Long interfacial waves inside an inclined permeable channel

The time evolution of superposed layers of fluid flowing down inside an inclined permeable channel is investigated. Using the Kármán-Pohlhausen approximation, the problem is reduced to the study of the evolution equation for the liquid–liquid interface of the liquids film derived through a long-wave approximation. A linear stability analysis of the base flow is performed. The solutions and stability of the non-linear stationary long waves are investigated. A special form of the stationary long waves (say Shkadov waves) is introduced.     

Long waves between a subsonic gas flow-film interface flowing down an inclined plate

The present work deals with temporal stability properties of a falling liquid film down an inclined plane in the presence of a parallel subsonic gas flow. The waves are described by evolution equation previously derived as a generalization of the model for the Newtonian liquid. We confirm linear stability results of the basic flow using the Orr–Sommerfeld analysis to that obtained by long wave approximation analysis. The non-linear stability criteria of the model are discussed analytically and stability branches are obtained. Finally, the solitary wave solutions at the liquid–gas interface are discussed, using specially envelope transform and direct ansatz approach to Ginzburg–Landau equation. The influence of different parameters governing the flow on the stability behavior of the system is discussed in detail.     

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.     

Nonlinear vibrations of a vapor film in the presence of intense heat fluxes

Waves propagating along the interface between a thin vapor film and a liquid layer in the presence of a heat flux are investigated. The boundary conditions on the vapor-liquid phase surface take into account the temperature dependence of the pressure and the possibilities of formation of the metastable state of the superheated liquid and mass flow. Variations in the saturation pressure as functions of the temperature and mass flux lead to generation of weakly damped periodic waves of low amplitude whose velocity can be much higher than the velocity of the gravity waves. The waves ensure stability of the vapor film beneath the liquid layer in the gravity field. The finite-amplitude waves on the surface of the vapor film differ from the Stokes surface waves on the free surface of isothermal fluid. Instability regimes related with superheating of the liquid ant its explosive boiling when the amplitude of an initially small wave increases to infinity in a finite time can develop in a certain working-parameter regime.     

固体薄膜/液体膜/基底层状结构中液体膜/基底界面不平整性对结构稳定性影响

考虑液体层较薄的情况下液体膜/基底间界面不平整对结构的稳定性,特别是固体薄膜稳定时的褶皱变形产生的影响。用一扰动函数模拟液体膜/基底间界面的不平整,计算固体膜受力变形前后结构能量的变化,进而分析其稳定性。取两个扰动函数的特例,具体分析了液体膜/基底间界面不平整对结构能量变化以及对结构平衡态的影响。结果表明,平衡时固体膜褶皱变形与原来假设液体膜/基底间界面为理想平整面所得的结果有很大不同。     

Waves on the surface of a thin layer of viscous liquid

Under the assumption that the boundary layer approximation for the original equations is valid, we show the possibility of the existence of progressive waves on the surface of a vertically flowing film when surface tension is neglected. From the system of equations obtained for a thin layer of viscous liquid flowing down an inclined plane, one equation for perturbations of a thin film follows. Steady solutions of this equation allow periodic discontinuous solutions of the roll-wave type.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 2, pp. 109–113-March–April, 1973.     

Stability Analysis of a Falling Film Flow Down a Plane with Sinusoidal Corrugations

The liquid viscous film falling down a vertical wall with sinusoidal relief is considered. The linear stability of steady-state flow with respect to time-periodic disturbances is studied using the Floquet theory. It is shown that in the case of applying corrugations the variation in the disturbance growth rate is proportional to the second power of their undulations. Depending on the relief parameters there exist two possibilities: the instability domain can expand or certain disturbances can be stabilized. The growth rates are obtained numerically and analytically in the approximation of low-amplitude corrugations. The development of waves from small disturbances is simulated within the framework of nonlinear equations and the formation of structures whose wavelength is significantly greater than the space relief period is found out.     

