Weakly nonlinear stability analysis of a falling film with countercurrent gas flow

Weakly nonlinear stability analysis of a falling film with countercurrent gas–liquid flow has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear and nonlinear stability solutions for the film flow system. The results show that both supercritical stability and subcritical instability are possible for a film flow system when the gas flows in the countercurrent direction. The stability characteristics of the film flow system are strongly influenced by the effects of interfacial shear stress when the gas flows in the countercurrent direction. The effect of countercurrent gas flow in a falling film is to stabilize the film flow system.     

Two-fluid modeling in analyzing the interfacial stability of liquid film flows

The area-averaged two-fluid model formulation of a separated two-phase flow system is used to investigate interfacial stability of liquid film flows. The analysis takes into account the effects of phase change at the interface as well as the dynamic effects of the adjacent vapor flow on the interfacial stability. Wave formation and instability criteria are established in terms of the generalized fluid and flow parameters. The criteria are applied to investigate the stability of laminar liquid film flow with interfacial shear and phase change. The influence of various dimensionless parameters characterizing film thickness, gravity, phase change and interfacial shear are studied with respect to the neutral stability, temporal growth factor and the wave propagation velocity. The results of the present study indicate that the interfacial stability analysis developed within the frame of the two-fluid model formulation proves to be quite accurate as judged by comparing its results with the available experimental data and with the results of much longer and more complex analytical investigations which are valid only for the liquid film free of interfacial shear.     

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.     

Nonlinear Stability Analysis of Thin Viscoelastic Film Flow Traveling Down along a Vertical Cylinder

This paper investigates the weakly nonlinear stability theoryof a thin viscoelastic liquid film flowing down along the outsidesurface of a vertical cylinder. The long-wave perturbation method isemployed to solve for generalized nonlinear kinematic equations withfree film interface. The normal mode approach is first used to computethe linear stability solution for the film flow. The method of multiplescales is then used to obtain the weak nonlinear dynamics of the filmflow for stability analysis. The modeling results indicate that both thesubcritical instability and supercritical stability conditions arepossible to occur in a viscoelastic film flow system. The degree ofinstability in the film flow is further intensified by the lateralcurvature of cylinder. This is somewhat different from that of theplanar flow. The modeling results also indicate that by increasing theviscoelastic effect and decreasing the radius of the cylinder the filmflow can become less stable as traveling down along the verticalcylinder.     

Nonlinear Stability Analysis of Thin Viscoelastic Film Flowing Down on the Inner Surface of a Rotating Vertical Cylinder

This paper investigates the stability of thin viscoelastic liquid film flowing down on the inner surface of a rotating vertical cylinder by means of the long wave perturbation. After proving the insufficiency of the linear model in characterization of certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. This model is solved through the following procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, a nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that with the increase in the rotation speed Ω and the radius of cylinder R, the film flow system will be more stable.     

Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

A thin film of a power–law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy’s law together with Beavers–Joseph boundary condition for a general power–law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as ‘permanent waves’. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology.     

The lowest stability and bifurcation in supercritical Taylor vortices

Here, we numerically investigate the lowest stability and bifurcation boundary of supercritical Taylor vortices in flows with different wavenumbers and for various radius ratios; the radius ratios range from those corresponding to axisymmetrical Taylor vortex flow (TVF) to those corresponding to wavy vortex flow (WVF). The variation in the wavenumber of a supercritical TVF is found to affect the stability of the flow, because the wavenumber of the Taylor vortices remains constant only when the flow is quasi-static. The variation in the wavenumber is examined and found to be significant when the radius ratio is less than 0.7842. The results for TVF are compared with those for the flow during the quasi-static transition from TVF to WVF.     

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.     

Investigation of hydrodynamic instability of CO<Subscript>2</Subscript> injection into an aquifer

The two-dimensional problem of supercritical carbon dioxide injection into an aquifer is solved. Shocks and rarefaction waves propagating in a sequence from an injection well into the formation are described within the framework of a complete nonisothermal model of flows in a porous medium. In the approximation of isothermal immiscible water and carbon dioxide flow the hydrodynamic stability of the leading displacement front is investigated for various reservoir pressures and temperatures. The parameters of unstable fronts are determined using a sufficient instability condition formulated in analytic form. The approximate analytic results are supported by the direct numerical simulation of CO₂ injection using the complete model in which thermal effects and phase transitions are taken into account.     

