable indicates that viscous dissipation plays a destabilizing part in a nonisothermal flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 145–148, July–August, 1979.     

Viscous incompressible flow over the surface of a rotary disk

Viscous incompressible film flow over the surface of an impermeable rotary disk is studied. An exact self-similar solution of the complete Navier-Stokes system of equations is obtained and the velocity and pressure fields together with the radial profiles of the fluid film are determined. A physical interpretation of the results obtained is given.Volgograd. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 39–43, November–December, 1995.