STABILITY OF ULTRATHIN LIQUID FILM EVOLUTION WITH SURFACTANT

针对固体基底上厚度小于100 nm的含活性剂超薄液膜演化过程, 基于润滑理论推导出包含分离压影响的液膜厚度和活性剂浓度的演化方程, 采用正则模态法导出了描述液膜线性稳定性的特征方程, 分析了多个特征参数对线性稳定性的影响, 数值模拟了液膜厚度和活性剂浓度演化历程, 对比了模拟所得非线性结果与线性分析预测结果的一致性.结果表明:范德华力具有促进扰动增长的作用, 较强的玻恩斥力促使扰动衰减, 使液膜趋于稳定;较小的毛细力数易使液膜凹陷处发生二次失稳, 并最终导致去润湿现象发生;液膜厚度和溶于液膜内部的活性剂浓度初值越大, 液膜稳定性越强, 液膜表面活性剂浓度影响则相反;增大吸附系数不利于液膜稳定性.     

基底弹性对蒸发超薄液膜去润湿过程的影响

研究了基底的弹性变形对蒸发超薄膜的稳定性和去润湿动力学过程的影响. 基于长波近似, 得到了关于液体薄膜厚度的演化方程. 运用线性稳定性理论和数值模拟两种方法, 研究了基底弹性、范德华力以及液体蒸发等因素对液体薄膜的稳定性和去润湿过程的影响. 研究结果表明增大基底的弹性系数或者减小液体的表面张力, 都能加速液膜的破碎, 并且能够影响气液界面波的波长; 液体蒸发能促进气液界面扰动的增长, 有助于液膜的破裂.      

Finite-amplitude stability analysis of liquid films down a vertical wall with and without interfacial phase change

The generalized kinematic equation for film thickness, taking into account the effect of phase change at the interface, is used to investigate the nonlinear stability of film flow down a vertical wall. The analysis shows that supercritical stability and subcritical instability are both possible for the film flow system. Applications of the result to isothermal, condensate and evaporate film flow show that mass transfer into (away from) the liquid phase will stabilize (destabilize) the film flow. Finally, we find that supercritical filtered waves are always linearly stable with regard to side-band disturbance.     

Spacetime Interpretation of Torsion in Prismatic Bodies

A non-linear theory for the plastic deformation of prismatic bodies is constructed which interpolates between Prandtl??s linear soap-film approximation and Nádai??s sand-pile model. Geometrically Prandtl??s soap film and Nádai??s wavefront are unified into a single smooth surface of constant mean curvature in three-dimensional Minkowski spacetime.     

An accurate model for the thin film flow

This paper deals with the linear stability of a liquid film flowing down an inclined plane. The Navier-Stokes equations were reduced into four evolution equations that describe the development of the film depth, the flow rate, the free surface velocity, and the wall shear stress, using the Karman-Polhausen boundary layer integral method. Thus, we were able to determine the stability threshold and approach well the critical wave number for long waves. The obtained results were found to be in good agreement with the experiments of Liu et al.     

Effect of liquid viscosity on the stratified-slug transition in horizontal pipe flow

The effect of liquid viscosity on the initiation of slug flow was studied in horizontal 2.52 and 9.53 cm pipelines. The results show the stabilizing effect of viscosity predicted by Lin & Hanratty, and are at variance with analyses which use a long-wavelength inviscid approximation. For very viscous liquids a stability analysis which recognizes that slugs originate from a train of small-wavelength sinusoidal waves seems consistent with the measurements.     

界面剪切力作用下波状液膜流的水动力稳定性

液膜流的水动力稳定性作为保障其高效传热传质性能的重要因素之一,受多种因素的制约和影响. 当气液界面处存在因气流流动而产生剪切力作用时,剪切力将通过改变界面处的边界条件,从而影响液膜流动的稳定性. 基于边界层理论,采用积分法建立了剪切力作用下降液膜表面波演化方程,分析了界面剪切力对水动力稳定性的影响. 研究表明,正向剪切力为不稳定性因素,反向剪切力在较小雷诺数时为不稳定因素,在大雷诺数时为稳定性因素;正向剪切力使临界波数和临界波速增大,反向剪切力使其减小;剪切力对临界波速的影响在不同雷诺数下也有所不同.      