Film thickness measurements in liquid–liquid slug flow regimes

At present there is significant interest in the development of small scale medical diagnostic equipment. These devices offer faster processing times and require smaller sample volumes than equivalent macro scale systems. Although significant attention has been focused upon their outputs, little attention has been devoted to the detailed fluid mechanics that govern the flow mechanisms within these devices. Conventionally, the samples in these small scale devices are segmented into distinct discrete droplets or slugs which are suspended in an organic carrier phase. Separating these slugs from the channel wall is a very thin film of the organic carrier phase.The magnitude of this film is the focus of the present study and the effects of sample slug length and carrier phase fluidic properties on the film are examined over a range of Capillary numbers. A non-intrusive optical technique was used to capture images of the flow from which the magnitude of the film was determined.The experimental results show that the film is not constant along the length of the slug; however above a threshold value for slug length, a region of constant film thickness exists. When compared with existing correlations in the literature, the experimental data showed reasonable agreement with the Bretherton model when the Capillary number was calculated based on the mean two phase flow velocity. However, significant differences were observed when the Capillary number was redefined to account for the mean velocity at the liquid interface, i.e., the mean slug velocity.Analysis of the experimental data revealed that it fell into two distinct flow regimes; a visco-capillary regime and a visco-inertial regime. A modified Taylor expression is presented to estimate the magnitude of the film for flows in the visco-capillary regime while a new model is put forward, based on Capillary and Weber numbers, for flows in the visco-inertial regime. Overall, this study provides some novel insights into parameters, such as aqueous slug length and carrier phase fluidic properties, that affect the thickness of the film in liquid–liquid slug flow regimes.     

单畴外延铁电薄膜的相结构与稳定性

本文采用动态金茨堡-朗道（DGL）方程研究了薄膜厚度与错配应变对 取向单畴外延PbTiO3（PTO）铁电薄膜相结构与稳定性的影响。结合平面内松弛应变（等效应变）、表面效应与退极化场等机电耦合边界条件,通过数值求解DGL方程获得外延单畴铁电薄膜错配应变-厚度相图和错配应变-温度相图。数值分析结果显示,由于生成的界面位错松弛了薄膜内错配应变,在理论高应变区相图与传统分析结果有较大差别,文中发现在更广的理论错配拉应变区出现稳定的四方相（c相）结构和单斜相（r相）结构。结果也显示,随着薄膜厚度的减小,表面效应与退极化效应会把顺电相扩展到更低温度区域,从而压缩稳定的铁电相存在的温度区域。     

Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics

Reformulating constitutive relation in terms of gradient dynamics (being derivative of a dissipation potential) brings additional information on stability, metastability and instability of the dynamics with respect to perturbations of the constitutive relation, called CR-stability. CR-instability is connected to the loss of convexity of the dissipation potential, which makes the Legendre-conjugate dissipation potential multivalued and causes dissipative phase transitions that are not induced by non-convexity of free energy, but by non-convexity of the dissipation potential. CR-stability of the constitutive relation with respect to perturbations is then manifested by constructing evolution equations for the perturbations in a thermodynamically sound way (CR-extension). As a result, interesting experimental observations of behavior of complex fluids under shear flow and supercritical boiling curve can be explained.     

Stability of high‐speed two‐layer film casting of Newtonian fluids

The steady‐state flow and its linear stability are investigated for the isothermal two‐layer film casting process. Newtonian fluids are considered in this study. The continuity of traction is ensured at the interface, and the axial velocity is assumed to be uniform across each film layer separately. The effects of inertia, gravity, fluid parameters and processing conditions on the steady‐state flow and its stability are studied. The results indicate that the fluid properties and the processing conditions have significant influence on the flow. The flow stability is strongly dependent on the layer layout with respect to the take‐up rolling process. The frequency of the (unstable) disturbance is insensitive to flow and processing parameters. Copyright © 2006 John Wiley & Sons, Ltd.     

Secondary vibrational convective motions in a flat vertical layer of liquid

Investigations of the stability of steady-state plane-parallel convective motion between vertical planes heated to different temperatures [1–5] have shown that this motion, depending on the value of the Prandtl number P, exhibits instability of two types. With small and moderate Prandtl numbers, the instability is of a hydrodynamic nature. It is brought about by monotonic perturbations which, in the supercritical region, develop into a periodic, with respect to the vertical, system of steady-state vortices at the interface between the opposing convective flows. Articles [6, 7] are devoted to the numerical investigation of nonlinear secondary steady-state flows. If P>11.4, there appears a new mode of instability, i.e., running thermal waves increasing in the flow; with P>12, this mode becomes more dangerous [4]. This instability is connected with the development of vibrational perturbations, and it can be considered that in the supercritical region the perturbations lead to the establishment of steady-state vibrations. Linear theory has made it possible to determine the boundaries of the regions of stability. In the present article a numerical investigation is made of nonlinear supercritical conditions developing as a result of a loss of stability of the steady-state flow with respect to vibrational perturbations.     

Transmural pressure effects on the stability of clamped cylindrical shells subjected to internal fluid flow: Theory and experiments

An experimental and theoretical parametric study is undertaken to investigate the effect of transmural pressure on the non-linear dynamics and stability of circular cylindrical shells with clamped ends subjected to internal fluid flow. The theoretical structural model is based on the Donnell non-linear shallow shell theory, and potential flow theory is employed to describe the fluid-structure interaction. It is found that, for low transmural pressures in the range investigated, the shell loses stability by static subcritical divergence, while for higher transmural pressures the loss of stability is supercritical. In addition, there are ranges of flow velocity in which the shell exhibits quasi-periodic or even chaotic behaviour.     

Model of a wavy flow in a falling film of a viscous liquid

A new model is developed for describing long-wave perturbations in a falling film of a viscous liquid. The model is based on an integral approach and an expansion of the velocity profile into a series in linearly independent basis functions of a boundary-value problem. A linear analysis of film flow stability is performed, and dispersion dependences are obtained. Results predicted by the new model are demonstrated to be in good agreement with available experimental data on the film flow over a gently sloping surface.     

离心惯性力效应对超临界二氧化碳干气密封流场与密封特性影响分析

为揭示离心惯性力效应对S-CO₂干气密封流场与密封特性的影响规律,以螺旋槽干气密封为研究对象,引用考虑离心惯性力效应的Reynolds方程,在考虑气膜真实气体效应、黏度随压力与温度双重变化的同时,基于N-S方程与能量守恒定律,建立了绝热状态下考虑离心惯性力效应作用的能量控制方程. 然后,采用有限差分法对压力控制方程与能量控制方程进行耦合求解,并对考虑离心惯性力效应与没有考虑离心惯性力效应下的压力分布、温度分布以及密封特性进行了分析讨论. 研究表明:离心惯性力效应具有削弱流场内压力与温度的作用;从避免凝结流动角度考虑,离心惯性力效应引起的温降将不利于S-CO₂干气密封;考虑离心惯性力效应作用时,气膜开启力在不同槽深与转速下存在最佳工况点,泄漏率随着转速的增加显著减小,而离心惯性力效应与膜厚之间没有强交互作用;考虑离心惯性力效应作用的气膜开启力、泄漏率、出口温度均比不考虑离心惯性力效应作用的小,且这种差异随着转速的增大而增加,而随着膜厚的变化没有改变. 这些结果为进一步研究S-CO₂干气密封奠定了一定的理论基础.      

On Three-Dimensional Non-linear Buoyant Convection in Ternary Solidification

We consider the problem of three-dimensional non-linear buoyant convection in ternary solidification. Under the limit of large far-field temperature, the convective flow is modeled to be in a rectangular cube composing of a horizontal liquid layer above a primary mushy layer, which itself is over a secondary mushy layer. We first apply linear stability analysis to calculate the conditions at the onset of motion. Next, we carry out weakly non-linear analyses to determine solutions in the form of hexagons and their possible stability and to obtain information about tendency for chimney formation. We find that if the flow is driven either from both mushy layers with equal critical conditions at the onset of motion or only by the primary mushy layer, then the flow can be in the form of a double-cell structure vertically with down-hexagons below or above up-hexagons. There is tendency for vertically oriented chimney formation at different horizontal locations in each mushy layer. For the cases where only the critical conditions at the onset of motion are equal in both mushy layers and depending on the values of the mush Rayleigh numbers, the flow can be subcritical (or supercritical) in both mushy layers or mixed subcritical in one layer and supercritical in another layer.