Surface instabilities of thin liquid film flow on a rotating disk

 Steady flow of a liquid jet from a nozzle onto the centre of a rotating disk is studied with a streak line method to determine the superficial velocity of the spreading liquid film. Good agreement is found with an asymptotic analysis of the unperturbed flow field. Experimentally, the liquid surface is always perturbed by surface waves which appear as regular spirals, steady in the laboratory system in the low Reynolds number range. It could be shown that wave formation is very sensitive to entrance conditions. Therefore, it is assumed that wave generation is an entrance effect which acts as periodic forcing on the forming liquid film. Wave velocities outside the entrance region are measured and proved to be in good agreement with the prediction of a linear stability theory, as long as the flow rate and entrance perturbations are small. At higher flow rates or stronger disturbances, the radial development of the wave velocities takes on the characteristics predicted by nonlinear stability theories and is in qualitative agreement with experiments performed on an inclined plane. Received: 15 January 1998/Accepted: 8 June 1998     

Energy Scaling of Compressed Elastic Films –Three-Dimensional Elasticity¶and Reduced Theories

We derive an optimal scaling law for the energy of thin elastic films under isotropic compression, starting from three-dimensional nonlinear elasticity. As a consequence we show that any deformation with optimal energy scaling must exhibit fine-scale oscillations along the boundary, which coarsen in the interior. This agrees with experimental observations of folds which refine as they approach the boundary. We show that both for three-dimensional elasticity and for the geometrically nonlinear Föppl-von Kármán plate theory the energy of a compressed film scales quadratically in the film thickness. This is intermediate between the linear scaling of membrane theories which describe film stretching, and the cubic scaling of bending theories which describe unstretched plates, and indicates that the regime we are probing is characterized by the interplay of stretching and bending energies. Blistering of compressed thin films has previously been analyzed using the Föppl-von Kármán theory of plates linearized in the in-plane displacements, or with the scalar eikonal functional where in-plane displacements are completely neglected. The predictions of the linearized plate theory agree with our result, but the scalar approximation yields a different scaling.     

Bifurcation analysis of the travelling waves on a falling power-law fluid film

Non-linear waves on the surface of a falling film of power-law fluid on a vertical plane are investigated. The waves are described by evolution equations previously derived as a generalization of the model for the Newtonian liquid. This paper presents a numerical bifurcation analysis of the steady travelling waves on a falling film described by an equation containing three parameters. It is shown that the wave regimes are sensitive to the power-law index as well as to the film parameter and the wavenumber that is typical for the Newtonian liquid.     

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.     

Primary instabilities of liquid film flow sheared by turbulent gas stream

Evolution of excited waves on a viscous liquid film has been investigated experimentally for the annular gas–liquid flow in a vertical tube. For the first time the dispersion relations are obtained experimentally for linear waves on liquid film surface in the presence of turbulent gas flow. Both cocurrent and countercurrent flow regimes are investigated. As an example of comparison with theory, the experimental data are compared to the results of calculations based on the Benjamin quasi-laminar model for turbulent gas flow. The calculation results are found to be in good agreement with experiments for moderate values of film Reynolds number.     

聚变堆相关的液态金属膜流初步实验研究

在磁约束核聚变堆的面对等离子部件设计中,液态金属锂膜流因具有带走杂质、保护面对等离子固壁等优点而被认为是优选方案之一. 然而,如何克服聚变堆中强磁场环境下产生的磁流体力学效应并形成大面积均匀铺展锂膜流动是目前亟需解决的问题.本文通过搭建室温液 态镓铟锡回路和高温液态锂回路,开展了两种不同特性的液态金属膜流实验, 并采用传统可视化方法获得了展向磁场存在时镓铟锡和锂在导电底板形成的液膜流动表面特征.实验结果 表明: 无磁场时,两种液态金属膜流流动表面波动特性与常规流体膜流均一致, 即随着流动雷诺数的增加表面波动变得更为混乱; 而展向磁场存在时,镓铟锡膜流表面波动变得更为规则, 且沿着磁场方向平行排列,表现为拟二维波动的特征; 而锂膜流却产生了明显的磁流体 力学阻力效应,表现为在流动方向局部产生锂滞留现象, 且滞留点随雷诺数增大向下游移动. 最后通过膜流受力分析,进一步阐述了锂膜流受到比镓铟锡膜流更为严重磁流体力学效应影响的原因.